Chi-square Test of Independence Steps in Testing Chi-square Test of Independence Hypotheses

Chi-square Test of Independence  The chi-square test of independence is a statistical test to determine whether there is a statistically significant association between variables (mostly categorical).  This test is probably the most frequently used hypothesis test in the customer research and marketing.

Independence Defined  Two variables are independent if, for all cases, the classification of a case into a particular category of one variable has no effect on the probability that the case will fall into any particular category of the second variable (the test variable).  When two variables are independent, there is no relationship between them. We would expect that the frequency breakdowns of the test variable to be similar for all groups.

Independence Demonstrated  Suppose we are interested in the relationship between dropouts and loan type.  If there is no relationship between dropouts and loan type and 30% of our total sample has Individual Loans (and 70% Group Loans), we would expect 30% of the dropouts in our sample has Individual Loans and 30% of the active clients has Individual Loans.  If there is a relationship between dropouts and loan size, we would expect a higher proportion of dropouts has Individual Loans, for example 80% IL.

Expected Frequencies versus Observed Frequencies, Null Hypothesis  The chi-square test of independence plugs the observed frequencies and expected frequencies into a formula which computes how the pattern of observed frequencies differs from the pattern of expected frequencies.  The null hypothesis is that the variables are independent. This will be true if the observed counts in the sample are similar to the expected counts. The alpha level of significance is either 0.05 or 0.01.

Decision and Interpretation  If the probability of the test statistic is less than or equal to the probability of the alpha error rate, we reject the null hypothesis. We conclude that there is a relationship between the variables.  If the probability of the test statistic is greater than the probability of the alpha error rate, we fail to reject the null hypothesis. We conclude that there is no relationship between the variables, i.e. they are independent.

Which Cell or Cells Caused the Difference  We are only concerned with this procedure if the result of the chi-square test are statistically significant.  One of the problems in interpreting chi-square tests is the determination of which cell or cells produced the statistically significant difference. How we determine?

Standardized Residuals  SPSS prints out the standardized residual (converted to a z-score) computed for each cell.  Compare the size of the standardized residuals to the critical values that correspond to an alpha of 0.05 (+/-1.96) or an alpha of 0.01 (+/-2.58).

Interpreting Standardized Residuals  Standardized residuals that have a positive value mean that the cell was over-represented in the actual sample, compared to the expected frequency, i.e. there were more subjects in this category than we expected.  Standardized residuals that have a negative value mean that the cell was under-represented in the actual sample, compared to the expected frequency, i.e. there were fewer subjects in this category than we expected.