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2. mathematics
3. statistics

UK FHS
Historical sociology
(2014+)
Quantitative Data Analysis I. & II.
Contingency tables:
multivariate analysis and
elaboration
– introduction to
3-fold of data sorting, ordinal correlations
Jiří Šafr
jiri.safr(AT)seznam.cz
updated 26/11/2014
® Jiří Šafr, 2014
Multivariate analysis:
threefold level of data sorting in
crosstabulation
→ enables
a) more detailed description and
b) elaboration
(introduction 1.)
Third level of data sorting in
contingency table
• A contingency table analysis is used to examine
the relationship between two categorical
variables (bivariate crosstabulation)
• but it can be organized within levels of a third
variable.
If our goal is elaboration (rather than detailed
description), we call it test variable or factor.
We aim at to control for its effects.
• If a third variable is introduced, it will form
separate layers or strata in the table.
3rd level of sorting data
in contingency table
• We analyse simultaneously relationships among
several variables (mostly more independent –
explanatory variables).
• The principle is identical as in bivariate analysis.
• The goal of 3rd level of sorting data is in principle:
– More detailed description (in sub/sub-groups)
– Elaboration of relationships → searching for
causal relations, deeper understanding of context,
distinguishing between substantive and false relations,
controlling for effect of the 3rd variable (X↔Y / Z)
• This is true also for any 3rd level of sorting data in general, i.e. also for
means in subgroups and linear association (scatter-plots, correlation,
regression). We will explain it on contingency tables first.
Principle of multivariate analysis: 3rd level of data sorting (2×2×2 table)
Church Attendance by gender and age, USA 1990
100%
90%
80%
Under 40
40 and older
Men Women Men Women
Difference 9 % points
Weekly
21%
Less often
79
100% =
(270)
Source: General Social Survey, NORC
100 %
Difference 16 % points
30% 34%
70
66
(332) (317)
100 %
50%
50
(414)
50%
70%
60%
70%
66%
30%
34%
Women
Men
79%
50%
40%
30%
50%
20%
10%
21%
0%
Men
Under 40
Women
40 and older
Weekly
Less often
Source: [Babbie 1997: 391]
Dependent variable: Attendance to religious service simultaneously by 2 independent vars: Age, Gender
Both older men and women go to church more frequently than young (i.e. religiosity
rises up with age).
In each age category women attend church more often than men.
It seems that gender has slightly larger effect on church attendance than age.
Age as well as gender have independent effect on church attendance. Within each
category of independent variable different attributes of the other one still influence
people‘s behaviour.
Similarly both independent variables have cumulative effect on behaviour:
Older women visit church the most, whereas young men the least.
[Babbie 1997: 391-392]
Simplification of the 2×2×2 table:
Under 40
40 and older
Men Women Men Women
Weekly
21%
Less often
79
100% =
(270)
Attend Church Weekly
Men Women
Under 40
21
30
(270)
(332)
40 and older
34
50
(317)
(414)
30% 34%
70
66
(332) (317)
50%
50
(414)
100 % → 70 % Less often
Source: General Social Survey, NORC [Babbie 1997: 391]
We show only „positive“ categories of the variable („attend weekly“).
However we are not losing any information. Frequencies in brackets report
the base for percent, from which we can complete a sum for omitted category.
[Babbie 1997: 391]
Threefold data sorting (2×2×2 table) → description/exploration
Do students living at a dormitory (kolej) fail in exams (propadl) more often than those
Propadají
studenti „kolejáci“ – muži nebo „kolejáci“ – ženy?
living
elsewherevíce
(jinde)? Is it true for male (muži) as well as for female (ženy) students?
Male
Female
Muži
propadl
nepropadl
Celkem
Kolej
4%
96%
100%
Jinde
19%
81%
100%
Celkem
17%
83%
100%
Ženy
propadla
nepropadla
Celkem
Kolej
30%
70%
100%
Jinde
31%
69%
100%
Celkem
30%
70%
100%
15 percent
difference
only 1
percent
difference
In comparison to male students, female students living at dormitory tend to fail in
exams more often. However their proportion is about the same as in case of those
female students living somewhere else (i.e. effect of staying at dormitory on grades
is most probably not presented in case of women; regarding men this effect is
positive: male students staying at dormitory are more successful in exams as well as
they are the most successful from all).
Source: adapted from [Kapr, Šafář 1969: 152]
Introduction into
elaboration
Threefold data sorting
→
Controlling for the factor
Testing / controlling effect of 3rd
variable - factor → Elaboration
• Constructing separated tables split by
categories of the third variable makes
the tested factor holding constant.
→ relationship between two variables is
net – cleaned of distorting effect of this
factor variable.
Threefold data sorting: controlling effect of the third
variable: interpretation and arrangement of (2x3x3) table
Is voting related to age, even when effect of education is controlled?
