Chi-Square Test

Test for Independence

Situation. One has a single random sample, and this sample is cross-categorized by two variables each with 2 or more categories. The null hypothesis is that there is no relationship between Variable 1 and Variable 2, that is, the two variables are independent of each other. The alternative hypothesis is that there is a relationship between the two variables, that is, the two variables are dependent.

For example, one has a random sample of people who are cross-categorized by race and blood type. The null hypothesis is that there is no relationship between race and blood type, and the alternative hypothesis is that there is a relationship between race and blood type.

Test for Equality of Proportions

Situation. One has a single variable of interest with 2 or more categories and multiple independent random samples. The null hypothesis is that there is no difference in the proportion of each category across the different populations. The alternative hypothesis is that there is at least one difference in a proportion across the different populations.

For example, one has four treatment groups. These treatment groups are the multiple independent random samples. The variable of interest has two categories survived or died. The null hypothesis is that the proportions that survived are the same for each treatment group and the proportion that died are the same for each treatment group. The alternative hypothesis is [Page 176]that at least one proportion is different between the treatment groups. The test for equality of proportions is analogous to the one-way analysis of variance test for equality of means for quantitative data.

Expected Values and Tests Statistic: Independence and Equality of Proportions

With the tests for independence and equality of proportions, the data should be given in table form where i represents the ith row and j represents the jth column of the table. One finds the expected value (Eij) for each ijth cell. The expected value for the ijth cell is the value that one would expect if the null hypothesis is true.

Reasoning behind the Expected Value for Test for Independence

For the test for equality of proportions, the null hypothesis states that Variable 1 and Variable 2 are independent. By independence,

So the best estimate of the P[ijth cell] is the estimate of the proportion of the ith row multiplied proportion of the jth column, that is,

To find the expected count, one would multiply by the grand total, which gives the above formula for Eij.

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