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Module 3
Covariance Pattern Models for Repeated Measures Designs
Recall from Module 2 that the GLM can be expressed as
๐ฆ๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐ + ๐ฝ2๐ฅ2๐ + โฆ + ๐ฝ๐๐ฅ๐๐ + ๐๐
where ๐ฆ๐ is the response variable score for participant i and ๐๐ is the prediction
error for participant i. In a random sample of n participants, there are n prediction
errors (๐1, ๐2, โฆ , ๐๐). The GLM assumes that the n prediction errors are
uncorrelated. This assumption is reasonable in most applications because each
prediction error corresponds to a different participant, and it usually easy to design
a study such that no participant influences the response of any other participant.
In a repeated measures design, each participant provides r โฅ 2 responses. The
longitudinal design, the pretest-posttest design, and the within-subjects
experimental design are all special cases of the repeated measured design. In a
longitudinal design, the response variable for each participant is measured on two
or more occasions. In a pretest-posttest design, the response variable is measured
on one or more occasions prior to treatment and then on one or more occasions
following treatment. In a within-subjects experimental design, the response
variable for each participant is measured under all treatment conditions (usually
in counterbalanced order).
In a study with repeated measurements, the relation between the response variable
(y) and q predictor variables (๐ฅ1, ๐ฅ2, โฆ ๐ฅ๐) for one randomly selected person can be
represented by the following covariance pattern model (CPM)
๐ฆ๐๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐๐ + โฆ + ๐ฝ๐ ๐ฅ๐ ๐๐ + ๐ฝ๐ +1๐ฅ๐ +1๐ โฆ + ๐ฝ๐๐ฅ๐๐ + ๐๐๐
where i = 1 to n and j = 1 to r. Note that the i subscript specifies a particular
participant and the j subscript specifies a particular occasion. Note also that
predictor variables ๐ฅ๐ +1 to ๐ฅ๐ do not have a j subscript, as in a GLM, and describe
differences among the participants. These predictor variables are called time-
invariant predictor variables because their values will vary across participants but
remain constant over the r repeated measurements.
Predictor variables ๐ฅ1 to ๐ฅ๐ in the CPM have both an i subscript and a j subscript.
These predictor variables are called time-varying predictor variables because they
can vary over time and across participants. A CPM can have all time-invariant
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predictors, all time-varying predictors, or a combination of time-invariant and
time-varying predictors.
The time-invariant and time-varying predictor variables can be indicator variables,
fixed or random quantitative variables, or any combination of indicator and
quantitative variables. The predictor variables can be squared variables to describe
quadratic effects or product variables to describe interaction effects. A product
variable can be a product of two time-varying predictor variables, a product of two
time-invariant predictor variables, or a product of a time-invariant predictor
variable and a time varying predictor variables.
The CPM has n x r prediction errors (๐11, โฆ ๐1๐ , ๐21, โฆ ๐2๐ , ๐๐1 โฆ ๐๐๐). The r
prediction errors for each participant in the CPM are assumed to be correlated and
possibly also have unequal variances. The prediction errors for different
participants are assumed to be uncorrelated as in the GLM. The variances and
covariances of the prediction errors in a CPM can be represented by a prediction
error covariance matrix as described below.
Prediction Error Covariance Matrices
A prediction error covariance matrix is a symmetric matrix with variances of the
prediction errors in the diagonal elements and covariances among pairs of
prediction errors in the off-diagonal elements. In the GLM where the n prediction
errors are assumed to be uncorrelated and have the same variance, the prediction
error covariance matrix for the n x 1 vector of prediction errors (e) has the
following diagonal structure
cov(e) = [
๐2 0 โฏ 00 ๐2 โฆ 0โฎ โฎ โฎ0 0 โฏ ๐2
]
which can be expressed more compactly as cov(e) = ๐2๐๐.
The prediction error covariance matrix for the prediction errors in a CPM has the
followed block matrix structure
cov(e) = [
๐บ ๐ โฏ ๐๐ ๐บ โฆ ๐โฎ โฎ โฎ๐ ๐ โฏ ๐บ
]
which can be expressed more compactly as cov(e) = In โจ ๐บ where ๐บ is an r x r
covariance matrix for the r prediction errors for a participant and each ๐ is an r x r
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matrix of zeros. The r x r covariance matrix (๐บ) is usually assumed to be identical
across the n participants.
The r variances and the r(r โ 1)/2 covariances of the r prediction errors for
participant i [๐๐1, โฆ , ๐๐๐] can be summarized in an r ร r covariance matrix denoted
as ๐บ. For example, the covariance matrix for r = 3 is
๐บ = [๐1
2
๐12
๐13
๐12
๐22
๐23
๐13
๐23
๐32]
where ๐12 is the prediction error variance for occasion 1, ๐2
2 is the prediction error
variance for occasion 2, ๐32 is the prediction error variance for occasion 3, ๐12 is the
covariance of prediction errors for occasions 1 and 2, ๐13 is the covariance of
prediction errors for occasions 1 and 3, and ๐23 is the covariance of prediction
errors for occasions 2 and 3.
The above covariance matrix is referred to as an unstructured covariance matrix
because there are no assumptions made regarding the values of the variances or
covariances. An unstructured covariance matrix requires the estimation of r
variances and r(r โ 1)/2 covariances, or a total of r(r + 1)/2 parameters.
A covariance matrix where all variances are assumed to be equal and all
correlations are assumed to be equal is called a compound-symmetric covariance
matrix (also called an exchangeable covariance matrix) and is illustrated below for
r = 4. A compound symmetric covariance matrix requires the estimation of two
parameters (๐2 and ๐). A compound symmetric covariance structure might be
justified in a within-subjects experiment where participants are measured under r
treatment conditions in random or counterbalanced order.
๐บ = ๐๐ [
1๐๐๐
๐ 1 ๐๐
๐ ๐ ๐ ๐ 1 ๐ ๐ 1
]
If the response variable is measured over r equally or nearly equally spaced time
intervals, as is often the case in a longitudinal or pretest-posttest design, a first-
order autoregressive covariance structure could be justified. This covariance
structure assumes equal variances with correlations that decrease exponentially in
magnitude as the separation in time increases. An example of this structure for
r = 4 is shown below.
