# Linear Mixed Models

The Linear Mixed Models procedure expands the general linear model so that the data are permitted to exhibit correlated and nonconstant variability. The mixed linear model, therefore, provides the flexibility of modeling not only the means of the data but their variances and covariances as well.

The Linear Mixed Models procedure is also a flexible tool for fitting other models that can be formulated as mixed linear models. Such models include multilevel models, hierarchical linear models, and random coefficient models.

- Example
- A grocery store chain is interested in the effects of various coupons on customer spending. Taking a random sample of their regular customers, they follow the spending of each customer for 10 weeks. In each week, a different coupon is mailed to the customers. Linear Mixed Models is used to estimate the effect of different coupons on spending while adjusting for correlation due to repeated observations on each subject over the 10 weeks.
- Methods
- Maximum likelihood (ML) and restricted maximum likelihood (REML) estimation.
- Statistics
- Descriptive statistics: sample sizes, means, and standard deviations of the dependent variable and covariates for each distinct level combination of the factors. Factor-level information: sorted values of the levels of each factor and their frequencies. Also, parameter estimates and confidence intervals for fixed effects and Wald tests and confidence intervals for parameters of covariance matrices. Type I and Type III sums of squares can be used to evaluate different hypotheses. Type III is the default.

## Linear Mixed Models data considerations

- Data
- The dependent variable should be quantitative. Factors should be categorical and can have numeric values or string values. Covariates and the weight variable should be quantitative. Subjects and repeated variables may be of any type.
- Assumptions
- The dependent variable is assumed to be linearly related to the fixed factors, random factors, and covariates. The fixed effects model the mean of the dependent variable. The random effects model the covariance structure of the dependent variable. Multiple random effects are considered independent of each other, and separate covariance matrices will be computed for each; however, model terms specified on the same random effect can be correlated. The repeated measures model the covariance structure of the residuals. The dependent variable is also assumed to come from a normal distribution.
- Related procedures
- Use the Explore procedure to examine the data before running an analysis. If you do not suspect there to be correlated or nonconstant variability, you can use the GLM Univariate or GLM Repeated Measures procedure. You can alternatively use the Variance Components Analysis procedure if the random effects have a variance components covariance structure and there are no repeated measures.