This document discusses Chi Square and related procedures for analyzing categorical data. It explains that Chi Square can be used for goodness of fit tests to check if a sample follows a particular distribution, and for tests of association to check if two categorical variables are related. It provides examples of how to conduct and interpret Chi Square goodness of fit and association tests using SPSS. Other related procedures discussed include Fisher's Exact Test for small sample sizes and McNamer's Test for analyzing changes in paired categorical data.

1. Chi Square & Related Procedure

2. Why do we need Chi Square Procedures?
Limitation of
Binomial Test
Qualitative
Association
Binomial Test is for Binomial
variables. If there are
multinomial variables then
1. Identical Trial
2. There are k different outcomes
3. Probability of outcomes do not change from
trail to trial
4. Independent trials
Measure of Correlation does not work in case of
qualitative variables
Multinomial
Distribution
Multinomial
Distribution is
required

3. What is Chi Square Distribution?
Distribution Z2 = χ2
(1)
Distribution ΣZ2 = Z1
2 + Z2
2 + Z3
2 + … + Zk
2 = χ2
(k)
Properties
Takes only positive Value
Skewed, but becomes normal with increasing n
Mean = k, Variance = 2k

4. Shape of Chi Square Distribution

5. Uses of Chi Square & Assumptions
Goodness of
Fit Test
To check whether the given data
follows a given distribution or not
Test of
Association
To check whether the two
variables are associated or not
Assumptions
Qualitative DataData Type
E > = 1
Expected
Frequency
At most 20% Es can be less than 5
Chi Square = Σ {(O – E)2 /E}

6. Goodness of Fit test
Hypothesis
The data follows the
said distribution
Test
Chi Square Calculated > Chi Square Tabulated (df)
Reject H0
The distribution of market
share is as it was last year
Chi Square Calculated = Σ {(O – E)2 /E}
Significance
level
1%, or 5%
E = np
Always Check the Assumptions
If Assumptions do not meet then what
P < α

7. Goodness of Fit test Using SPSS
How the Data
Looks
Issue
Whether the market share of three firms has changed or not.
Last year market share distribution was as follows
(A = 25%, B = 40%, C =35%)
Sample Size = 52

8. Goodness of Fit test Using SPSS

9. Press
Goodness of Fit test Using SPSS

10. Brand
Observed N Expected N Residual
A 12 13.0 -1.0
B 24 20.8 3.2
C 16 18.2 -2.2
Total 52
Test Statistics
Brand
Chi-Square .835
a
df 2
Asymp. Sig. .659
a. 0 cells (0.0%) have expected frequencies less than 5.
The minimum expected cell frequency is 13.0.
Goodness of Fit test Using SPSS

11. Test of Association
Hypothesis
Variables are associated
or dependent
Test
Chi Square Calculated > Chi Square Tabulated (df)
Reject H0
Result is associated with
Gender
Chi Square Calculated = Σ {(O – E)2 /E}
Significance
level
1%, or 5%
E = CR/T
Always Check the Assumptions
If Assumptions do not meet then what
P < α

12. Test of Association Using SPSS
How the Data
Looks
Issue Result is associated with Gender
Sample Size = 40

13. Test of Association

14. Test of Association

15. Test of Association

16. Test of Association
Press

17. Result
TotalPass Fail
Gender Male 12 8 20
Female 14 6 20
Total 26 14 40
Test of Association
Chi-Square Tests
Value df
Asymp. Sig.
(2-sided)
Exact Sig. (2-
sided)
Exact Sig. (1-
sided)
Pearson Chi-Square .440
a
1 .507
a. 0 cells (0.0%) have expected count less than 5. The minimum expected
count is 7.00.
b. Computed only for a 2x2 table

18. Fisher Exact Test
Fisher
Probability
If Assumptions do not meet then what
It is the substitute of Chi Square of Association when Sample size is
small

19. McNamer Test for Dependent Association
McNamer Those who were arrested stealing will be rearrested stealing
Test of
Association
changes in related or paired binomial attributes, whether
changes in one direction is significantly greater than that in
the opposite direction.
Working
Compares the ratio of change with the total who changed.

20. Opinion After
Favor Not Favor Total
Opion
Before
Favor a b a+b
Not Favor c d c+d
Total a+c b+d a+b+c+d
Opinion After
Favor Not Favor Total
Opion
Before
Favor a b a+b
Not Favor c d c+d
Total a+c b+d a+b+c+d

21. McNamer Test Using SPSS
Issue
Is there any association in the opinion before and
after the speech given be a political leader?
How the Data
Looks
Sample Size = 20

25. Opinion_bf & Opinion_after
Opinion_bf
Opinion_after
Favorable Unfavorable
Favorable 7 4
Unfavorable 5 4
Test Statisticsa
Opinion_bf &
Opinion_after
N 20
Exact Sig. (2-tailed) 1.000
b
a. McNemar Test
b. Binomial distribution used.

26. Hypothesis Testing of Categorical Data
Single Variable Two Variables
One
Population
Two
Population
Binary
Values
Multiple
Values
Goodness
of Fit Test
Independent
Values
Paired Values
Chi Square Test
of Association
Fisher
McNamer
Test
Z Test of
Proportion
Binomial Test
Large Sample
Normal Dist
Small Sample
Chi Square
Assumptions Met
YesNo