1. science
2. mathematics
3. statistics

SAMPLE SIZE
DETERMINATION
IN HEALTH STUDIES
A Practical Manual
s.
K. Lwanga
Epidemiological and Statistical Methodology
World Health Organization
Geneva, Switzerland
and
S. Lemeshow
Division of Public Health
University of Massachusetts at Amherst
MA USA
World Health Organization
Geneva
1991
WHO Library Cataloguing in Publication Data
Lwanga, S K.
Sample size determination in health studies' a practical
manual
1 Sampling studies 2.Health surveys I. Lemeshow, S
II.Title
ISBN 92 4 154405 8 (NLM Classification WA 950)
©
World Health Organization 1991
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Contents
Introduction
One-sample situations
Estimating a population proportion with specified absolute
preclslOn
Estimating a population proportion with specified relative
preClSlon
Hypothesis tests for a population proportion
Two-sample situations
Estimating the difference between two population
proportions with specified absolute precision
Hypothesis tests for two population proportions
Vll
1
2
3
6
6
7
Case-control studies
Estimating an odds ratio with specified relative precision
Hypothesis tests for an odds ratio
9
10
Cohort studies
Estimating a relative risk with specified relative precision
Hypothesis tests for a relative risk
12
12
13
Lot quality assurance sampling
Accepting a population prevalence as not exceeding a specified
value
Decision rule for "rejecting a lot"
15
Incidence-rate studies
Estimating an incidence rate with specified relative precision
Hypothesis tests for an incidence rate
Hypothesis tests for two incidence rates in follow-up (cohort)
studies
17
17
17
Definitions of commonly used terms
21
Tables of minimum sample size
1. Estimating a population proportion with specified absolute
preclslOn
2. Estimating a population proportion with specified relative
preClSlon
3. Hypothesis tests for a population proportion
4. Estimating the difference between two population proportions with specified absolute precision
23
iii
9
15
15
18
25
27
29
33
Sample size determination
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
iv
Hypothesis tests for two population proportions
Estimating an odds ratio with specified relative precision
Hypothesis tests for an odds ratio
Estimating a relative risk with specified relative precision
Hypothesis tests for a relative risk
Acceptjng a population prevalence as not exceeding a
specified value
Decision rule for "rejecting a lot"
Estimating an incidence rate with specified relative precision
Hypothesis tests for an incidence rate
Hypothesis tests for two incidence rates in follow-up (cohort)
studies (study duration not fixed)
36
42
50
52
60
63
69
72
73
77
Preface
In many of WHO's Member States, surveys are being undertaken to
obtain information for planning, operating, monitoring and evaluating
health services. Central to the planning of any such survey is the decision
on how large a sample to select from the population under study, and it is
to meet the needs of health workers and managers responsible for making
that decision that this manual has been prepared. It is essentially a revised
and expanded version of a popular unpublished document on sample size
that has been widely used in WHO's field projects and training courses.
The examples and tables presented, which have been selected to cover
many of the approaches likely to be adopted in health studies, will not only
be of immediate practical use to health workers but also provide insight
into the statistical methodology of sample size determination.
The authors would like to thank Dr B. Grab, formerly Statistician, WHO,
Geneva, Dr R. J. Hayes, London School of Hygiene and Tropical
Medicine, and colleagues in the Unit of Epidemiological and Statistical
Methodology, the Diarrhoeal Diseases Control Programme and the
Expanded Programme on Immunization of WHO for their comments.
The financial support of the UNDP/World Bank/WHO Special Programme for Research and Training in Tropical Diseases is gratefully
acknowledged.
v
Introduction
Among the questions that a health worker should ask when planning a
surveyor study is "How large a sample do I need?" The answer will depend
on the aims, nature and scope of the study and on the expected result, all of
which should be carefully considered at the planning stage.
For example, in a study of the curative effect of a drug on a fatal disease
such as the acquired immunodeficiency syndrome (AIDS), where a single
positive result could be important, sample size might be considered
irrelevant. In contrast, if a new malaria vaccine is to be tested, the number
of subjects studied will have to be sufficiently large to permit comparison of
the vaccine's effects with those of existing preventive measures.
The type of "outcome" under study should also be taken into account.
There are three possible categories of outcome. The first is the simple case
where two alternatives exist: yes/no, dead/alive, vaccinated/not vaccinated,
existence of a health committee/lack of a health committee. The second
category covers multiple, mutually exclusive alternatives such as religious
beliefs or blood groups. For these two categories of outcome the data are
generally expressed as percentages or rates. The third category covers
continuous response variables such as weight, height, age and blood
pressure, for which numerical measurements are usually made. In this case
the data are summarized in the form of means and variances or their
derivatives. The statistical methods appropriate for sample size determination will depend on which of these types of outcome the investigator is
interested in.
Only once a proposed study and its objectives have been clearly defined
can a health worker decide how large a sample to select from the
population in question. This manual is intended to be a practical guide to
making such decisions. It presents a variety of situations in which sample
size must be determined, including studies of population proportion, odds
ratio, relative risk and incidence rate. 1 In each case the information needed
is specified and at least one illustrative example is given. All but one
example are accompanied by tables of minimum sample size for various
study conditions so that the reader may obtain solutions to problems of
sample size without recourse to calculations (more extensive tables are
available in the publication by Lemeshow et al. mentioned below).
Random sampling is assumed for all examples, so that if the sample is not
to be selected in a statistically random manner the tables are not valid.
1 Continuous response variables are not considered in this manual because of the wide range of possible
parameter values.
vii
Sample size determination
The manual is designed to be used in "cookbook" fashion; it neither helps
the reader to decide what type of study, confidence level or degree of
precision is most appropriate, nor discusses the theoretical basis of sample
size determination. Before using the manual, therefore, the investigator
should have decided on the study design, made a reasonable guess at the
likely result, determined what levels of significance, power and precision
(where relevant) are required and considered operational constraints such
as restrictions on time or resources. The reader who wishes to learn more
about the statistical methodology of sample size determination is referred
to Lemeshow, S. et aI., Adequacy of sample size in health studies (Chichester,
John Wiley, 1990; published on behalf of the World Health Organization)
or to any standard textbook on statistics.
viii
One-sample situations
Estimating a population proportion with specified
absolute precision
Required information
and notation
(a) Anticipated population proportion
(b) Confidence level
(c) Absolute precision required on either side
of the proportion (in percentage points)
P
100(1-1X)%
d
A rough estimate of P will usually suffice. If it is not possible to estimate P,
a figure of 0.5 should be used (as in Example 2); this is the "safest" choice for
the population proportion since the sample size required is largest when
P = 0.5. If the anticipated proportion is given as a range, the value closest
to 0.5 should be used.
Tables 1a and 1b (pages 25-26) present mInImUm sample SIzes for
confidence levels of 95% and 90%, respectively.
Simple random sampling is unlikely to be the sampling method of choice in
an actual field survey. If another sampling method is used, a larger sample
size is likely to be needed because of the "design effect". For example, for a
cluster sampling strategy the design effect might be estimated as 2. This
would mean that, to obtain the same precision, twice as many individuals
would have to be studied as with the simple random sampling strategy. In
Example 2, for instance, a sample size of 192 would be needed.
Example 1
Solution
A local health department wishes to estimate the prevalence of tuberculosis among children under five years of age in its locality. How many
children should be included in the sample so that the prevalence may be
estimated to within 5 percentage points of the true value with 95%
confidence, if it is known that the true rate is unlikely to exceed 20%?
(a) Anticipated population proportion
(b) Confidence level
(c) Absolute precision (15%-25%)
20%
95%
5 percentage points
Table 1a (page 25)shows that for P = 0.20 and d = 0.05 a sample size of 246
would be needed.
If it is impractical, with respect to time and money, to study 246 children,
the investigators should lower their requirements of confidence to, per-
Sample size determination
haps, 90%. Table 1b (page 26) shows that, in this case, the required sample
size would be reduced to 173.
Example 2
Solution
An investigator working for a national programme of immunization seeks
to estimate the proportion of children in the country who are receiving
appropriate childhood vaccinations. How many children must be studied
if the resulting estimate is to fall within 10 percentage points of the true
proportion with 95% confidence? (It is not possible to make any assumption regarding the vaccination coverage.)
(a) Anticipated population proportion
("safest" choice, since P is unknown)
(b) Confidence level
(c) Absolute precision (40%-60%)
50%
95%
10 percentage points
Table 1a (page 25) shows that for P=0.50 and d=O.1O a sample size of96
would be required.
Estimating a population proportion with specified
relative precision
Required information
and notation
(a) Anticipated population proportion
(b) Confidence level
(c) Relative precision
P
100(1-IX)%
E
The choice of P for the sample size computation should be as "conservative" (small) as possible, since the smaller P is the greater is the minimum
sample size.
Tables 2a and 2b (pages 27-28) present mmImum sample SIzes for
confidence levels of 95% and 90%, respectively.
Example 3
Solution
An investigator working for a national programme of immunization seeks
to estimate the proportion of children in the country who are receiving
appropriate childhood vaccinations. How many children must be studied
if the resulting estimate is to fall within 10% (not 10 percentage points) of
the true proportion with 95% confidence? (The vaccination coverage is not
expected to be below 50%.)
(a) Anticipated population proportion
(conservative choice)
(b) Confidence level
(c) Relative precision (from 45% to 55%)
50%
95%
10% (of 50%)
Table 2a (page 27) shows that for P = 0.50 and E = 0.10 a sample size of 384
would be needed.
If it is impractical, with respect to time and money, to study 384 children,
the investigators should lower their requirements of confidence to, per2
One-sample situations
haps, 90%. Table 2b (page 28) shows that, in this case, the required sample
size would be reduced to 271.
Simple random sampling is unlikely to be the sampling method of choice in
an actual field survey. If another sampling method is used, a larger sample
size is likely to be needed because of the "design effect". For example, for a
cluster sampling strategy the design effect might be estimated as 2. This
would mean that, to obtain the same precision, twice as many individuals
would have to be studied as with the simple random sampling strategy. In
this example, therefore, for a confidence level of 95%, a sample size of 768
would be needed.
Example 4
Solution
How large a sample would be required to estimate the proportion of
pregnant women in a population who seek prenatal care within the first
trimester of pregnancy, to within 5% of the true value with 95%
confidence? It is estimated that the proportion of women seeking such care
will be between 25% and 40%.
(a) Anticipated population proportion
(b) Confidence level
(c) Relative precision
25%-40%
95%
5% (of 25%-40%)
Table 2a (page 27) presents the following sample sizes for
population proportions in the range 25%-40%.
P
Sample size
0.25
0.30
0.35
0.40
4610
3585
2854
2305
E=
0.05 and for
Therefore a study of roughly 4610 women might be planned to satisfy the
stated objectives. If necessary a smaller sample size could be used, but this
would result in a loss of precision or confidence or both if the true value of
P was close to 25%.
Hypothesis tests for a population proportion
This section applies to studies designed to test the hypothesis that the
proportion of individuals in a population possessing a given characteristic
is equal to a particular value.
Required information
and notation
(a) Test value of the population proportion
under the null hypothesis
(b) Anticipated value of the population proportion
(c) Level of significance
(d) Power of the test
(e) Alternative hypothesis: either
or
Po
Pa
1000(%
100(1- fJ)%
Pa>P o or Pa<P o
(for one-sided test)
Pai=P o
(for two-sided test)
3
Sample size determination
Tables 3a-d (pages 29-32) present minimum sample sizes for a level of
significance of 5%, powers of90% and 80%, and both one-sided and twosided tests. For Tables 3c and 3d the complement of Po should be used as
the column value whenever Po> 0.5.
Example 5
Solution
The five-year cure rate for a particular cancer (the proportion of patients
free of cancer five years after treatment) is reported in the literature to be
50%. An investigator wishes to test the hypothesis that this cure rate
applies in a certain local health district. What minimum sample size would
be needed if the investigator was interested in rejecting the null hypothesis
only if the true rate was less than 50%, and wanted to be 90% sure of
detecting a true rate of 40% at the 5% level of significance?
(a) Test cure rate
(b) Anticipated cure rate
( c ) Level of significance
( d) Power of the test
(e) Alternative hypothesis (one-sided test)
50%
40%
5%
90%
cure rate < 50%
Table 3a (page 29) shows that for Po = 0.50 and P a = 0.40 a sample size of
211 would be needed.
Example 6
Solution
Previous surveys have demonstrated that the usual prevalence of dental
caries among schoolchildren in a particular community is about 25%.
How many children should be included in a new survey designed to test for
a decrease in the prevalence of dental caries, if it is desired to be 90% sure
of detecting a rate of 20% at the 5% level of significance?
(a) Test caries rate
(b) Anticipated caries rate
( c ) Level of significance
( d) Power of the test
(e) Alternative hypothesis (one-sided test)
25%
20%
5%
90%
caries rate < 25%
Table 3a (page 29) shows that for Po = 0.25 and P a = 0.20 a sample size of
601 would be needed.
If the investigators use this sample size, and if the actual caries rate is less
than 20%, then the power of the test will be larger than 90%, i.e. they will
be more than 90% likely to detect that rate.
Example 7
Solution
The success rate for a surgical treatment of a particular heart condition is
widely reported in the literature to be 70%. A new medical treatment has
been proposed that is alleged to offer equivalent treatment success. A
hospital without the necessary facilities or staff to provide the surgical
treatment has decided to use the new medical treatment for all new
patients presenting with this condition. How many patients must be
studied to test the hypothesis that the success rate of the new method of
treatment is 70% against an alternative hypothesis that it is not 70% at the
5% level of significance? The investigators wish to have a 90% power of
detecting a difference between the success rates of 10 percentage points or
more in either direction.
(a) Test success rate
(b) Anticipated success rate
4
70%
80% or 60%
One-sample situations
( c ) Level of significance
( d) Power of the test
( e ) Alternative hypothesis (two-sided test)
5%
90%
success rate #- 70%
Table 3c (page 31) shows that for (1- Po) = 0.30 and iFa - Pol = 0.1 0 a
sample size of 233 would be needed.
Example 8
Solution
In a particular province the proportion of pregnant women provided with
prenatal care in the first trimester of pregnancy is estimated to be 40% by
the provincial department of health. Health officials in another province
are interested in comparing their success at providing prenatal care with
these figures. How many women should be sampled to test the hypothesis
that the coverage rate in the second province is 40% against the alternative
that it is not 40%? The investigators wish to be 90% confident of detecting
a difference of 5 percentage points or more in either direction at the 5%
level of significance.
(a) Test coverage rate
(b) Anticipated coverage rate
( c ) Level of significance
( d) Power of the test
(e) Alternative hypothesis (two-sided test)
40%
35% or 45%
5%
90%
coverage rate #- 40%
Table 3c (page 31) shows that for Po =0.40 and iFa-Pol=0.05 a sample
size of 1022 would be needed.
5
Two-sample situations
Estimating the difference between two population
proportions with specified absolute precision
Required information
and notation
(a) Anticipated population proportions
( b) Confidence level
(c) Absolute precision required on either side
PI and P z
100(1-a)%
of the true value of the difference between
the proportions (in percentage points)
d
(d) Intermediate value
V=P I (1-Pd+P 2 (1-P z )
For any value of d, the sample size required will be largest when both PI
and P z are equal to 50%; therefore if it is not possible to estimate either
population proportion, the "safest" choice of 0.5 should be used in both
cases.
The value of V may be obtained directly from Table 4a (page 33) from the
column corresponding to P z (or its complement) and the row corresponding to PI (or its complement).
Tables 4b and 4c (pages 34-35) present mInImUm sample SIzes for
confidence levels of 95% and 90%, respectively.
Example 9
Solution
What size sample should be selected from each of two groups of people to
estimate a risk difference to within 5 percentage points of the true
difference with 95 % confidence, when no reasonable estimate of PI and P z
can be made?
(a) Anticipated population proportions
("safest " choice)
( b ) Confidence level
( c ) Absolute precision
(d) Intermediate value
50%,50%
95%
5 percentage points
0.50
Table 4b (page 34) shows that for d = 0.05 and V = 0.50 a sample size of 769
would be needed in each group.
Example 10
In a pilot study of 50 agricultural workers in an irrigation project, it was
observed that 40% had active schistosomiasis. A similar pilot study of 50
agricultural workers not employed on the irrigation project demonstrated
that 32% had active schistosomiasis. If an epidemiologist would like to
carry out a larger study to estimate the schistosomiasis risk difference to
6
Two-sample situations
within 5 percentage points of the true value with 95% confidence, how
many people must be studied in each of the two groups?
Solution
(a)
( b)
( c)
( d)
Anticipated population proportions
Confidence level
Absolute precision
In termedia te value
40%,32%
95%
5 percentage points
0.46
Table 4b (page 34) shows that for d = 0.05 and V = 0.46 a sample size of 707
would be needed in each group.
Hypothesis tests for two population proportions
This section applies to studies designed to test the hypothesis that two
population proportions are equal. For studies concerned with very small
proportions, see Example 13.
Required information
and notation
(a) Test value of the difference between the population
proportions under the null hypothesis
(b) Anticipated values of the population
proportions
PI and P 2
( c) Level of significance
1000:%
( d) Power of the test
100(1- /3)%
( e) Alternative hypothesis: either
P 1 - P 2 > 0 or P 1 - P 2 < 0
(for one-sided test)
or
P 1 -P 2 ¥0
(for two-sided test)
Tables 5a-h (pages 36-41) present minimum sample sizes for a level of
significance of 5 %, powers of 90% and 80%, both one-sided and two-sided
tests, and the special case of very small proportions. Tables 5e-h should be
used whenever the proportion under consideration is less than 5%.1
Example 11
It is believed that the proportion of patients who develop complications
after undergoing one type of surgery is 5% while the proportion of patients
who develop complications after a second type of surgery is 15%. How
large should the sample size be in each of the two groups of patients if an
investigator wishes to detect, with a power of 90%, whether the second
procedure has a complication rate significantly higher than the first at the
5% level of significance?
