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Multivariate Descriptive Research • In the previous lecture, we discussed ways to quantify the relationship between two variables when those variables are continuous. • What do we do when one or more of the variables is categorical? Categorical Variables • Fortunately, this situation is much easier to deal with because we can use the same techniques that we’ve discussed already. • Let’s consider a situation in which we are interested in how one continuous variable varies as a function of a categorical variable. • Example: How does mood vary as a function of sex (male vs. female)? • In this case, we want to know how the average woman’s score compares to that of the average man’s score. • level of a categorical variable Participants Mood score Males A 4 B 3 C 4 D 3 M = 3.5, SD = .5 Females A 5 B 4 C 5 D 4 M = 4.5 , SD = .5 First, find the average score for each level of the categorical variable separately. (Also find the SD.) Second, find the difference between the means of each group. This is called a mean difference. (4.5 – 3.5 = 1.0) Third, express this mean difference relative to the SD. This is called a standardized mean difference. 1/.5 = 2 In this example, women score 2 SD higher than the men. Participants Mood score Males A 4 B 3 C 4 D 3 M = 3.5, SD = .5 Females A 5 B 4 C 5 D 3 M = 4.25 , SD = .83 Note: If the SD’s for the two groups are different, you can simply average the two SD’s. Here, the two SD’s are .5 and .83. Averaged, these are (.5 + .83)/2 = .66. The standardized mean difference is (4.25 – 3.5)/.66 = .75/.66 = 1.13 Thus, on average, women score 1.13 SD’s higher than men on this mood variable. Cohen’s d • If we divide the mean difference by the average SD of the two groups, we obtain a standardized mean difference or Cohen’s d. d MA MB SD 2 A SD / 2 2 B Pooled standard deviation Mood Bargraph 5 4.5 4 3.5 3 2.5 2 1.5 1 Men Women Sex Bargraph: More than two categorical variables 7 Mood 6 5 4 3 2 1 Men Women Sex Non-bereaved Bereaved Both variables are categorical • When two variables are categorical, it is sometimes most useful to express the data as percentages. • Example: Let’s assume that depression is a categorical variable, such that some people are depressed and others are not. • What is the relationship between biological sex and depression? Depression status Sex Not Depressed Depressed row total Male 600 60 660 Female 40 300 340 column total 640 360 1000 Depression status Sex Not Depressed Depressed row total Male .60 .06 .66 Female .04 .30 .34 column total .64 .36 1.00 In this table, we’ve expressed each cell as a proportion of the total. Depression status Sex Not Depressed Depressed row total Male .60 .06 .66 Female .04 .30 .34 column total .64 .36 1.00 .60/.64 = .94 .06/.36 = .16 Here, we’ve expressed the association with respect to sex. For example, we can see here that 16% of people who are depressed are male. Moreover, 94% of people who are not depressed are male. Depression status Sex Not Depressed Depressed row total Male .60 .06 .66 .06/.66 = .09 Female .04 .30 .34 .30/.34 = .88 column total .64 .36 1.00 Here, we’ve expressed the association with respect to depression status. For example, we can see here that 9% of men are depressed and 88% of women are depressed. Phi • It is possible to quantify the association among these variables using a correlation coefficient when the two variables are binary. • This statistic is sometimes referred to as phi. • (Phi is + .78 in this example) Variable 1 Variable 2 0 1 row total 0 a b n3 1 c d n4 Col total n1 n2 Phi = (a*d) – (b*c) / sqrt(n1*n2*n3*n4) Online calculator at: http://www.quantitativeskills.com/sisa/statistics/twoby2.htm