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๐บ = ๐๐
[
1๐
๐2
๐3
๐ 1 ๐
๐2
๐2 ๐3
๐ ๐2
1 ๐๐ 1
]
Like a compound-symmetric covariance matrix, a first-order autoregressive
covariance matrix requires the estimation of only two parameters (๐2 and ๐).
A more general covariance structure for longitudinal data with equally or nearly
equally spaced time intervals is the Toeplitz matrix, illustrated below for r = 4.
๐บ = ๐๐ [
1๐1
๐2
๐3
๐1
1 ๐1
๐2
๐2 ๐3
๐1 ๐2
1 ๐1
๐1 1
]
A total of r parameters (๐2 and ๐1 โฆ ๐๐โ1) must be estimated in a Toeplitz
covariance structure.
If the r prediction errors are assumed to be uncorrelated but have unequal
variances, these assumptions imply a diagonal prediction error covariance matrix
illustrated below for r = 3,
๐บ = [๐1
2
00
0๐2
2
0
00๐3
2]
Compound symmetric, autoregressive, and Toeplitz structures that allow unequal
variances also can be specified. These structures require the estimation of an
additional r โ 1 variance parameters. For example, a compound symmetric
prediction error covariance matrix with unequal variances is given below for r = 3
๐บ = [๐1
2
๐๐1๐2
๐๐1๐3
๐๐1๐2
๐22
๐๐2๐3
๐๐1๐3
๐๐2๐3
๐32
]
where all correlations are assumed to equal ๐.
Generalized Least Squares Estimation
The CPM in Equation 3.2 can be expressed in matrix form for a random sample of
n participants as
y = X๐ท + e (3.1)
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where y is an nr x 1 vector of observations, X is a nr x q + 1 design matrix, ๐ท is a
q + 1 x 1 vector of parameters (containing one y-intercept and q slope coefficients),
and e is nr x 1 vector of prediction errors.
The OLS estimate of ๐ท (Equation 2.19) is appropriate if the prediction errors are
uncorrelated and have a common variance (i.e., cov(e) = ๐๐2๐๐). In the CPM, the
prediction error covariance matrix is cov(e) = In โจ ๐บ. Let ๏ฟฝฬ๏ฟฝ= In โจ ๏ฟฝฬ๏ฟฝ where ๏ฟฝฬ๏ฟฝ is the
sample estimate of ๐บ. The sample estimate of ๐บ is obtained by computing an OLS
estimate of ๐ท (Equation 2.19) and the vector of estimated prediction errors using
Equation 2.20. The sample variances and covariances are then computed from
these estimated prediction errors. When the prediction errors are not assumed to
be uncorrelated or have equal variances, ๐ท in Equation 3.1 is usually estimated
using generalized least squares (GLS) rather than OLS. The GLS estimate of ๐ท is
๏ฟฝฬ๏ฟฝGLS = (๐โฒ๏ฟฝฬ๏ฟฝโ1๐)โ1๐โฒ๏ฟฝฬ๏ฟฝโ1๐ฒ. (3.2)
The GLS estimate of ๐ท can be used to obtained a revised estimate of the estimated
prediction errors and a revised estimate of ๐บ is computed from these revised
prediction errors. Equation 3.2 is recomputed using the revised ๏ฟฝฬ๏ฟฝ= In โจ ๏ฟฝฬ๏ฟฝ. This
process is continued until the GLS estimate of ๐ท stabilizes.
The covariance matrix of ๏ฟฝฬ๏ฟฝGLS is
cov(๏ฟฝฬ๏ฟฝGLS) = (๐โฒ๏ฟฝฬ๏ฟฝโ1๐)โ1 (3.3)
and the standard error of a particular slope estimate (๐ฝ๐) is equal to the square root of the kth diagonal element in Equation 3.3. An approximate 100(1 โ ๐ผ)% confidence interval for ๐ฝ๐ is given below.
๏ฟฝฬ๏ฟฝ๐ ยฑ ๐ก๐ผ/2;๐๐๐๐ธ๏ฟฝฬ๏ฟฝ๐ (3.4)
The recommended df for Equation 3.4 uses a Satterthwaite df that has a
complicated formula. SAS and SPSS can be used to compute Equation 3.4 with a
Satterthwaite df.
A confidence interval for ๐ฝ๐ can be used to test H0: ๐ฝ๐ = b, where b is some numeric
value specified by the researcher. A directional two-sided test can be used to choose
H1: ๐ฝ๐ > b or H2: ๐ฝ๐ < b, or declare the results to be inconclusive.
Unlike the GLM, confidence interval methods are not currently available for
standardized slopes or semi-partial correlations in the CPM. Therefore, it is
especially important for the researcher to have a clear understanding of the metrics
D.G. Bonett (8/2018)
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of all variables in the CPM in order to properly interpret the scientific meaning or
practical implications of a confidence interval for ๐ฝ๐.
Centering the Predictor Variables
Consider a simple CPM that has only time (๐ฅ1) as a predictor variable
๐ฆ๐๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐๐ + ๐๐๐ (Model 1)
where ๐ฝ1 is the slope of the line relating ๐ฅ1 to y and ๐ฝ0 is the y-intercept. Suppose
๐ฅ1 was coded 1, 2, 3, 4, and 5 to represent five possible weeks when a participant
could be measured. With ๐ฅ1 coded this way, ๐ฝ0 describes the predicted y-score for
๐ฅ1 = 0 which would correspond to one week prior to the start of the study. If ๐ฅ1 had
instead been baseline centered so that ๐ฅ1 was coded 0, 1, 2, 3, and 4, then ๐ฝ0 would
describe the predicted y-score for the first week of the study. The time variable also
could be mean centered. If ๐ฅ1 is coded -2, -1, 0, 1, and 2 then ๐ฝ0 would describe the
predicted y-score for the third week of the study.
It is usually a good idea to mean center all time-invariant predictor variables.
Consider the following CPM that has one time-invariant predictor variable
๐ฆ๐๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐๐+ ๐ฝ2๐ฅ2๐ + ๐๐๐ (Model 2)
where ๐ฅ2๐ is the time-invariant score for participant i. As an example, if ๐ฅ1 is
baseline centered and ๐ฅ2๐ is the ACT score for participant i then ๐ฝ0 describes the
predicted y-score at week 1 for participants with an ACT score of 0. This y-intercept
is not meaningful because an ACT score of 0 is impossible. However, if the ACT
scores are mean centered, then ๐ฝ0 describes the predicted y-score at week 1 for
participants with an average ACT score. If a product of two time-invariant
predictor variables is included in the model to assess an interaction effect, the
time-invariant predictor variables should be mean centered before computing
product variable.