Solution
0%
(a) Test value of difference in complication rates
5%,15%
(b) Anticipated complication rates
( c ) Level of significance
5%
( d ) Power of the test
90%
(e) Alternative hypothesis
risk difference (P 1 - P 2) < 0%
(one-sided test)
1 For further discussion of small proportions, see Lemeshow, S. et ai., Adequacy of sample size in health
studies (Chichester, John Wiley, 1990; published on behalf of the World Health Organization).
7
Sample size determination
Table 5a (page 36) shows that for PI = 0.05 and P 2 = 0.15 a sample size of
153 would be needed in each group.
Example 12
Solution
In a pilot survey in a developing country, an epidemiologist compared a
sample of 50 adults suffering from a certain neurological disease with a
sample of 50 comparable control subjects who were free of the disease.
Thirty of the subjects with the disease (60%) and 25 of the controls (50%)
were involved in fishing-related occupations. If the proportion of people
involved in fishing-related occupations in the entire population is similar
to that observed in the pilot survey, how many subjects should be included
in a larger study in each of the two groups if the epidemiologist wishes to be
90% confident of detecting a true difference between the groups at the 5%
level of significance?
(a) Test value of difference between proportions involved in
fishing-related occupations
0%
(b) Anticipated proportions involved in fishing-related
occupations
( c ) Level of significance
( d ) Power of the test
(e) Alternative hypothesis (two-sided test)
60%,50%
5%
90%
risk difference # 0%
The required sample size is obtained from Table 5c (page 38) from the
column corresponding to the smallest of PI' P 2 and their complements
and the row corresponding to IP 2 - P 11. In this case, for (1- PI) = 0.40
and I P 2 - P 11 = 0.10, the required sample size would be 519 in each group.
Example 13
Solution
Two communities are to participate in a study to evaluate a new screening
programme for early identification of a particular type of cancer. In one
community the screening programme will include all adults over the age of
35, whereas in the second community the procedure will not be used at all.
The annual incidence of the type of cancer under study is 50 per 100000
( = 0.0005) in an unscreened popUlation. A drop in the rate to 20 per
100000 (= 0.0002) would justify using the procedure on a widespread
basis. How many adults should be included in the study in each of the two
communities if the investigators wish to have an 80% probability of
detecting a drop in the incidence of this magnitude at the 5% level of
significance?
(a) Test difference in cancer rates
( b ) Anticipated cancer rates
( c ) Level of significance
( d) Power of the test
(e) Alternative hypothesis
(one-sided test)
0%
0.05 %, 0.02 %
5%
80%
risk difference (P 1 - P 2) > 0%
Table 5f (page 40) shows that for PI = 0.0005 and P 2 = 0.0002 a sample
size of 45 770 would be needed in each group.
8
Case-control studies
Examples 14 and 15 concern the odds ratio, which is the ratio ofthe odds of
occurrence of an event in one set of circumstances to the odds of its
occurrence in another. For example, if the "event" is a disease, people with
and without the disease may be classified with respect to exposure to a
given variable:
Disease
No disease
Exposed
Unexposed
a
c
b
d
The odds ratio is then ad/be.
Estimating an odds ratio with specified relative
precision
Required information
and notation
(a) Two of the following should be known:
• Anticipated probability of "exposure"
for people with the disease [a/(a + b)]
• Anticipated probability of "exposure"
for people without the disease [c/(c + d)]
• Anticipated odds ratio
(b) Confidence level
(c) Relative precision
Pi
Pi
OR
100(1 - Ct)%
[;
When the number of people in the population who are affected by the
disease is small relative to the number of people unaffected:
e~(a+c)
and
d ~(b + d).
In this case, therefore, the probability of "exposure" given "no disease"
(P i) is approximated by the overall exposure rate.
Tables 6a-h (pages 42-49) present minimum sample sizes for confidence
levels of 95% and 90% and relative precisions of 10%, 20%, 25% and
50%.
F or determining sample size from Table 6 when 0 R ~ 1, the values of both
Pi and OR are needed. Either of these may be calculated, if necessary,
9
Sample size determination
provided that Pi is known:
OR
=
[Pf/(l- Pi)]/[Pi/(1 - P!)]
P!
=
Pf/[OR(l- Pi) + PiJ.
and
If OR < 1, the values of Pi and 1/0R should be used instead.
Example 14
Solution
In a defined area where cholera is posing a serious public health problem,
about 30% of the population are believed to be using water from
contaminated sources. A case-control study of the association between
cholera and exposure to contaminated water is to be undertaken in the
area to estimate the odds ratio to within 25% of the true value, which is
believed to be approximately 2, with 95% confidence. What sample sizes
would be needed in the cholera and control groups?
Anticipated probability of "exposure" given "disease"
Anticipated probability of "exposure" given "no disease"
(approximated by overall exposure rate)
Anticipated odds ratio
( b) Confidence level
(c) Relative precision
(a)
Table 6c (page 44) shows that for OR
would be needed in each group.
=
2 and P!
=
?
30%
2
95%
25%
0.3 a sample size of 408
Hypothesis tests for an odds ratio
This section outlines how to determine the minimum sample size for
testing the hypothesis that the population odds ratio is equal to one.
Required information
and notation
(a) Test value of the odds ratio under the null hypothesis
(b) Two of the following should be known:
• Anticipated probability of "exposure"
for people with the disease [al(a + b)]
• Anticipated probability of "exposure"
for people without the disease [cl(c + d)]
• Anticipated odds ratio
( c) Level of significance
( d) Power of the test
(e) Alternative hypothesis
(for two-sided test)
ORo=1
P*1
P*2
ORa
100a%
100(1- fJ)%
Tables 7a and 7b (pages 50-51) present minimum sample sizes for a level of
significance of 5% and powers of 90% and 80% in two-sided tests.
For determining sample size from Table 7 when ORa> 1, the values of
both P! and OR are needed. Either of these may be calculated, if necessary,
10
Case--control studies
provided that Pi is known:
ORa
=
[Pi/(I - Pi)]/[P!I(I- Pi)]
Pi
=
Pi/[ORa(1- Pi)
and
+ PiJ.
If ORa < I, the values of Pi and IIOR a should be used instead.
Example 15
The efficacy of BeG vaccine in preventing childhood tuberculosis is in
doubt and a study is designed to compare the vaccination coverage rates in
a group of people with tuberculosis and a group of controls. Available
information indicates that roughly 30% of the controls are not vaccinated.
The investigators wish to have an 80% chance of detecting an odds ratio
significantly different from 1 at the 5 % level. If an odds ratio of 2 would be
considered an important difference between the two groups, how large a
sample should be included in each study group?
Solution
1
(a) Test value of the odds ratio
?
(b) Anticipated probability of "exposure" given "disease"
Anticipated probability of "exposure" given "no disease"
30%
2
Anticipated odds ratio
(c) Level of significance
5%
80%
(d) Power of the test
odds ratio # 1
(e) Alternative hypothesis
Table 7b (page 51) shows that for OR
130 would be needed in each group.
11
=
2 and Pi
=
0.30 a sample size of
Cohort studies
Estimating a relative risk with specified relative
precision
Required information
and notation
(a) Two of the following should be known:
• Anticipated probability of disease in people
exposed to the factor of interest
• Anticipated probability of disease in people
not exposed to the factor of interest
• Anticipated relative risk
(h) Confidence level
(c) Relative precision
P2
RR
100(I-IX)%
B
Tables 8a-h (pages 52-59) present minimum sample sizes for confidence
levels of95% and 90%, and levels of precision of 10%, 20%, 25% and
50%.
For determining sample size from Table 8 when RR ~ 1, the values of both
P 2 and RR are needed. Either of these may be calculated, if necessary,
provided that P I is known:
and
If RR< 1, the values of PI and I/RR should be used instead.
Example 16
Solution
An epidemiologist is planning a study to investigate the possibility that a
certain lung disease is linked with exposure to a recently identified air
pollutant. What sample size would be needed in each of two groups,
exposed and not exposed, if the epidemiologist wishes to estimate the
relative risk to within 50% of the true value (which is believed to be
approximately 2) with 95% confidence? The disease is present in 20% of
people who are not exposed to the air pollutant.
(a) Anticipated probability of disease given "exposure"
Anticipated probability of disease given "no exposure"
Anticipated relative risk
(b) Confidence level
(c) Relative precision
?
20%
2
95%
50%
Table 8d (page 55) shows that for RR = 2 and P 2 = 0.20 a sample size of 44
would be needed in each group.
12
Cohort studies
Hypothesis tests for a relative risk
This section outlines how to determine the minimum sample size for
testing the hypothesis that the population relative risk is equal to one.
Required information
and notation
(p) Test value of the relative risk under the null hypothesis
(h) Two of the following should be known:
RRo = 1
• Anticipated probability of disease in people exposed
PI
to the variable
• Anticipated probability of disease in people not
exposed to the variable
Pz
• Anticipated relative risk
RRa
( c) Level of significance
100()(%
(d) Power of the test
100(1-rn%
( e) Alternative hypothesis
(for two-sided test)
Tables 9a~c (pages 60~62) present minimum sample sizes for a level of
significance of 5% and powers of 90%, 80% and 50% in two-sided tests.
For determining sample size from Table 9 when RRa > 1, the values of both
P z and RRa are needed. Either of these may be calculated, if necessary,
provided that P I is known:
RRa= PI/P z
and
If RRa < 1, the values of PI and l/RRa should be used instead.
Example 17
Two competing therapies for a particular cancer are to be evaluated by a
cohort study in a multicentre clinical trial. Patients will be randomized to
either treatment A or treatment B and will be followed for 5 years after
treatment for recurrence of the disease. Treatment A is a new therapy that
will be widely used if it can be demonstrated that it halves the risk of
recurrence in the first 5 years after treatment (i.e. RRa = 0.5); 35 %
recurrence is currently observed in patients who have received treatment
B. How many patients should be studied in each of the two treatment
groups if the investigators wish to be 90% confident of correctly rejecting
the null hypothesis (RRo = 1), if it is false, and the test is to be performed at
the 5 % level of significance?
Solution
1
?
35%
Anticipated probability of recurrence given treatment B
0.5
Anticipated relative risk
5%
( c ) Level of Significance
90%
( d) Power of the test
rela
ti
ve
risk
=I- 1
(e) Alternative hypothesis
(a) Test value of the relative risk
(b) Anticipated probability of recurrence given treatment A
13
Sample size determination
Table 9a (page 60) shows that for RRa = 0.5 (1/ RRa = 2) and P 2 = 0.35 (P 1
= 0.175) a sample size of 135 would be needed in each group (figure
obtained by interpolation; the exact sample size is 131 by computation).
14
Lot quality assurance sampling
Accepting a population prevalence as not exceeding
a specified value
This section outlines how to determine the minimum sample size that
should be selected from a given population so that, if a particular
characteristic is found in no more than a specified number of sampled
individuals, the prevalence of the characteristic in the population can be
accepted as not exceeding a certain value.
Required information
and notation
(a) Anticipated population prevalence
P
(b) Population size
(c) Maximum number of sampled individuals showing
N
characteristic
( d) Confidence level
d*
100(1-C()%
Tables lOa-j (pages 63-68) present minimum sample sizes for confidence
levels of 95% and 90% and values of d* of 0-4.
Example 18
Solution
In a school of 2500 children, how many children should be examined so
that if no more than two are found to have malaria parasitaemia it can be
concluded, with 95% confidence, that the malaria prevalence in the school
is no more than 10%?
(a) Anticipated population prevalence
( h) Population size
(c) Maximum number of malaria cases in the sample
( d) Confidence level
10%
2500
2
95%
Table 10c (page 64) shows that for P = 0.1 0 and N = 2500 a sample size
of 61 children would be needed.
Decision rule for "rejecting a lot"
This section applies to studies designed to test whether a "lot" ( a sampled
population) meets a specified standard. The null hypothesis is that the
proportion of individuals in the population with a particular characteristic
is equal to a given value, and a one-sided test is set up such that the lot is
accepted as meeting the specified standard only if the null hypothesis can
15
Sample size determination
be rejected. For this purpose a "threshold value" of individuals with the
characteristic (d*) is computed as a basis for a decision rule; if the number
of sampled individuals found to possess the characteristic does not exceed
the threshold, the null hypothesis is rejected (and the lot is accepted),
whereas if the threshold is exceeded, the lot is rejected.
Required information
and notation
(a) Test value of the population proportion under
the null hypothesis
(b) Anticipated value of the population proportion
( c) Level of significance
( d) Power of the test
Po
Pa
1001):%
100(1-(1)%
Tables lla-c (pages 69-71) present minimum sample sizes for a level of
significance of 5% and powers of 90%, 80% and 50% in one-sided tests.
Example 19
Solution
In a large city, the local health authority aims at achieving a vaccination
coverage of 90% of all eligible children. In response to concern about
outbreaks of certain childhood diseases in particular parts of the city, a
team of investigators from the health authority is planning a survey to
identify areas where vaccination coverage is 50% or less so that appropriate action may be taken. How many children should be studied, as a
minimum, in each area and what threshold value should be used if the
study is to test the hypothesis that the proportion of children not
vaccinated is 50% or more, at the 5% level of significance? The investigators wish to be 90% sure of recognizing areas where the target
vaccination coverage has been achieved (i.e. where only 10% of children
have not been fully vaccinated).
(a) Test value of the population proportion
(b) Anticipated value of the population proportion
( c ) Level of significance
( d) Power of the test
50%
10%
5%
90%
Because the mistake of accepting groups of children as adequately
vaccinated, when in fact the coverage is 50% or less, is the more important,
Po=0.50 and Pa=O.lO. Table lla (page 69) shows that in this case a
sample size of 10 and a threshold value of 2 should be used.
Therefore, a sample of 10 children should be taken from each of the areas
under study. If more than 2 children in a sample are found not to have been
adequately vaccinated, the lot (the sampled population) should be "rejected", and the health authority may take steps to improve vaccination
coverage in that particular area. If, however, only 2 (or fewer) children are
found to be inadequately vaccinated, the null hypothesis should be rejected
and the group of children may be accepted as not being of immediate
priority for an intensified vaccination campaign.
16
Incidence-rate studies
Estimating an incidence rate with specified relative
precision
Required information
and notation
(a) Relative precision
(b) Confidence level
f:
100(1- et)%
Table 12 (page 72) presents minimum sample sizes for confidence levels of
99%,95% and·90%.
Example 20
Solution
How large a sample of patients should be followed up if an investigator
wishes to estimate the incidence rate of a disease to within 10% of its true
value with 95% confidence?
(a) Relative precision
(b) Confidence level
10%
95%
Table 12 shows that for [; = 0.10 and a confidence level of 95% a sample
size of 385 would be needed.
Hypothesis tests for an incidence rate
This section applies to studies designed to test the hypothesis that the
incidence rate of a characteristic is equal to a particular value.
Required information
and notation
(a) Test value of the popUlation incidence rate under
the null hypothesis
(b) Anticipated value of the population incidence rate
(c) Level of significance
(d) Power of the test
Ao
Aa
100et%
100(1- fJ)%
(e) Alternative hypothesis: either
or
Aa > Ao or Aa < Ao
(for one-sided test)
Aa #- Ao (for two-sided test)
Tables 13a-d (pages 73-76) present minimum sample sizes for a level of
significance of 5 %, powers of 90% and 80% and both one-sided and twosided tests.
17
Sample size determination
Example 21
Solution
On the basis of a 5-year follow-up study of a small number of people, the
annual incidence rate of a particular disease is reported to be 40%. What
minimum sample size would be needed to test the hypothesis that the
population incidence rate is 40% at the 5% level of significance? It is
desired that the test should have a power of90% of detecting a true annual
incidence rate of 50% and the investigators are interested in rejecting the
null hypothesis only if the true rate is greater than 40%.
(a) Test value of the incidence rate
(b) Anticipated incidence rate
( c ) Level of significance
( d) Power of the test
(e) Alternative hypothesis (one-sided test)
Table 13a (page 73) shows that for ..1.0
sample size of 169 would be needed.
=
40%
50%
5%
90%
incidence rate> 40%
0.40 and Aa
=
0.50 a minimum
Hypothesis tests for two incidence rates in followup (cohort) studies
This section applies to studies designed to test the hypothesis that the true
incidence rates of a disorder or characteristic in two groups of individuals
are equal. Subjects either have a common date of entry into the study and
are followed up until they develop the characteristic in question or cannot
be followed up any more (Example 22), or are inducted into the study as
they become available but are followed up only until a specified date
(Example 23).
Required information
and notation
(a) Test value of the difference between the popu-
lation incidence rates under the null hypothesis
. 1. 1 -..1. 2=0
Anticipated values of the incidence rates
Al and ..1.2
Level of significance
100a%
Power of the test
100(1-[3)%
Alternative hypothesis: either
)'1 - )'2 > 0 or Al - ..1.2 < 0
(for one-sided test)
or
..1.1-..1.2 #0
(for two-sided test)
(f) Duration of study (if fixed)
T
(b)
( c)
( d)
( e)
If the study is terminated at a fixed point in time, before all the subjects
have necessarily experienced the end-point of interest, the observations are
said to be censored. The values of A then have to be modified according to
the formula
as in Example 23.
Tables 14a-d (pages 77-80) present minimum sample sizes for a level of
significance of 5%, powers of 90% and 80% and both one-sided and two18
Incidence-rate studies
sided tests, when the duration of the study is not fixed and the two groups
studied are of equal size. No tables are given for studies of fixed duration
because too many parameters are involved to permit easy tabulation.
Example 22
Solution
As part of a study of the long-term effect of noise on workers in a
particularly noisy industry, it is planned to follow up a cohort of people
who were recruited into the industry during a given period of time and to
compare them with a similar cohort of individuals working in a much
quieter industry. Subjects will be followed up for the rest of their lives or
until their hearing is impaired. The results of a previous small-scale survey
suggest that the annual incidence rate of hearing impairment in the noisy
industry may be as much as 25%. How many people should be followed up
in each of the groups (which are to be of equal size) to test the hypothesis
that the incidence rates for hearing impairment in the two groups are the
same, at the 5% level of significance and with a power of 80%? The
alternative hypothesis is that the annual incidence rate for hearing
impairment in the quieter industry is not more than the national average of
about 10% (for people in the same age range), whereas in the noisy
industry it differs from this.
