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More Chapter About Categorical Variables
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Principal Question:
Is there a relationship between the two variables, so that the category into which individuals fall for one variable seems to depend on the category they are in for the other variable?
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Recall:
Data displayed in a contingency or two-way table. Each combination of row/column is a cell of table. Two types of conditional percents: row and column. Row percents: percents across a row, based on total number in the row. Column percents: percents down a column, based on total number in the column. If one variable is explanatory, use it to define rows and use row percents.
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15.1 Chi-Square Test for Two-Way Tables
Recall there are five steps for assessing statistical significance. Step 1: Determine null and alternative hypotheses
H0: The two variables are not related. Ha: The two variables are related. Sometimes associated is used instead of related.
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Example 15.1 Ear Infections and Xylitol
Experiment: n = 533 children randomized to 3 groups Group 1: Placebo Gum; Group 2: Xylitol Gum; Group 3: Xylitol Lozenge Response = Did child have an ear infection?
Only 16.2% of children in Xylitol Gum group had infection.
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Example 15.1 Infections and Xylitol (cont)
Let p1 = p2 = p3 = proportion who would get an ear infection in a population given placebo gum proportion who would get an ear infection in a population given xylitol gum proportion who would get an ear infection in a population given xylitol lozenges
(no relationship between trt and
H0: p1 = p2 = p3
outcome)
Ha: p1, p2 , p3 are not all the same (there is a relationship)
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Example 15.2 Making Friends
Q: With whom do you find it easiest to make friend opposite sex or same sex or no difference?
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
H0: No difference in distribution of responses of men and women (no relationship between gender and response) Ha: There is a difference in distribution of responses of men and women (is a relationship between gender and response)
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Tech Note: Homogeneity and Independence
Two variations of the general hypothesis statements which depend on the method of sampling. If samples have been taken from separate populations, the null hypothesis statement is a statement of homogeneity (sameness) among the populations. If a sample has been taken from a single population, and two categorical variables measured for each individual, the statement of no relationship is a statement of independence between the two variables.
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Step 2: Chi-square Statistic and Necessary Conditions
Guidelines for large sample:
1. All expected counts should be greater than 1. 2. At least 80% of the cells should have an expected count greater than 5. Compute expected count for each cell:
Expected count = (Row total) (Column total) Total n
Compute test statistic by totaling over all cells:
=
2
(Observed Expected)2 Expected
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Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
More on the Chi-square Statistic
Chi-square statistic measures the difference between the observed counts and the counts that would be expected if there were no relationship (i.e. if the null hypothesis were true). Large difference => evidence of a relationship.
Chi-square probability distribution used to find p-value. Degrees of freedom df = (Rows 1)(Columns 1).
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Example 15.1 Infections and Xylitol (cont)
Output for testing significance of the relationship:
p-value = 0.035 which is < 0.05 There is a statistically significant relationship between the risk of an ear infection and the preventative treatment.
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Example 15.1 Infections and Xylitol (cont)
Expected count for Placebo Gum, Yes Infection cell:
Expected Counts:
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Example 15.1 Infections and Xylitol (cont)
Chi-square Test Statistic:
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Step 3: p-value of Chi-square Test
Large test statistic => evidence of a relationship. So how large is enough to declare significance? p-value = probability the chi-square test statistic could have been as large or larger if the null hypothesis were true.
Chi-square probability distribution used to find p-value. Degrees of freedom df = (Rows 1)(Columns 1) = (r 1)(c 1)
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Chi-square Distributions
Skewed to the right distributions. Minimum value is 0. Indexed by the degrees of freedom.
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Example 15.1 Infections and Xylitol (cont)
Chi-square statistic was 6.69 df = (3-1)(2-1) = 2 p-value = 0.035
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Finding the p-value from Table A.5:
Look in the corresponding df row of Table A.5. Scan across until you find where the statistic falls.
If value of statistic falls between two table entries, p-value is between values of p (column headings) for these two entries. If value of statistic is larger than entry in rightmost column (labeled p = 0.001), p-value is less than 0.001 (written as p < 0.001). If value of statistic is smaller than entry in leftmost column (labeled p = 0.50), p-value is greater than 0.50 (written as p > 0.50).
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Example 15.3
Table has three rows and three columns. The computed chi-square statistic is 8.12. Degrees of freedom are df = (3 1)(3 1) = 4. Finding the p-value: Scan the df = row 4 in Table A.5 and the value of 8.12 is between the entries 7.78 (p = 0.10) and 8.50 (p = 0.075). Thus, the p-value is between 0.075 and 0.10. 0.075 < p-value < 0.10
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Step 4: Making a Decision
Large test statistic => small p-value => evidence a real relationship exists in the population.
Two equivalent rules: Reject H0 when
p-value 0.05 Chi-square statistic is greater than the entry in the 0.05 column of Table A.5 (the critical value).
Note: For 2 2 tables, a test statistic of 3.84 or larger is significant.
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Step 5: Reporting a Conclusion
Example: Testing whether there is a relationship between smoking (yes or no) and drinking alcohol (never, occasionally, often).
Ways to write do not reject H0
The relationship between smoking and drinking alcohol is not statistically significant. The proportions of smokers who never drink, drink occasionally, and drink often are not significantly different from the proportions of non-smokers who do so. There is insufficient evidence to conclude that there is a relationship in the population between smoking and drinking alcohol.
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Step 5: Reporting a Conclusion
Example: Testing whether there is a relationship between smoking (yes or no) and drinking alcohol (never, occasionally, often).
Ways to write reject H0
There is a statistically significant relationship between smoking and drinking alcohol. The proportions of smokers who never drink, drink occasionally, and drink often are not the same as the proportions of non-smokers who do so. Smokers have significantly different drinking behavior than non-smokers.
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Example 15.2 Making Friends (cont)
Q: With whom do you find it easiest to make friend opposite sex or same sex or no difference?
df = (2 1)(3 1) = 2. Table A.5: value of 8.515 falls between the entries in the 0.025 column (7.38) and the 0.01 column (9.21). 0.01 < p-value < 0.025 There is statistically significant relationship at the 0.05 level. There appears to be a a difference in distribution of responses of men and women if the populations were asked this question.
Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
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Supporting Analyses
To learn about the specific nature of the relationship:
Description of row (or column) percents. Bar chart of counts or percents. Examination each cells contribution to chisquare. Cells with largest values have contributed most to significance of the relationship => deserve attention in any description of the relationship. Confidence intervals for important proportions or for differences between proportions.
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15.2 Analyzing 2 2 Tables
Shortcut Formula:
The test statistic formula is below, based on df = 1.
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Example 6.10
Randomly Pick S or Q (cont)
College students asked: Randomly choose one of the letters S or Q, or Randomly choose one of the letters Q or S.
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Chi-Square Test or Z-Test for Difference in Two Proportions?
Does it make a difference?
If desired Ha has no specific direction (two-sided), the two tests give exactly the same p-value. The squared value of the z-statistic equals the chi-square statistic. If desired Ha has a direction (one-sided), the z-test should be used.
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