In SPSS linear mixed model (analyze->mixed model->linear), one can opt for predicted values. In the SPSS data file, a column is added with new data, i.e., the predicted values. When I run a linear model via generalized linear mixed model in SPSS (analyze->mixed model->generalized linear, which is basically the same as the first model when you opt for a linear model), the estimated means are given in a plot which is automatically generated by SPSS. Does anybody know the difference between predicted values and estimated means?
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Possibly [this](http://stats.stackexchange.com/questions/41789/what-makes-a-glm-estimate-the-means-differently-from-the-actual-sample-means) question is similar to yours. – ttnphns Nov 28 '12 at 12:38
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I haven't used SPSS before, but the mixed model prediction probably uses the BLUP (http://en.wikipedia.org/wiki/Best_linear_unbiased_prediction), whereas the estimated mean is only the plugin estimated regression coefficients.
If your mixed model is: \begin{equation} y_{ij}=\beta_0 + \beta_1 x_{ij} + u_i + \epsilon_{ij} \end{equation} where $y_{ij}$ is you response and $x_{ij}$ is a predictor, then the BLUP is \begin{equation} E(y_{ij}|\hat u_i)=\hat\beta_0 + \hat\beta_1 x_{ij} + \hat u_i \end{equation} where $\hat u_i$ is typically estimated as its posterior mean using Bayes rule.
In contrast, the estimated mean is simply \begin{equation} E(y_{ij})=\hat\beta_0 + \hat\beta_1 x_{ij} \end{equation} Note that this is marginal over the $u_i$.
Hope that helps!
Ian Fiske
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