(a) Test value of the difference in incidence rates
(b) Anticipated incidence rates
( c ) Level of significance
( d) Power of the test
( e ) Alternative hypothesis (two-sided test)
(f) Duration of study
o
25% and 10%
5%
80%
Al =P )'2
not applicable
Table 14d (page 80) shows that for Al = 0.25 and A2 = 0.10 a sample size of
23 would be required in each group.
Example 23
Solution
A study similar to that outlined in Example 22 is to be undertaken, but the
duration of the study will be limited to 5 years. How many subjects should
be followed up in each group?
o
(a) Test value of the difference in incidence rates
(b) Anticipated incidence rates
25% and 10%
5%
80%
Al =P A2
5 years
( c) Level of significance
( d) Power of the test
(e) Alternative hypothesis (two-sided test)
(f) Duration of study
The values of ), must be modified according to the formula for f(A) given
on page 18:
f(I =0.175) = 0.0918 where X = (AI
f(AI =0.25) = 0.1456
f(A2 =0.10) = 0.0469.
+ A2 )/2
The appropriate sample size formula is
where k is the ratio of the sample size for the second group of subjects (n 2 )
to that for the first group (nd (in this example k = 1).
19
Sample size determination
Thus
n1
= {1.96)[2(0.0918)J + 0.842)(0.1456 + 0.0469)}2/(0.25 - 0.10)2
= 1.462/0.023
=
65.0.
A sample size of 65 would therefore be needed for each group.
For a one-sided test the corresponding sample size formula is
20
Definitions of commonly used
terms
The brief definitions listed here are intended to serve only as reminders for
the reader. Fuller explanations of statistical terms and a discussion of the
statistical theory relevant to sample size determination are to be found in
Lemeshow, S. et al., Adequacy of sample size in health studies (Chichester,
John Wiley, 1990; published on behalf of the World Health Organization).
a
The significance level of a test the probability of rejecting the null
hypothesis when it is true (or the probability of making a Type I error).
p
The probability of failing to reject the null hypothesis when it is false (or
the probability of making a Type II error).
Case--control studies
Studies in which subjects are selected on the basis of their status with
respect to a given characteristic (such as the presence of a disease); the
"cases" show the characteristic and the "controls" do not. Both groups are
studied with respect to their prior and current exposure to suspected risk
factors.
Cluster sampling
A sampling process in which sampling units are made up of clusters or
groups of study units.
Cohort studies
Studies in which subjects are selected with respect to the presence and
absence of a characteristic (such as exposure to a given factor) suspected of
being associated with the particular outcome of interest (for example a
disease). Both groups of subjects are followed up for development of the
outcome.
Confidence level
The probability that an estimate of a population parameter is within
certain specified limits of the true value; commonly denoted by "I-a".
Design effect
In cluster sampling, the design effect is an indication of the variation due to
clustering. It is estimated by the ratio of the variance when cluster sampling
is used to the variance when simple random sampling is used.
Incidence rate
The number of specific events (for example new cases of a disease)
occurring in a specified population per unit time.
Lot quality assurance
sampling
Sampling techniques, with industrial origins, designed to ascertain
whether batches of items meet specified standards.
Null hypothesis
A statement concerning the value of a population parameter. It is the
hypothesis under test in a test of significance, for example the hypothesis
that an observed difference is entirely due to sampling errOr.
21
Sample size determination
Odds ratio
The ratio of the odds of occurrence of an event in one set of circumstances
to the odds of its occurrence in another (see also page 9).
One-sided test
In hypothesis testing, when the difference being tested is directionally
specified beforehand (for example when Xl < X 2, but not Xl> X 2, is being
tested against the null hypothesis Xl = X 2).
Population proportion
The proportion of individuals in a population possessing a given characteristic.
Power of a test
The probability of correctly rejecting the null hypothesis when it is false;
commonly denoted by "1 - {J".
Precision
A measure of how close an estimate is to the true value of a population
parameter. It may be expressed in absolute terms or relative to the
estimate.
Prevalence
The number of cases of a disease (or people with a particular characteristic)
existing in a specified population at a given point in time.
Relative risk
The ratio of the risk (probability) of an outcome (for example disease or
death) among people exposed to a given factor to the risk among people
not exposed.
Significance level
See definition of iY..
Simple random sampling
Sampling procedure in which every study unit has the same chance of being
selected and every sample of the same size has the same chance of being
chosen.
Study units
The individual members of a population whose characteristics are to be
measured.
Two-sided test
In hypothesis testing, when the difference being tested for significance is
not directionally specified beforehand (for example when the test takes no
account of whether Xl> X 2 or Xl < X 2).
Represent the number of standard errors from the mean; Zl-" and Zl-,,/2
are functions of the confidence level and Z 1 _ p is a function of the power of
the test.
22
Tables of minimum sample size
Table 1. Estimating a population proportion with specified absolute precision
n = zi -a/2P(1- P)/d
2
(a) Confidence level 95%
p
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.20
0.25
1825
456
203
114
73
51
37
29
23
18
15
13
11
9
8
5
3457
864
384
216
138
96
71
54
43
35
29
24
20
18
15
9
6
4898
1225
544
306
196
136
100
77
60
49
40
34
29
25
22
12
8
6147
1537
683
384
246
171
125
96
76
61
51
43
36
31
27
15
10
7203
1801
800
450
288
200
147
113
89
72
60
50
43
37
32
18
12
8067
2017
896
504
323
224
165
126
100
81
67
56
48
41
36
20
13
8740
2185
971
546
350
243
178
137
108
87
72
61
52
45
39
22
14
9220
2305
1024
576
369
256
188
144
114
92
76
64
55
47
41
23
15
9508
2377
1056
594
380
264
194
149
117
95
79
66
56
49
42
24
15
9604
2401
1067
600
384
267
196
150
119
96
79
67
57
49
43
24
15
9508
2377
1056
594
380
264
194
149
117
95
79
66
56
49
42
24
15
9220
2305
1024
576
369
256
188
144
114
92
76
64
55
47
41
23
15
8740
2185
971
546
350
243
178
137
108
87
72
61
52
45
39
22
14
8067
2017
896
504
323
224
165
126
100
81
67
56
48
41
36
20
7203
1801
800
450
288
200
147
113
89
72
60
50
43
37
32
18
12
6147
1537
683
384
246
171
125
96
76
61
51
43
36
31
27
15
10
4898
1225
544
306
196
136
100
77
60
49
40
34
29
25
22
12
8
3457
864
384
216
138
96
71
54
43
35
29
24
20
18
15
9
6
1825
456
203
114
73
51
37
29
23
18
15
13
11
9
8
5
d
~I
'Sample size less than 5.
13
I~
Table 1
( continued)
(b) Confidence level 90%
p
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.20
0.25
1285
321
143
80
51
36
26
20
16
13
11
9
8
7
6
2435
609
271
152
97
68
50
38
30
24
20
17
14
12
11
6
3450
863
383
216
138
96
70
4330
1082
481
271
173
120
88
68
53
43
36
30
26
22
19
11
7
5074
1268
564
317
203
141
104
79
63
51
42
35
30
26
23
13
8
5683
1421
631
355
227
158
116
89
70
57
47
39
34
29
25
14
9
6156
1539
684
385
246
171
126
96
76
62
51
43
36
31
27
15
10
6494
1624
722
406
260
180
133
101
80
65
54
45
38
33
29
16
10
6697
1674
744
419
268
186
137
105
83
67
55
47
40
34
30
17
11
6765
1691
752
423
271
188
138
106
84
68
56
47
40
35
30
17
11
6697
1674
744
419
268
186
137
105
83
67
55
47
40
34
30
17
11
6494
1624
722
406
260
180
133
101
80
65
54
45
38
33
29
16
10
6156
1539
684
385
246
171
126
96
76
62
51
43
36
31
27
15
10
5683
1421
631
355
227
158
116
89
70
57
47
39
34
29
25
14
9
5074
1268
564
317
203
141
104
79
63
51
42
35
30
26
23
13
8
4330
1082
481
271
173
120
88
68
53
43
36
30
26
22
19
11
7
3450
863
383
216
138
96
70
2435
609
271
152
97
68
50
54
38
43
35
29
24
20
18
15
9
6
30
1285
321
143
80
51
36
26
20
16
13
11
9
8
7
6
d
~I
'Sample size less than 5.
54
43
35
29
24
20
18
15
9
6
24
20
17
14
12
11
6
*
*
en
III
3
"CI
iD
1/1
jij'
CD
Q.
....
...
CD
CD
3
S'
....
III
0'
:s
Table 2. Estimating a population proportion with specified relative precision
n=zi _>/z(l-P)/sz P
(a) Confidence level 95%
p
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
31431
7858
3492
1964
1257
873
641
491
388
314
140
79
50
35
26
20
13
25611
6403
2846
1601
1024
711
523
400
316
256
114
64
41
28
21
16
10
20686
5171
2298
1293
827
575
422
323
255
207
92
52
33
23
17
13
8
16464
4116
1829
1029
659
457
336
257
203
165
73
41
26
18
13
10
7
12805
3201
1423
800
512
356
261
200
158
128
57
32
20
14
10
8
5
9604
2401
1067
600
384
267
196
150
119
96
43
24
15
11
8
6
6779
1695
753
424
271
188
138
106
84
68
30
17
11
8
6
4268
1067
474
267
171
119
87
67
53
43
19
11
7
5
2022
505
225
126
81
56
41
32
25
20
9
5
G
tjl
0.01 729904 345744
0.02 182476 86436
0.03 81100 38416
0.04 45619 21609
0.05 29196 13830
0.06 20275 9604
0.07 14896 7056
0.08 11405 5402
0.09
9011
4268
0.10
3457
7299
0.15
3244
1537
864
0.20
1825
0.25
1168
553
0.30
811
384
282
0.35
596
216
0.40
456
138
0.50
292
• Sample size less than 5.
217691 153664 115248 89637 71344 57624 46953 38416
54423 38416 28812 22409 17836 14406 11738 9604
7927
5217
4268
24188 17074 12805 9960
6403
2935
2401
13606 9604
7203 5602 4459
3602
2854
6147
4610 3585
2305
1878 1537
8708
1304 1067
6047
4268 3201 2490 1982 1601
784
2352 1829 1456 1176
958
4443 3136
1115
734
600
3401
2401
1801 1401
900
474
881
711
580
2688 1897 1423 1107
713
2177
1537
1152
576
470
384
896
317
171
968
683
512
398
256
209
224
178
144
117
96
544
384
288
61
246
143
114
92
75
348
184
79
242
171
128
100
64
52
43
31
178
125
94
73
58
47
38
45
136
96
72
56
36
29
24
29
19
15
87
61
46
36
23
.
-I
III
cr
CD
N
Table 2
( continued)
(b) Confidence level 90%
p
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
33074
8268
3675
2067
1323
919
675
517
408
331
147
83
53
37
27
21
13
27060
6765
3007
1691
1082
752
552
423
334
271
120
68
43
30
22
17
11
22140
5535
2460
1384
886
615
452
346
273
221
98
55
35
25
18
14
9
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
11597
2899
1289
725
464
322
237
181
143
1'6
52
29
19
13
9
7
5
9020
2255
1002
564
361
251
184
141
111
90
40
23
14
10
7
6
6765
1691
752
423
271
188
138
106
4775
1194
531
298
191
133
97
75
59
48
21
12
8
5
3007
752
334
188
120
1424
356
158
89
57
40
29
22
18
14
6
e
~I
0.01 514145 243542 153341 108241 81181
0.02 128536 60886 38335 27060 20295
0.03 57127 27060 17038 12027 9020
0.04 32134 15221
9584
6765 5074
0.05 20566 9742
6134
4330 3247
0.06 14282 6765 4259 3007 2255
0.07 10493 4970 3129 2209 1657
0.08
8034 3805 2396 1691 1268
0.09
6347 3007 1893 1336 1002
0.10
5141
2435 1533 1082
812
0.15
682
2285 1082
481
361
0.20
271
1285
609
383
203
0.25
823
245
173
390
130
0.30
571
271
170
120
90
0.35
125
199
66
420
88
0.40
321
152
96
68
51
0.50
61
97
43
206
32
'Sample size less than 5.
63141 50255 40590
15785 12564 10148
5584 4510
7016
3141
3946
2537
2526
2010
1624
1754 1396 1128
1289 1026
828
987
785
634
620
780
501
631
503
406
223
281
180
126
158
101
101
80
65
70
56
45
41
52
33
31
39
25
25
20
16
18040 14571
4510
3643
1619
2004
1128
911
722
583
501
405
368
297
228
282
180
223
180
146
80
65
45
36
29
23
16
20
15
12
11
9
7
6
84
68
30
17
11
8
6
84
61
47
37
30
13
8
5
en
CI
3
'C
i'
III
j;j'
CD
....
..o·=
Q.
CD
CD
3
CI
=
Table 3. Hypothesis tests for a population proportion
For a one-sided test
11= [Zl-",/[Po(l-PO)]+Zl-p,/[Pa(l-Pa)J}2/(Po-Pa)2.
For a two-sided test
11 = {z 1 -,.'2 J [P 0(1 - P o)J + z 1 -pJ[P a(1 - P a)J}2 I(P 0 - Pa)2.
(a) Level of significance 5%, power 90%, one-sided test
pol
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
239
76
378
40
109
498
25
54
137
601
18
33
66
161
686
13
22
39
75
180
754
10
16
26
44
83
195
804
8
12
19
29
48
88
205
837
6
10
14
20
31
50
92
211
853
5
8
11
15
21
32
52
93
213
852
0.60
0.65
6
8
11
16
22
33
52
93
210
834
5
7
9
12
16
22
32
51
91
203
799
0.70
0.75
0.80
0.85
0.90
0.95
Pa
~I
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
221
67
34
21
15
11
8
7
5
362
102
49
30
20
14
11
9
7
6
5
• Sample size less than 5.
485
131
62
36
24
17
13
10
8
6
5
589
156
72
42
27
19
14
11
8
7
5
676
176
80
46
30
21
15
11
9
7
5
746
191
87
49
31
22
16
12
9
7
5
799
203
91
51
32
22
16
12
9
7
5
834
210
93
52
33
22
16
11
8
6
852
213
93
52
32
21
15
11
8
5
853
211
92
50
31
20
14
10
6
837
205
88
48
29
19
12
8
804
195
83
44
26
16
10
754
180
75
39
22
13
5
7
9
12
16
22
31
49
87
191
746
686
161
66
33
18
5
7
9
11
15
21
30
46
80
176
676
601
137
54
25
5
7
8
11
14
19
27
42
72
156
589
498
109
40
5
6
8
10
13
17
24
36
62
131
485
378
76
Ii
*
5
6
7
9
11
14
20
30
49
102
362
239
5
7
8
11
15
21
34
67
221
Table 3
( continued)
(b) Level of significance 5%, power 80%, one-sided test
,
Po I 0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
184
60
283
32
83
368
21
42
103
441
15
26
50
119
501
11
18
30
56
133
548
8
13
20
33
61
143
584
7
10
14
22
35
65
149
607
5
8
11
15
23
37
67
153
617
5
6
8
11
16
24
38
68
154
615
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
Pa
W
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
*
150
44
22
14
9
7
5
253
69
33
20
13
10
7
6
5
*
Sample size less than 5.
342
91
43
25
16
12
9
7
5
419
109
50
29
19
13
10
8
6
5
483
125
57
32
21
15
11
8
6
5
535
137
62
35
23
16
11
9
7
5
574
145
65
37
23
16
12
9
7
5
601
151
67
38
24
16
12
9
7
5
615
154
68
38
24
16
11
8
6
5
617
153
67
37
23
15
11
8
5
607
149
65
35
22
14
10
7
5
7
9
12
16
24
38
67
151
601
584
143
61
33
20
13
8
5
7
9
12
16
23
37
65
145
574
548
133
56
30
18
11
en
3
'C
;I»
5
7
9
11
16
23
35
62
137
535
501
119
50
26
15
5
6
8
11
15
21
32
57
125
483
441
103
42
21
5
6
8
10
13
19
29
50
109
419
368
83
32
III
N'
III
*
7
5
9
12
16
25
43
91
342
6
283
60
7
10
13
20
33
69
253'
184
...
..o·=
Q.
5
III
III
*
5
7
9
14
22
44
150
~.
I»
=
Table 3 ( continued)
(c) Level of significance 5%, power 90%, two-sided test
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.01
0.02
0.03
0.04
0.05
5353
1423
668
395
264
9784
2524
1155
667
438
13686
3490
1580
905
589
17061
4324
1947
1109
718
189
19911
5026
2255
1279
826
22234
5597
2504
1417
912
24032
6036
2695
1522
978
25305
6344
2827
1594
1022
26052
6521
2901
1633
1045
26273
6565
2916
1639
1047
87
214
97
233
105
248
111
62
257
114
259
113
62
38
25
X
~
I Pa - Pol
~I
0.10
79
122
0.15
40
69
0.20
0.25
0.30
25
17
12
35
24
17
1.
74
43
29
20
0.35
0.40
0.45
0.50
10
8
6
5
13
10
8
6
15
12
9
7
X is the smaller of Po and (1 - Pol·
• Sample size less than 5.
50
56
60
64
261
115
63
33
23
36
25
38
26
40
40
40
27
27
27
17
13
10
8
18
14
11
8
19
14
11
8
19
14
11
8
19
14
10
8
19
13
10
7
17
12
8
I!
