Predicted Values of the Mean

Chapter Contents

The GENMOD Procedure

Predicted Values of the Mean

Predicted Values

A predicted value, or fitted value, of the mean $\mu_i$ corresponding to the vector of covariates x_i is given by

$\hat{\mu_i} = g^{-1}(x^'_i \hat{{\beta}})$

where g is the link function, regardless of whether 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 x_i. In the case where x_i does not correspond to a valid observation, x_i is not checked for estimability. You should check the estimability of 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 $\eta_i = x^'_i \hat{{\beta}}$ is estimated by

$\sigma^2_x = x^'_i {{\Sigma}}x_i$

where ${{\Sigma}}$ is the estimated covariance of $\hat{{\beta}}$ .

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

$g^{-1} (x^'_i \hat{\beta} +- z_{1-\alpha/2} \sigma_{x} )$

where z_p is the 100p 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.

Chapter Contents
Previous
Next
Top