Odds Ratio and Relative Risks for 2×2 Tables

The FREQ Procedure

Odds Ratio and Relative Risks for 2×2 Tables

Odds Ratio (Case-Control Studies)

The odds ratio is a useful measure of association for a variety of study designs. For a retrospective design called a case-control study, the odds ratio can be used to estimate the relative risk when the probability of positive response is small (Agresti 1990). In a case-control study, two independent samples are identified based on a binary (yes-no) response variable, and the conditional distribution of a binary explanatory variable is examined, within fixed levels of the response variable. Refer to Stokes, Davis, and Koch (1995) and Agresti (1996).

The odds of a positive response (column 1) in row 1 is n₁₁ / n₁₂. Similarly, the odds of a positive response in row 2 is n₂₁ / n₂₂. The odds ratio is formed as the ratio of the row 1 odds to the row 2 odds. The odds ratio for 2 ×2 tables is defined as

OR = [(n₁₁/n₁₂)/(n₂₁/n₂₂)] = [(n₁₁ n₂₂)/(n₁₂ n₂₁)]

The odds ratio can be any nonnegative number. When the row and column variables are independent, the true value of the odds ratio equals 1. An odds ratio greater than 1 indicates that the odds of a positive response are higher in row 1 than in row 2. Values less than 1 indicate the odds of positive response are higher in row 2. The strength of association increases with the deviation from 1.

The transformation G = ( OR-1)/( OR+1) transforms the odds ratio to the range (-1,1) with G = 0 when OR = 1; G = -1 when OR = 0; and G approaches 1 as OR approaches infinity. G is the gamma statistic, which PROC FREQ computes when you specify the MEASURES option.

The asymptotic $100(1 - \alpha)$ % confidence limits for the odds ratio are

$( {OR} \cdot \exp ( -z \sqrt{v} ), {OR} \cdot \exp ( z \sqrt{v} ) )$

where

v = var (ln OR) = [1/(n₁₁)] + [1/(n₁₂)] + [1/(n₂₁)] + [1/(n₂₂)]

and z is the $100(1 - \alpha/2)$ percentile of the standard normal distribution. If any of the four cell frequencies are zero, the estimates are not computed.

When you specify option OR in the EXACT statement, PROC FREQ computes exact confidence limits for the odds ratio using an iterative algorithm based on that presented by Thomas (1971). Because this is a discrete problem, the confidence coefficient for these exact confidence limits is not exactly $1-\alpha$ but is at least $1-\alpha$ .Thus, these confidence limits are conservative. Refer to Agresti (1992).

Relative Risks (Cohort Studies)

These measures of relative risk are useful in cohort (prospective) study designs, where two samples are identified based on the presence or absence of an explanatory factor. The two samples are observed in future time for the binary (yes-no) response variable under study. Relative risk measures are also useful in cross-sectional studies, where two variable are observed simultaneously. Refer to Stokes, Davis, and Koch (1995) and Agresti (1996).

The column 1 relative risk is the ratio of the column 1 risks for row 1 to row 2. The column 1 risk for row 1 is the proportion of the row 1 observations classified in column 1,

$p_{1| 1} = n_{11} / n_{1 \cdot}$

Similarly, the column 1 risk for row 2 is

$p_{1| 2} = n_{21} / n_{2 \cdot}$

The column 1 relative risk is then computed as

${RR}_1 = \frac{p_{1| 1}}{p_{1| 2}}$

A relative risk greater than 1 indicates that the probability of positive response is greater in row 1 than in row 2. Similarly, a relative risk less than 1 indicates that the probability of positive response is less in row 1 than in row 2. The strength of association increases with the deviation from 1.

The asymptotic $100(1 - \alpha)$ % confidence limits for the column 1 relative risk are

$( {RR}_1 \cdot \exp ( -z \sqrt{v} ) , {RR}_1 \cdot \exp ( z \sqrt{v} ) )$

where

$v = var (\ln {RR}_1) = \frac{1-p_{1| 1}}{n_{11}} + \frac{1-p_{1| 2}}{n_{21}}$

and z is the $100(1 - \alpha/2)$ percentile of the standard normal distribution. If either n₁₁ or n₂₁ is zero, the estimates are not computed.

PROC FREQ computes the column 2 relative risks in a similar manner.

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