Table 3
( continued)
(d) Level of significance 5%, power 80%, two-sided test
~x
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.01
0.02
0.03
0.04
0.05
3933
1031
478
280
185
7250
1856
844
485
316
10172
2582
1164
664
430
12701
3209
1440
818
528
14837
3737
1673
947
610
16580
4167
1861
1052
676
19626
4905
2179
1225
783
53
26
16
86
114
131
53
53
172
18
82
85
31
36
41
44
46
20
14
24
17
9
7
11
8
7
5
12
9
7
6
18
13
10
8
28
20
14
11
8
7
30
20
15
11
8
7
48
30
21
15
11
8
181
86
48
114
41
24
16
12
161
11
18888
4732
2107
1188
761
111
19453
4868
2165
1219
780
0.10
17930
4499
2006
1132
727
184
lPa- Pol
w
N
0.11
0 ••
0.21
0.30
0.35
0.40
0.45
0.50
en
11
8
6
5
X is the smaller of Po and (1 - Po)'
• Sample size less than 5.
6
5
a:
6
6
30
20
15
11
8
6
II
41
»
20
14
10
7
I»
3
;-
"0
1/1
i:r
CD
Q.
CD
....
2.
..o·
CD
::::I
I»
::::I
Table 4. Estimating the difference between two population proportions with specified absolute precision
n =zI ~,dP dl- PI) + P z (1- Pz)]/d z
or
n=zi~"lz V/d
z
where
V=P 1 (l-Pd+P 2 (l-P Z )
(a) Values of V
X
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.08
0.09
0.10
0.12
0.13
0.14
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.23
0.24
0.25
0.25
0.25
0.26
0.26
0.26
0.26
0.03
0.04
0.05
0.06
0.07
0.08
0.08
0.09
0.10
0.11
0.13
0.14
0.15
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.24
0.25
0.26
0.26
0.26
0.27
0.27
0.27
0.27
0.04
0.05
0.06
0.07
0.08
0.09
0.09
0.10
0.11
0.12
0.13
0.15
0.16
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.28
0.28
0.05
0.06
0.07
0.08
0.09
0.09
0.10
0.11
0.12
0.13
0.14
0.16
0.17
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.28
0.29
0.29
0.29
0.06
0.07
0.08
0.09
0.10
0.10
0.11
0.12
0.13
0.14
0.15
0.17
0.18
0.20
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.29
0.30
0.30
0.30
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.16
0.17
0.18
0.20
0.21
0.22
0.24
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.31
0.32
0.33
0.33
0.33
0.34
0.34
0.34
0.34
0.14
0.15
0.16
0.17
0.18
0.18
0.19
0.20
0.21
0.22
0.23
0.25
0.26
0.28
0.29
0.30
0.31
0.32
0.33
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.37
0.38
0.38
0.38
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.23·
0.24
0.25
0.27
0.28
0.29
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.38
0.39
0.40
0.40
0.40
0.41
0.41
0.41
0.41
0.20
0.21
0.22
0.23
0.24
0.24
0.25
0.26
0.27
0.28
0.29
0.31
0.32
0.34
0.35
0.36
0.37
0.38
0.39
0.40
0.41
0.41
0.42
0.42
0.43
0.43
0.43
0.44
0.44
0.44
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.28
0.29
0.30
0.32
0.33
0.34
0.36
0.37
0.38
0.39
0.40
0.41
0.42
0.43
0.43
0.44
0.45
0.45
0.45
0.46
0.46
0.46
0.46
0.24
0.25
0.26
0.27
0.28
0.28
0.29
0.30
0.31
0.32
0.33
0.35
0.36
0.38
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.47
0.48
0.48
0.48
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.31
0.32
0.33
0.35
0.36
0.37
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.46
0.47
0.48
0.48
0.48
0.49
0.49
0.49
0.49
0.26
0.27
0.28
0.29
0.30
0.30
0.31
0.32
0.33
0.34
0.35
0.37
0.38
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.49
0.50
0.50
0.50
0.26
0.27
0.28
0.29
0.30
0.31
0.32
0.32
0.33
0.34
0.36
0.37
0.38
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.47
0.48
0.49
0.49
0.49
0.50
0.50
0.50
0.50
Y
~I
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
Xis the smaller of P2 and (1-P2 )·
Y is the smaller of P, and (1 - P,).
I~
Table 4
( continued)
(b) Sample size for confidence level 95%
d
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
385
769
1153
1537
1921
2305
2690
3074
3458
3842
4610
5379
6147
6915
7684
8452
9220
9989
10757
11525
12294
13062
13830
14599
15367
16135
16904
17672
18440
19209
97
193
289
385
481
577
673
769
865
961
1153
1345
1537
1729
1921
2113
2305
2498
2690
2882
3074
3266
3458
3650
3842
4034
4226
4418
4610
4803
43
86
129
171
214
257
299
342
385
427
513
598
683
769
854
940
1025
1110
1196
1281
1366
1452
1537
1623
1708
1793
1879
1964
2049
2135
25
49
73
97
121
145
169
193
217
241
289
337
385
433
481
529
577
625
673
721
769
817
865
913
961
1009
1057
1105
1153
1201
16
31
47
62
77
93
108
123
139
154
185
216
246
277
308
339
369
400
431
461
492
523
554
584
615
646
677
707
738
769
8
12
16
20
24
27
31
35
39
47
54
62
70
77
85
93
100
108
116
123
131
139
146
154
162
170
177
185
193
6
7
9
11
12
14
16
18
21
24
28
31
35
38
41
45
48
52
55
59
62
65
69
72
76
79
82
86
5
6
7
8
9
10
12
14
16
18
20
22
24
25
27
29
31
33
35
37
39
41
43
45
47
49
5
5
6
7
8
9
10
12
13
14
15
16
18
19
20
21
23
24
25
26
28
29
30
31
0.30
0.35
0.40
0.45
0.50
V
w
~
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
'Sample size less than 5.
*
*
*
en
III
*
5
6
6
7
8
9
10
11
12
12
13
14
15
16
17
18
18
19
20
21
22
*
3
'tI
ii'
III
5
6
6
7
7
8
9
9
10
11
11
12
12
13
14
14
15
16
16
N'
III
Q.
.....3
III
III
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
5
5
5
6
6
7
7
7
8
8
8
9
9
10
10
5'
*
*
5
5
5
6
6
6
7
7
7
8
8
8
...0'
III
:::I
Table 4
( continued)
(c) Sample size for confidence level 90%
d
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
271
542
812
1083
1354
1624
1895
2165
2436
2707
3248
3789
4330
4871
5413
5954
6495
7036
7577
8119
8660
9201
9742
10283
10825
11366
11907
12448
12989
13531
68
136
203
271
339
406
474
542
609
677
812
948
1083
1218
1354
1489
1624
1759
1895
2030
2165
2301
2436
2571
2707
2842
2977
3112
3248
3383
31
61
91
121
151
181
211
241
271
301
361
421
482
542
602
662
722
782
842
903
963
1023
1083
1143
1203
1263
1323
1384
1444
1504
17
34
51
68
85
102
119
136
153
170
203
237
271
305
339
373
406
440
474
508
542
576
609
643
677
711
745
778
812
846
11
22
33
44
55
65
76
87
98
109
130
152
174
195
217
239
260
282
304
325
347
369
390
412
433
455
477
498
520
542
6
9
11
14
17
19
22
25
28
33
38
44
49
55
60
65
71
76
82
87
93
98
103
109
114
120
125
130
136
5
7
8
9
10
11
13
15
17
20
22
25
27
29
32
34
37
39
41
44
46
49
51
53
56
58
61
5
5
6
7
7
9
10
11
13
14
15
17
18
19
21
22
24
25
26
28
29
30
32
33
34
0.25
0.30
0.35
0.40
0.45
0.50
V
~I
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
'Sample size less than 5.
*
5
6
7
7
8
9
10
11
12
13
13
14
15
16
17
18
19
20
20
21
22
5
5
6
7
7
8
8
9
10
10
11
11
12
13
13
14
14
15
16
*
5
5
6
6
7
7
8
8
8
9
9
10
10
11
11
12
I~
5
5
5
6
6
6
7
7
7
8
8
8
9
9
5
5
5
5
6
6
6
6
7
7
7
*
5
5
5
5
5
6
6
Table 5. Hypothesis tests for two population proportions
For a one-sided test
n = {z 1-,j[2P(1- P)] + zl_pj[P 1(1- P tl + P 2(1- P 2)]}2 I(P 1 - P 2 )2
where
P=(P 1+P 2 )/2.
For a two-sided test
n= {z 1-"!2j[2P(1- P)] +Zl -pJ[P 1(1- P tl+ P 2(1- P 2)]}2 I(P 1 - P2)2.
For a one-sided test for small proportions
n ={z 1 _, + z 1 _p)2 1[0.00061 (arcsinJ P 2 - arcsinJ p 1)2].
For a two-sided test for small proportions
n = {Z 1 -2/2 + Z1 _p)2 1[0.00061(arcsinJ P 2 - arcsinJp 1)2].
I~
3
I'E.
CD
(a) Level of significance 5%, power 90%, one-sided test
P,
~I
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
P2
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1/1
;:i'
CD
.....
..o·=
Q.
474
474
153
82
53
38
29
23
19
15
13
11
9
8
7
6
5
748
217
109
67
46
34
26
21
17
14
12
10
8
7
6
5
* Sample size less than 5.
153
748
988
273
131
79
53
39
29
23
18
15
12
10
9
7
6
5
82
217
988
1194
320
150
89
59
42
31
24
19
16
13
10
9
7
6
53
109
273
1194
1365
358
166
96
63
44
33
25
20
16
13
10
8
7
38
67
131
320
1365
1502
388
177
101
66
46
33
25
20
16
12
10
8
29
46
79
150
358
1502
1605
410
185
105
67
46
33
25
19
15
12
9
23
34
53
89
166
388
1605
1674
423
189
106
67
46
33
24
18
14
11
19
26
39
59
96
177
410
1674
1708
427
189
105
66
44
31
23
17
13
15
21
29
42
63
101
185
423
1708
1708
423
185
101
63
42
29
21
15
13
17
23
31
44
66
105
189
427
1708
1674
410
177
96
59
39
26
19
11
14
18
24
33
46
67
106
189
423
1674
1605
388
166
89
53
34
23
9
12
15
19
25
33
46
67
105
185
410
1605
1502
358
150
79
46
29
8
10
12
16
20
25
33
46
66
101
177
388
1502
1365
320
131
67
38
7
8
10
13
16
20
25
33
44
63
96
166
358
1365
1194
273
109
53
6
7
9
10
13
16
19
24
31
42
59
89
150
320
1194
988
217
82
5
6
7
9
10
12
15
18
23
29
39
53
79
131
273
988
748
153
CD
CD
5
6
7
8
10
12
14
17
21
26
34
46
67
109
217
748
474
3
5
6
7
8
9
11
13
15
19
23
29
38
53
82
153
474
III
=
Table 5
( continued)
(b) Level of significance 5%, power 80%, one-sided test
\
P2 \
~I
p,i
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.05
343
111
60
39
28
21
17
14
12
10
8
7
6
5
5
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
343
111
541
60
157
714
39
79
197
862
28
49
95
231
986
21
34
57
109
259
1085
17
25
39
64
120
281
1159
14
20
28
43
70
128
296
1209
12
16
22
31
46
74
134
306
1233
10
13
8
11
14
18
24
33
49
77
137
306
1209
7
9
11
14
19
25
34
49
76
134
296
1159
6
8
9
12
15
19
25
33
48
74
128
281
1085
541
157
79
49
34
25
20
16
13
11
9
8
7
6
5
•
'Sample size less than 5.
714
197
95
57
39
28
22
17
14
11
9
8
7
6
5
862
231
109
64
43
31
23
18
14
12
10
8
7
6
5
986
259
120
70
46
32
24
19
15
12
10
8
7
5
1085
281
128
74
48
33
25
19
15
12
9
8
6
1159
296
134
76
49
34
25
19
14
11
9
7
1209
306
137
77
49
33
24
18
14
11
8
1233
309
137
76
48
32
23
17
13
10
1233
306
134
74
46
31
22
16
12
17
23
32
48
76
137
309
1233
1209
296
128
70
43
28
20
14
1159
281
120
64
39
25
17
1085
259
109
57
34
21
986
231
95
49
28
0.75
0.80
5
5
6
7
8
10
12
14
18
23
31
43
64
109
231
862
7
8
10
12
15
19
24
32
46
70
120
259
986
862
197
79
39
714
157
60
0.85
5
6
7
8
9
11
14
17
22
28
39
57
95
197
714
541
111
0.90
5
6
7
8
9
11
13
16
20
25
34
49
79
157
541
343
0.95
5
5
6
7
8
10
12
14
17
21
28
39
60
111
343
I~
Table 5
( continued)
(c) Level of significance 5%, power 90%, two-sided test
~I
w
(Xl
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.05
0.10
0.15
0.20
0.25
0.30
0.35
50654
12711
5669
3199
2053
52126
13053
5809
3271
2095
52546
13131
5832
3278
2095
519
524
231
128
81
5f9
ai?
54
39
28
21
16
62
36
26
19
14
0.40
0.45
0.50
"en
III
19753
5143
2376
1386
918
27531
7062
3216
1852
1212
34258
8717
3940
2253
1465
39933
10110
4548
2588
1675
44558
11239
5038
2857
1843
48132
12107
5412
3061
1969
336
393
477
503
181
1.
440
203
97
109
118
47
266
133
82
57
85
36
42
47
28
23
19
16
33
26
21
17
36
28
23
19
72
52
39
30
24
19
10924
2962
1418
854
582
188
101
65
0.30
0.35
0.40
0.45
0.50
x is the smallest of P"
(1 - P,). P2 and (1 - P2 ).
217
2T1
231
128
77
125
81
54
40
31
24
19
56
41
31
24
19
57
130
82
68
41
31
24
19
40
30
23
17
82
125
77
3
'tI
CD
1/1
iii'
CD
.....
..cj'
Q.
CD
CD
3
S·
III
=
( continued)
Table 5
(d) Level of significance 5%, power 80%, two-sided test
~x I
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
8161
2213
1060
638
435
141
76
49
36.
27
22
18
15
12
14756
3842
1775
1036
686
200
100
62
43
32
25
20
16
14
20566
5275
2403
1384
906
251
121
73
49
36
27
22
17
14
25590
6512
2944
1683
1095
294
138
82
54
39
29
23
18
15
29830
7552
3398
1934
1252
329
152
89
33284
8396
3764
2135
1377
357
163
94
61
42
31
24
19
15
35954
9044
4043
2287
1471
376
170
96
62
43
31
24
18
14
37838
9495
4235
2390
1534
388
173
97
62
42
31
23
17
14
38937
9751
4340
2444
1566
392
173
96
61
41
29
22
16
12
39251
9809
4357
2449
1566
388
170
94
I P2 - P,1
0.01
0.02
0.03
0.04
0.05
0.10
0.15
4).20
0.25
0.30
0.35
0.40
0.45
0.50
w
co
58
41
31
24
19
15
58
39
27
20
15
11
X is the smallest of Pl' (1 - P,). P2 and (1 - P2 ).
"-I
=
cr
iD
en
(e) Level of significance 5%, power 90%, one-sided test. small proportions
"-
P, I 0.0001
0.0002
0.0003
0.0004
0.0005
0.0010
0.0025
0.0050
0.0075
0.0100
0.0200
0.0300
0.0400
0.0500
249634
79919
423934
42827
124798
596429
28029
63398
168559
768327
9158
14011
20928
31688
49897
2675
3328
4006
4753
5600
12662
1160
1336
1500
1662
1828
·2796
9960
728
813
890
963
1035
1412
3182
16856
527
579
624
667
708
912
1703
4957
23657
246
262
276
288
300
362
607
847
1407
2460
159
167
174
180
186
211
278
401
661
784
4135
117
122
126
130
134
149
187
251
326
418
1212
5758
92
96
99
101
104
114
13B
179
222
273
613
1619
7341
P2
0.0001
0.0002
0.0003
0.0004
0.0005
0.0010
0.0025
0.0050
0.0075
0.0100
0.0200
0.0300
0.0400
0.0500
249634
79919
42827
28029
9158
2676
1160
728
527
246
159
117
92
423934
124798
63398
14011
3328
1336
813
579
262
167
122
96
596429
168559
20928
4006
1500
890
624
276
174
126
99
768327
31688
4763
1662
963
667
288
180
130
101
49897
6600
1828
1035
708
300
186
134
104
12662
2796
1412
912
352
211
149
114
9950
3182
1703
507
278
187
139
16856
4957
847
401
251
179
23657
1407
561
326
222
2460
784
418
273
4135
1212
613
5758
1619
7341
Table 5
( continued)
(h) Level of significance 5%, power 80%, two-sided test, small proportions
P1 i 0.0001
0.0002
0.0003
0.0004
0.0005
0.0010
0.0025
0.0050
0.0075
0.0100
228767
73239
388498
39247
114367
546575
25686
58099
154470
704104
8392
12840
19179
29040
45727
2451
3050
3671
4356
5132
1063
1224
1374
1523
1675
667
745
815
882
948
11604
2562
9118
1294
2916
15447
483
530
572
611
648
836
0.0200
0.0300
0.0400
226
241
253
264
275
146
153
160
165
171
107
112
116
119
123
85
88
91
93
95
323
465
776
1289
2254
193
255
368
614
719
136
104
127
0.0500
P2
.j::.
.....
0.0001
0.0002
0.0003
0.0004
0.0005
0.OQ10
228767
73239
39247
25686
().002$
8392
2451
0.0050
1063
0.0075
667
0.0100
0.0200
0.0300
0.0400
0.0500
483
226
146
107
85
388498
114367
58099
546575
154470
3050
1224
745
630
19179
3671
1374
816
572
241
153
112
88
253
160
116
91
12840
704104
29040
4356
1523
882
611
264
165
119
93
457.27
5132
1675
948
648
275
171
123
95
11604
2562
1294
836
323
193
136
104
9118
2916
1561
465
255
171
127
15447
4542
776
368
230
164
21680
1289
514
299
204
1561
4542
21680
2254
719
383
250
3789
3789
1110
561
5277
1484
171
230
299
383
1110
5277
6728
164
204
250
561
1484
6728
I!
