using hierarchical linear models to measure growth · using hierarchical linear models to measure...
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Using Hierarchical Linear Models to Measure Growth
Measurement Incorporated Hierarchical Linear Models Workshop
Chapter 6Chapter 6MotivationLinear Growth CurvesLinear Growth CurvesQuadratic Growth CurvesSome other Growth CurvesSome other Growth CurvesCenteringHLM/Repeated Measures/SEMHLM/Repeated Measures/SEMPower
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MotivationMotivationHLM is useful when we have observations (Level-1) nested within a Level-2 variable.nested within a Level 2 variable.
Think students nested within classes.
Useful because HLM:Explains dependencies.C Computes relationship within each group.
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MotivationMotivationThese models can easily apply to repeated measures:
Individual Change/Growth Models.Individual Change/Growth Models.
We have repeated measures observed within a student.
Assuming that we use appropriate measures, HLM can provide an integrated approach for studying the can provide an integrated approach for studying the structure and indicators of individual growth.
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MotivationMotivationSo now we will have data like
Level-2 AnalysisBecause this is
HLM, we can also e e a ys s(e.g. Character level) have additional
covariates at each level
Level-1 Analysis(e.g. Repeated
Measures of Time)
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MotivationMotivationIn this context we can use time as a level-1 covariate.
This allows us to also look at change across time .
Or even growth rates at a specific time.
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ObjectiveObjectiveAfter this chapter you should be able to:
Fit a basic (2-level) HLM to repeated measures data.Linear growth curves.Quadratic growth curves.
Understand their basic interpretation.
Understand the importance of centering and its effects.
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ObjectiveObjectiveYou will also:
Understand the difference between HLM and other Understand the difference between HLM and other approaches.
Have a better feel for power considerations.
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The ModelThe ModelFor the model we will have a different notation.
Which is consistent with the notation introduced for the 3Which is consistent with the notation introduced for the 3-level analyses.
We will have observations of examinee i at time point t for some variable Yti :ti
Observations are nested within examinee.
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The ModelThe Model
For the repeated measures growth curve we will For the repeated measures growth curve we will have
a is usually time points
tiPtiPitiitiiiti eaaaY +π++π+π+π= L2
210
For the examinee levelQ
These are individual characteristics
∑=
+β+β=πpQ
qpiqipqppi rX
10
All of the assumptions still applyAll of the assumptions still apply
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Brief ExamplesBrief ExamplesStudents who have taken several tests (which are on the same scale)the same scale).
An athlete training for the olympics has repeated An athlete training for the olympics, has repeated measures of the time it takes on some event.
Individuals who are in some kind of counseling with repeated observations of behavior.repeated observations of behavior.
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Linear Growth ModelsLinear Growth ModelsWe begin with a basic linear growth model.
In this case, the rate of change is constant across timetime.
While a linear model may not be the true model it While a linear model may not be the true model it works well.
For short intervals of time.For short intervals of time.Few observations.
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Linear Growth ModelsLinear Growth Models
Time
titiiiti eaY +π+π= 10Repeated measures
E
Q
Error
What is this?
∑=
+β+β=πpQ
qoiqioqoi rX
100
Level-2 and
∑=
+β+β=πpQ
qiqiqi rX
111101What is this?
errors (we let them correlate)
=q 1
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Linear Growth ModelsLinear Growth Models
Using this model we can:g
Estimate a mean growth slope (curve).
Determine the reliability of status and change.y g
Estimate relationship between initial status and rate of change.
Provide some general descriptive statistics.Provide some general descriptive statistics.
Model relations of person-level variables to status and growth rate.
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Working Example (p. 164)Working Example (p. 164)Study of children’s growth during preschool and early elementary grades.elementary grades.
Outcome measure is IRT examinee parameter value from pa test of natural science knowledge.
143 examinees in a Head Start program.
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DataDataSo our data looks like
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Working ExampleWorking Example
First let’s assume that we want to:
Estimate the average increase in ability per unit of time f h for the participants.
Determine the reliability of status and changeDetermine the reliability of status and change.
Estimate relationship between initial status and rate of change.
E i h i bili i d Estimate the variability across intercepts and rates.
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Random Coefficient ModelRandom Coefficient ModelThe first model that we deal with is the random coefficient modelcoefficient model
How would we know the How would we know the
titiiiti eaY +π+π= 10
average increase? variability?
ioi r000 +β=π
r+β=π
How would we know the relationship b/t starting
stress and rate?ii r1101 +β=π
What do these parameters mean for our problem?
How would we know the reliability?
stress and rate?
What do these parameters mean for our problem?
