Cross-Tabulations

1. Cross-Tabulations • We have been looking at these for some time already. • An arrangement of two categorical variables into rows and columns. • Row variable • Column variable • Tells about relationships between two categorical variables

2. Depression and a new baby, Fathers | depress baby | 0 1 | Total -----------+----------------------+---------- 0 | 92 59 | 151 | 60.93 39.07 | 100.00 | 75.41 71.95 | 74.02 -----------+----------------------+---------- 1 | 30 23 | 53 | 56.60 43.40 | 100.00 | 24.59 28.05 | 25.98 -----------+----------------------+---------- Total | 122 82 | 204 | 59.80 40.20 | 100.00 | 100.00 100.00 | 100.00

3. Stress and social class | class stress | Low Middle Upper | Total -----------+---------------------------------+---------- Low | 246 90 55 | 391 | 62.92 23.02 14.07 | 100.00 | 59.42 64.75 80.88 | 62.96 -----------+---------------------------------+---------- High | 168 49 13 | 230 | 73.04 21.30 5.65 | 100.00 | 40.58 35.25 19.12 | 37.04 -----------+---------------------------------+---------- Total | 414 139 68 | 621 | 66.67 22.38 10.95 | 100.00 | 100.00 100.00 100.00 | 100.00

4. What goes into the cells? • Frequencies • Cell • Margin • Total • Row percentages • Column percentages • Total percentages

5. Percentages • Independent variable - suspected cause • Dependent variable - suspected effect • Percentages should be based on the independent or causal variable

6. Stress and social class | class stress | Low Middle Upper | Total -----------+---------------------------------+---------- Low | 246 90 55 | 391 | 62.92 23.02 14.07 | 100.00 | 59.42 64.75 80.88 | 62.96 -----------+---------------------------------+---------- High | 168 49 13 | 230 | 73.04 21.30 5.65 | 100.00 | 40.58 35.25 19.12 | 37.04 -----------+---------------------------------+---------- Total | 414 139 68 | 621 | 66.67 22.38 10.95 | 100.00 | 100.00 100.00 100.00 | 100.00

7. Make comparisons • Compare categories of the independent variable • To see effect on proportion in one category of the dependent variable • To make comparisons we must be sure the comparisons make sense -- are of the same thing: not apples with oranges!

8. Independence • Two variables, A and B, are independent if p(A) = p(A|B) • p(Stress) = .37, p(Stress|Hi class) = .19 • Also, note • p(s|low) = .41 p(s|mid) = .35 p(s|hi) = .19 • Also note, these are from the appropriate percentages, since class causes stress.

9. Independence • If there is independence, then • p(s) = p(s|lo) = p(s|mid) = p(s|hi) • What would the frequencies be if there was independence? • p(s) = .37 = p(s|lo) = p(s|mid) = p(s|hi) • This .37 is taken from the margin (unconditional probability of stress)

10. Apply this | class stress | Low Middle Upper | Total -----------+---------------------------------+---------- Low | 246 90 55 | 391 | 62.96 62.96 62.96 | 62.96 | 260.65 87.52 42.81 | -----------+---------------------------------+---------- High | 168 49 13 | 230 | 37.04 37.04 37.04 | 37.04 | 153.35 51.48 25.19 | -----------+---------------------------------+---------- Total | 414 139 68 | 621 | 100.00 100.00 100.00 | 100.00

11. Observed and Expected • Are they the same? • Then p(s) = p(s|class) -- Independence • Are they different? • Then p(s) ‡ p(s|class) -- Relationship • How can we tell?

12. Look at parts of formula What if we just sum difference without squaring? How big is a difference of 5 points? What happens when there are lots of cells in the table we are looking at?