Table 6. Estimating an odds ratio with specified relative precision
n = zi -a/2 {1 /[Pi( 1 - Pi)] + 1/[P!(1- P!)]} /[loge(1- e)Y
(a) Confidence level 95%, relative precision 10%
'\oRI
P*
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
69912
35313
23785
18025
14572
7691
5429
4326
3692
3296
3043
2884
2797
2769
2797
2884
3296
4326
7691
63061
31923
21550
16367
13261
7078
5052
4071
3513
3172
2961
2838
2783
2786
2846
2968
3469
4655
8462
58494
29664
20061
15263
12389
6672
4806
3908
3403
3101
2922
2827
2798
2827
2914
3067
3651
4990
9235
55232
28051
18998
14476
11767
6385
4634
3798
3333
3062
2907
2835
2828
2880
2993
3175
3838
5327
10010
52786
26842
18201
13886
11302
6172
4510
3721
3288
3041
2908
2856
2869
2942
3078
3288
4030
5667
10786
50883
25901
17582
13429
10941
6009
4416
3665
3259
3033
2919
2884
2916
3009
3168
3405
4223
6009
11563
49361
25149
17087
13063
10654
5880
4344
3626
3242
3034
2937
2919
2968
3080
3262
3525
4419
6351
12340
48116
24535
16683
12765
10419
5776
4288
3597
3233
3042
2960
2958
3023
3154
3357
3646
4615
6694
13117
47079
24023
16347
12516
10225
5691
4244
3576
3230
3055
2987
3000
3081
3230
3454
3769
4812
7037
13894
46202
23589
16063
12307
10061
5621
4209
3563
3233
3071
3017
3044
3141
3308
3553
3893
5011
7381
14672
45449
23219
15819
12128
9921
5562
4181
3554
3239
3090
3050
3090
3203
3387
3652
4017
5209
7725
15450
44798
22897
15609
11974
9800
5513
4159
3549
3248
3112
3084
3138
3265
3467
3752
4143
5408
8070
16228
44228
22616
15425
11839
9695
5470
4141
3548
3259
3136
3120
3187
3329
3548
3854
4269
5608
8414
17006
43725
22369
15263
11720
9603
5434
4128
3549
3273
3161
3157
3237
3394
3629
3955
4395
5807
8759
17784
43278
22149
15120
11615
9521
5403
4117
3552
3288
3187
3195
3288
3459
3711
4057
4522
6007
9104
18562
42878
21952
14991
11522
9449
5376
4110
3558
3305
3215
3234
3340
3525
3794
4160
4649
6207
9449
19340
42518
21776
14876
11438
9384
5353
4104
3565
3323
3244
3274
3392
3591
3876
4262
4776
6408
9794
20118
2
~I
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
0.90
For OR < 1, use the colur.m value corresponding to 1lOR and the row value corresponding to Pi.
en
I»
3
'tI
CD
III
j;j'
CD
...
..
Q.
CD
CD
3
5'
I»
0'
::I
Table 6
( continued)
(b) Confidence level 95%, relative precision 20%
\oRI
P*
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
15587
7873
5303
4019
3249
1715
1211
965
823
735
679
643
624
618
624
643
735
965
1715
14059
7117
4805
3649
2957
1578
1127
908
784
708
660
633
621
622
635
662
774
1038
1887
13041
6614
4473
3403
2762
1488
1072
872
759
692
652
631
624
631
650
684
814
1113
2059
12314
6254
4236
3228
2624
1424
1034
847
744
683
649
632
631
643
668
708
856
1188
2232
11768
5984
4058
3096
2520
1376
1006
830
733
678
649
637
640
656
687
733
899
1264
2405
11344
5775
3920
2994
2440
1340
985
818
727
677
651
643
650
671
707
760
942
1340
2578
11005
5607
3810
2913
2376
1311
969
809
723
677
655
651
662
687
728
786
985
1416
2751
10727
5470
3720
2846
2323
1288
956
802
721
679
660
660
674
704
749
813
1029
1493
2925
10496
5356
3645
2791
2280
1269
946
798
721
681
666
669
687
721
770
841
1073
1569
3098
10301
5259
3581
2744
2243
1254
939
795
721
685
673
679
701
738
792
868
1117
1646
3271
10133
5177
3527
2704
2212
1240
932
793
722
689
680
689
714
755
815
896
1162
1723
3445
9988
5105
3480
2670
2185
1229
928
792
724
694
688
700
728
773
837
924
1206
1799
3618
9860
5042
3439
2640
2162
1220
924
791
727
699
696
711
743
791
859
952
1251
1876
3792
9748
4987
3403
2613
2141
1212
921
792
730
705
704
722
757
809
882
980
1295
1953
3965
9649
4938
3371
2590
2123
1205
918
792
733
711
713
733
772
828
905
1008
1340
2030
4139
9560
4894
3343
2569
2107
1199
917
794
737
717
721
745
786
846
928
1037
1384
2107
4312
9479
4855
3317
2550
2093
1194
915
795
741
724
730
757
801
865
951
1065
1429
2184
4486
2
~
w
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
0.90
For OR < 1, use the column value corresponding to 1/ OR and the row value corresponding to Pi.
I~
Table 6 (continued)
(c) Confidence level
95%, relative precision 25%
""'p.OR I
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
0.01
0.02
0.03
0.04
0.05
(UO
9378
4737
3191
2418
1955
8459
4282
2891
2196
1779
960
7846
3979
2691
2048
1662
6315
3223
2193
1679
1372
764
6198
3165
2155
1651
1350
6097
3115
2122
1627
1331
6009
3072
2094
1606
1315
5933
3034
2069
1588
1301
5805
2971
2028
1558
1278
5752
2945
2011
1546
1268
5703
2921
1996
1535
1259
1M
566
M8
625
510
499
472
425
467
448
441
438
411
407
398
381
374
374
382
399
392
380
376
380
391
390
381
380
387
402
408
390
383
385
395
413
435
407
4M
416
392
387
392
404
425
394
392
399
414
438
478
434
4'2
7.
5.
476
734
570
741
.1
477
435
415
5865
3001
2048
1572
1288
729
593
492
6621
3374
2292
1753
1429
789
583
487
6454
3291
2238
1713
1398
&46
7081
3601
2442
1863
1516
828
006
6825
3475
2359
1802
1468
806
078
7409
3763
2549
1942
1579
867
822
412
490
670
1239
426
615
715
1343
441
541
457
587
761
1447
1551
473
593
852
1658
2
tl
0.16
0;20
0.28
0. •
0.35
0.40
0.45
0.50
0.55
0.80
0~70
0.80
0.90
1()32
721
M1
443
409
387
376
372
376
387
443
581
1032
...
625
1135
896
806
775
576
493
4M
408
397
397
406
424
451
489
619
4'0
401
403
414
434
464
506
646
898
944
1760
1864
For OR< 1, use the column value corresponding to 1 lOR and the row value corresponding to Pi.
...
4,a
405
409
422
444
477
523
409
415
430
455
490
539
672
699
414
421
438
465
504
556
725
1.
1OIl
2073
1083
2177
-
-- -.. - ..
726
m
118
.2
.1
478
419
441
444
448
4D
432
476
514
478
417
421
424
419
428
447
476
517
424
435
456
487
531
429
441
464
498
545
434
448
473
509
558
440
455
482
520
572
014
041
1.
2_
1t14
·2611
566
573
590
753
119
1121
2111
1116
2316
1m
2490
(I)
III
3
;-
"CI
UI
jij'
CD
....
..o·s·
CI.
CD
CD
3
III
=
Table 6
( continued)
(d) Confidence level 95%, relative precision 50%
pRi
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
1616
816
550
417
337
118
1458
738
498
379
307
164
1352
686
464
353
287
1277
649
439
335
272
1220
621
421
321
262
1176
599
407
311
253
1141
582
395
302
247
1112
567
386
295
241
1088
556
378
290
237
1068
546
372
285
233
1051
537
366
281
230
1036
530
361
277
227
1022
523
357
274
224
1011
517
353
271
222
1000
512
350
269
220
991
508
347
267
219
983
504
344
265
217
143
105
134
100
125
125
124
95
71
72
75
71
75
71
82
76
13
82
76
14
96
83
82
74
128
97
82
76
96
84
130
98
83
75
71
96
84
132
99
83
75
71
129
97
86
139
103
85
76
71
136
101
0.25
0.30
148
108
88
78
71
126
117
155
112
91
79
127
126
100
0.35
0.40
0.45
0.50
0.55
71
67
65
64
65
69
66
65
65
66
68
66
65
66
68
68
66
66
67
70
68
66
67
68
68
68
69
72
76
69
69
70
73
78
70
70
72
75
80
70
71
73
77
83
71
72
72
72
68
67
68
70
74
74
79
85
73
76
81
87
0.60
0.70
0.80
0.90
61
77
100
178
69
81
108
196
71
85
116
214
74
89
124
232
76
94
131
250
79
98
139
268
82
103
147
286
85
107
155
88
112
163
322
93
121
179
357
96
304
90
116
171
339
73
74
77
82
90
99
125
187
375
130
195
393
P*2
0.01
0.02
0.03
0.04
0.05
0~10
~I
0.15
0.20
95
86
76
71
For OR< 1, use the column value corresponding to 1/ OR and the row value corresponding to P~.
83
75
72
72
76
74
77
95
83
77
75
75
73
75
79
84
92
74
76
80
86
94
75
78
82
88
97
76
79
83
90
99
102
135
203
411
105
139
211
429
108
144
219
447
111
149
227
465
83
I~
Table 6
( continued)
(e) COi'lfidence level 90%, relative precision 10%
~oRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
49246
24875
16754
12697
10264
5418
3824
3048
2601
2322
2144
2032
1970
1951
1970
2032
2322
3048
5418
44421
22487
15180
11529
9341
4986
3559
2868
2475
2234
2086
1999
1961
1963
2005
2091
2443
3279
5961
41203
20896
14131
10752
8727
4700
38906
19759
13382
10197
8289
4498
3265
2675
2348
2157
2048
1997
1992
2029
2108
2236
2704
3753
7051
37182
18907
12821
9782
7961
4348
3177
2621
2316
2142
2048
2012
2021
2073
2169
2316
2839
3992
7598
35842
18245
12385
9459
7707
4233
3111
2582
2296
2137
2056
2032
2054
2120
2232
2399
2975
4233
8145
34770
17715
12037
9202
7505
4142
3060
2554
2284
2138
2069
2056
2091
2170
2298
2483
3113
4474
8692
33893
17282
11752
8992
7339
4069
3021
2534
2278
2143
2085
2084
2130
2222
2365
2568
3251
4715
9240
33163
16922
11515
8817
7202
4009
2989
2519
2276
2152
2104
2113
2171
2276
2433
2655
3390
4957
9787
32545
16617
11315
8669
7087
3960
2965
2510
2277
2163
2125
2144
2213
2330
2503
2742
3530
5199
10335
32015
16355
11143
8543
6988
3918
2945
2503
2281
2177
2148
2177
2256
2386
2573
2830
3669
5442
10883
31556
16129
10995
8434
6903
3883
2930
2500
2288
2192
2172
2211
2300
2442
2643
2918
3810
5684
11431
31154
15931
10866
8339
6829
3853
2917
2499
2296
2209
2198
2245
2345
2499
2715
3007
3950
5927
11979
30800
15757
10752
8256
6764
3828
2908
2500
2306
2227
2224
2280
2391
2556
2786
3096
4091
6170
12527
30485
15602
10650
8182
6707
3806
2900
2503
2316
2245
2251
2316
2437
2614
2858
3185
4232
6413
13075
30203
15463
10560
8116
6656
3787
2895
2506
2328
2265
2278
2353
2483
2672
2930
3275
4373
6656
13623
29950
15339
10479
8057
6610
3771
2891
2511
2341
2285
2306
2389
2530
2731
3003
3364
4514
6899
14172
P*
2
~I
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
O.~
0.45
0.50
0.55
0.60
0.70
0.80
0.90
33~
2753
2398
2185
2058
1991
1971
1991
2053
2161
2572
3515
6505
For OR < 1, use the column value corresponding to 1/ OR and the row value corresponding to Pi.
en
=
3
"CI
CD
1/1
i;r
CD
Q.
CD
....
.o·==
CD
3
=
Table 6
( continued)
(f) Confidence level 90%, relative precision 20%
,PRJ
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
10979
5546
3736
2831
2289
1208
853
680
580
518
478
453
440
435
440
453
518
680
1208
9903
5014
3385
2571
2083
1112
794
640
552
499
465
446
437
438
447
467
545
731
1329
9186
4659
3151
2397
1946
1048
755
614
535
487
459
444
440
444
458
482
574
784
1451
8674
4406
2984
2274
1848
1003
728
597
524
481
457
446
445
453
470
499
603
837
1572
8290
4216
2859
2181
1775
970
709
585
517
478
457
449
451
462
484
517
633
890
1694
7991
4068
2761
2109
1719
944
694
576
512
477
459
453
458
473
498
535
664
944
'i816
7752
3950
2684
2052
1673
924
683
570
510
477
462
459
466
484
513
554
694
998
1938
7557
3853
2620
2005
1637
907
674
565
508
478
465
465
475
496
528
573
725
1052
2060
7394
3773
2567
1966
1606
894
667
562
508
480
470
471
484
508
543
592
756
1106
2182
7256
3705
2523
1933
1580
883
661
560
508
483
474
478
494
520
558
612
787
1160
2304
7138
3647
2485
1905
1558
874
657
559
509
486
479
486
503
532
574
631
818
1214
2427
7035
3596
2452
1881
1539
866
654
558
510
489
485
493
513
545
590
651
850
1268
2549
6946
3552
2423
1860
1523
859
651
558
512
493
490
501
523
558
606
671
881
1322
2671
6867
3513
2397
1841
1508
854
649
558
514
497
496
509
533
570
622
691
912
1376
2793
6797
3479
2375
1824
1496
849
647
558
517
501
502
517
544
583
638
711
944
1430
2915
6734
3448
2355
1810
1484
845
646
559
519
505
508
525
554
596
654
730
975
1484
3038
6677
3420
2337
1797
1474
841
645
560
522
510
515
533
564
609
670
750
1007
1538
3160
p*2
0.01
0.02
0.03
0.04
0.05
0.10
0.15
~I 0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
0.90
For OR < 1, use the column value corresponding to 1/ OR and the row value corresponding to Pi.
I~
Table 6
( continued)
(g) Confidence level 90%, relative precision 25%
\
~
(Xl
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
6606
3337
2248
1703
1377
727
513
409
349
312
288
273
265
262
265
273
312
409
727
5959
3017
2037
1547
1253
669
478
385
332
300
280
269
263
264
269
281
328
440
800
5527
2803
1896
1443
1171
631
455
370
322
293
277
268
265
268
276
290
345
472
873
5219
2651
1795
1368
1112
4988
2537
1720
1312
1068
584
427
352
311
288
275
270
271
278
291
311
381
536
1020
4808
2448
1662
1269
1034
568
418
347
308
287
276
273
276
285
300
322
399
568
1093
4664
2377
1615
1235
1007
556
411
343
307
287
278
276
281
292
309
333
418
600
1166
4547
2319
1577
1206
985
546
406
340
306
288
280
280
286
298
318
345
436
633
1240
4449
2270
1545
1183
966
538
401
338
306
289
283
284
292
306
327
357
455
665
1313
4366
2229
1518
1163
951
532
398
337
306
291
286
288
297
313
336
368
474
698
1387
4295
2194
1495
1146
938
526
395
336
306
292
289
292
303
320
346
380
493
730
1460
4233
2164
1475
1132
926
521
393
336
307
294
292
297
309
328
355
392
511
763
1534
4179
2137
1458
1119
916
517
392
336
308
297
295
302
315
336
365
404
530
795
1607
4132
2114
1443
1108
908
514
390
336
310
299
299
306
321
343
374
416
549
828
1681
4089
2093
1429
1098
900
511
389
336
311
302
302
311
327
351
384
428
568
861
1754
4052
2075
1417
1089
893
4018
2058
1406
1081
887
506
388
337
314
307
310
321
340
367
403
452
606
926
1901
604
438
359
315
290
275
268
268
273
283
300
363
504
946 .
For OR < 1. use the column value corresponding to 1/ OR and the row value corresponding to Pi.
508
389
337
313
304
306
316
333
359
393
440
587
893
1828
rn
DI
3
~
i'
III
N'
CD
....
..o·
CI.
CD
CD
3
:i'
DI
:::I
Table 6
( continued)
(h) Confidence level 90%, relative precision 50%
"
~I
0.Q1
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
0.90
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
1138
575
388
294
238
126
89
71
61
54
50
47
46
46
46
47
1027
520
351
267
216
116
83
67
58
52
49
47
46
46
47
49
57
76
138
952
483
327
249
202
109
79
64
66
51
48
46
46
46
48
50
60
82
151
899
457
310
236
192
104
76
62
55
50
48
47
47
47
49
52
63
87
163
860
437
297
226
184
101
74
61
829
422
287
219
179
98
12
60
804
410
279
213
174
96
71
59
53
50
48
48
49
51
54
58
72
104
784
400
272
208
170
94
70
59
53
50
49
49
50
52
55
60
76
109
214
767
391
267
204
167
93
70
69
63
50
49
49
51
53
57
62
79
115
227
752
384
262
201
164
92
69
68
63
50
50
50
52
54
58
64
82
121
239
740
378
258
198
162
91
69
68
63
51
50
51
53
56
60
66
85
126
252
730
373
255
195
160
90
68
68
53
51
51
52
54
57
62
68
89
132
265
720
369
252
193
158
90
68
68
54
52
51
52
55
58
63
70
92
137
277
712
365
249
191
157
89
68
58
54
52
52
53
56
60
65
72
95
143
290
705
361
247
190
155
88
68
68
698
358
244
188
154
88
67
68
54
53
53
55
58
62
68
76
102
154
315
692
355
243
187
153
88
67
59
55
63
54
56
59
64
70
78
105
160
328
54
71
126
54
54
50
48
47
47
48
51
54
66
93
176
50
48
47
48
49
52
56
69
98
189
201
For OR < 1, use the column value corresponding to 1/ OR and the row value corresponding to Pi.
