Download Contingency Table Analysis & Review

Contingency Table Analysis
Mary Whiteside, Ph.D.
Overview
Hypotheses of equal proportions
 Hypotheses of independence
 Exact distributions and Fisher’s test
 The Chi squared approximation
 Median test
 Measures of dependence
 The Chi squared goodness-of-fit test
 Cochran’s test

Contingency Table Examples

Countries - religion by government
 States – dominant political party by
geographic region
 Mutual funds - style by family
 Companies - industry by location of
headquarters
More examples 
Countries - government by GDP
categories
 States - divorce laws by divorce rate
categories
 Mutual funds - family by Morning Star
rankings
 Companies - industry by price earnings
ratio category
Statistical Inference hypothesis of
equal proportions
H0: all probabilities (estimated by
proportions, relative frequencies) in
the same column are equal,
H1:at least two of the probabilities in
the same column are not equal
Here, for an r x c contingency table, r
populations are sampled with fixed
row totals, n1, n2, … nr.
Hypothesis of independence
H0: no association
i.e. row and column variable are independent,
H1: an association,
i.e. row and column variable are not independent
Here, one populations is sampled with
sample size N. Row totals are random
variables.
Exact distribution for 2 x 2 tables:
hypothesis of equal proportions; n1 =
n2 = 2
2
0
2
0
2
0
2
0
0
2
1
1
0
2
0
2
0
2
2
0
0
2
1
1
Fisher’s Exact Test
For 2 x 2 tables assuming fixed row
and column totals r, N-r, c, N-c:
 Test statistic = x, the frequency of
cell11
 Probability = hyper-geometric
probability of x successes in a
sample of size r from a population of
size N with c successes

Large sample approximation for
either test
Chi squared
= S [Observed - Expected]2 /Expected
 Observed frequency for cell ij comes
from cross-tabulation of data
 Expected frequency for cell ij

= Probability Cell ij * N

Degrees of freedom (r-1)*(c-1)
Computing Cell Probabilities
Assumes independence or equal
probabilities (the null hypothesis)
 Probability Cell ij = Probability Row i
* Probability Column j
= (R i/N) * (C j/N)
 Expected frequency ij = (R/N)*(C/N)*N
= R*C/N.
Distribution of the Sum
Chi Square with (r-1)*(c-1) degrees of
freedom
 Assumes

[Observed - Expected]2 /Expected
is standard normal squared

Implies
[Observed - Expected] /Square root[Expected]
is standard normal

Implies
 = s2 and Observed is a Poisson RV
Poisson is approximately normal if  > 5,
traditional guideline
 Conover’s relaxed guideline page 201

Measures of Strength:
Categorical Variables
 Phi
2x2
 Cramer's V for rxc
 Pearson's Contingency
Coefficient
 Tschuprow's T
Measures of Strength:
Ordinal Variables
 Lambda A ..
Rows dependent
 Lambda B .. Columns dependent
 Symmetric Lambda
 Kendall's tau-B
 Kendall's tau-C
 Gamma
Steps of Statistical Analysis
Significance - Strength
1- Test for significance of the observed
association
2 - If significant, measure the strength
of the association
Consider the correlation
coefficient
a measure of association (linear relationship
between two quantitative variables)
significant but not strong
 significant and strong
 not significant but “strong”
 not significant and not strong

r and Prob (p-value)
r = .20
 r = .90
 r = .90
 r = .20

p-value < .05
p-value < .05
p-value > .05
p-value > .05
Concepts
Predictive associations must be both
significant and strong
 In a particular application, an
association may be important even if
it is not predictive (I.e. strong)

More concepts
Highly significant , weak
associations result from large
samples
 Insignificant “strong” associations
result from small samples - they may
prove to be either predictive or weak
with larger samples

Examples
Heart attack Outcomes by
Anticoagulant Treatment
 Admission Decisions by Gender

Summary
 Is
there an association?
– Investigate with Chi square p-value
 If
so, how strong is it?
– Select the appropriate measure of
strength of association
 Where
does it occur?
– Examine cell contributions