• West, Welch, and Galecki (2007, chap. 7)analyze data studying the effect of ceramic dental veneer placement on gingival(gum) health. Data on 55 teeth located in the maxillary arches of 12 patients wereconsidered.
• use http://www.stata-press.com/data/r13/veneer, clear (Dentalveneer data)
• describe
  

• Veneers were placed to match the original contour of thetooth as closely as possible, and researchers were interested in how contourdifferences (variable cda) impacted gingival health. Gingival health was measured asthe amount of gingival cervical fluid (GCF) at each tooth, measured at baseline(variable base gcf) and at two posttreatment follow-ups at 3 and 6 months.The variable gcf records GCF at follow-up, and the variable followuprecordsthe follow-up time. Because two measurements were taken for each tooth andthere exist multiple teeth per patient, we fit a three-level model with thefollowing random effects: a random intercept and random slope on follow-up timeat the patient level, and a random intercept at the tooth level. For the ithmeasurement of the jth tooth from the kth patient, we have
• gcfijk = β0 + β1followupijk + β2base gcfijk + β3cdaijk + β4ageijk+u0k + u1kfollowupijk + v0jk + Єijk
• mixed gcf followup base_gcf cdaage || patient: followup, cov(un) || tooth:, reml nolog
  

• We modify the above to estimate two distinct errorvariances: one for the 3-month follow-up and one for the 6-month follow-up. Tofit this model, we add the residuals(independent, by(followup)) option,which maintains independence of residual errors but allows forheteroskedasticity with respect to follow-up time.
• mixed gcf followup base_gcf cda age || patient:followup, cov (un) || tooth: residuals (independent, by (followup)) reml nolog
  

• The default residual-variance structure is independent, and when specified without by() is
  

equivalent to the defaultbehavior of mixed: estimating one overall residual standard variance for the entiremodel

• The xi: (interaction expansion) option is used before invoking the xtmixed command to create dummy variables for the categories of the fixed factors TREATMENT and SEX, and for the corresponding interaction terms. The i. notation (i.treatment*i.sex) automatically specifies that fixed effects associated with the interaction between TREATMENT and SEX should be included in the model, along with the main effects for both of these terms.
• This tells us that Control is the reference level of TREATMENT (Control omitted), i.e., no dummy variable is created for the control treatment, and that Male is the reference level of SEX (Male omitted).
• The xtmixed command syntax has three parts, separated by commas. The first part specifies the fixed effects, the second part specifies the random effects, and the third part specifies the covariance structure for the random effects, in addition to miscellaneous options. We discuss these parts of the syntax in detail later. The first variable listed after the xtmixed command is the continuous dependent variable, WEIGHT. The variables following the dependent variable are the terms that will have associated fixed effects in the model. We include i.treatment*i.sex and litsize.
• The two vertical bars (||) precede the variable that defines clusters of observations (litter:).
• The absence of additional variables after the colon indicates that there will only be a single random effect associated with the intercept for each litter in the model.
• The covariance option after an additional comma specifies the covariance structure for the random effects (or the D matrix). Because Model 3.1 includes only a single random effect associated with the intercept (and therefore a single variance parameter associated with the random effects), it has an identity covariance structure. The covariance option is actually not necessary in this simple case.
• Finally, the variance option requests that the estimated variances of the random effects and the residuals be displayed in the output, rather than their estimated standard deviations, which is the default. The xtmixed procedure uses REML estimation by default.
• The AIC and BIC information criteria can be obtained by using the following command after the xtmixed command has finished running:
  

• By default, the xtmixed command does not display F-tests for the fixed effects in the model. Instead, omnibus Wald chi-square tests for the fixed effects in the model can be performed using the test command. For example, to test the overall significance of the fixed treatment effects, the following command can be used:
  

• The two terms listed after the test command are the dummy variables automatically generated by Stata for the fixed treatment effects (we obtain the names for these dummy variables from the initial output for the model shown above). The fixed effects associated with these variables will be displayed in the output, and the indicator variables for the high and low levels of TREATMENT will automatically be saved in the data set. The test command is testing the null hypothesis that the two fixed effects associated with these terms are equal to zero (i.e., the null hypothesis that the treatment means are all equal).
• Similar omnibus tests may be obtained for the fixed SEX effect, the fixed LITSIZE effect, and the interaction between TREATMENT and SEX:
  

• The dummy variables that Stata generates for the interaction effects (_ItreXsex_2_1 and _ItreXsex_3_1) can be somewhat confusing to identify, but they are located in the Variables window in Stata and are also displayed in the initial model output. Once a model has been fitted using the xtmixed command, EBLUPs of the random effects associated with the levels of the random factor (LITTER) can be saved in a new variable (named EBLUPS) using the following syntax:
  

• The saved EBLUPs can then be used to check for random effects that may be outliers.