1. science
2. mathematics
3. statistics

Hypothesis Testing
Objectives:

Students should be able to identify the null and alternative (research)
hypotheses in a statistical test

Students should know the difference between one-and two-directional
hypothesis testing

Students should know what alpha, beta, power, and p-values are

Students should be able to identify/define type I and type II errors

Students should understand the differences between statistical
significance and clinical importance

Students should know how to determine statistical significance given
alpha and a calculated p-value OR given alpha and a corresponding
confidence interval
Hypothesis Testing
The second type of inferential statistics
Hypothesis testing is a statistical method used to make
comparisons between a single sample and a population, or
between 2 or more samples.
The result of a statistical hypothesis test is a probability, called a
p-value, of obtaining the results (or more extreme results) from
tests of samples, if the results really weren’t true in the
population.
Commonly Used Statistical Tests
Tests for quantitative data (i.e. comparing means):
 Two groups: t-test (paired or 2-sample)
 Two or more groups (ANOVA: analysis of variance)
Tests for categorical (nominal, ordinal) data (i.e. comparing
proportions):
 Chi-square, Fisher’s exact test
Tests for association between two quantitative variables:
 Correlation and regression
Hypothesis Testing
In all hypothesis testing, the numerical result from the statistical test is
compared to a probability distribution to determine the probability of
obtaining the result if the result is not true in the population.
Examples of two
probability
distributions:
normal
distribution
t distribution
the normal and
t-distributions
-4
-3
-2
-1
0
1
2
3
4
Steps in Statistical
Hypothesis Testing
1. Formulate null and research hypotheses
2. Set alpha error (Type I error) and beta error (Type
II error)
3. Compute statistical test and determine statistical
significance
4. Draw conclusion
Step 1: Formulate Null and
Research Hypotheses
Null Hypothesis (H0):
There is no difference between groups;
there is no relationship between the independent and
dependent variable(s).
Research Hypothesis (HR):
There is a difference between groups;
there is a relationship between the independent
and dependent variable(s).
Directional vs
Non-directional Hypotheses
Null and research hypotheses are either non-directional (two-tailed) or directional
(one-tailed):
Non-directional (two-tailed):
H0: Drug A = Drug B
HR: Drug A  Drug B
Rejection
Region
2.5%
NonRejection
Region
Rejection
Region
2.5%
Directional (one-tailed):
H0: Drug A  Drug B
HR: Drug A > Drug B
or
H0: Drug A  Drug B
HR: Drug A < Drug B
NonRejection
Region
Rejection
Region
5.0%
Example:
Directional vs Non-directional
Research question: Does age of onset of paranoid schizophrenia differ for
males and females?
Non-directional (two-tailed):
H0: Male Age = Female Age
HR: Male Age  Female Age
Rejection
Region
2.5%
NonRejection
Region
Rejection
Region
2.5%
Directional (one-tailed):
H0: Male Age  Female Age
HR: Male Age > Female Age
(or the opposite)
NonRejection
Region
Rejection
Region
5.0%
Step 2: Set Alpha (Type I) and
Beta (Type II) Errors
Alpha () is the level of significance in hypothesis testing:
Alpha is a probability specified before the test is performed.
Alpha is the probability of rejecting the null hypothesis
when it is true.
By convention, typical values of alpha specified in medical
research are 0.05 and 0.01.
Alphas have corresponding critical values, the same ones used
to calculate confidence intervals – 0.05/1.96, 0.01/2.575
Step 2: Set Alpha (Type I) and
Beta (Type II) Errors
Beta () is the probability of accepting the null hypothesis
when it is false.
Typical values for beta are 0.10 to 0.20
Beta is directly related to the power of a statistical test:
Power is the probability of correctly rejecting the null
hypothesis when it is false. Power = 1 - Beta
A type II error occurs when a false null hypothesis is accepted.
P-values
P-values are the actual probabilities calculated from a
statistical test, and are compared against alpha to determine
whether to reject the null hypothesis or not.
Example:
alpha = 0.05; calculated p-value = 0.008; reject null hypothesis
alpha = 0.05; calculated p-value = 0.110; do not reject null
hypothesis
A type I error occurs when a true null hypothesis is rejected.
Possible Outcomes in
Statistical Testing
Null Hypothesis
(Treatment A = Treatment B)
POPULATION
Decision Based
on Inferential
Statistical Test
Accept H0
(No difference)
Reject H0
(Difference)
True
(No difference)
False
(Difference)
Correct
Decision
Type II Error
(beta () error)
Type I Error
(alpha ()
error)
Correct
Decision
Power (1-)
Post treatment mortality in CABG/PTCA study:
What are the null and alternative hypotheses for a
two-tailed test?
Null Hypothesis (H0)
There is no difference in posttreatment mortality
between the CABG and PTCA groups
(the post treatment mortality is equal, i.e. P1 = P2)
Research Hypothesis (HR)
There is a difference in posttreatment mortality
between the CABG and PTCA groups (the post
treatment mortality is not equal, i.e. P1 P2)
Hypothesis Testing
Step 3
Compute statistical test and determine statistical
significance