Regarding ordinal independent variables we compare percentage differences between the
extreme categories separately among categories of controlling variable (the factor).
Základní vzdělání
Střední vzdělání
< 39 let
40-59
18%
24%
32%
36%
34%
49%
Nevolil
82
76
68
64
66
Celkem
100 %
100 %
100 %
100 %
N
(109)
(202)
(45)
(97)
Volil
> 60 let < 39 let
40-59
Vysokoškolské vzdělání
> 60 let < 39 let
40-59
> 60 let
40%
50%
70%
51
60
50
30
100 %
100 %
100 %
100 %
100 %
(271)
(139)
(27)
(62)
(50)
Differences between extreme categories of age in percentage points:
14 %
We ask:
13 %
30 %
Whereas in case of Elementary education (ZŠ) and Secondary (SŠ) there are differences
between youngest and oldest about the same, in case of University (VŠ) the difference is about
twice. → Thus Education partly intervenes into the relationship between voting and age.
1. Are there differences of Y (voting) along X (age) within categories of
controlling variable Z (education)? We compare it with bivariate crosstabulation
(Y by X).
2. Are differences between the extreme categories X (age) within categories of
controlling variable Z (education) approximately the same?
Interaction and additive effect
Interaction effect – effect of one variable on
another is contingent on the value of third variable
Note: plus % Didn‘t vote we get complete a sum of 100%.
VOLIL
mladí
starší
ZŠ
vzdělání
SŠ
VŠ
31
33
29
37
51
50
younger
older
45
40
37
35
33
31
29
30
31
25
Elem.
Secn.
Univ.
31
51
Different effect of age in categories of education on voting: for juniors no difference, for seniors %
difference in voting is rising with higher education. The highest voting is among older university graduates.
Additive effect – effects of both variables add
together to produce the additional
final result
vzdělání
VOLIL
mladí
starší
ZŠ
Still the same
percentage point
difference
between
categories of age
in categories of
education
SŠ
30
40
75
75
65
younger
older
65
55
45
45
40
VŠ
35
45
Similar effect of age in categories of education, only on „different level“
35
35
30
25
Elem.
Secn.
Univ.
65
75
[Treiman 2009: 26-28]
Testing the effect of further factor
(then in bivariate relationship)
• We compare intensity of relationship in original
bivariate table with relationships in new tables with
third variable-controlling factor (now split into its
categories).
• If in new tables the association between original
variables disappears or is substantially weaken
→ the association in the original (bivariate)
table is function of the third variable (controlling
factor)
•
Further you will see, how to detect hidden relationship quickly using association coefficients within
subgroups of the third controlling factor (for nominal variables Phi, CramV, Lambda, and ordinal
correlation).
•
Later in QDA II. We will also learn how to standardize (weight) the table along the controlling factor Z,
i.e. as if all cases in categories of variable X have the same proportion within categories of Z (e.g. the
same education).
Why we conduct elaboration?
1. To detect and describe
interaction (additive) effects
and when doing this we can reveal
2. Spurious association
(false association/correlation)
3. Suppressed – hidden
association
The aim is net relationship between two variables when controlled for effect of 3rd variable.
Following two examples will explain it.
Coefficients of association (e.g. Lambda used here) are explained in
later or in 3. Contingency tables and analysis of categorical data .
Example I.: Spurious association
(false association/correlation)
1. bivariate relationship
Preference for meal
Religiosity
HAMBURGER
Total
CAVIAR
High
Low
Total
Source: [Disman 1993: 219-223]
Seemingly strong association, but …
2. After controlling for effect of Education
(Threefold data sorting)
People with low education
Preference for meal
Religiosity
HAMBURGER
Total
CAVIAR
High
Low
Total
No association for people with low education; 0 % point difference (also Lambda=0).
Source: [Disman 1993: 219-223]
2. After controlling for effect of
Education (3rd level of data sorting)
People with high education
Preference for meal
Religiosity
HAMBURGER
Total
CAVIAR
High
Low
Total
Association disappears when we control effect of education → factor
behind which influences both religiosity and preference for food.
Source: [Disman 1993: 219-223]
Example II.: Suppressed – hidden
association
1. bivariate relationship
Package A Package B
Total
Would
buy
Would
not buy
Total
Source: [Disman 1993: 219-223]
Na první pohled žádná souvislost, ale …
2. when gender controlled for
(Threefold data sorting)
men
Package A
women
Package B
Total
Package A
Would
buy
Would
buy
Would
not buy
Would
not buy
Total
Package B
Total
Total
Source: [Disman 1993: 219-223]
Controlling for 3rd variable – factor revealed suppressed
association (false independency) between the two variables.
Reason for this bias → the relationship between the variables
exists only in a part of the population (within women).
