The GENMOD Procedure

Predicted Values of the Mean

Subsections:

Predicted Values

A predicted value, or fitted value, of the mean $\mu _ i$ corresponding to the vector of covariates $\mb{x}_ i$ is given by

\[ \hat{\mu _ i} = g^{-1}(\mb{x}^{\prime }_ i \hat{\bbeta }) \]

where g is the link function, regardless of whether $\mb{x}_ i$ corresponds to an observation or not. That is, the response variable can be missing and the predicted value is still computed for valid $\mb{x}_ i$. In the case where $\mb{x}_ i$ does not correspond to a valid observation, $\mb{x}_ i$ is not checked for estimability. You should check the estimability of $\mb{x}_ i$ in this case in order to ensure the uniqueness of the predicted value of the mean. If there is an offset, it is included in the predicted value computation.

Confidence Intervals on Predicted Values

Approximate confidence intervals for predicted values of the mean can be computed as follows. The variance of the linear predictor $\eta _ i = \mb{x}^{\prime }_ i \hat{\bbeta }$ is estimated by

\[ \sigma ^2_ x = \mb{x}^{\prime }_ i \bSigma \mb{x}_ i \]

where $\bSigma $ is the estimated covariance of $\hat{\bbeta }$. The robust estimate of the covariance is used for $\bSigma $ in the case of models fit with GEEs.

Approximate $100(1-\alpha )\% $ confidence intervals are computed as

\[ g^{-1} \left(\mb{x}^{\prime }_ i \hat{\beta } \pm z_{1-\alpha /2} \sigma _{x} \right) \]

where $z_ p$ is the $100p$th percentile of the standard normal distribution and g is the link function. If either endpoint in the argument is outside the valid range of arguments for the inverse link function, the corresponding confidence interval endpoint is set to missing.