## Linear of a factor can be thought of

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.The name linear mixed models comes from the fact that these models are linear inthe parameters and that the independent variables may involve a mix of fixed andrandom effects.

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Fixed effects are unknown constant parameters associated with either continuouscovariates or the levels of categorical factors in an LMM. Estimation of theseparameters in LMMs is generally of intrinsic interest, because they indicate the relationshipsof the covariates with the continuous outcome variable (West, Welch, andGalecki, 2006).When the levels of a factor can be thought of as having been sampled from asample space, such that each particular level is not of intrinsic interest (e.g., classroomsor clinics that are randomly sampled from a larger population of classroomsor clinics), the effects associated with the levels of those factors can be modeled asrandom effects in an LMM. In contrast to fixed effects, which are represented byconstant parameters in an LMM, random effects are represented by (unobserved)random variables, which are usually assumed 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 measure of the continuousresponse variable Y taken on the t-th occasion for the i-th subject, can be written as:Yti = ?1 × X(1)ti + ?2 × X(2)ti + ?3 × X(3)ti + . . . ?p × X(p)ti+u1i × Z(1)ti + · · · + uqi × Z(q)ti + etiwhere the upper part of the formula is for fixed effects and latter is the randomeffects of the model.

The value of t(t = 1, . . . , ni), indexes the nilongitudinal observationson the dependent variable for a given subject, and i(i = 1, . . . , m) indicatesthe i-th subject (unit of analysis). The model involves two sets of covariates, namelythe X and Z covariates.

The first set contains p covariates, X(1), . . . , X(p), associatedwith the fixed effects ?1, . . . , ?p (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 termsX(1)ti , .

. . , X(p)ti represent the t-th observed value of the corresponding covariate forthe i-th subject.

The assumptions is that the p covariates may be either time-invariantcharacteristics of the individual subject (e.g., gender) or time varying for each measurement(e.

g., time of measurement, or weight at each time point) (West, Welch,and Galecki, 2006).Each ? parameter represents the fixed effect of a one-unit change in the correspondingX covariate on the mean value of the dependent variable, Y, assumingthat the other covariates remain constant at some value. These ? parameters arefixed effects that need to be estimated, and their linear combination with the X covariatesdefines the fixed portion of the model (West, Welch, and Galecki, 2006).

The effects of the Z covariates on the response variable are represented in therandom portion of the model by the q random effects, u1i, . . . , uqi, associated withthe i-th subject.

In addition, eti represents the residual associated with the t-th observationon the i-th subject. The assumption here is that for a given subject, theresiduals are independent of the random effects (West, Welch, and Galecki, 2006)

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