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Applied Statistics Using SAS and SPSS
Topic: Chi-square tests
By Prof Kelly Fan, Cal. State Univ., East Bay
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Outline ALL variables must be categoricalGoal one: verify a distribution of Y
One-sample Chi-square test (SPSS lesson 40; SAS handout)
Goal two: test the independence between two categorical variablesChi-square test for two-way contingency table (SPSS
lesson 41; SAS section 3.G)McNemar’s test for paired data (SPSS lesson 44; SAS
section 3.L) Measure the dependence (Phil and Kappa coefficients)
(SPSS lesson 41, 44; SAS section 3.G, 3.M)
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Example: Postpartum Depression Study
Are women equally likely to show an increase, no change, or a decrease in depression as a function of childbirth?
Are the proportions associated with a decrease, no change, and an increase in depression from before to after childbirth the same?
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Example: Postpartum Depression Study
Depression after birth in comparison with before birth
Observed frequencies
Hypothesized proportions
Expected frequencies
Less depressed (-1) 14 1/3 20
Neither less nor more depressed (0)
33 1/3 20
More depressed (1) 13 1/3 20
From a random sample of 60 women
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One-sample Chi-Square Test
Must be a random sample
The sample size must be large enough so that expected frequencies are greater than or equal to 5 for 80% or more of the categories
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One-sample Chi-Square Test
Test statistic:
Oi = the observed frequency of i-th category
ei = the expected frequency of i-th category
i i
ii
e
eo 22 )(
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SPSS Output
1. Weight your data by count first
2. Analyze >> Nonparametric Tests >> Legacy Dialogs >> Chi Square, count as test variable
Postpartum Depression
14 20.0 -6.0
33 20.0 13.0
13 20.0 -7.0
60
less depressed
same
more depressed
Total
Observed N Expected N Residual
Test Statistics
12.700
2
.002
Chi-Square a
df
Asymp. Sig.
PostpartumDepression
0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 20.0.
a.
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Conclusion
Reject Ho
The proportions associated with a decrease, no change, and an increase in depression from before to after childbirth are significantly different to 1/3, 1/3, 1/3.
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Example: Postpartum Depression Study
Are the proportions associated with a change and no change from before to after childbirth the same?
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Example: Postpartum Depression Study
Depression after birth in comparison with before birth
Observed frequencies
Hypothesized proportions
Expected frequencies
Same amount of depression (0)
33 1/2 30
More or less depressed (1)
27 1/2 30
From a random sample of 60 women
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SPSS Output
Postpartum Depression--Recoded
33 30.0 3.0
27 30.0 -3.0
60
same
more or less depressed
Total
Observed N Expected N Residual
Test Statistics
.600
1
.439
Chi-Square a
df
Asymp. Sig.
PostpartumDepression--Recoded
0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 30.0.
a.
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Two-way Contingency Tables
Report frequencies on two variables
Such tables are also called crosstabs.
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Contingency Tables (Crosstabs)
1991 General Social Survey
Frequency Party Identification
Democrat Independent Republican
Race White 341 105 405
Black 103 15 11
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Crosstabs Analysis (Two-way Chi-square test) Chi-square test for testing the
independence between two variables:
1. For a fixed column, the distribution of frequencies over rows keeps the same regardless of the column
2. For a fixed row, the distribution of frequencies over columns keeps the same regardless of the row
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Measure of dependence for 2x2 tables
The phi coefficient measures the association between two categorical variables
-1 < phi < 1 | phi | indicates the strength of the
association If the two variables are both ordinal, then
the sign of phi indicate the direction of association
SPSS OutputP. 332 – 333
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SAS Output
Statistic DF Value ProbChi-Square 2 79.4310 <.0001
Likelihood Ratio Chi-Square 2 90.3311 <.0001Mantel-Haenszel Chi-Square 1 79.3336 <.0001
Phi Coefficient 0.2847 Contingency Coefficient 0.2738 Cramer's V 0.2847
Sample Size = 980
Measure of dependence for non-2x2 tables
Cramers V
Range from 0 to 1V may be viewed as the association between
two variables as a percentage of their maximum possible variation.
V= phi for 2x2, 2x3 and 3x2 tables
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Fisher’s Exact Test for Independence
The Chi-squared tests are ONLY for large samples:
The sample size must be large enough so that expected frequencies are greater than or equal to 5 for 80% or more of the categories
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SAS/SPSS Output
• SAS output: Fisher's Exact Test Table Probability (P) 3.823E-22 Pr <= P 2.787E-20
• SPSS output: in “crosstabs” window, click “exact”, then tick “exact”:
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Matched-pair Data
Comparing categorical responses for two “paired” samples
When eitherEach sample has the same subjects (or say
subjects are measured twice)
OrA natural pairing exists between each subject in
one sample and a subject form the other sample (eg. Twins)
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Example: Rating for Prime Minister
Second Survey
First Survey Approve Disapprove
Approve 794 150
Disapprove 86 570
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Marginal Homogeneity
The probabilities of “success” for both samples are identical
Eg. The probability of approve at the first and 2nd surveys are identical
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McNemar Test (for 2x2 Tables only)
SAS: Section 3.L; SPSS: Lesson 44
Ho: marginal homogeneity
Ha: no marginal homogeneity
Exact p-valueApproximate p-value (When n12+n21>10)
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SAS Output
McNemar's Test Statistic (S) 17.3559 DF 1 Asymptotic Pr > S <.0001 Exact Pr >= S 3.716E-05
Simple Kappa Coefficient Kappa 0.6996 ASE 0.0180 95% Lower Conf Limit 0.6644 95% Upper Conf Limit 0.7348
Sample Size = 1600Level of agreement
SPSS Output
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• SPSS: p. 361 and in “two-samples tests” window tick McNemar and click “exact”, then tick “exact”:
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