Time-varying predictor variables should be person centered rather than mean
centered. Consider the following CPM that adds a time-varying predictor variable
to Model 1
๐ฆ๐๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐๐ + ๐ฝ2๐ฅ2๐๐ + ๐๐๐ (Model 3)
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where ๐ฅ1 is the time variable and ๐ฅ2๐๐ is a time-varying predictor score for
participant i on occasion j. In this model, ๐ฝ2 describes the slope of the line relating
y to ๐ฅ2 across all time periods and all participants. If participants have substantially
different ๐ฅ2 scores, ๐ฝ2 will be a misleading description of the relation between y
and ๐ฅ2 within each person. Consider the following scatterplot for two participants
who have substantially different ๐ฅ2 scores. The two thin lines represent the within-
person slopes and the thick line represents the overall slope that would be
represented by ๐ฝ2 in Model 3. In this example, the within-person slopes are
positive but ๐ฝ2 is negative.
. .
.
y *
*
*
0 1 2 3 4 5 6 7 8 9 10
๐ฅ2
The ๐ฅ2 scores have been person centered in the following plot and ๐ฝ2 in Model 3
now describes the within-person slopes.
. .
.
y *
*
*
-5 -4 -3 -2 -1 0 1 2 3 4 5
๐ฅ2
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When a time-varying predictor variable (๐ฅ2) has been person centered, the slope
coefficient for ๐ฅ2 describes the within-person relation between y and ๐ฅ2. Some of
the variability in y can usually be predicted by between-person differences in ๐ฅ2,
but these between-person differences are lost when ๐ฅ2 is person centered. This lost
information can be recovered by simply adding another predictor variable to the
model that represents the mean time-varying predictor score for each participant
as shown below in Model 4
๐ฆ๐๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐๐ + ๐ฝ2๐ฅ2๐๐ + ๐ฝ3๐ฅ3๐ + ๐๐๐ (Model 4)
where ๐ฅ2๐๐ is the person-centered time-varying predictor variable score for
participant i on occasion j and ๐ฅ3๐ is the mean time-varying predictor score for
participant i. Note that ๐ฅ3 is a time-invariant predictor variable. In Model 4, ๐ฝ2
describes the slope of the line relating y to ๐ฅ1 across time and within persons and
๐ฝ3 describes the slope of the line relating y to ๐ฅ2 across persons.
To illustrate the computation of ๐ฅ2 and ๐ฅ3 in Model 4, considered the following
hypothetical data for the first two participants where ๐ฅ1 is the time variable
(baseline centered) and ๐ฅ2 is a time-varying predictor variable. Participant 1 was
measured on three occasions and participant 2 was measured on four occasions.
Participant y x1 x2
1 15 0 7
1 19 1 9
1 22 2 11
2 23 0 16
2 27 1 19
2 34 2 25
2 35 3 28
The mean of the ๐ฅ2 scores for participant 1 is (7 + 9 + 11)/3 = 9 and the mean of the
๐ฅ2 scores for participant 2 is (16 + 19 + 25 + 28)/4 = 22. Subtract 9 from the ๐ฅ2 scores
for participant 1 and subtract 22 from the ๐ฅ2 scores for participant 2. The person
centered ๐ฅ2 scores are given below along with a new time-invariant variable (๐ฅ3)
that has the person means of ๐ฅ2.
Participant y x1 x2 x3
1 15 0 -2 9
1 19 1 0 9
1 22 2 2 9
2 23 0 -6 22
2 27 1 -3 22
2 34 2 3 22
2 35 3 6 22
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Wide and Long Data Formats
To analyze a CPM, the data need to be in a "long format" rather than a "wide
format". All of the within-subjects methods described in PSYC 204 required the
data to be in a wide format. To illustrate the difference between these two types of
data formats, consider a study with n = 4 participants who are each measured on
three occasions. Hypothetical data for a wide format is shown below.
Participant Time 1 Time 2 Time 3
1 10 14 15
2 15 18 17
3 12 13 19
4 14 20 22
The long format for these same data is shown below.
Participant Time Score
1 1 10
1 2 14
1 3 15
2 1 15
2 2 18
2 3 17
3 1 12
3 2 13
3 3 19
4 1 14
4 3 20
4 3 22
Programs that analyze data in wide format will delete any row (participant) for
which any column has missing data (listwise deletion). With long format, only
specific occasions with missing data are lost. In the following example, participant
1 had a missing observation at time 2, and participants 2 and 4 had a missing
observation at time 1.
Participant Time 1 Time 2 Time 3
1 10 -- 15
2 -- 18 17
3 12 13 19
4 -- 20 22
In this example, participants 1, 2, and 4 would be deleted in a statistical analysis
that used the wide format leaving only one participant for analysis. The long format
for these data is shown below.
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Participant Time Score
1 1 10
1 3 15
2 2 18
2 3 17
3 1 12
3 2 13
3 3 19
4 3 20
4 3 22
With a long format, all nine available observations are used in the analysis. One
advantage of a CPM for analyzing repeated measures data is that all available data
will be used when one or more participants are randomly missing one or more
observations. In a CPM, a participant is dropped only if that participant is missing
all r observations.
Modeling the Repeated Measurements
As noted above, all of the predictor variables that were used in the GLM to
represent differences among the n participants also can be included as time-
invariant predictor variables in a CPM. Time-varying predictor variables can be
included in a CPM to represent differences among the r repeated measures for each
participant. Several basic types of time-varying predictor variables are described
below for longitudinal designs, pretest-posttest designs, and within-subjects
experimental designs.
In a longitudinal design, the CPM could include a time-varying predictor variable
that represents points in time. For example, suppose a social skill score is obtained
for a sample of kindergarten students during the first week of four consecutive
months. Hypothetical data for the first two students are shown below.
Participant Month SocSkill
1 0 24
1 1 35
1 2 28
1 3 19
2 0 30
2 1 39
2 2 32
2 3 29
โฎ โฎ โฎ
If social skill is assumed to have a quadratic trend over time, Month2 could be
included as an additional time-varying predictor variable. The mean-centered
month variable for the above two students is shown below.
