Problem 7.2: Risk Ratios and Odds Ratios with SPSS

When you have two dichotomous variables and thus a 2 x 2 contingency table or cross-tabulation, you can compute risk ratios and odds ratios instead of chi-square and phi. These ratios are commonly used to report results in medical, applied health, and prevention science fields. The assumptions for odds ratios are listed under Problem 7.1.

• What is the relative risk of having low math grades for students who didn’t or did take algebra 2? And, what is the odds ratio?

To compute these measures, you again use Fig. 7.1, 7.2, and 7.3, but this time click on Risk in the Statistics window in Fig. 7.2.

• Click on Analyze →Descriptive Statistics → Crosstabs…
  
• Click on Reset.
• Move algebra 2 to the Rows box and math grades (not grades in h.s.) to the Columns (See Fig. 7.1.)
• Click on Statistics to get Fig. 7.2.
• Check Risk in the lower right part of the window. Click off Chi-Square and Phi and Cramer’s V if they are checked.
• Click on Continue to get Fig. 7.1 again.
• Click on Cells to get Fig. 7.3.
• Check Observed and Row. Make sure Expected is not
• Click on Continue, then OK.
• Compare your output and syntax to Output 7.2.

Output 7.2: Crosstabs with Risk Ratios and Odds Ratios

CROSSTABS

/TABLES=alg2 BY mathgr

/FORMAT= AVALUE TABLES

/STATISTICS=RISK

/CELLS= COUNT ROW

/COUNT ROUND CELL.

Crosstabs

Interpretation of Output 7.2

The first two tables are very similar to those in Output 7.1, except that the variables are different and there are no Expected Counts in the Crosstabulation table. Note that all 75 participants had both variables. Forty-four students had low math grades (few As and Bs). Of those 44, 28 had not taken algebra 2 and 16 had taken it. The Risk Estimate table shows the odds ratio, the two risk ratios, and confidence intervals for each. The first risk ratio is 1.53, which is computed by dividing 70% by 45.7%. This risk ratio can be interpreted to mean that students who didn’t take algebra 2 are 1.53 times as likely to have low math grades as students who did take algebra 2. Conversely, of the 31 students with high math grades (mostly As and Bs), 12 didn’t take algebra 2 and 19 did. The second risk ratio is .553, which is 30% divided by 54.3%. This risk ratio is interpreted as students who didn’t take algebra 2 are only about half as likely to have high math grades as those who took algebra 2. Notice that the 95% Confidence Interval for each ratio does not include 1.0. That is, the Lower and Upper bounds are either greater thanl.0 (i.e., 1.012 and 2.317) or less than 1.0 (.315 and .970). This indicates that the risk ratios are statistically significant. A risk ratio of 1.0 would indicate that, for example, those who didn’t take algebra 2 are equally likely to have low math grades as students who did take algebra 2.

The Odds Ratio (OR) of 2.77 is a ratio of ratios. In this case, 1.53/.55 = 2.77. This OR means that the odds of getting low math grades for those who didn’t take algebra 2 are 2.77 times as high as the odds of getting high math grades if one didn’t take algebra 2.

Odds ratios and risk ratios are common examples of a third group or family of effect size measures, called risk potency measures. Remember that we discussed in Chapter 6 the r family of effect sizes, including phi, Cramer’s V, and eta used in this chapter, and the d family, which is used to indicate effect size in Chapters 9 and 10. Although odds ratios and risk ratios are common effect size measures when both variables are dichotomous (also called binary), especially in the health-related literature, they are somewhat difficult to interpret clearly. Furthermore, there are no agreed-upon standards for what represents a large ratio because the ratio may approach infinity if the outcome is very rare or very common, even when the association is near random.

How to Write About Output 7.2

Results

Because whether or not students took algebra 2 and whether their math grades were high or low were both binary variables and neither alternative was rare, an odds ratio was computed. The OR was 2.77, indicating that the odds of students getting low math grades if they didn’t take algebra 2 were 2.77 times as high as the odds for those who did take algebra 2.

Source: Morgan George A, Leech Nancy L., Gloeckner Gene W., Barrett Karen C.

(2012), IBM SPSS for Introductory Statistics: Use and Interpretation, Routledge; 5th edition; download Datasets and Materials.