54
52
52
54
57
61
67
74
98
149
303
I~
Table 7. Hypothesis tests for an odds ratio
n = {Zl-a/2J[2Pi(1-Pi)] +Zl-flJ[Pt(1- Pt)+ Pi(l- P!)]}2 /(Pt -Pi)2
In this formula the term 2PH1 - P~) is used instead of 21'*(1 - 1'*) because the study
population is likely to be made up of many more controls than cases, and the exposure rate
among the controls is often known with a high degree of precision; under the null
hypothesis this is the exposure rate for the cases as well. If the investigator is in doubt
about the exposure rate among the controls, however, the formula should be modified and
the term 21'*(1- 1'*) used, where 1'*= (Pf+ ~)/2.
(a) Level of significance 5%, power 90%, two-sided test
len
III
ORa I
P:~
~I
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
Il
5.00
III
0.01
0.02
0.03
0.04
0.05
35761
18133
12261
9327
7569
9375
4771
3237
2472
2013
4355
2224
1514
1160
948
2554
1308
894
687
563
1700
873
598
461
379
1225
631
434
335
276
932
482
332
257
213
738
383
264
205
170
602
313
217
169
140
503
262
182
142
118
428
224
156
122
102
370
194
135
106
89
0.10
0.1:6
0.20
0.26
0.30
4072
1102
807
666
689
318
161
124
1Q8
101
97
81
79
68
67
61
68
68
72
68
63
61
61
63
61
47
46
61
41
86
101
80
71
86
68
204
186
176
217
1.
143
131
126
126
644
627
392
329
296
276
46
41
42
0.35
0.40
0.45
0.50
0.55
1769
1707
1686
1699
1747
519
509
510
522
544
267
265
269
279
294
172
173
177
185
198
125
127
131
138
149
97
100
104
111
120
80
82
87
93
101
68
70
75
80
88
59
62
66
71
78
52
55
59
64
71
47
50
54
59
66
0.60
0.70
0.80
0.90
.1834
2170
679
704
317
394
660
1076
216
272
393
766
163
132
172
9&
88
118
179
80
108
166
74
101
366
3i4O
2929
2379
2068
1881
2948
982
6421
1861
240
ee
307
266
112
141
221
606
609
445
209
130
197
399
For ORa < 1, use the column value corresponding to 1/ ORa and the row value corresponding to Pi.
166
319
324
170
119
94
79
4$
287
151
106
83
70
41
3'
37
31
14
35
32:
.32
33.
43
46
50
55
61
40
43
46
51
57
37
40
43
48
54
35
38
41
46
51
69
94
141
66
89
139
.62
85
133
303
289
2.'18
42
46
• •
257
135
95
75
63
40
34:. ..
,.
;:i'
231
122
86
68
57
~
Q.
~
3
:37'
iJ:
..•• 3I:j.:.
3I:j
:1'
33
36
39
43
49
......
.··.1•.·••
1. ······1···
69
.82
268
..
S'
:
III
0'
:::I
Table 7
( continued)
(b) Level of significance 5%, power 80%, two-sided test
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
26421
13400
9063
6896
5598
6858
3492
2371
1811
1476
3157
1614
1100
843
690
1836
942
644
496
407
1213
624
429
331
272
656
340
235
183
151
516
269
186
145
121
419
219
152
119
99
348
182
127
100
83
295
155
108
85
71
254
134
94
74
62
221
117
82
65
55
195
103
73
58
49
174
92
65
52
44
156
83
59
47
40
"':1
810
231
1$7
868
448
309
239
198
f.f'
'''<ORal
p*2
0.01
0.02
0.03
0.04
0.05
..
.
.>
,.
e.· !: . . .~. ······>j:i;\i
•
.,:., ..- - '.
... .., 1..1 4. ••.._. - .2,. - •••• =! .
O?f1J
~I
..
0.:··:
~.I·
~.lD.
0. •
0.35
0.40
0.45
0.50
0.55
.....
~:'l2
1400
1318
1273
1259
1270
1307
404
387
380
381
391
408
1._ .742
t.lO·
,-",.
1373
0.•
.
~,
For
1~
t~.
ltf
",,,,,"C
...
"lj...
J4-
.«).
... f't:
·21.
2GI
1'37·
110
II
128
129
133
139
149
t4
93
95
98
104
112
12
199
198
202
209
221
73
75
78
84
91
60
62
65
70
238
.12
113
100
1~1
218:
820
·fT
:1.·.
a
1
iG
77
51
53
56
61
67
44
46
50
54
60
85
11
n~:
'111
OCI
U
11
1.'
·1M
I~
.
341
ORa < 1, use the column value corresponding to 1/ ORa and the row value corresponding to
P~.
*
39
42
45
49
54
'fu
1.····
36
38
41
45
50
32
35
38
42
47
17D
30
32
35
39
44
;~E
28
30
33
37
42
26
29
31
35
40
25
27
30
33
38
Table 8. Estimating a relative risk with specified relative precision
n=zi -~/2[(1-Pl)/Pl +(1-P2)/P2]/[loge(1-e)]2
(a) Confidence level 95%, relative precision 10%
~RRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
0.01
0.02
0.03
0.04
0.05
68521
33915
22379
16612
13151
56986
28147
18534
13728
10844
53690
26499
17436
12904
10185
51218
25263
16612
12286
9690
49295
24302
15971
11805
9306
47757
23533
15458
11421
8998
46499
22904
15039
11106
8746
43802
21555
14140
10432
8207
43143
21226
13920
10267
8075
42566
20937
13728
10123
7960
42057
20683
13558
9996
7858
41605
20457
13407
9883
7768
41200
20254
13272
9781
7687
40836
20072
13151
9690
7614
6230
3923
2769
2077
1615
4747
2984
2027
1484
1121
4499
2769
1904
4807
2641
4027
2454
1308
975
4153
2538
1131
1246
923
3Ia4
2326
1511
1119
817
3768
2275
1133
1088
3692
2231
1100
t0l2
36M
3461
2423
1800
1_
6076
3164
2192
1615
1231
45450
22379
14689
10844
8537
a923
44563
21936
14393
10622
8359
0.10
~I C:UI
0;20
0.21
61600
30454
20072
14881
11767
1638
1286
1039
846
693
567
1088
866
693
554
441
956
750
590
462
357
862
668
517
396
297
737
558
693
520
462
297
174
847
270
P2
0.30
0.35
0.40
0.45
0.50
0.55
(l.GO
0.70
0;80
0.90
1_
1039
792
606
462
347
,.-
,.
1196
1i1
657
198
81
77
Since RR= P,/P2 • RR~ 1/P2 •
For RR < 1. use the column value corresponding to 1/ RR and the row value corresponding to P,.
2384
1615
1164
146
2192
1411
1039
1.
aE
·21.
aas .34JI
2128
1423
2t.~
1483
14&1
,.
2077·
en
DI
3
~
i"
en
j;j'
..
.o·
CD
Q.
CD
CD
3
:i'
DI
=
Table 8
( continued)
(b) Confidence level 95%, relative precision 20%
~RRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
0.01
0.02
0.03
0.04
0.05
15276
7561
4990
3704
2932
13733
6790
4475
3318
2624
12705
6275
4132
3061
2418
11970
5908
3887
2877
2271
11419
5633
3704
2739
2161
10990
5418
3561
2632
2075
10647
5247
3447
2546
2006
10367
5107
3353
2476
1950
10133
4990
3275
2418
1904
9935
4891
3209
2368
1864
9766
4806
3153
2326
1830
9490
4668
3061
2257
1775
9377
4611
3023
2229
1752
9276
4561
2989
2204
1732
9186
4516
2959
2181
1714
9104
4475
2932
2161
1698
0.10
13.
815
618
463
.361
1236
112
U)I$
1003
6 ••
ill
618
501
342
488
541
481
1132
103
481
361
9619
4732
3104
2289
1801
.123
DO
243
250
287
232
189
155
127
243
193
155
124
99
216
214
168
132
103
80
P2
<U6
~I tUO
e,a
0:.
0.35
0.40
0.45
0.50
0.55
qQ .....
,~,..
~.'.:
..
eo'
-
1. .,.
••
11
.'
154
*2
331
193
149
116
89
67
9. .. .,--&. .,- ..
•
2.
,.
911
426
m
177
136
103
78
it.
218
165
125
206
155
116
372
287
111
147
6'
20
11
Since RR=P,/P2, RR~1/P2'
For RR < 1, use the column value corresponding to 1/ RR and the row value corresponding to P"
'82
331
131
•• 1"
811
32&
131
412
323.·
...- ...•
ttl
·31.3
111
..1
I!
Table 8
( continued)
(c) Confidence level 95%, relative precision 25%
~RRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
0.01
0.02
0.03
0.04
0.05
9191
4549
3002
2229
1764
8263
4085
2693
1996
1579
7644
3776
2486
1842
1455
7202
3555
2339
1731
1367
6612
3260
2143
1584
1249
6406
3157
2074
1532
1207
6237
3073
2018
1490
1174
6097
3002
1971
1455
1145
5978
2943
1931
1425
1122
5710
2809
1842
1358
1068
5642
2775
1819
1341
1054
5581
2744
1799
1326
1042
5478
2693
1764
1300
1022
636
527
372
279
217
743
465
325
242
186
661
423
294
217
166
637
576
355
243
176
131
556
341
541
330
224
161
119
527
320
515
312
211
150
110
5876
2892
1897
1400
1101
504
306
5787
2847
1868
1378
1084
0.10
0.15
~I 0.. 20
0.25
0.30
6870
3389
2229
1648
1300
604
496
488
2M
461.
206
146
.202
143
.1
290<,.;;;;211
1M· ·;·"1!I~:~:1.11
0.35
0.40
0.45
0.50
0.55
173
140
114
93
76
146
117
93
75
60
129
101
80
62
48
99
75
93
70
0.60
0.70
0.80
0.90
62
47
27
12
37
4.75
5.00
P2
40
24
11
394
272
199
151
116
90
70
54
40
372
256
186
140
107
82
62
47
233
168
124
89
Since RR=P,/ P2' RR ,;;;1/P2 ·
For RR < 1, use the column value corresponding to 1/ RR and the row value corresponding to P, .
217
155
114
300
196
~'~~~
5527
2717
1781
1312
1031
~;; .....;410.
"o,"\W~O",,.
o~;
:;:
",
140
'
..
,
~
~,
'
.. ~j
-"
en
3
DI
"CI
CD
1/1
N'
..
..
CD
a.
CD
CD
~
3
5'
DI
S'
::;,
Table 8
(continued)
(d) Confidence level 95%, relative precision 50%
"(RRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
0.01
0.02
0.03
0.04
0.05
1584
784
518
384
304
1424
704
464
344
272
1317
651
429
318
251
1241
613
403
299
236
1184
584
384
284
224
1139
562
369
273
215
1104
544
358
264
208
1075
530
348
257
203
1051
518
340
251
198
1030
507
333
246
194
1013
499
327
242
190
984
484
318
234
184
962
473
310
229
180
952
468
307
226
178
944
464
304
224
176
144
128
118
104
100
$4
17
$1
88
81
53
12
81
54
37
•
972
478
314
231
182
(UO
0.'11
0'.21.
997
491
322
238
187
16
Sf
34
Qi2$
48
26
26
24
P2
t1I
t1I
91
M
to
•
42
0.3G
31·
32
0.35
0.40
0.45
0.50
0.55
30
24
20
16
14
11
26
20
16
13
11
0;71.
,.•
oi9l··
•
1;.
O~.L
8
1
•
73
no
68
51
47
31
35
28
23
18
14
11
9
2.20
16
12
10
7
G4
$2
44
42
M
31
23
·31
19
14
11
8
18
13
" •
a
59
~
2$
22
16
12
21
16
7
·Sample size less than 5.
Since RR=P,/P2 , RR~ 1/P2 •
For RR< 1, use the column value corresponding to 1/ RR and the row value corresponding to P,.
.
27
.21
26
1$
-
•
13
.JQ;. ,
II·'·
I~
Table 8
( continued)
(e) Confidence level 90%, relative precision 10%
~RRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
48266
23890
15764
11701
9264
43391
21452
14139
10483
8289
3901
2438
1707
1268
976
767
610
488
391
311
244
140
61
40141
19827
13056
9670
7639
3676
2221
37819
18666
12282
9090
7174
33442067
1428
1045
790
607
471
364
279
209
36078
17796
11701
8654
6826
3169
1961
1341
976
732
558
427
326
244
34724
17118
11250
8316
6555
3034
33640
16577
10889
8045
6338
2926
32754
16133
10593
7823
6161
2837
1729
1175
843
621
463
32015
15764
10347
7639
6013
2763
31390
15452
10139
7482
5888
2701
1a1
110:7
788
576
30855
15184
9960
7348
5781
2647
1602.
3.75
4.00
4.25
4.50
4.75
5.00
29625
14569
9550
7041
5535
29307
14410
9444
6961
5472
24'2
29022
14267
9349
6890
5415
2464.
28765
14139
9264
6826
5363
P2
01
0>
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
0.90
4388
2763
1961
1463
1138
906
732
596
488
399
326
209
122
1544
1138
867
674
529
416
326
252
190
,.
1274
921
&87
519
393
1788
1219
878
&51
488
366
65
Since RR=P,/P2' RR~1/P2'
For RR< 1, use the column value corresponding to 1/ RR and the row value corresponding to P,.
1660
1138
813
59&
1610:
7&7
30390 29984
14952 14748
9670
9805
7131
7232
5688
5607
2601
2160
1571 . 1544
1f.)$?
1637
732.
7.
2524
1520
10U.
'.9·
1003
.,.
'...
2411
9~
tn
III
3
'C
ii'
UI
j;j'
CD
..
..o·
Q.
CD
CD
~
3
5i'
III
:::s
Table 8 ( continued)
(f)
Confidence level 90%. relative precision 20%
~RRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
0.01
0.02
0.03
0.04
0.05
10761
5326
3515
2609
2066
9674
4783
3153
2337
1848
8949
4421
2911
2156
1703
8432
4162
2738
2027
1600
8044
3968
2609
1930
1522
7742
3817
2508
1854
1462
7500
3696
2428
1794
1413
7303
3597
2362
1744
1374
7138
3515
2307
1703
1341
6999
3445
2261
1668
1313
6879
3385
2221
1639
1289
6776
3334
2186
1613
1269
6685
3288
2156
1590
1250
6605
3248
2130
1570
1234
0.10
0.15
870
798
496
677
416
284
616
375
254
182
133
591
563
346
339
335
247
176
129
241
171
580
351
236
167
571
358
231
164
227
224
221
153
633
386
262
188
139
366
164
653
399
272
196
145
602
254
194
707
436
299
218
6413
3153
2066
1522
1196
544-
381
283
218
746
461
319
233
176
6470
3181
2085
1536
1208
550
330
0.25
0.30
979
616
435
327
254
6534
3213
2106
1552
1220
656
0.35
0.40
0.45
0.50
0.55
202
164
133
109
89
171
136
109
87
70
151
118
93
73
56
136
105
82
63
47
125
96
73
55
116
88
109
82
104
73
55
32
14
43
P2
c..n
-...J
0.20
0.60
0.70
0.80
0.90
47
28
544-
345
206
13
Since RR= P,/ P2 • RR~ 1/ P2 •
For RR< 1. use the column value corresponding to 1/ RR and the row value corresponding to P,.
327
218
I~
Table 8 ( continued)
(g)
Confidence level 90%, relative precision 25%
,\RRI
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
0.01
0.02
0.03
0.04
0.05
6474
3205
2115
1570
1243
5821
2878
1897
1406
1112
5073
2504
1648
1220
963
4840
2387
1570
1161
916
4658
2297
1509
1116
880
4513
2224
1461
1079
851
4394
2164
1421
1050
827
4295
2115
1388
1025
807
4211
2073
1360
1004
790
4139
2037
1336
986
776
4077
2006
1316
971
763
4022
1979
1297
957
753
3974
1955
1281
945
743
3931
1933
1267
934
734
3893
1914
1254
925
727
3859
1897
1243
916
720
0.10
0.15
0..20
0.25
0.30
589
298
208
153
117
449
218
192
141
106
426
262
180
131
407
250
171
124
93
393
240
164
381
232
158
113
84
371
226
153
109
80
363
220
149
355
215
145
103
349
211
142
101
344
208
139
99
339
204
137
335
202
135
331
199
133
327
197
131
197
153
524
327
229
171
131
5385
2660
1752
1297
1025
480
0.35
0.40
0.45
0.50
0.55
122
99
80
66
54
44
103
82
66
53
42
33
91
71
56
44
34
82
64
49
38
29
70
53
66
50
29
17
8
19
9
P2
(J1
00
0.60
0.10
0.80
0.90
371
262
99
75
58
44
33
118
88
62
26
Since RR= P,/ P2 , RR";; 1/ P2 .
For RR< 1, use the column value corresponding to 1/ RR and the row value corresponding to P,.
106
78
en
DI
3
'C
CD
1/1
j;j'
CD
...