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Participant Month Month2 SocSkill
1 -1.5 2.25 24
1 -0.5 0.25 35
1 0.5 0.25 28
1 1.5 2.25 19
2 -1.5 2.25 30
2 -0.5 0.25 39
2 0.5 0.25 32
2 1.5 2.25 29
โฎ โฎ โฎ โฎ
A CPM model with only Month as a predictor variable implies a linear relation
between month and social skill as illustrated in Figure 3a. If Month2 is added to the
model, then the model implies a quadratic relation between month and social skill
as illustrated in Figure 3b.
In pretest-posttest designs, a dummy variable could be added to code treatment.
For example, suppose the social skill of each kindergarten student in the sample is
measured every month for two months before exposure to a social skill training
program and then every month for two months following training. Hypothetical
data for the first two students are shown below. Note that a dummy variable for
treatment is equal to 0 in months 1 and 2 and equal to 1 in months 3 and 4.
Participant Month Treatment SocSkill
1 0 0 24
1 1 0 35
1 2 1 28
1 3 1 24
2 0 0 30
2 1 0 39
2 2 1 32
2 3 1 30
โฎ โฎ โฎ โฎ
If a CPM for a longitudinal or pretest-posttest design only includes the Treatment
dummy variable as a predictor variable, the model implies a horizontal trend prior
to treatment and a jump after treatment that remains horizontal as illustrated in
Figure 3c. If both Month and the Treatment dummy variable are included as
predictor variables, the model implies a linear trend prior to treatment with a jump
following treatment with a linear trend following treatment that has the same slope
as during pretreatment time periods (see Figure 3d). If the pretreatment and
posttreatment slopes are expected to differ (see Figure 3e), then the product of
Month and Treatment can be added to the model as shown below.
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Participant Month Treatment Month x Treatment SocSkill
1 0 0 0 24
1 1 0 0 35
1 2 0 0 28
1 3 1 3 24
1 4 1 4 35
2 0 0 0 30
2 1 0 0 39
2 2 0 0 32
2 3 1 3 30
2 4 1 4 39
โฎ โฎ โฎ โฎ โฎ
In Figure 3e, there is a shift in the trend lines following treatment. If the
pretreatment slope is assumed to differ from the posttreatment slope but the two
lines are assumed to connect (see Figure 3f), this pattern can be modeled by
including only Month and Month x Treatment in the model.
(a) (b) (c)
Time Time Time
(d) (e) (f)
Time Time Time ________________________________________________
Figure 3. Examples of time trends
In within-subject experiments where participants are measured under all
treatment conditions in random or counterbalanced order, k โ 1 dummy variables
are needed to represent the k-level treatment factor. For example, with k = 3
treatments, the data file would include two dummy variables as shown below with
hypothetical response variable scores for the first two participants.
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Participant dummy1 dummy2 Score
1 1 0 61
1 0 1 57
1 0 0 78
2 1 0 54
2 0 1 62
2 0 0 48
โฎ โฎ โฎ โฎ
Time-varying covariates are random quantitative predictor variables that can be
included in longitudinal, pretest-posttest or within-subjects experimental designs.
For example, in the above within-subjects experiment, suppose the response
variable is the number of questions answered correctly after reading three short
reports. Participants vary in the length of time they read each report. If reading
time is related to reading comprehension, reading time could be included as a
time-varying covariate as illustrated below for the first two participants.
Participant dummy1 dummy2 Minutes Score
1 1 0 3.5 61
1 0 1 4.8 57
1 0 0 6.1 78
2 1 0 4.6 54
2 0 1 5.9 62
2 0 0 4.1 48
โฎ โฎ โฎ โฎ โฎ
In longitudinal designs, lagged time-varying covariates are sometimes useful. A
one-period lagged covariate uses the value of the covariate at time t โ 1 as the
predictor variable value at time t. For example, suppose a sample of first-year
college students agree to report their number of good friends and their loneliness
each month for six months. The researcher believes that the number of close
friends reported in the prior month is a predictor of loneliness in the current
month. Hypothetical friend and loneliness data are given below where the friend
variable has been lagged one month. With a one period lagged predictor variable,
the first time period (month = 0) is excluded from the analysis because the lagged
predictor variable value is usually unavailable at time 1.
Participant Month FriendsL1 Loneliness
1 1 2 46
1 2 3 44
1 3 2 44
1 4 4 38
1 5 6 30
2 1 3 37
2 2 3 35
2 3 4 30
2 4 4 28
2 5 5 20
โฎ โฎ โฎ โฎ
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In all of the above examples, time-invariant predictor variables can be added to the
model to describe quantitative or qualitative participant characteristics. For
example, participant gender could be included as a time-invariant predictor
variable in the above loneliness study. Suppose the first participant is male and the
second participant is female. The data file could include a dummy variable coding
gender as shown below. Note that the value of the time-invariant dummy variable
coding gender is unchanged across all five time periods (because gender does not
change over time) and differs only across participants.
Participant Month Gender FriendsL1 Loneliness
1 1 1 2 46
1 2 1 3 44
1 3 1 2 44
1 4 1 4 38
1 5 1 6 30
2 1 0 3 37
2 2 0 3 35
2 3 0 4 30
2 4 0 4 28
2 5 0 5 20
โฎ โฎ โฎ โฎ โฎ
Multi-level Models for Repeated Measures Designs
A multi-level model (MLM), which is also referred to as a mixed linear statistical
model, is an alternative to the CPM for repeated measures designs. Like the CPM,
a MLM can have time-varying predictor variables, time-invariant predictor
variables, or both types of predictor variables. All of the methods for modeling the
repeated measurements in a CPM can be applied to the MLM. All of the methods
for centering predictor variables in a CPM apply to a MLM.
In repeated measures designs, a MLM can be expressed in terms of a level-1 model
that includes only time-varying predictor variables and one or more level-2 models
that include only time-invariant predictor variables. Unlike a CPM with a first-
order autoregressive or Toeplitz prediction covariance matrix, a MLM does not
require equally or nearly equally spaced time intervals and participants need not
all be measured on the same set of time points.
The MLM is more complicated than a CPM and the MLM can be most easily
understood by starting with a very simple type of MLM. Consider a longitudinal
study where the researcher believes that the response variable (y) changes linearly
over time. A level-1 model that includes only the time predictor variable (๐ฅ1) is
given below
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15
๐ฆ๐๐ = ๐0๐ + ๐1๐๐ฅ1๐๐ + ๐๐๐ (Model 5)
where the values of ๐ฅ1๐๐ are the time points at which participant i was measured.