• Calculations for statistical tests are different depending
on the type of test
• All involve determining a value of a test statistic that is
then converted to a probability of obtaining that test
statistic if the null hypothesis is true.
• The value of a test statistic is determined from the
measurement being tested, and the variability of the
measurement in the sample (the SE of the
measurement).
Example of a statistical test: two-sample t test
Does age of onset of paranoid schizophrenia differ for
males and females?
H0: Male Age = Female Age
HR: Male Age  Female Age
Male
Female
Test statistic:
n
12
12
mean age
26.8
29.6
(x1  x 2 )
t
SE(x1  x 2 )
SD
5.8
6.2
Example of a statistical test: two-sample t test
Does age of onset of paranoid schizophrenia differ for males
and females?
calculated test statistic: t = -1.142
Critical value of t for alpha = 0.05: + 1.960
The computed value of t does not exceed the critical value
so the null hypothesis of no difference in age is not
rejected (the p value is greater than 0.05)
Conclusion:
The mean age of onset is not different for males versus
females
Is the post treatment mortality different for patients
receiving CABG compared to patients receiving
PTCA?
There are a number of statistical tests that can be
used:
2 examples are 1) chi-square test, or 2) z test for
proportions. The resulting p values will be the
same regardless of the test used.
The researchers used a z test:
the p value from the test was 0.3508.
If alpha = 0.05, what did they conclude?
Is the post treatment mortality different for patients
receiving CABG compared to patients receiving
PTCA?
The p value is 0.3508… this is
>0.05, so the conclusion from the
study is that there is no difference
If there is truly no difference between
CABG and PTCA, the probability of
obtaining the difference of 0.6% is
~35%
Hypothesis Testing
Step 4
Draw conclusion about the population
based on the results of the statistical test
on the sample
Statistical conclusion: the results either are
or are not statistically significant………
BUT
You need to interpret the results in a
meaningful (and not just statistical) way
Principles for Statistical Significance
1. The size of a p value does not indicate importance of the
result.
2. Interpret nonsignificance cautiously.
a. finding no difference may be important
b. statistically nonsignificant  clinically unimportant
3. Results may be statistically significant but clinically trivial.
P Values vs. Confidence Intervals
•
There is a direct relationship between levels of alpha set for a statistical
test and the level set for constructing a confidence interval.
For example, alpha = 0.05 for a 2-sided statistical test is equivalent to a 95%
confidence interval
Rejection
Region
2.5%
NonRejection
Region
95% confidence interval
Rejection
Region
2.5%
P Values vs. Confidence Intervals
Statistical significance can be obtained from a confidence interval as
well as a hypothesis test
AND
Confidence intervals convey more information than p values
For this reason, most medical journals now prefer that results be
presented with confidence intervals rather than p values.
If the NULL VALUE for a statistical hypothesis test using alpha = 0.05
is contained within the 95% confidence interval,
we can conclude NO statistical significance at alpha = 0.05
without doing the hypothesis test:
P Values vs. Confidence Intervals
Example:
For differences between means or proportions, the null hypothesis is
that the difference is equal to zero:
If the 95% CI includes the value zero, the differences are not statistically
significant at alpha = 0.05.
For the test comparing the ages of males and females for onset of
paranoid schizophrenia, the null hypothesis is that the difference in age
is zero years.
P Values vs. Confidence Intervals
Example:
Male
Female
n
12
12
mean age
26.8
29.6
SD
5.8
6.2
The difference in age obtained from the sample is:
26.8-29.6 = -2.8 years
The standard error of the difference is 2.45
(calculation not shown)
The 95% confidence interval is:
-2.8 +/- 2(2.45) = -2.8 +/- 4.9 = -7.7 to 2.1
years
P Values vs. Confidence Intervals
The 95% confidence interval is:
-2.8 +/- 2(2.45) = -2.8 +/- 4.9 = -7.7 to 2.1 years
This means that the true population mean difference in
age is somewhere between males being 7.7 years
younger to males being 2.1 years older than females
The 95% CI includes 0 years, so there is no statistically
significant difference in age. In addition, we have
information about the precision of our estimate of the
difference, which cannot be obtained from p values
alone.
Note: This is a relatively wide confidence interval
because the sample size is small
P Values vs. Confidence Intervals
For the CABG/PTCA result:
The 95% CI is –0.6% to 1.7%
We can be 95% confident that the true difference in
mortality between CABG and PTCA is between –0.6%
and +1.7%
This confidence interval contains the value zero;
therefore, we could have concluded that the mortality
is not different based on the confidence interval alone.
P Values vs. Confidence Intervals
For ratio variables, such as relative risk and
odds ratio, the value one represents equality.
The null hypothesis is that the ratio is equal to
one:
If the 95% CI includes the value one, the
difference is not statistically significant at
alpha = 0.05.