Linear of a factor can be thought of

Linear Mixed Models (LMMs) are used for continuous dependent variables in which
the residuals are normally distributed but may not correspond to the assumptions
of independence or equal variance. LMMs can be used to analyze datasets that have
been 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 repeatedly
over time or under different conditions.
The name linear mixed models comes from the fact that these models are linear in
the parameters and that the independent variables may involve a mix of fixed and
random effects.
Fixed effects are unknown constant parameters associated with either continuous
covariates or the levels of categorical factors in an LMM. Estimation of these
parameters in LMMs is generally of intrinsic interest, because they indicate the relationships
of the covariates with the continuous outcome variable (West, Welch, and
Galecki, 2006).
When the levels of a factor can be thought of as having been sampled from a
sample space, such that each particular level is not of intrinsic interest (e.g., classrooms
or clinics that are randomly sampled from a larger population of classrooms
or clinics), the effects associated with the levels of those factors can be modeled as
random effects in an LMM. In contrast to fixed effects, which are represented by
constant 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 background
1.4.1 General specification of the model
The general formula of an LMM, where Yti represents the measure of the continuous
response variable Y taken on the t-th occasion for the i-th subject, can be written as:
Yti = ?1 × X
ti + ?2 × X
ti + ?3 × X
ti + . . . ?p × X
+u1i × Z
ti + · · · + uqi × Z
ti + eti
where the upper part of the formula is for fixed effects and latter is the random
effects of the model. The value of t(t = 1, . . . , ni), indexes the ni
longitudinal observations
on the dependent variable for a given subject, and i(i = 1, . . . , m) indicates
the i-th subject (unit of analysis). The model involves two sets of covariates, namely
the X and Z covariates. The first set contains p covariates, X
, . . . , X
, associated
with the fixed effects ?1, . . . , ?p (West, Welch, and Galecki, 2006).
The second set contains q covariates, Z
, . . . , Z
, associated with the random
effects u1i
, . . . , uqi that are specific to subject i. The X and/or Z covariates may be
continuous or indicator variables. For each X covariate, X
, . . . , X
, the terms
ti , . . . , X
ti represent the t-th observed value of the corresponding covariate for
the i-th subject. The assumptions is that the p covariates may be either time-invariant
characteristics 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 corresponding
X covariate on the mean value of the dependent variable, Y, assuming
that the other covariates remain constant at some value. These ? parameters are
fixed effects that need to be estimated, and their linear combination with the X covariates
defines the fixed portion of the model (West, Welch, and Galecki, 2006).
The effects of the Z covariates on the response variable are represented 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 associated with the t-th observation
on the i-th subject. The assumption here is that for a given subject, the
residuals are independent of the random effects (West, Welch, and Galecki, 2006)