For example, if participant 1 (i = 1) was measured on weeks 1, 2, 4, and 5 then ๐ฅ1๐1
would have values of ๐ฅ111 = 1, ๐ฅ112 = 2, ๐ฅ113 = 4, and ๐ฅ114 = 5; and if participant
2 (i = 2) was measured on weeks 1, 4, 6, 8, and 9 then ๐ฅ1๐2 would have values of
๐ฅ121 = 1, ๐ฅ122 = 4, ๐ฅ123 = 6, ๐ฅ124 = 8, and ๐ฅ125 = 9. Note that the time points need
not be equally spaced and different participants can be measured at different sets
of time points and different numbers of time points. Note also that the parameters
of the level-1 model contain an i subscript to indicate that each participant has their
own y-intercept (๐0๐) and slope (๐1๐) values. The prediction errors (๐๐๐) in the
level-1 model are typically assumed to be uncorrelated among participants and
have equal variances across participants and time (but this assumption can be
relaxed). Assuming equal variances, the variance of ๐๐๐ for all i and j is equal to ๐๐2.
The n participants are assumed to be a random sample from some specified study
population of N people. The level-1 model indicates that each of the N persons have
their own y-intercept and slope value. The level-1 model describes a random
sample of n participants and thus the ๐0๐ and ๐1๐ values (i = 1 to n) are a random
sample of a population of y-intercept and slope values. In the same way that a
statistical model describes a random sample of y scores, statistical models can be
used to describe a random sample of ๐0๐ and ๐1๐ values. The statistical models for
๐0๐ and ๐1๐ are called level-2 models.
The following level-2 models for ๐0๐ and ๐1๐ are the simplest type because they have
no predictor variables
๐0๐ = ๐ฝ0 + ๐ข0๐ (Model 6a)
๐1๐ = ๐ฝ1 + ๐ข1๐ (Model 6b)
where ๐ข0๐ and ๐ข1๐ are parameter prediction errors for the random values of ๐0๐
and ๐1๐, respectively. These parameter prediction errors can be correlated with
each other but they are assumed to be uncorrelated with the level-1 prediction
errors (๐๐๐). The n parameter prediction errors for ๐0๐ are assumed to be
uncorrelated with each other and have variances equal to ๐๐ข02 . Likewise, the n
parameter prediction errors for ๐1๐ are assumed to be uncorrelated with each other
and have variances equal to ๐๐ข12 . The value of ๐๐ข1
2 describes the variability of the ๐1๐
values in the population, and the value of ๐๐ข02 describes the variability of the ๐0๐
D.G. Bonett (8/2018)
16
values in the population. The variability of the y-intercept values (๐0๐) is usually
interesting only if the variability of the slope values (๐1๐) is small. The graphs below
illustrate a sample of n = 5 participants where the slope variability is large (top)
and the slope variability is small (bottom).
Time
Time
In Model 6a, ๐ฝ0 is the population mean of the y-intercepts, and in Model 6b, ๐ฝ1 is
the population mean of the slope coefficients. MLM computer programs will
compute estimates of ๐ฝ0, ๐ฝ1, ๐๐ข02 , and ๐๐ข1
2 along with confidence intervals for ๐ฝ0, ๐ฝ1,
๐๐ข02 , and ๐๐ข1
2 . Confidence intervals for ๐๐ข0 and ๐๐ข1
, which are easier to interpret, are
obtained by taking the square roots of the confidence interval endpoints for ๐๐ข02
and ๐๐ข12 .
If the estimate of the slope variability (๐๐ข12 ) is small or uninteresting, this variance
can be constrained to equal 0 and then Model 6b reduces to ๐1๐ = ๐ฝ1. This level-2
model implies that the slope coefficient relating time to y is the same for everyone
in the population and equal to ๐ฝ1. If the confidence interval for ๐๐ข12 suggests that
๐๐ข12 is not small, this indicates that there is potentially interesting variability in the
slope coefficients among people in the study population. One or more predictor
variables can be included in Model 6b in an effort to explain some of the variability
in the slope coefficients.
In a MLM, the y-intercepts (๐0๐) are almost always assumed to be random. If the
confidence interval for ๐๐ข02 suggests that ๐๐ข0
2 is not small, this indicates that there
is potentially interesting variability in the y-intercepts among people in the study
D.G. Bonett (8/2018)
17
population. One or more predictor variables can be included in Model 6a in an
effort to explain some of the variability in the y-intercepts.
The level-2 models can be substituted into the level-1 model to give the following
composite model
๐ฆ๐๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐๐ + ๐๐๐โ (Model 7)
where ๐๐๐โ = ๐ข0๐ + ๐ข1๐๐ฅ1๐๐ + ๐๐๐ is the composite prediction error for participant i at
time j. Although it was assumed that the level-1 prediction errors were
uncorrelated and homoscedastic, the composite prediction errors will be
correlated and could have unequal variances. Using covariance algebra (Appendix
of Module 2), the variance of ๐๐๐โ at time j is
var(๐๐๐โ ) = var(๐ข0) + 2๐ฅ1๐cov(๐ข0, ๐ข1) + ๐ฅ1๐
2 var(๐ข1) + var(e) (3.5)
and the covariance between two composite prediction errors at time ๐ and time ๐โฒ
is
cov(๐๐๐
โ , ๐๐๐โฒโ ) = var(๐ข0) + (๐ฅ1๐ + ๐ฅ1๐โฒ)cov(๐ข0, ๐ข1) (3.6)
The y-intercept (๐ฝ0) and the slope (๐ฝ1) in the composite model are identical to the
y-intercept (๐ฝ0) and slope (๐ฝ1) in a CPM that has only ๐ฅ1๐๐ as a time-varying
predictor variable. Unlike the CPM where the researcher can specify any type of
prediction error covariance structure (e.g., unstructured, Toeplitz, first-order
autoregressive), the MLM where the level-1 prediction errors are uncorrelated and
have equal variances has a composite prediction error covariance structure given
by Equations 3.5 and 3.6.
