The GLIMMIX Procedure

Odds and Odds Ratio Estimation

In models with a logit, generalized logit, or cumulative logit link, you can obtain estimates of odds ratios through the ODDSRATIO options in the PROC GLIMMIX, LSMEANS, and MODEL statements. This section provides details about the computation and interpretation of the computed quantities. Note that for these link functions the EXP option in the ESTIMATE and LSMESTIMATE statements also produces odds or odds ratios.

Consider first a model with a dichotomous outcome variable, linear predictor $\eta = \mb {x}’\bbeta + \mb {z}’\bgamma $, and logit link function. Suppose that $\eta _0$ represents the linear predictor for a condition of interest. For example, in a simple logistic regression model with $\eta = \alpha + \beta x$, $\eta _0$ might correspond to the linear predictor at a particular value of the covariate—say, $\eta _0 = \alpha + \beta x_0$.

The modeled probability is $\pi = 1/(1+\exp \{  -\eta \} )$, and the odds for $\eta = \eta _0$ are

\[  \frac{\pi _0}{1-\pi _0} = \frac{1 / (1+\exp \{ -\eta _0\} )}{\exp \{ -\eta _0\}  / (1+\exp \{ -\eta _0\} )} = \exp \{ \eta _0\}   \]

Because $\eta _0$ is a logit, it represents the log odds. The odds ratio $\psi (\eta _1,\eta _0)$ is defined as the ratio of odds for $\eta _1$ and $\eta _0$,

\[  \psi (\eta _1, \eta _0) = \exp \{ \eta _1 - \eta _0\}   \]

The odds ratio compares the odds of the outcome under the condition expressed by $\eta _1$ to the odds under the condition expressed by $\eta _0$. In the preceding simple logistic regression example, this ratio equals $\exp \{  \beta (x_1 - x_0)\} $. The exponentiation of the estimate of $\beta $ is thus an estimate of the odds ratio comparing conditions for which $x_1 - x_0 = 1$. If x and x + 1 represent standard and experimental conditions, for example, $\exp \{ \beta \} $ compares the odds of the outcome under the experimental condition to the odds under the standard condition. For many other types of models, odds ratios can be expressed as simple functions of parameter estimates. For example, suppose you are fitting a logistic model with a single classification effect with three levels:

proc glimmix;
   class A;
   model y = A / dist=binary;

The estimated linear predictor for level j of A is $\widehat{\eta }_ j = \widehat{\beta } + \widehat{\alpha }_ j$, $j=1,2,3$. Because the $\mb {X}$ matrix is singular in this model due to the presence of an overall intercept, the solution for the intercept estimates $\beta + \alpha _3$, and the solution for the jth treatment effect estimates $\alpha _ j - \alpha _3$. Exponentiating the solutions for $\alpha _1$ and $\alpha _2$ thus produces odds ratios comparing the odds for these levels against the third level of A.

Results designated as odds or odds ratios in the GLIMMIX procedure might reduce to simple exponentiations of solutions in the Parameter Estimates table, but they are computed by a different mechanism if the model contains classification variables. The computations rely on general estimable functions; for the MODEL, LSMEANS, and LSMESTIMATE statements, these functions are based on least squares means. This enables you to obtain odds ratio estimates in more complicated models that involve main effects and interactions, including interactions between continuous and classification variables.

In all cases, the results represent the exponentiation of a linear function of the fixed-effects parameters, $\eta = \mb {l}’\bbeta $. If $L_{\eta }$ and $U_{\eta }$ are the confidence limits for $\eta $ on the logit scale, confidence limits for the odds or the odds ratio are obtained as $\exp \{ L_{\eta }\} $ and $\exp \{ U_{\eta }\} $.