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The LOGISTIC Procedure |
The odds ratio is defined as the ratio of the odds for those with the risk factor (X=1) to the odds for those without the risk factor (X=0). The log of the odds ratio is given by
The parameter, , associated with X represents the change in the log odds from X = 0 to X = 1. 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 event change as you change X from 0 to 1. For instance, means that the odds of an event when X=1 are twice the odds of an event when X=0.
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 become , and the odds when X = b become . The odds ratio corresponding to an increase in X from a to b is
For a polytomous risk factor, the computation of odds ratios depends on how the risk factor is parameterized. For illustration, suppose that Race is a risk factor with four categories: White, Black, Hispanic, and Other.
For the effect parameterization scheme (PARAM=EFFECT) with White as the reference group, the design variables for Race are as follows.
Design Variables | |||
Race | X_{1} | X_{2} | X_{3} |
Black | 1 | 0 | 0 |
Hispanic | 0 | 1 | 0 |
Other | 0 | 0 | 1 |
White | -1 | -1 | -1 |
The log odds for Black is
The log odds for White is
Therefore, the log odds ratio of Black versus White becomes
For the reference cell parameterization scheme (PARAM=REF) with White as the reference cell, the design variables for race are as follows.
Design Variables | |||
Race | X_{1} | X_{2} | X_{3} |
Black | 1 | 0 | 0 |
Hispanic | 0 | 1 | 0 |
Other | 0 | 0 | 1 |
White | 0 | 0 | 0 |
The log odds ratio of Black versus White is given by
For the GLM parameterization scheme (PARAM=GLM), the design variables are as follows.
Design Variables | ||||
Race | X_{1} | X_{2} | X_{3} | X_{4} |
Black | 1 | 0 | 0 | 0 |
Hispanic | 0 | 1 | 0 | 0 |
Other | 0 | 0 | 1 | 0 |
White | 0 | 0 | 0 | 1 |
The log odds ratio of Black versus White is
Consider the hypothetical example of heart disease among race in Hosmer and Lemeshow (1989, p 44). The entries in the following contingency table represent counts.
Race | ||||
Disease Status | White | Black | Hispanic | Other |
Present | 5 | 20 | 15 | 10 |
Absent | 20 | 10 | 10 | 10 |
The computation of odds ratio of Black versus White for various parameterization schemes is tabulated in the following table.
Odds Ratio of Heart Disease Comparing Black to White | |||||
Parameter Estimates | |||||
PARAM | Odds Ratio Estimation | ||||
EFFECT | 0.7651 | 0.4774 | 0.0719 | exp(2 ×0.7651 + 0.4774 + 0.0719) = 8 | |
REF | 2.0794 | 1.7917 | 1.3863 | exp(2.0794) = 8 | |
GLM | 2.0794 | 1.7917 | 1.3863 | 0.0000 | exp(2.0794) = 8 |
Since the log odds ratio () is a linear function of the parameters, the Wald confidence interval for can be derived from the parameter estimates and the estimated covariance matrix. Confidence intervals for the odds ratios are obtained by exponentiating the corresponding confidence intervals for the log odd ratios. In the displayed output of PROC LOGISTIC, the "Odds Ratio Estimates" table contains the odds ratio estimates and the corresponding 95% Wald confidence intervals. For continuous explanatory variables, these odds ratios correspond to a unit increase in the risk factors.
To customize odds ratios for specific units of change for a continuous risk factor, 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. Let (L_{j},U_{j}) be a confidence interval for . The corresponding lower and upper confidence limits for the customized odds ratio 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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