To illustrate the covariance structure implied by Equations 3.5 and 3.6, suppose
the estimates of var(๐ข0), var(๐ข1), cov(๐ข0, ๐ข1) and var(e), are 5.3, 2.4, -1.3, and 3.8,
respectively. Next, assume that the participants were measured on four equally
spaced time points with baseline centering so that ๐ฅ1๐1 = 0, ๐ฅ1๐2 = 1, ๐ฅ1๐3 = 2, and
๐ฅ1๐4 = 3. Plugging these values into Equations 3.5 and 3.6 gives the following 4 x 4
composite prediction error covariance matrix.
๏ฟฝฬ๏ฟฝ = [
9.14.02.71.4
4.0 8.9 1.40.1
2.7 1.4 1.4 0.1 13.5 โ 1.2โ1.2 22.9
]
D.G. Bonett (8/2018)
18
With the assumption of a random y-intercept and a random slope, the resulting
composite prediction error covariance matrix assumes unequal variances that
decreases from 9.1 to 8.9 in periods 1 and 2 and then increases to 13.5 and 22.9 in
periods 3 and 4, respectively. Note also that the covariance between the measures
in periods 3 and 4 is assumed to be negative. This pattern of variances and
covariances could be very difficult to justify.
Treating the slope coefficient as random rather than fixed in a MLM can produce
a composite prediction error covariance structure that poorly approximates the
true composite prediction error covariance structure which in turn will give
misleading confidence interval and hypothesis testing results. The consequences
of treating a slope coefficient as random rather than fixed is most pronounced
when var(๐ข1) is large. One strategy is to reduce the value of var(๐ข1) by including
explanatory variables in the level-2 model. For example, suppose vocabulary size
is measured in a sample of preschool children each month for five consecutive
months. The slope coefficient for each child could vary considerable across
children (i.e., some children show large gains, some moderate gains, and others
very little gain) which would result in a large value of var(๐ข1). The researcher
suspects that younger children are more likely to have larger gains than older
children. The following level-2 models for Model 1 could then be specified.
๐0๐ = ๐ฝ0 + ๐ฝ02๐ฅ2๐ + ๐ข0๐ (Model 8a)
๐1๐ = ๐ฝ1 + ๐ฝ12๐ฅ2๐ + ๐ข1๐ (Model 8b)
where ๐ฅ2๐ is a time-invariant predictor variable that is equal to the age of child i at
the beginning of the study.
Any predictor variable that is used in the level-2 slope model is almost always used
in the level-2 y-intercept model because any variable that is related to the slope is
almost always related to the y-intercept. If age is a good predictor of the individual
slopes, then the var(๐ข1) could become much smaller compared to a level-2 model
that does not include age as a predictor variable.
Substituting Models 8a and 8b into the level-1 model (Model 7) gives the following
composite model
๐ฆ๐๐ = ๐ฝ0 + ๐ฝ1๐ฅ1๐๐ + ๐ฝ2๐ฅ2๐ + ๐ฝ12๐ฅ1๐๐๐ฅ2๐ + ๐๐๐โ (Model 9)
where ๐๐๐โ = ๐ข0๐ + ๐ข1๐๐ฅ1๐๐ + ๐๐๐. Note that the composite prediction error is the same
in Model 7. Note also that a level-2 predictor variable of a random slope coefficient
produces an interaction effect in the composite model. In this example, the
D.G. Bonett (8/2018)
19
composite model includes an age x time interaction effect which describes a
relation between vocabulary size and time that depends on the child's age. The
parameters in Model 9 are identical to the parameters in a CPM (๐ฝ12 would be
labeled ๐ฝ3 in a CPM).
Suppose that when age is added to the level-2 models the estimates of var(๐ข0๐),
var(๐ข1), cov(๐ข0, ๐ข1) and var(e) are 3.5, 0.4, -0.05, and 1.7. Adding age in the level-2
models has reduced the estimate of var(๐ข1) from 2.4 to 0.4. The composite
prediction error covariance matrix with a baseline centered time variable is
๏ฟฝฬ๏ฟฝ = [
5.23.53.43.3
3.5 5.5 3.43.4
3.4 3.3 3.4 3.3 6.6 3.2 3.2 8.5
]
which now has similar variances and similar covariances. This covariance structure
assumes the variances increase over time from 5.2 to 8.5 which seems more
realistic than the previous example where the variances decreased and then
increased over time. This covariance structure, which also assumes that all
covariances are similar, would not be realistic in longitudinal and pretest-posttest
designs where the measurements obtained in adjacent time points are usually
more highly correlated than measurements separated by longer periods of time.
If the prediction errors in the level-1 model are assumed to be uncorrelated and
have equal variances, the composite prediction error covariance matrix in a MLM
will often be a poor approximation to the true composite prediction error
covariance matrix. To address this problem, MLM programs have options to
specify more realistic level-1 predictor error covariance structures. In longitudinal
and pretest-posttest designs, a first-order autoregressive prediction error structure
for the level-1 prediction errors usually gives a more realistic composite prediction
error covariance matrix. The level-1 prediction errors also could be assumed to
have unequal variances. Although MLM programs have options to specify more
general level-1 prediction error covariance structures, the parameter estimates of
these covariance structures are sometimes so highly correlated with the estimates
of ๐๐ข02 and ๐๐ข1
2 that the MLM program will not be able to provide unique estimates
of the covariance structure parameters and variances of the random y-intercepts
or random slopes. One strategy is to assume random y-intercepts and no random
slopes and then use the most general level-1 covariance structure that can be
estimated.
D.G. Bonett (8/2018)
20
A Comparison Multi-level and Covariance Pattern Models
In a general MLM, the level-1 model can have multiple time-varying predictor
variables, and a level-2 model is specified for every parameter of the level-1 model.
Furthermore, each level-2 model can have no time-invariant predictor variables or
multiple time-invariant predictor variables. After the level-2 models are
substituted into the level-1 model, the resulting composite model will have the
same predictor variables as an equivalent CPM. Thus, MLM can estimate the same
parameters as a CPM. Unlike a CMP, a MLM also provides variance estimates of
the random coefficients. If the variance of any slope parameter in a MLM is large,
this suggest that the model is missing important interaction effects. Both the CPM
and the MLM can be implemented using "mixed linear model" programs in SAS,
SPSS, and R.
As noted previously, a CPM assumes every participant can be measured on the
same set of time points, and the first-order autoregressive and Toeplitz covariance
structures assume that the time points are equally or nearly equally spaced. In a
MLM, the time points can be unequally spaced and the time points need not be the
same for each participant.
