Chi-square Test of Independence

The chi-square goodness of fit test and test for independence are both available on SPSS. Recall that chi-square is useful for analyzing whether a frequency distribution for a categorical or nominal variable is consistent with expectations (a goodness of fit test), or whether two categorical or nominal variables are related or associated with each other (a test for independence). Categorical or nominal variables assign values by virtue of being a member of a category. Sex is a nominal variable. It can take on two values, male and female, which are usually coded numerically as 1 or 2. These numerical codes do not give any information about how much of some characteristic the individual possesses. Instead, the numbers merely provide information about the category to which the individual belongs. Other examples of nominal or categorical variables include hair color, race, diagnosis (e.g., ADHD vs. anxiety vs. depression vs. chemically dependent), and type of treatment (e.g., medication vs. behavior management vs. none). Note that these are the same type of variables that can be used as independent variables in a t-test or ANOVA. In the latter analyses, the researcher is interested in the means of another variable measured on a interval or ratio scale. In chi-square, the interest is in the frequency with which individuals fall in the category or combination of categories.

The chi-square test for independence is a test of whether two categorical variables are associated with each other. For example, imagine that a survey of approximately 200 individuals has been conducted and that 120 of these people are females and 80 are males. Now, assume that the survey includes information about college major. To keep the example simple, assume that each person is either a psychology or a biology major. It might be asked whether males and females tend to choose these two majors at about the same rate or does one of the majors have a different proportion of one sex than the other major. The table below shows the case where males and females tend to be about equally represented in the two majors. In this case college major is independent of sex. Note that the percentage of females in psychology and biology is 59.8 and 60.2, respectively. Another way to characterize these data is to say that sex and major are independent of each other because the proportion of males and females remains the same for both majors.

The next example shows the same problem with a different result. In this example, the proportion of males and females depends upon the major. Females compose 79.6 percent of psychology majors and only 39.2 percent of biology majors. Clearly, the proportion of each sex is different for each major. Another way to state this is to say that choice of major is strongly related to sex, assuming that the example represents a statistically significant finding. It is possible to represent the strength of this relationship with a coefficient of association such as the contingency coefficient or Phi. These coefficients are similar to the Pearson correlation and interpreted in roughly the same way.

The method for obtaining a chi-square test for independence is a little tricky. Begin by clicking Analyze > Summarize > Crosstabs.... Transfer the variables to be analyzed to the Row(s) and Column(s) boxes. Then go to the Statistics... button and check the Chi-square box and anything that looks interesting in the Nominal Data box, followed by the Continue button. Next, click the Cells... button and check any needed descriptive information. Row, column, and total Percentages are particularly useful for interpreting the data. Finally, click OK and the output will quickly appear.