Inference on Categorical Data

Overview

Thus far in the course, statistical inference has been limited to inference on qualitative data with two possible outcomes (using proportions) and inference on quantitative data (using means).   This chapter focuses exclusively on inference for qualitative (or categorical) data.  In Section 12.1, we discuss the chi-square goodness-of-fit test, which is used to test the hypothesis that data follow a specified distribution.  Section 12.2 presents the chi-square test for independence and the chi-square test for homogeneity of proportions.  Finally, Section 12.3 presents McNemar’s Test for comparing two dependent proportions.

What to Emphasize

This is a very important chapter for students.  Most of the data analysis our students will do or be exposed to is based on qualitative data.  Throughout the chapter, emphasize the requirements of each model and the types of conclusions that one may make based on the results of the inference (accept versus do not reject, for example).

As with the other inferential chapters, emphasize the interpretation of the P­-value.

Goodness-of-Fit Test – The goodness-of-fit test is used to test a hypothesis regarding a particular probability distribution.

• The section begins with a discussion of the chi-square distribution. Emphasize the properties of the chi-square distribution – in particular, how the chi-square distribution has only non-negative values and is skewed right yet, as the degrees of freedom increase, the distribution becomes more symmetric and bell-shaped.   If you decide to use Table VIII (the Critical Values of the Chi-Square Distribution), be sure to emphasize that it is structured similar to Student’s t
• The goodness-of-fit test is an inferential procedure to determine whether it is reasonable to conclude that a frequency distribution follows a specified distribution. The main point to emphasize is that the test statistic is based on comparing observed counts to counts we would expect if the statement (that is, distribution) if the null hypothesis were true.  Again, the null hypothesis is a statement of no change, no effect, or no difference.  Here, we assume there is no difference in the distribution of the sample data and the distribution stated in the null hypothesis.  So, the null hypothesis is used to determine expected counts.
• A key area of emphasis is after the hypothesis test. Spend time comparing expected counts to observed counts to identify the major contributors to the chi-square test statistic.
• Finally, be sure students understand that if the sample evidence does not allow one to reject the null hypothesis, we are not saying the statement in the null hypothesis is true. The only conclusion we may make is that the sample data are consistent with the distribution stated in the null hypothesis.

Tests for Independence and the Homogeneity of Proportions – This section presents two different hypothesis tests that happen to have the exact same procedure for conducting the test. This represents a source of confusion in determining the type of hypothesis test actually being conducted. At the start of the presentation on the test for homogeneity of proportions (and after the test for independence), it is a good idea to explain the difference between the two tests.  A test for independence essentially requires the measurement of two variables collected from a single population.  In Example 1 of this section, we compare marital status and happiness.  Each count represents the marital status and happiness of a single individual from the population of adult Americans.  A test for homogeneity of proportions compares the proportion of individuals from three or more populations who have some characteristic.  Example 4 compares the population of individuals receiving Zocor, the population of individuals receiving a placebo, and the population of individuals receiving Cholestyramine.  The characteristic for each population is whether the subject experiences abdominal pain or not.  Require students to explain which test applies to a given problem and emphasize this with your examples.

• In the test for independence, emphasize how the Multiplication Rule for Independent Events is used to determine expected counts. Again, the null hypothesis is assumed true until the evidence indicates otherwise.  So, the assumption of independence is used to find expected counts.  These expected counts are then compared to those observed.  Also emphasize the graphical representation as visual support of the hypothesis test.
• In the test for homogeneity of proportions, emphasize that the procedures of the test are the same as those for the test for independence. However, in the test for homogeneity of proportions, we are comparing the proportion of individuals that have a specified characteristic from three or more populations.  In addition, this test is simply an extension of the two-sample Z test comparing two populations.  Again, visual support of the hypothesis test through the conditional distribution bar graph is a good idea.

Inference about Two Population Proportions: Dependent Samples – This section continues the theme from Chapter 11. Here, we are comparing two population proportions from dependent samples.  Begin by reminding students of the difference between a dependent and independent sample.

• Emphasize that we are collecting information on two qualitative variables with two outcomes (success/failure; yes/no; and so on) for each individual in the study. Therefore, the sampling method is dependent.
• The test statistic is not intuitive. However, explain that large values of the test statistic represent evidence against the statement in the null because this would suggest different proportions for the two variables.