The confidence interval and hypothesis testing methods in a CPM and MLM
require larger sample sizes in models where the prediction error covariance matrix
(in a CPM) or the composite prediction error covariance matrix (in a MLM) contain
many variance and covariance parameters to be estimated. In a MLM model with
only a random y-intercept, the composite prediction error covariance matrix has a
compound symmetric structure that requires the estimation of only two
parameters and this type of MLM can be applied in small samples even if the
number of repeated measurements is large. Of course, the compound symmetry
assumption could be unrealistic. Treating a slope coefficient as random will add
only two additional parameters (the variance of the random slope and the
covariance between the random y-intercept and the random slope) and could
produce a more realistic composite prediction error covariance matrix.
Alternatively, assuming a first-order autoregressive prediction error covariance
structure in the level-1 model could produce a more realistic composite prediction
error covariance structure.
If the covariance structure of the composite prediction errors in a MLM are a poor
approximation to the correct covariance structure, then the hypothesis testing and
confidence interval results for the parameters of the composite model could be
misleading.
D.G. Bonett (8/2018)
21
The Akaike Information Criterion (AIC) can be used to assess the effects of
treating one or more random slope coefficients as random rather than fixed. The
AIC also can be used to assess the effect of using different level-1 prediction error
covariance structures in a MLM or different prediction error covariance structures
in a CPM. When comparing two models with the same predictor variables, the
model with the smaller AIC value suggests a more appropriate composite
prediction error structure. For example, if the AIC in a CPM for a first-order
autoregressive structure (with require the estimation of only two parameters) is
smaller than the AIC for an unstructured prediction error matrix, this could justify
the use of the first-order autoregressive covariance structure. If the true prediction
error covariance structure can be closely approximated by a simple covariance
structure, then the hypothesis testing and confidence interval results in a CPM
should perform better in small samples than a CPM that uses an unstructured
prediction error covariance matrix.
In repeated measures designs where a CPM and a MLM are both appropriate, one
recommendation is to use both models. First, the parameters of the CMP are
estimated using the most realistic prediction covariance matrix, such as an
unstructured, heteroscedastic Toeplitz, or heteroscedastic compound symmetric
prediction error covariance matrix. Next, a MLM could be used to obtain estimates
and confidence intervals for the variances of the random y-intercept and any
random slope coefficients. If the variance of any random slope coefficient is large,
this suggests the need for additional predictor variables in the level-2 models
which implies a need for additional interaction effects in the CPM.
Random Factors
All of the factors considered in Module 2 of PSYC 204 and Module 2 of this course
have been fixed factors because it was assumed that the factor levels used in the
study were deliberately selected and were the only factor levels of interest. In
comparison, the levels of a random factor are randomly selected from a large
superpopulation of M possible factor levels. The appeal of using a random factor
is that the statistical results apply to all M levels of the random factor even though
only a small subset of the factor levels are actually used in the study. With a fixed
factor, the statistical results apply only to the factor levels included in the study.
Recall that a factor can be a classification factor or a treatment factor. With a
classification factor, participants are classified into the levels of the factor based on
some existing characteristic of the participant. The levels of a classification factor
define different subpopulations of people. With a treatment factor, participants are
randomly assigned to the levels of the factor. A random factor can be a random
D.G. Bonett (8/2018)
22
classification factor or a random treatment factor. Most random factors are
classification factors and only random classification factors will be illustrated here.
In studies where a large number (M) of subpopulations could be examined, such
as all schools in a state, all neighborhoods in a large city, or all branch offices of a
large organization, it could be costly or impractical to take a random sample of
participants from each of the M subpopulations. In these situations, the researcher
could randomly select k subpopulations from the superpopulation of M
subpopulations (e.g., schools, neighborhoods, branch offices) and then take a
random sample of ๐๐ participants from each of the k subpopulations. This type of
sampling is called two-stage cluster sampling.
A linear statistical model with one random factor and no other factors or covariates
is called a one-way random effects ANOVA model and the population means of
interest are ๐1, ๐2, โฆ , ๐๐. Although only k of these M population means will be
estimated in the study, it is possible to obtain a confidence interval for the
superpopulation mean ๐ = (๐1 + ๐2 + โฆ + ๐๐)/๐. With equal sample sizes per
group (and equal to n), a 100(1 โ ๐ผ)% confidence interval for ๐ is
๏ฟฝฬ๏ฟฝ ยฑ ๐ก๐ผ/2;(๐โ1)โ๐๐๐ด/๐๐ (3.7)
where ๏ฟฝฬ๏ฟฝ = (๏ฟฝฬ๏ฟฝ1 + ๏ฟฝฬ๏ฟฝ2 + โฆ + ๏ฟฝฬ๏ฟฝ๐)/๐ and MSA is the mean square estimate for the
between-subjects factor in a one-way ANOVA (see Module 2 of PSYC 204).
The standard deviation of the M population means is ๐๐ =โโ (๐๐ โ ๐)2๐๐=1 /๐ , which
is a measure of effect size because larger values of ๐๐ represent larger differences
among the population means. A standardized measure of effect size in designs with
a random factor is ๐2 = ๐๐2/(๐๐
2 + ๐๐2), where ๐๐
2 is the within-group error variance.
An estimate of ๐๐2 is
๏ฟฝฬ๏ฟฝ๐2 = (๐๐A โ ๐๐E)/๐, (3.8)
an estimate of ๐2 is
๏ฟฝฬ๏ฟฝ2 = (๐๐A โ ๐๐E)/[๐๐A + (๐ โ 1)๐๐E], (3.9)
and an estimate of ๐๐2 is MSE where MSA and MSE are the mean square estimates
from a one-way ANOVA table. Recall from PSYC 204 that a confidence interval for
๐2 in the one-way ANOVA involved complicated computations. Surprisingly, a
D.G. Bonett (8/2018)
23
confidence interval for ๐2 in the one-way random effects ANOVA can be hand
computed. The 100(1 โ ๐ผ)% lower (L) and upper (U) confidence limits for ๐2 are
L = (F/๐น๐ผ/2; ๐๐1,๐๐2 โ 1)/( ๐ + F/๐น๐ผ/2; ๐๐1,๐๐2 โ 1) (3.10a)
U = (F/๐น1โ๐ผ/2; ๐๐1,๐๐2 โ 1)/(๐ + F/๐น1โ๐ผ/2; ๐๐1,๐๐2 โ 1) (3.10b)
where F = MSA/MSE, ๐น๐ผ/2; ๐๐1,๐๐2 and ๐น1โ๐ผ/2; ๐๐1,๐๐2 are critical F values with df1 =
k โ 1 and df2 = k(n โ 1). The qf function in R can be used to obtain these critical F
values.
