# Linear follow a normal distribution (West,Welch, and Galecki,

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Last updated: June 24, 2019

Linear Mixed Models (LMMs) are used for continuous dependent variables in whichthe residuals are normally distributed but may not correspond to the assumptionsof independence or equal variance. LMMs can be used to analyze datasets that havebeen collected with the following study designs:1. studies with clustered data, like students in classrooms;2. longitudinal or repeated-measures studies, in which subjects are measured repeatedlyover time or under different conditions.LMMs are models that are linear in the parameters as are the more common linearmodels presented earlier, but the difference comes from that LMMs may includeboth fixed and random effects. By adding random effects the model dealswith datasets that have several responses for one subject. Fixed effects are unknownconstant parameters associated with either continuous covariates or the levels of categoricalfactors in an LMM. Estimation of these parameters in LMMs is generally ofunderlying interest as is with also linear models (West,Welch, and Galecki, 2006).

Random effects can be modeledWhen the levels of a factor can be thought of as having been sampled from a samplespace, such that each particular level is not of intrinsic interest, the effects associatedwith the levels of those factors can be modeled as random effects in an LMM. Incontrast to fixed effects, which are represented by constant parameters in an LMM,random effects are represented by (unobserved) random variables, which are usuallyassumed to follow a normal distribution (West,Welch, and Galecki, 2006).4 Chapter 1. Theoretical background1.4.1 General specification of the modelThe general formula of an LMM, where Yti represents the continuous response variableY taken on the t-th occasion for the i-th subject, can be written as:where the upper part of the formula defines the fixed effects and latter the randomeffects of the model. The value of t(t = 1, . .

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. , ni), indexes the ni longitudinal observationson the dependent variable for a given subject, and i(i = 1, . . . ,m) indicatesthe i-th subject. The model involves two sets of covariates, namely the X and Z covariates.

The first set contains p covariates, X(1), . . . , X(p), associated with the fixedeffects b1, . . .

, bp (West,Welch, and Galecki, 2006).The second set contains q covariates, Z(1), . . . , Z(q), associated with the randomeffects u1i, . . .

, uqi that are specific to subject i. The X and/or Z covariates may becontinuous or indicator variables. For each X covariate, X(1), . . . , X(p), the termsti represent the t-th observed value of the corresponding covariate forthe i-th subject (West,Welch, and Galecki, 2006).

Each b parameter represents the fixed effect as defined in the linear model formulamentioned above. The effects of the Z covariates on the response variable arerepresented in the random portion of the model by the q random effects, u1i, . .

. , uqi,associated with the i-th subject. In addition, eti represents the residual associatedwith the t-th observation on the i-th subject. The assumption here is that for agiven subject, the residuals are independent of the random effects (West,Welch, andGalecki, 2006).