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Statistical Analysis [6-7]

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Chi-square Test of Independence

The Chi-square test is used when examining the association and difference between two categorical variables.

Example of application of chi-square test:

1. Compare the smoking behavior between Male and Female
2. Compare education level by different race group
3. Compare the voting preference by income level

Example

Data Smoke; Infile datalines dsd; Input Index Gender :$6. Smoker : $20.; Datalines; 1,Female,Occasional 2,Male,Non-smoker 3,Female,Occasional 4,Male,Occasional 5,Male,Non-smoker 6,Female,Frequency Smoker 7,Female,Non-smoker 8,Female,Non-smoker 9,Female,Non-smoker 10,Male,Non-smoker 11,Male,Frequency Smoker 12,Female,Non-smoker 13,Female,Non-smoker 14,Female,Non-smoker 15,Male,Non-smoker 16,Male,Non-smoker 17,Male,Non-smoker 18,Female,Frequency Smoker 19,Male,Non-smoker 20,Female,Non-smoker 21,Male,Frequency Smoker 22,Male,Non-smoker 23,Female,Occasional 24,Male,Non-smoker 25,Male,Occasional 26,Male,Occasional 27,Male,Non-smoker 28,Female,Non-smoker 29,Female,Non-smoker 30,Female,Non-smoker 31,Female,Occasional 32,Male,Occasional 33,Female,Non-smoker 34,Male,Non-smoker 35,Male,Non-smoker 36,Female,Non-smoker 37,Female,Non-smoker 38,Male,Occasional 39,Male,Non-smoker 40,Male,Non-smoker ; Run;

The SMOKE data set contains a list of survey participants on their smoking behavior.

The data set has 40 observations with information about the gender (Male vs. Female) and the 3 types of smoker:

• Non-smoker
• Occasional (<=5 per week)
• Frequent (>5 per week)

A research on the smoking behavior has been conducted to find out whether men smoke significantly more than women.

The following hypothesis is being tested with the chi-square test:

H0: Smoking behavior is independent of gender
H1: Smoking behavior is dependent on gender

Example

Proc Freq Data=Smoke;
Table Gender * Smoker / Chisq;
Run;

A chi-square test can be done by using Proc Freq with the CHISQ option.

Run the code on SAS Studio and the following results will be generated:

1. Two-way Crosstabulation Table

The standard two-way crosstabulation table is created showing the counting statistics from each category.

2. Chi-square Test Results

The p-value is 0.7241. There is not sufficient evidence to reject the null hypothesis.

We conclude that there is no significant difference in the smoking behavior between the two genders.


Assumption for Chi-square Test

One of the major assumptions for chi-square test is that each cell count has to be at least 5 or above.

In our example, some of the cell counts are less than 5:

A warning is shown on the chi-square table:

Fisher's Exact Test

When some of the cell counts are less than 5, the Fisher's Exact test is the better test for the hypothesis testing.

Example

​Proc Freq Data=Smoke;
Table Gender * Smoker / Fisher;
Run;

Similar to the chi-square test, simply add the keyword FISHER to the table statement to get the Fisher's exact test results:

The p-value is very high. There is no significant difference between the two genders on the smoking behaviour.

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Exercise

Copy and run the VOTER data set from the yellow box below:

Data Voter; Infile datalines dsd; Input Voter Party : $12. Income : $20.; Datalines; 1,Democrat,< 70K 2,Republican,70K - 120K 3,Democrat,70K - 120K 4,Republican,70K - 120K 5,Republican,< 70K 6,Democrat,70K - 120K 7,Republican,>120 K 8,Democrat,>120 K 9,Republican,70K - 120K 10,Republican,>120 K 11,Republican,>120 K 12,Republican,>120 K 13,Democrat,>120 K 14,Republican,< 70K 15,Republican,< 70K 16,Democrat,>120 K 17,Republican,< 70K 18,Republican,70K - 120K 19,Republican,70K - 120K 20,Democrat,>120 K 21,Democrat,70K - 120K 22,Republican,>120 K 23,Democrat,< 70K 24,Republican,>120 K 25,Republican,< 70K 26,Democrat,< 70K 27,Republican,>120 K 28,Republican,70K - 120K 29,Republican,>120 K 30,Democrat,70K - 120K 31,Democrat,< 70K 32,Democrat,>120 K 33,Republican,>120 K 34,Democrat,< 70K 35,Democrat,>120 K 36,Republican,< 70K 37,Democrat,>120 K 38,Republican,>120 K 39,Republican,70K - 120K 40,Democrat,>120 K 41,Democrat,70K - 120K 42,Democrat,70K - 120K 43,Democrat,>120 K 44,Democrat,70K - 120K 45,Republican,>120 K 46,Democrat,>120 K 47,Republican,70K - 120K 48,Democrat,>120 K 49,Republican,< 70K 50,Democrat,>120 K 51,Democrat,>120 K 52,Democrat,70K - 120K 53,Democrat,>120 K 54,Democrat,< 70K 55,Republican,< 70K 56,Republican,< 70K 57,Democrat,>120 K 58,Democrat,< 70K 59,Republican,>120 K 60,Democrat,>120 K 61,Democrat,70K - 120K 62,Republican,< 70K 63,Democrat,70K - 120K 64,Republican,70K - 120K 65,Democrat,< 70K 66,Republican,>120 K 67,Democrat,>120 K 68,Democrat,< 70K 69,Republican,70K - 120K 70,Democrat,70K - 120K 71,Democrat,>120 K 72,Republican,< 70K 73,Democrat,>120 K 74,Democrat,70K - 120K 75,Republican,< 70K 76,Republican,< 70K 77,Republican,70K - 120K 78,Republican,>120 K 79,Democrat,< 70K 80,Democrat,70K - 120K 81,Democrat,>120 K 82,Democrat,< 70K 83,Republican,>120 K 84,Republican,< 70K 85,Democrat,< 70K 86,Republican,>120 K 87,Democrat,< 70K 88,Democrat,>120 K 89,Democrat,70K - 120K 90,Republican,< 70K 91,Republican,>120 K 92,Democrat,< 70K 93,Republican,70K - 120K 94,Democrat,< 70K 95,Democrat,< 70K 96,Democrat,>120 K 97,Republican,70K - 120K 98,Republican,70K - 120K 99,Republican,70K - 120K 100,Democrat,>120 K ; Run;

The VOTER data set contains the voting preference among the 3 income groups.

The income groups can be classified as "<70K", "70K to 120K" and ">120K". 

Perform a statistical hypothesis test and find out if there is any significance difference in voting preference among the 3 income groups.

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