..o·
Do
CD
CD
3
S·
DI
=
Table 9. Hypothesis tests for a relative risk
n= {zl-a/zJ[2P(l- P)] +Zl-pJ[P l (l-Pd+Pz(l-Pz)]Y /(P 1 -Pzf
where
P=(P 1 +P z )/2
(a) Level of significance 5%, power 90%, two-sided test
RRa)
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
37411
18492
12185
9032
7140
3357
2095
1465
1086
834
654
519
414
329
261
203
113
46
10378
5122
3371
2495
1969
918
568
393
287
217
167
130
101
5066
2497
1641
1212
955
442
270
185
133
99
75
56
42
30
21
3104
1528
1002
739
582
266
161
109
2149
1056
692
509
400
182
109
72
50
36
25
17
1605
787
515
379
297
133
79
52
35
24
16
11
1261
618
403
296
232
103
60
39
26
17
11
1028
503
328
240
188
82
47
30
19
12
862
421
274
200
156
68
38
24
15
9
738
360
234
171
133
57
32
19
12
643
313
203
148
115
568
276
178
130
101
42
23
13
.7
507
246
159
115
89
37
457
221
143
103
80
416
201
129
93
381
184
118
85
65
33
29
26
19
11
11
1i
7
13
P2 ""
~I
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
77
58
42
77
56
41
30
21
14
Since RRa=P,/P2' RRa ~1/P2·
For RRa < 1. use the column value corresponding to 1/ RRa and the row value corresponding to P"
4$
27
16
9
&
72
6
I:3
'C
ii'
1/1
N'
CD
....
..S'
Q.
CD
CD
3
III
0'
;:,
Table 9
( continued)
(b) Level of significance 5%, power 80%, two-sided test
""'-,RRal
P2
0)1
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
27946
13814
9103
6747
5334
2508
1566
1095
812
623
489
388
309
247
195
152
85
35
7752
3827
2518
1864
1471
686
425
294
215
163
125
97
76
58
44
32
3785
1866
1226
906
714
330
202
138
100
74
56
42
32
23
16
2319
1142
749
553
435
200
121
82
58
42
31
23
16
11
1606
789
517
381
299
136
82
1199
589
385
283
222
100
59
39
27
19
13
8
943
462
302
222
174
769
376
245
180
141
62
36
23
15
10
644
315
205
150
117
51
29
18
12
7
552
269
175
128
100
481
234
152
111
86
37
20
12
7
425
207
134
97
76
32
17
10
6
379
184
119
87
67
28
15
8
342
166
107
78
60
25
13
7
311
151
97
70
54
22
11
6
285
138
89
64
49
20
10
5
54
38
27
19
14
77
45
29
20
13
9
Since RRa= P,/ P2' RRa~1/P2'
For RRa< 1. use the column value corresponding to 1/ RRa and the row value corresponding to P,.
43
24
15
9
I~
Table 9
( continued)
(c) Level of significance 5%, power 50%, two-sided test
""!1
P2
Ol
N
0.01
0.02
0.03
0.04
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.70
0.80
R
a
I
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
13675
6760
4455
3302
2611
1228
767
536
398
306
240
191
152
122
96
75
42
18
3794
1873
1233
913
721
337
209
102
66
49
38
17
11
9
168
82
53
39
30
13
7
153
75
48
35
27
12
6
140
68
44
32
25
11
6
8
5
7
6
186
91
59
43
34
15
8
5
28
22
16
12
9
462
227
148
109
86
39
23
15
10
7
5
236
115
75
55
43
19
37
588
289
189
139
110
50
30
20
14
10
7
5
271
133
86
63
50
22
13
62
49
38
29
22
787
387
254
187
147
67
41
27
19
14
10
316
155
101
74
58
26
15
10
7
81
1136
559
367
271
214
98
60
.41
29
22
16
12
9
6
377
185
121
89
70
31
145
106
1853
914
601
444
350
162
100
69
50
209
7
18
12
8
6
17
'Sample size less than 5.
Since RRa=P,/P2• RRa~1/P2.
For RRa < 1. use the column value corresponding to 1/ RRa and the row value corresponding to P,.
*
•
en
III
3
'C
CD
III
jij'
CD
....
..o·
Q.
CD
CD
3
S·
III
:::I
Table 10. Accepting a population prevalence as not exceeding a specified value
The value of n is obtained by solution of the inequality
x~o Cx
d*
M
(N-M)
C(n-x)
I Cn<a
N
where M = N P, for a finite population; or
Prob {d :::;;d*} < a
i.e.
d*
I
Prob(d)<a
d=O
or
d*
I
nCdpd(l- p)"-d < a
d=O
I~...
for an infinite population.
~I
CD
10
(a) Confidence level 95%, d*=O
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
100
200
1 000
2000
2500
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
9
9
9
9
9
13
13
14
14
14
25
27
29
29
29
45
51
57
58
58
82
90
112
115
116
5000
1Qooo
15000
20000
25000
2
2;
2
:3
3
3.
4
4
4
4
9
9
"9
9
9
14
14
14
14
14
29
29
29
29
4
5
5
5
5.
5
6
2
2
2
58
69
59
59
59
118
118
118
111
119
96
140
212
225
228
234
286
4
4
5
5
9
9
14
14
59
59
119
119
P
N
50000
Infinite
2;
2
2
2
2
2
2
2
3.
3.
3
3
6
6
6
6
6
6
29
29
29
237
236
238
239
239
Table 10
( continued)
(b) Confidence level 95%, d* = 1
P
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
8
8
8
8
8
8
8
8
8
8
8
8
10
10
10
10
10
10
10
10
10
10
10
10
14
14
14
14
14
14
14
14
14
14
14
14
20
21
22
22
22
22222222
2222
22
38
42
45
46
46
46
46
46
46
46
46
46
64
95
127
174
181
183
186
187
187
188
188
188
188
100
191
324
348
356
367
312
314
316
319
379
379
N
100
200
1 000
2000
2500
5000
10000
15000
20000
25000
50000
Infinite
77
90
92
92
93
83
93
93
93
94
94
~I
(I)
11/
3
"CI
i'
III
p:j'
CD
....
aCD
CD
I :::I~,
(c) Confidence level 95%, d* = 2
P
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
4
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
10
11
11
11
11
11
11
11
11
11
11
11
13
14
14
14
14
14
14
14
14
14
14
14
18
19
19
19
19
19
19
19
19
19
19
19
27
28
29
30
30
30
48
54
60
61
61
61
61
61
61
62
62
62
77
100
155
227
238
242
246
"246
100
200
417
455
467
N
100
200
1000
2000
2500
5000
10000
15000
2()000
25000
50000
Infinite
8
9
9
9
9
9
9
30
30
30
30
30
30
98
118
122
122
123
123
124
124
124
125
125
.,- ..,201
251
251
486
43
502
502
502
11/
Ig'
Table 10
( continued)
(d) Confidence level 95%, d* = 3
P
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
9
9
9
9
9
9
9
9
9
9
9
9
10
11
11
11
11
11
11
11
11
11
11
11
13
13
13
13
13
13
13
13
13
13
13
13
16
17
17
17
17
17
17
17
17
17
17
17
22
23
24
24
24
24
24
24
24
24
24
24
32
34
36
37
37
37
37
37
37
37
37
37
58
66
74
75
75
75
75
75
75
76
76
76
88
116
145
150
150
152
152
152
153
153
155
155
100
176
275
291
297
303
305
306
307
307
309
309
100
200
501
552
571
596
607
610
614
618
619
619
N
100
200
1 000
2000
2500
5000
10000
15000
20000
25000
50000
~I
Infinite
Ie
(e) Confidence level 95%, d* = 4
p
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
7
7
7
7
7
7
7
7
7
7
7
7
8
9
10
10
10
10
10
10
10
10
10
10
10
10
12
13
13
13
13
13
13
13
13
13
13
13
15
16
16
16
16
16
16
16
16
16
16
16
19
20
20
21
21
21
21
21
21
21
21
21
26
27
28
28
28
28
28
28
28
28
28
28
38
41
43
43
43
43
44
66
77
87
88
89
89
89
95
132
170
176
177
179
180
180
180
181
181
181
100
191
321
342
349
357
361
361
362
363
363
364
100
200
578
643
669
701
715
720
724
728
728
730
N
100
200
1 000
2000
2500
5000
10000
15000
20000
25000
50000
Infinite
9
9
9
9
9
9
9
9
9
9
Ii...
44
44
44
44
44
89
89
90
90
90
Table 10
( continued)
(f) Confidence level 90%, d* = 0
P
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
7
7
7
7
7
10
11
11
11
11
20
21
22
22
22
37
41
44
45
45
78
78
87
89
90
94
120
168
175
177
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
4
4
4
5
5
5
5
5
7
7
7
7
7
11
11
11
11
11
22
22
2
3
3
3
3
3
22
23
45
45
45
45
45
91
91
91
91
92
181
182
182
184
184
2
2
2
2
2
2
3
3
4
4
5
5
7
7
11
11
23
23
46
46
92
92
184
184
N
100
200
1 000
2000
2500
5000
10000
15000
20000
25000
50000
Infinite
22
(J)
(J)
en
QI
3
"C
iD
1/1
j;j'
~
....
CI.
CD
CD
I~,
(g) Confidence level 90%, d* = 1
:::I
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
100
200
1 000
2 000
2 500
3
3
3
3
3
4
4
4
4
4
4
4
5
5
5
5
5
7
7
7
7
7
8
9
9
9
9
11
12
12
12
12
17
18
18
18
18
32
35
37
38
38
56
65
74
76
76
93
112
145
149
151
100
188
274
290
296
5000
10000
15000
20000
25000 .
3
3
3
3
3
4
4
4
4
4
4
4
5
5
5
5
5
7
7
7
7
7
9
9
9
9
9
12
12
12
12
12
18
19
19
19
19
38
38
38
38
38
76
76
76
76
77
153
154
154
154
155
303
306
308
311
311
50000
Infinite
3
3
5
5
7
7
9
9
12
12
19
19
38
38
77
77
155
155
311
311
N ",P
I
4
4
4
4
4
4
4
4
4
4
It
Table 10 ( continued)
(h) Confidence level 90%, d*
=2
0.90
0.80
0.70
0.60
0.50
OAO
0.30
0.20
0.10
0.05
0.025
0.0125
100
200
1000
2000
2500
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
9
9
9
9
9
12
12
12
12
12
16
16
16
16
16
23
24
25
25
25
43
47
51
52
52
71
86
101
104
104
100
141
195
203
206
100
199
366
391
401
5000
10000
15000
20000
25000
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
9
7
9
7
7
8
9
9
9
12
12
12
12
12
16
16
16
16
17
25
25
25
25
25
52
52
52
52
52
105
105
105
105
105
209
·210
211
211
212
414
418
42'0
426
427
50000
4
4
5
5
6
6
8
8
9
9
12
12
17
17
25
25
52
52
106
106
212
212
427
427
P
N
~l
Infinite
I
(i) Confidence level 90%, d* = 3
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
8
8
8
8
8
9
9
9
9
9
11
12
12
12
12
14
15
15
15
15
19
20
21
21
21
29
30
32
32
32
52
58
64
65
65
82
104
126
130
130
100
164
241
253
258
100
200
449
484
500
6
6
6
6
6
8
8
8
8
8
9
12
12
12
12
12
15
15
15
15
15
21
21
21
21
21
32
32
32
32
32
65
65
65
65
66
131
132
132
132
132
262
10
10
10
10
265
265
267
518
526
527
531
535
7
7
8
8
10
10
12
12
15
15
21
21
32
32
66
66
135
135
267
267
535
535
0.90
0.80
0.70
100
200
1000
2000
2500
5
5
5
5
5
6
6
6
6
6
5000
10000
15000
20000
25000
5
5
5
5
5
50000
5
5
P
N
Infinite
Ii...
264
Q
Table 10 ( continued)
( i) Confidence level 90%, d* = 4
~I
~I
0.90
0.80
100
200
1 000
2000
2500
7
7
7
7
7
8
8
8
8
8
5000
10000
1&cOOO
JOcOOO
25cOOO
7
7
7
7
7
8
8
50000
7
7
Infinite
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.025
0.0125
9
9
9
9
9
11
11
11
11
11
14
14
14
14
14
17
18
18
18
18
23
24
25
25
25
34
36
38
38
38
60
69
77
78
78
90
121
150
155
156
100
180
285
302
308
100
200
527
572
595
11
12
12
12
12
14
14
14
14
14
18
18
18
18
25
25
38
25
79
157
158
168
168
159
314
316
316
317
318
619
628
628
18
38
38
38
38
78
78
78
8
9
9
10
10
10
8
8
10
10
12
12
14
14
18
18
25
25
39
39
79
79
159
159
318
318
I:3
"C
8
8
25
25
78
637
637
637
638
CD
en
iii'
CD
CI.
CD
CD
...
..c)'
3
S'
DI
~
I~_I""
.,.
_ ......................................
• .... J ..............
~
.............
n= [Zl-aJ {P o(1- Po)} +Zl-pJ {P a(1- P a n]2 /(P o - Pa)2
d* = [nPO-z1-aJ {nP o(1- PO)}]
The value of d* is always rounded down to the nearest integer (for example 5.8 would become 5).
(a) Level of significance 5%, power 90%, one-sided test (Pa< Po)
\Pol
Pa
0.10
~
•.
0.05
0.10
0.15
0.20
.;
0.25
0.20
0.15
0.30
0.35
0.45
0.40
0.50
d*
n
d*
n
d*
n
18
33
66
161
2
5
13
38
13
22
39
75
1
4
8
19
10
16
26
44
1
3
6
12
8
12
19
29
1
2
4
8
6
10
14
20
0
2
3
6
686
1.
180
753
25
242
4S
81.
206
11
31
80
312
31
10
10
11
38,
n
d*
n
d*
n
d*
n
d*
n
239
16
76
378
6
45
40
109
498
3
14
84
25
54
137
601
2
8
25
132
F:
62
till
,.
13
ee
218
804
..",'
d*
137
Oi.
d*
n
•
'
211
813
t3
402
, - - - - , - _ . _ -
I
-I
III
ffi I
\pJ
Pa
0.55
~
n
0.05
0.10
0.15
0.20
5
8
11
15
d*
0
2
3
5
d*
n
1,.
'i\".a'• •
"
,22
d*
n
t
6
8
11
0.70
0.65
0.60
5
7
9
1
2
3
I
12
1:1
U
I
t
~,2I.:
,'1.
I:~
21
It
II
7
11
18
48
11
27
852
444
210
834
114
477
91
203
798
51
120
496
0.50
0.55
0.60
0.65
0.70
I'"
t Sample size less than 5.
1.
d*
n
t
1
2
3
0.75
d*
n
5
7
1
2
5
2
I
t2
1•
22
3.1
4
7
I:
I
I
3
4
17
49
87
191
746
29
53
123
501
I
n
,
1,
15
21
6
8
11
12
14
30
46
80
176
676
18
29
53
122
488
19
27
42
72
156
•
d*
n
t
t
t
t
t
0.85
0.80
t
t
t
t
, .
a
3.
&:
4
12
18
29
52
116
d*
n
.I
0.90
d*
n
t
t
t
t
t
t
t
t
t
t
t
t
I
10
I
6
7
13
17
24
36
62
8
12
17
27
48
9
11
14
20
30
6
8
10
15
24
131
104
101
362
18
----
t
t
,I.
I.
6
I
•
d*
n
t
t
t
t
.I
I~
0.95
t
4
4
-to
311
t
5
7
8
11
15
3
5
6
9
12
21
11
3D
I»
M
17
221
2t4
( continued)
Table 11
(b) Level of significance 5%, power 80%, one-sided test (Pa < Po)
0.10
Pol
0.15
0.20
0.30
0.25
0.35
0.40
0.45
0.50
Pa
0.05
0.10
0.15
0.20
0,2$
0.30
0.35
0.40
0.45
n
d*
n
d*
n
d*
n
d*
n
d*
n
d*
n
d*
n
d*
n
184
11
60
283
4
32
32
83
368
2
10
60
21
42
103
441
1
5
18
95
15
26
50
119
501
1
3
9
27
133
11
18
30
56
133
1
2
6
13
37
173
8
13
20
33
61
142
683
0
2
4
8
18
47
213
7
10
14
22
35
65
149
0
1
3
6
10
22
57
252
5
8
11
15
23
37
67
548
606
d*
0
1
2
4
7
13
26
66
288
163
617
len
I»
3
~
CD
251
~
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.60
0.55
0.75
0.70
0.65
0.80
0.85
0.90
1/1
i:r
0.95
CD
...
Q.
n
d*
5
6
8
11
16
24
38
68
154
615
0
1
2
3
5
9
15
30
74
317
t Sample size less than
d*
n
1
2
2
4
6
10
17
33
80
340
5
7
9
12
16
23
37
65
145
573
1
2
3
5
7
11
19
35
84
353
d*
n
t
t
t
5
7
9
12
16
24
38
67
151
600
d*
n
1
2
4
5
8
11
20
37
86
356
5
6
8
11
15
21
32
57
125
483
d*
n
t
t
t
t
t
t
t
5
7
9
11
16
22
35
62
136
534
d*
n
t
5
6
8
.10
13
19
29
50
109
419
2
3
4
5
8
12
19
35
80
321
2
4
5
8
11
18
32
71
279
5
6
7
10
13
20
33
69
253
t
t
t
t
3
4
4
7
9
15
26
68
219
t
t
t
t
t
t
t
5
7
9
14
22
44
CD
CD
3
If
t
t
t
t
150
5.
d*
n
t
t
t
t
t
t
t
5
7
9
12
16
25
43
91
342
d*
n
t
t
t
t
t
t
t
t
2
2
3
5
8
12
19
37
85
346
d*
n
3
5
7
11
19
39
138
( continued)
Table 11
(c) Level of significance 5%, power 50%, one-sided test (Pa < Po)
" pJ
Pa
~
0.05
0.10
0.15
0.20
0.10
0.15
0.25
0.20
0.30
0.35
0.40
n
d*
n
d*
n
d*
n
d*
n
d*
n
98
4
35
138
13
20
44
174
4
26
13
23
51
203
0
2
7
40
10
15
26
57
0
1
3
11
7
10
16
28
0
1
2
5
6
8
11
17
0
0
1
3
228
57
62
247
15
74
29
65
260
7
19
91
0.25
0.30
0.35
0.40
0.45
d*
0.45
d*
n
0.50
d*
n
5
6
8
11
0
0
1
2
17
4
9
23
107
30
67
268
d*
n
t
5
6
8
0
0
1
11
17
2
5
10
27
121
31
68
271
tSample size less than 5.
