chi square & related procedure
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Chi Square & Related Procedure
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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 Trial2. There are k different outcomes 3. Probability of outcomes do not change from
trail to trial4. Independent trials
Measure of Correlation does not work in case of qualitative variables
Multinomial Distribution
Multinomial Distribution is
required
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What is Chi Square Distribution?
Distribution Z2 = χ2(1)
Distribution ΣZ2 = Z12 + Z2
2 + Z32 + … + Zk
2 = χ2(k)
Properties
Takes only positive Value
Skewed, but becomes normal with increasing n
Mean = k, Variance = 2k
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Shape of Chi Square Distribution
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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 > = 1Expected
FrequencyAt most 20% Es can be less than 5
Chi Square = Σ {(O – E)2 /E}
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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
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Goodness of Fit test Using SPSS
How the Data Looks
IssueWhether 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
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Goodness of Fit test Using SPSS
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Press
Goodness of Fit test Using SPSS
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Brand Observed N Expected N ResidualA 12 13.0 -1.0B 24 20.8 3.2C 16 18.2 -2.2
Total 52
Test Statistics BrandChi-Square .835a
df 2Asymp. Sig. .659a. 0 cells (0.0%) have expected frequencies less than 5. The minimum expected cell frequency is 13.0.
Goodness of Fit test Using SPSS
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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
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Test of Association Using SPSS
How the Data Looks
Issue Result is associated with Gender
Sample Size = 40
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Test of Association
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Test of Association
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Test of Association
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Test of Association
Press
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Result
TotalPass FailGender Male 12 8 20
Female 14 6 20
Total 26 14 40
Test of Association
Chi-Square Tests
Value dfAsymp. Sig.
(2-sided)Exact Sig. (2-
sided)Exact Sig. (1-
sided)Pearson Chi-Square .440a 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
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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
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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.
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Opinion AfterFavor Not Favor Total
Opion Before
Favor a b a+bNot Favor c d c+dTotal a+c b+d a+b+c+d
Opinion AfterFavor Not Favor Total
Opion Before
Favor a b a+bNot Favor c d c+dTotal a+c b+d a+b+c+d
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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
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Opinion_bf & Opinion_after
Opinion_bfOpinion_after
Favorable UnfavorableFavorable 7 4Unfavorable 5 4
Test Statisticsa
Opinion_bf & Opinion_after
N 20Exact Sig. (2-tailed) 1.000b
a. McNemar Testb. Binomial distribution used.
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Hypothesis Testing of Categorical Data
Single Variable Two Variables
One Population
Two Population
Binary Values
Multiple Values
Goodness of Fit Test
Independent ValuesPaired Values
Chi Square Test of AssociationFisher
McNamer Test
Z Test of Proportion Binomial Test
Large Sample Normal Dist Small Sample
Chi Square Assumptions Met
YesNo