When examining relationships in
elaboration coefficients of
association/ordinal correlation
can help us find interaction or
suppressed effects
Ordinal correlation for ordinal variables –
bivariate „zero order“ table/correlation (4o×4o table)
When our data is from random sample (i.e. not whole population) we have
to in addition first test statistical hypothesis, that the coefficient is not
zero (i.e. it is not zero in the whole population and not only in our sample).
Approx. Significance (also p) is here < 5% → we reject the null hypothesis
that Gamma/TauB is zero in whole population). More on this in QDA II.
Source: data [ISSP 2007, ČR]
CROSSTABS income4 BY edu4 /STATISTICS GAMMA BTAU.
Is the strength of relationship
(ordinal correlation) identical for
men and women?
→ we can compute conditional
association/correlation coefficients separately
in categories of control variable – factor
(gender)
Here 4o×4o×2 table.
Ordinal correlation for ordinal variables in 3rd level of data
sorting (separately for men and women) → gender [s30] is controlling factor
First order conditional table/ correlation
CROSSTABS prijem4 BY vzd4 BY s30 /STATISTICS GAMMA BTAU.
Among women
education has a
a little stronger
effect, but on the
whole women
earn less than
men regardless
of education
level (see also the
graph with means
of income).
Source: data [ISSP 2007, ČR]
In QDA II. we will further compute partial ordinal correlation (GAMMA).
Types of contingency tables with 3 variables
and coefficients of association/ correlation
Generally you can always use association (no direction just
strength of mutual dependence) → coefficients of association.
• 2×2×2 (similarly 2×2×3n) – all dichotomous → coefficients
association and also special point biserial correlation or tetrachoric
correlation
• 2×3o×3n or 2×3o×2 – dependent variable dichotomous,
independent ordinal, control nominal → ordinal correlation in
groups of control factor (without eventuality of considering linear
trends in strength of association/correlation)
• 2×3n×3o – dependent variable dichotomous, independent nominal,
control factor ordinal → only coefficients of association (but we
can consider linear trend in strength of association between
categories of control factor)
• 3o×3o×3o (similarly 2×2×3o) – all ordinal → ordinal correlation
(we can consider linear trend in strength of correlation between
categories of control factor) + coefficients of partial correlation (i.e.
net correlation of X↔Y when effect of Z is controlled; more on this in QDA II.)
It stands also for more than 3 categories (e.g. 4o or 4n).
Coefficients of association in (bivariate) multivariate
analysis in SPSS within CROSSTABS
•
Within CROSSTABS we can compute several measures of association and correlation for
variables Y x X (bivariate) as well as separately in categories of controlling factor Z →
this can help us quickly assess interaction and reveal „false“ relationship.
•
For nominal variables (Y, X, Z-controlling factor) coefficients of association
(they range 0-1 → no direction):
CROSSTABS var1 BY var2 BY var3-controlling /CELLS COL
/STATISTICS CC PHI.
Coefficients of association: CC = Contingency coefficient, PHI = Cramer V (+ equivalent for
dichotomised variables is Phi); there are also other coefficients of association and correlation (e.g. Lambda).
•
for ordinal variables (Y, X) and nominal/ordinal controlling factor (Z) in
addition of association coeff. ordinal correlation (they range -1–0–1 →
determine direction):
CROSSTABS var1 BY var2 /CELLS COL
/STATISTICS CC PHI GAMMA CORR BTAU.
Correlation coefficients: GAMMA = Goodman&Kruskal Gamma, BTAU = Kendaull Tau B,
CORR = Spearman Rho (+ Pearson correl. coef. R for ratio variables)
•
Notice, if we don‘t find correlation, it doesn't mean that, there is no (strong) relationship–association.
Moreover with ordinal variables comparison of correlations and coefficients of association can help us indicate what is the relationship (nonlinearity).
•
Notice: in case of means in subgroups (MEANS) we van compute coefficient Eta2 (for ratio x nominal variable):
MEANS var1-dependet-numeric BY var2-independent-categ. BY var3-controlling-categorial /CELLS
MEAN STDDEV COUNT /STATISTICS ANOVA.
More on coeficients of association and correlation can be found in 2. Korelace a asociace: vztahy mezi
kardinálními/ ordinálními znaky (in Czech only) na http://metodykv.wz.cz/AKD2_korelace.ppt
Notice: First, check counts
(absolute frequency) when sorting
data in higher level
(namely (but not only) in crosstabulation)
• When doing 3rd level of data sorting always
check counts in v individual cells of the table
with caution, notably in small samples.
CROSSTABS var1 BY var2 BY var3
/CELLS COL COUNT.
• If frequencies are too small, then interpretation
of the table makes no sense from the statistical
as well as substantive point of view.
→ You can collapse (recode) sparse cell entries.
More examples will be added later …