-...II
Pa
~I
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
I~.......
CD
d*
n
d*
n
t
t
t
5
6
0
1
5
1
0.25
0.30
0.35
0.40
0.45
8
11
17
30
67
2
3
5
12
30
6
8
11
17
29
1
2
3
13
0.50
0.55
0.60
0.65
0.70
268
134
65
260
32
143
t
0.75
0.80
0.85
0.90
tSample size less than 5.
6
6
7
10
16
1
2
4
7
28
62
247
14
34
148
5
7
10
15
26
57
228
1
2
4
7
14
34
148
5
6
9
13
23
51
203
4
7
13
33
142
t
t
t
t
2
3
6
13
30
174
130
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
6
9
16
3
5
11
35
138
26
110
t
t
7
11
25
98
d*
n
t
t
t
t
5
7
11
20
44
d*
n
t
t
t
t
t
t
2
2
d*
n
t
t
t
t
t
t
t
t
d*
n
t
t
t
t
t
t
t
d*
n
t
t
t
t
t
t
d*
n
t
0.05
0.10
0.15
0.20
t
d*
n
4
8
20
83
6
. 13
52
t
4
11
46
Sample size determination
Table 12. Estimating an incidence rate
with specified relative precision
n = (Zl_~/2/f.)2
Confidence level
E
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
99%
95%
90%
66358
16590
7374
4148
2655
1844
1355
1031
820
664
461
339
260
205
166
138
38417
9605
4269
2402
1537
1068
785
601
475
385
267
197
151
119
97
80
67
57
50
43
38
34
30
27
25
22
20
19
11
16
27061
6766
3007
1692
1083
152
553
423
335
211
188
139
106
84
68
56
41
41
35
116
99
85
74
65
58
52
46
42
38
35
32
29
27
72
31
27
24
21
19
17
16
14
13
12
11
Table 13. Hypothesis tests for an incidence rate
F or a one-sided test
n =(Zl-aAO + ZI_pAa)2 /{}-o - )~a)2.
For a two-sided test
n = (Zl-a/2AO + Zl-pAaf /(A o -
Aa)2.
(a) Level of significance 5%, power 90%, one-sided test
,d
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
21
10
57
7
21
109
6
13
37
179
5
10
21
57
266
5
8
15
31
81
369
5
7
12
21
43
109
490
5
7
10
16
29
57
142
629
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
6
9
13
21
37
6
8
12
17
27
46
90
220
956
5
7
10
14
21
33
57
109
266
1146
5
7
9
13
18
26
41
68
130
315
1352
5
6
8
11
15
21
31
48
81
154
369
1576
5
6
8
10
13
18
25
37
57
94
179
428
1817
5
6
7
9
12
16
21
30
43
66
109
206
490
2075
5
6
7
9
11
14
19
25
34
50
76
125
235
557
2351
5
5
7
8
10
13
16
21
29
39
57
86
142
266
629
2643
0.95
Aa
-...J
w
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
18
8
6
5
*
•
51
18
11
8
7
6
5
5
102
33
18
13
10
8
7
6
6
5
5
5
*
*
.
.
.
• Sample size less than 5.
*
.
.
169
51
27
18
14
11
9
8
7
7
6
6
5
5
5
254
74
38
25
18
14
12
10
9
8
7
7
6
6
356
102
51
33
23
18
15
13
11
10
9
8
7
474
133
66
42
29
23
18
15
13
12
10
9
610
169
83
51
36
27
22
18
16
14
12
72
179
784
764
209
102
62
43
33
26
21
18
16
934
254
122
74
51
38
30
25
21
1121
303
145
88
60
45
35
29
1326
356
169
102
70
51
40
1548
413
196
117
80
59
1786
474
224
133
90
2042
540
254
151
2315
610
286
2606
685
2913
~
5
6
8
9
12
15
19
24
33
45
64
97
160
298
704
2952
CT
CD
....
Co)
Table 13
( continued)
(b) Level of significance 5%, power 80%, one-sided test
Ao
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
18
9
44
7
18
83
6
11
29
135
5
9
18
44
199
5
7
13
25
62
5
7
10
18
34
6
9
14
23
6
8
11
18
5
7
10
14
5
7
9
12
5
6
8
11
5
6
7
10
5
6
7
9
5
5
7
8
5
6
8
5
6
7
5
6
7
275
83
44
108
29
56
135
578
22
36
69
165
704
18
27
15
21
32
53
99
13
9
11
44
20
28
14
18
23
8
10
12
16
62
10
13
18
24
34
10
12
15
38
11
15
21
29
116
275
1155
72
135
318
1330
51
83
155
364
1518
. 39
58
95
176
412
31
44
66
108
199
26
35
50
75
121
465
223
Aa
0.05
0.10
0.15
0.20
0.25
0.30
0.36
~I 0.40
0.45
0.50
12
5
.
.
"
.
"
34
12
7
5
.
"
69
21
12
8
6
5
"
0.55
0.60
0.65
0.70
0.75
0••
0,85
6,;90'
"
"
"
117
34
18
12
9
7
177
6
5
"
"
"
• Sample size less than 5.
"
"
50
249
25
16
12
69
364
465
34
21
334
92
45
431
117
540
9
7
6
5
5
15
12
9
8
7
27
19
15
12
10
56
34
24
18
14
145
69
42
29
21
.
6
5
5
8
7
6
662
177
84
50
34
44
83
199
842
796
211
100
60
12
17
25
40
10
14
20
9
12
16
30
23
942
249
117
69
47
34
235
992
1101
290
136
80
64
18
25
1272
334
155
92
1456
381
177
171'
1652
431
1930
1860
.It
-
20
520
2154
...
en
III
3
"CD
III
j:r
CD
...
..cr
Q-
CD
CD
3
:i"
III
:=
Table 13
( continued)
(c) Level of significance 5%, power 90%, two-sided test
\
AO I 0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
28
13
10
28
137
8
17
47
223
7
13
28
7
11
20
40
102
459
6
10
16
28
55
137
608
6
9
13
21
37
72
178
779
6
8
12
17
28
47
91
223
970
6
8
11
15
22
35
59
113
274
1182
6
7
10
13
19
28
43
72
137
331
1416
5
7
9
12
16
23
33
52
86
163
392
1670
5
7
9
11
15
20
28
40
61
102
192
459
1946
5
7
8
10
13
17
24
33
47
72
119
223
531
2242
5
6
8
10
12
16
21
28
38
55
83
137
257
608
2560
5
6
8
9
11
14
18
24
32
44
63
95
157
293
691
2898
5
6
7
9
11
13
5
6
7
8
10
12
15
19
24
31
42
57
81
122
200
371
872
3639
Aa
~l
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
21
9
6
5
72
61
21
12
9
7
6
5
5
5
• Sample size less than 5.
122
38
21
14
11
9
8
7
6
6
5
5
5
204
61
32
21
16
12
10
9
8
7
7
6
6
5
5
72
331
306
89
45
29
21
16
13
11
10
9
8
7
7
6
430
122
61
38
27
21
17
14
12
11
10
9
8
575
160
79
49
34
26
21
17
15
13
12
11
741
204
99
61
42
32
25
21
18
16
14
928
252
122
74
51
38
30
25
21
18
1136
306
147
89
61
45
35
29
24
1365
366
174
104
71
53
41
33
1615
430
204
122
83
61
47
1886
500
236
140
95
69
2179
575
270
160
108
2492
656
306
181
2826
741
345
3181
832
17
21
28
37
50
72
108
178
331
779
3258
3557
~
DI
C'
iD
....
c..I
Table 13
( continued)
(d) Level of significance 5%, power 80%, two-sided test
\
,d
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
23
12
58
9
23
108
8
15
38
174
7
12
23
58
256
6
10
17
33
81
353
6
9
14
23
44
108
466
6
8
12
18
30
58
139
595
6
8
10
15
23
38
73
174
739
6
7
10
13
19
29
48
69
213
899
5
7
9
12
16
23
35
56
5
7
8
11
14
20
28
42
69
128
302
1267
5
6
8
10
13
17
23
33
50
81
150
353
1474
5
6
8
9
12
15
20
27
38
58
94
174
407
1697
5
6
7
9
11
14
18
23
31
44
67
108
199
466
1936
5
6
7
8
10
13
16
20
27
36
51
76
123
227
528
5
6
7
8
10
12
16
16
23
30
41
58
86
139
256
595
5
6
7
8
9
Aa
0.05
0.10
0.15
0.20
0.25
0.30
~I
0.36
0.40
14
6
•
42
14
8
6
0.46
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.65
0.90
0.95
86
26
14
9
7
6
5
*
•
*
•
*
• Sample size less than 5.
*
146
42
21
14
10
8
6
6
5
221
62
31
19
14
11
9
7
6
6
5
5
•
312
66
42
26
18
14
11
9
8
7
6
6
5
419
114
65
34
23
17
14
11
10
8
7
7
541
146
70
42
29
21
17
14
12
10
9
680
181
86
52
35
26
20
16
14
12
834
221
104
62
42
31
24
19
16
1003
265
124
74
50
36
28
23
108
256
1075
1188
312
146
86
58
42
32
1389
364
169
99
67
48
1606
419
194
114
76
1839
478
221
129
2190
2087
541
250
2460
2350
609
2630
(I)
III
3
'C
CD
11
III
13
16
21
26
34
46
65
97
156
286
665
2746
.
j;j'
CD
a..
CD
CD
~
3
..o·
:I
III
:I
Table 14. Hypothesis tests for two incidence rates in follow-up (cohort) studies (study duration not fixed)
F or a one-sided test
n 1 = {Zl-,j[(1 + k)I2] + Z l-pj(U i + ;"~)}2/kVl - )"2f·
For a two-sided test
n 1 = {Zl-,/2j[(1 +k)I2]+zl-pj(Ui+}"nV/ k()'1-A 2f
where
A=(A 1 +)"2)/2
and k is the ratio of the sample size for the second group of subjects (n 2 ) to that for the first group (n 1 ).
For Tables 14 a-d, k= 1.
(a) Level of significance 5%, power 90%, one-sided test
A,
-....I
-....I
.le2
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
41
19
109
14
41
212
12
26
71
349
11
19
41
109
521
10
16
29
60
157
726
9
14
23
41
83
212
966
9
13
19
31
55
109
277
1240
9
12
8
11
15
22
33
52
89
174
431
1891
8
11
14
19
28
41
64
109
212
521
2268
8
10
13
18
24
34
50
78
132
254
619
2680
8
10
13
16
21
29
41
8
10
12
15
19
26
35
49
71
109
183
349
842
3605
8
9
11
14
18
23
30
41
57
83
127
212
403
966
4119
8
9
11
13
8
9
11
13
16
19
24
31
41
55
76
109
167
277
521
1240
5250
8
9
10
12
15
18
22
28
36
47
63
86
124
189
312
585
1390
6867
'\
0.05
0.10
0.15
0.20
0.25
0.30
0.36
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.86
0.90
0.95
41
19
14
12
11
10
9
9
9
8
8
8
8
8
8
8
8
8
109
41
26
19
16
14
13
12
11
11
10
10
10
9
9
9
9
212
71
41
29
23
19
17
15
14
13
13
12
11
11
'1
10
349
109
60
41
31
26
22
19
18
16
15
14
13
13
12
521
157
83
55
41
33
28
24
21
19
18
17
16
15
726
212
109
71
52
41
34
29
26
23
21
19
18
966
277
140
89
64
50
41
35
30
27
24
22
1240
349
174
109
78
60
49
41
36
31
28
1549
431
212
132
93
71
57
48
41
36
17
26
41
71
140
349
1549
1891
521
254
157
109
83
66
55
47
2268
619
300
183
127
96
76
63
2680
726
349
212
146
109
86
3125
842
403
243
167
124
60
93
157
300
726
3125
3605
966
460
277
189
4119
1099
521
312
4667
1240
585
17
21
27
35
48
66
96
146
243
460
1099
4667
5250
1390
5867
Ii.01:0
Table 14
( continued)
(b) Level of significance 5%, power 80%, one-sided test
).,1 0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
29
14
79
10
29
153
8
18
51
252
8
14
29
79
376
7
12
21
43
113
524
7
10
16
29
60
153
697
6
9
14
22
39
79
199
895
6
8
12
18
29
51
101
252
1118
6
8
11
16
24
37
64
126
311
1365
6
8
10
14
20
29
46
79
153
376
1638
6
7
9
13
17
24
36
56
95
183
447
1934
0.70
0.75
0.80
0.85
0.90
0.95
6
5
7
8
11
14
18
25
35
51
79
132
252
608
2602
5
7
8
10
13
16
22
29
41
60
92
153
291
697
2974
5
6
8
10
12
15
19
25
34
48
69
106
176
332
793
3369
5
6
8
9
11
14
5
6
7
9
11
13
16
20
26
34
45
62
90
136
225
422
1004
4235
)'2
~I
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
a.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
29
14
10
8
8
7
7
6
6
6
6
6
6
5
5
5
5
5
79
29
18
14
12
10
9
8
8
8
7
7
7
7
6
6
6
153
51
29
21
16
14
12
11
10
9
9
8
8
8
8
7
252
79
43
29
22
18
16
14
13
12
11
10
10
9
9
376
113
60
39
29
24
20
17
15
14
13
12
11
11
524
153
79
51
37
29
24
21
18
16
15
14
13
697
199
101
64
46
36
29
25
22
19
17
16
895
252
126
79
56
43
35
29
25
22
20
1118
311
153
95
67
51
41
34
29
26
1365
376
183
113
79
60
48
39
34
1638
447
216
132
92
69
55
45
1934
524
252
153
106
79
62
2256
608
291
176
120
90
7
9
12
15
21
29
43
67
113
216
524
2256
2602
697
332
199
136
2974
793
376
225
3369
895
422
3790
1004
17
22
29
39
55
79
120
199
376
895
3790
4235
(I)
al
3
'C
ii'
III
j;j'
ID
...
..o·
0-
lD
ID
3
5'
al
:=
Table 14
( continued)
(c) Level of significance 5%, power 90%, two-sided test
\
.
11.2
U3i
\
,1,1
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.46
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0;85
0.90
0.95
0.05
50
24
17
14
13
12
11
11
10
10
10
10
9
9
9
9
9
9
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
50
24
134
17
50
260
14
31
87
428
13
24
50
134
638
12
20
35
73
192
891
11
17
28
50
101
260
1185
11
15
24
38
67
134
339
1521
10
14
21
31
50
87
171
428
1900
10
13
19
27
40
63
109
213
528
2320
10
13
17
24
34
50
78
134
260
638
2783
10
12
16
21
29
41
61
9
12
15
20
26
35
50
73
114
192
368
891
3834
9
11
14
18
24
31
42
69
87
134
225
428
1033
4422
9
11
14
17
22
28
37
50
70
101
156
260
494
1185
5053
9
11
13
16
20
25
33
43
58
81
117
179
298
564
1348
6726
9
11
13
15
19
24
30
38
50
67
93
134
205
339
638
1521
6440
9
10
12
15
18
22
27
34
44
57
76
106
152
231
382
718
1705
7197
134
50
31
24
20
17
15
14
13
13
12
12
11
11
11
11
10
260
87
50
35
28
24
21
19
17
16
15
14
14
13
13
12
428
134
73
50
38
31
27
24
21
20
18
17
16
15
15
638
192
101
67
50
40
34
29
26
24
22
20
19
18
891
260
134
87
63
50
41
35
31
28
26
24
22
1185
339
171
109
78
61
50
42
37
33
30
27
1521
428
213
134
95
73
59
50
43
38
34
1900
528
260
162
114
87
70
58
50
44
2320
638
311
192
134
101
81
67
57
2783
759
368
225
166
117
93
76
3287
891
428
260
179
134
106
95
162
311
759
3287
3834
1033
494
298
205
152
4422
1185
564
339
231
6053
1348
638
382
5726
1521
718
6440
1705
7197
-I
III
r:T
....ca
CD
Table 14
( continued)
(d) Level of significance 5%, power 80%, two-sided test
A,I 0.05
0.10
0.15
0.20
0.25
17
13
37
194
10
23
64
320
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
9
9
14
26
54
143
665
8
13
21
37
75
194
885
8
11
17
28
50
100
253
1136
7
10
15
23
37
64
128
320
1419
7
10
14
20
30
47
81
159
394
1733
7
9
13
17
25
37
58
100
194
477
2078
7
9
12
16
22
31
45
7
9
11
14
19
26
37
54
85
143
274
665
2863
7
8
10
13
17
23
31
7
8
10
13
16
21
27
37
52
75
116
194
369
885
3774
7
8
10
12
15
19
24
32
43
60
87
134
222
421
1007
4277
6
8
9
11
14
17
22
28
37
50
69
100
153
253
477
1136
4811
6
8
9
11
13
16
20
25
32
42
57
79
113
173
285
536
1274
5376
A2
~I
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
37
37
100
13
10
100
37
23
9
17
17
9
8
8
7
7
7
7
7
7
7
7
6
6
14
13
11
10
10
9
9
9
8
8
8
8
8
194
64
37
26
21
17
15
14
13
12
11
10
10
10
9
9
320
100
54
37
28
23
20
17
16
14
13
13
12
11
11
477
143
75
50
37
30
25
22
19
17
16
15
14
13
17
37
100
477
665
194
100
64
47
37
31
26
23
21
19
17
16
885
253
128
81
58
45
37
31
27
24
22
20
1136
320
159
100
71
54
44
37
32
28
25
1419
394
194
120
85
64
52
43
37
32
1733
477
232
143
100
75
60
50
42
2078
567
274
168
116
87
69
57
2455
665
320
194
134
100
79
71
120
232
567
2455
2863
771
369
222
153
113
3303
885
421
253
173
44
64
100
168
320
771
3303
3774
1007
477
285
4277
1136
536
4811
1274
5376
(I)
III
3
'C
iD
1/1
jij'
CD
....
.
o·
CI.
CD
CD
~.
::I
III
::I