# Computing a multivariate odds ratio

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The odds ratio defined in Section 3.7.2 loses its meaning for a multivariate model regardless of whether the predictive variables are of indicator, categorical or continuous type. Fortunately, (3.43) can be generalized to the case of a multivariate model once we realize that all terms in a multivariate logistic equation except $$a$$ vanish the same way as they did in (3.43) once we construct the “incremental” odds ratio. In view of this, our algorithm will proceed as follows:

1. construct a multivariate logistic regression model for the variable in question as
 $$\ln \left ( \frac{p(x)}{1-p(x)} \right ) = b_0 + \sum_{i=1}^N a_i x_i \; , i=\overline{1, N} \;$$ (3.51)

where $$x = ( x_1, \dots, x_n )^T$$ is the vector of predictive variables;

1. observe that
 $$\ln \left ( \frac{p(x_i+1)}{1-p(x_i+1)} \right ) = b_0 + \sum_{k=1}^{i-1} a_i x_i + a_i (x_i+1) + \sum_{k=i+1}^N a_i x_i \; ,$$ (3.52)

and hence

 $$\ln \left ( \frac{p(x_i+1)}{1-p(x_i+1)} \right ) - \ln \left ( \frac{p(x_i)}{1-p(x_i)} \right ) = \nonumber \\ \ln \left ( \frac{p(x_i+1)[1-p(x_i)]}{p(x_i)[1-p(x_i+1)]} \right ) = a_i = \ln \left ( OR_{multi}^{cont} \right ) \; .$$ (3.53)

Exponentiating both sides, we obtain

 $$OR_{multi}^{cont} = e^{a_i} \; .$$ (3.54)

The odds ratio defined by (3.54) represents a proportional increase in the odds of encountering an outcome of interest corresponding to a unitary increase in the value of the respective (continuous) predictive variable of interest. The same note of caution with respect to the domain of the “incremental” multivariate odds ratio applies here as in the univariate case above.

In the example in Section 3.7.1, the odds ratio matrix computed for both continuous / interval and indicator variables is presented in Table 3.18.

Table 3.18: Example: Multivariate odds ratio statistics for COPD logistic regression model

As can be seen from Table 3.18, the most significant predictive variables with respect to their odd ratios are NUM_HOSP_30_DAYS_COPD, NUM_ER_VISITS_30_DAYS_PNEU, and NUM_ER_VISITS_30_DAYS_COPD. We can also observe that PAT_AGE_YRS appears to be insignificant from the point of view of the corresponding odds ratio. In spite of this, we need to bear in mind that, as pointed out in Section 3.7.2, a one year increase in patient age does not change the odds of hospitalization significantly and thus patient’s age cannot be automatically discarded from the final model.