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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”: