Data Analysis for Two-Way Tables

The Basics Two-way table of counts Organizes data about 2 categorical variables Row variables run across the table Column variables run down the table Row and Column totals provide the marginal distributions of the two variables SEPERATELY

To describe the association between the row and the column variables, use conditional distributions. To find the conditional distribution of the row variable for one specific value of the column variable, look only at that one column in the table. Find each entry in the column as a % of the total. Very useful if the column variable is the explanatory or independent variable.

A three-way table reports the frequencies of each combination of levels of three categorical variables. A two-way table can be obtained by adding the corresponding entries in these tables. This is known as aggregating the data.

Bar graphs can also be used to describe an association between 2 categorical variables. This can be produced in Minitab, under graph, Chart

Simpson’s Paradox An association or comparison that holds for all of several groups can reverse direction when the data are combine to form a single group. Fact that observed associations can be misleading when there are lurking variables.

Inference for Two Way Tables The null hypothesis, Ho, of interest in a two-way table is that there is no association between the row variable and the column variable. The alternative hypothesis, Ha, is that there is an association. For an row (r) X column (c) table Ho states that the c distributions of the row variable are identical. Ha states that the distributions are not all the same.

Expected Cell Counts To test the null hypothesis in r x c tables, we compare the observed cell counts with the expected cell counts calculated under the assumption that the null is TRUE. The test statistic becomes the numerical summary. Expected cell count =

Chi-Square Statistic The chi-square statistic is a measure of how much the observed cell counts in a 2-way table diverge from the expected cell counts. where “observed” is the sample count

The Chi-square Distribution Again, it is an approximation related to the normal approximation for the binomial distribution Like the t distributions, the Chi-square distributions form a family described by a single parameter, degrees of freedom. df = (r – 1) X (c – 1) See the density curve on page 625. Table F provides upper critical values.

The Chi-square test and the z test A comparison of the proportions of “successes” in two populations leads to a 2 X 2 table. We can compare two populations proportions either by the chi-square test or by the 2-sample z test (proportion test) These test will always give the exact same result, because the chi-square is equal to the squares of the corresponding N(0,1) critical values. The advantage of the z test is that it allows us have either a 1 or 2-sided test