Odds Ratio Estimation

The LOGISTIC Procedure

Odds Ratio Estimation

Consider a dichotomous response variable Y that takes the value `A' for an event and `B' for a nonevent. Suppose the data set also includes a dichotomous risk factor variable that takes the value 1 if the risk factor is present and 0 if the risk factor is absent. According to the logistic model,

$\Pr( Y = {`A'} | X = 1 ) & = & \frac{\exp(\alpha + \beta)}{1 + \exp(\alpha + \beta)} \ \Pr( Y = {`A'} | X = 0 ) & = & \frac{\exp(\alpha)}{1 + \exp(\alpha)} \$

where $\alpha$ is the intercept and $\beta$ is the coefficient for the risk factor. The odds of an event for those individuals with the risk factor is defined as

$\frac{\Pr( Y = {`A'} | X = 1 )} {\Pr( Y = {`B'} | X = 1 )} & = & \frac{\Pr( Y ... ...+\beta)} ) / ( \frac{1}{1 + \exp(\alpha+\beta)}) \ & = & \exp(\alpha + \beta) \$

A similar calculation shows that the odds of an event for those without the risk factor is $\exp(\alpha)$ . The odds ratio is defined as the ratio of the odds for those with the risk factor to the odds for those without the risk factor, and it is given by

$\psi = \frac{\exp(\alpha + \beta)}{\exp(\alpha)} = \exp(\beta)$

So, the odds ratio is obtained by simply exponentiating the value of the parameter associated with the risk factor. The odds ratio indicates how the odds of an event change as you change X from 0 to 1. For instance, $\psi=2$ means that the odds of an event when X=1 are twice the odds of an event when X=0. Also notice that the parameter, $\beta$ ,represents the change in the log odds from X = 0 to X = 1.

Suppose the values of the dichotomous risk factor are coded as constants a and b instead of 0 and 1. The odds when X = a becomes $\exp(\alpha + a \beta)$ , and the odds when X = b becomes $\exp(\alpha + b \beta)$ . The odds ratio corresponding to an increase in X from a to b is

$\psi = \exp[(b - a) \beta] = [\exp(\beta)]^{b-a} \equiv [\exp(\beta)]^c$

Note that for any a and b such that $c=b-a=1, \psi=\exp(\beta)$ .So the odds ratio can be interpreted as the change in the odds for any increase of one unit in the corresponding risk factor. However, the change in odds for some amount other than one unit is often of greater interest. For example, a change of one pound in body weight may be too small to be considered important, while a change of 10 pounds may be more meaningful. The odds ratio for a change in X_j from a to b is estimated by raising the odds ratio estimate for a unit change in X_j to the power of c=b-a as shown previously.

In the displayed output of PROC LOGISTIC, the "Analysis of Maximum Likelihood Estimates" table contains an Odds Ratio column with values $\exp(\hat{\beta_i})$ .That is, these odds ratios correspond to a unit increase in the risk factors. To customize odds ratios for specific units of change, you can use the UNITS statement to specify a list of relevant units for each explanatory variable in the model. Estimates of these customized odds ratios are given in a separate table.

Confidence intervals for the odds ratios are derived from those for the corresponding parameter estimates. Let (L_j,U_j) be either the likelihood ratio-based or Wald confidence interval for $\beta_{j}$ . The corresponding lower and upper confidence limits for the customized odds ratio $\exp(c\beta_j)$ are exp[cL_j] and exp[cU_j], respectively (for c>0), or exp[cU_j] and exp[cL_j], respectively (for c<0). You use the CLODDS= option to request the confidence intervals for the odds ratios.

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