The one-way random effect ANOVA can be expressed as special type of MLM. The
level-1 model can be expressed as
๐ฆ๐๐ = ๐๐ + ๐๐๐ (3.11)
where ๐๐ is the subpopulation mean for level j of the random factor. With randomly
selected factor levels, the ๐๐ values (j = 1 to k) are a random sample from the
superpopulation of ๐๐ values. A level-2 model for the random ๐๐ values is
๐๐ = ๐ + ๐ข๐ (3.12)
where ๐ is defined above and ๐ข๐ is a parameter prediction error that is assumed to
be uncorrelated with ๐๐๐. The variance of ๐๐ is ๐๐2 which also was defined above.
Substituting the level-2 model into the level-1 model gives the following composite
model
๐ฆ๐๐ = ๐ + ๐๐๐โ (3.13)
where ๐๐๐โ = ๐๐๐ + ๐๐. Using covariance algebra, the variance of ๐๐๐
โ (for every value
of i and j) is equal to ๐๐2 + ๐๐
2, and the covariance between any two participants
within the same factor level is equal to ๐๐2. Thus, the correlation between two
participants within the same factor level is equal to ๐๐2/(๐๐
2 + ๐๐2), which was
defined above as ๐2 but is also called an intraclass correlation because it describes
the correlation between any two participant scores within the same factor level.
Mixed linear model programs can be used to obtain hypothesis tests and
confidence intervals for ๐ and ๐๐2.
One or more person-level predictor variables can be added to Equation 3.11. The
predictor variables can be indicator variables or quantitative variables. When all of
D.G. Bonett (8/2018)
24
the predictor variables are quantitative, the resulting composite model is referred
to as a one-way random effect ANCOVA model. One or more group-level predictor
variables, which can be indicator variables or quantitative variables, can be added
to Equation 3.12 to explain some of the variability in the group means. Mixed linear
model programs are required to obtain hypothesis tests and confidence intervals
in these more general random factor models.
Assumptions
Hypothesis tests and confidence interval for the parameters of a CPM assume the
n participants have been randomly sampled from some population (the random
sampling assumption) and the responses from one participant are uncorrelated
with the responses of any other participant (the independence assumption). The r
responses from any single participant are not required to be uncorrelated, but any
structured prediction error covariance matrix in a CPM (e.g., compound
symmetric, first-order autoregressive, Toeplitz) specified by the researcher must
closely approximate the true variances and covariances among the r responses. The
first-order autoregressive and Toeplitz covariance structures that could be used in
a CPM assume equally or nearly equally spaced time points in longitudinal and
pretest-posttest designs. If the sample size is large enough, specifying an
unstructured prediction error covariance matrix is usually recommended. The
prediction errors are assumed to have an approximate normal distribution in the
population (the prediction error normality assumption). The prediction error
normality assumption is usually not a concern if the number of participants (n) is
greater than about 20.
The random sampling assumption, the independence assumption, and the
prediction error normality assumption are also required in a MLM. The composite
prediction error normality assumption is usually not a concern if the number of
participants (n) is greater than about 20. The variances of the random coefficients
are additional parameters in a MLM. Hypothesis tests and confidence intervals for
the random coefficient variances assume that the coefficient values in the
population have an approximate normal distribution (the random coefficient
normality assumption). Hypothesis tests and confidence intervals for the random
coefficient variances are very sensitive to minor violations of the random
coefficient normality assumption and increasing the sample size will not mitigate
the problem. Specifically, a confidence interval for a random coefficient variance
can have a true coverage probability that is far less than 1 โ ๐ผ if the distribution of
person level coefficient values in the population are leptokurtic, regardless of
sample size. The widths of the confidence intervals for the variances of the random
parameters depend primarily on the sample size (n) rather than r. A large sample
D.G. Bonett (8/2018)
25
size is usually needed to obtain a usefully narrow confidence interval for a random
coefficient variance. One of the advantages of the MLM over the CPM is the ability
to assess the variability of the random intercept and slopes but this advantage is
diminished given the random coefficient normality assumption of the MLM.
The assumptions for a random effects ANOVA model include all the assumptions
for the fixed-x GLM described in Module 2 in addition to several other important
assumptions. The consequences of violating the GLM assumptions also hold in the
random effects ANOVA model. In addition to the GLM assumptions, the random
effects ANOVA model assumes that the factor levels have been randomly selected
from a definable superpopulation of factor levels. If this assumption cannot be
justified, then the confidence interval for the superpopulation mean (Equation 3.7)
will be uninterpretable. The random effects ANOVA model also assumes that the
superpopulation distribution of means has an approximate normal distribution.
Violating this assumption is usually not a problem for Equation 3.7 if k (the number
of factor levels) greater than about 30. However, the confidence interval for ๐2 can
have a true coverage probability that is far less than 1 โ ๐ผ if the superpopulation
means are leptokurtic regardless of sample size. The parameter prediction errors
(๐ข๐) are assumed to be uncorrelated with the person-level prediction errors (๐๐๐).
Violating this assumption will introduce bias into the estimate of ๐2 regardless of
sample size.
In repeated measures studies where either a CPM or a MLM could be used, a CPM
with the least restrictive prediction error covariance matrix (e.g., heteroscedastic
Toeplitz or unstructured) should be used to estimate the composite model
parameters. If the sample size is sufficiently large and the random coefficient
normality assumption is plausible, a MLM could be used next to obtain confidence
intervals for the variances of the random coefficients. A large variance for any
random slope indicates the omission of important interaction effects and the need
for additional research to discover these interaction effects. When designing a
longitudinal or pretest-posttest, the researcher should plan to measure each
participant on the same set of equally or nearly equally spaced time points, so that
a CPM can be used, and obtain a sample size that is large enough to use an
unrestricted prediction error covariance matrix in the CMP. Using the smallest
number of time points needed to assess the effects of the time-varying predictor
variables will improve the small-sample performance of the confidence interval
and hypothesis tests when using an unrestricted prediction error covariance
structure.