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HLM ProgramHLM Program
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HLM Model ParametersHLM Model ParametersThe following parameters are estimated:
Fixed intercept: β00Fixed intercept: β00 Fixed slope: β10 Variance of random intercept: τ00p 00
Variance of random slope: τ11Covariance of random intercept and slope: τ01Level-1 error variance: σ2
The following slides dissect the HLM output for each of The following slides dissect the HLM output for each of these terms.
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Fixed Effects EstimatesFixed Effects Estimates
Fixed intercept: β00 Fixed slope: β1100 p β11
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Variance Component EstimatesVariance Component Estimates
Variance of random intercept: τ
Level-1 error variance: σ2
Variance of random intercept: τ00
Variance of random slope: τ11
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Random Coefficient ModelRandom Coefficient ModelSo we see that this model is no different from what we have seen in the past.
We make a basic assumption that dependencies in our observations are due to the second level of our observations are due to the second level of analysis (people).
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Intercepts/Slopes as OutcomesIntercepts/Slopes as OutcomesNow we will use the model
eaY +π+π=Same withinperson level
titiiiti eaY +π+π= 10
W
pmodel
iWioi rX 00100 )( +β+β=π
iWii rX 111101 )( +β+β=π
Now our intercept and slope are modeled as a iii 111101 )(ββπmodeled as a
function of what they are wearing So just think about what we are saying
here…here…
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Intercepts/Slopes as OutcomesIntercepts/Slopes as OutcomesAfter looking at the results we have seen how a basic growth model could be useful.
In addition, I think they have a really nice interpretationinterpretation.
BUT, growth and change are not always linear., g g yLearningTrainingTestingTesting
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A Quadratic Growth CurveA Quadratic Growth CurveFor that reason we can expand our model (as was shown in the general case) shown in the general case) .
In expanding the model we allow for the growth rate In expanding the model we allow for the growth rate to change across time.
Do you remember what parameter was used to indicate our growth rate in the linear growth model?
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A Quadratic Growth CurveA Quadratic Growth Curve
So to expand our model to the quadratic we will use
20 1 2( ) ( )ti i i ti i ti tiY a L a L eπ π π= + − + − +
∑ ++=pQ
iqiqi rX1
00000 ββπExpected score at time L
=q 1
∑ ++=pQ
iqiqi rX 11101 ββπThe growth rate at
time L=q 1
Characterizes the acceleration over time ∑ ++=
pQ
iqiqi rX 22202 ββπ=q 1
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A Quadratic Growth CurveA Quadratic Growth CurveIntercept gives level at time L of the study.
The second coefficient gives the spontaneous growth rate at time L.
The third coefficient gives the acceleration at which the change is occurring.
This means that the growth rate actually depends on timeThis means that the growth rate actually depends on time.
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A Quadratic Growth CurveA Quadratic Growth Curve
To determine the growth rate at any given time we To determine the growth rate at any given time we use the first derivative of our level-1 model:
titiitiiiti eLaLaY +−π+−π+π= 2210 )()(
Level-1 Model
)(2 21 LaGrowthRate tiii −π+π=Growth Rate=1st
derivative withderivative with respect to time
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ExampleExampleSo now we go to the example that is nearly the same as is given in the book.
We have verbal scores that are recorded at different ages.
Level-1 variables:Level 1 variables:Verbal ability.Age (we will actually use age-12).
Level-2:Study.Sex.MomSpeak (or the log of this variable)MomSpeak (or the log of this variable).
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Data SetData Set
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Quadratic Growth CurveQuadratic Growth CurveThe nice thing about this example is that it gives us a feel about how to complete a basic analysis with a feel about how to complete a basic analysis with a quadratic growth curve.
I will follow the example and we can go over the results.
We will also discuss the interpretation of these We will also discuss the interpretation of these variables.
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Quadratic Growth CurveQuadratic Growth Curve
So the procedure is:p
Look at the data (ask: do we need quadratic?).
Fit random coefficients data.
Simplify (Do we need certain coefficients?)Fixed versus random effects.Intercepts versus notIntercepts versus not.Associations versus not between random effects.
Fit second level modelFit second level model.
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Graph DataGraph DataLook at the data.
In HLM this is also fairly easy.y
File…Graph Data…line plots, scatter plots.P bl XPut time variable on X-axis.Choose line plot.
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Graph Data ResultGraph Data Result
T d t
608 60
824.20
Trend seems to indicate some type of curved learning taking place.
393.00
608.60
VOC
AB
p
-0.70 3.15 7.00 10.85 14.70-38.20
177.40
AGE12
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Fit Level-1 AnalysisFit Level 1 AnalysisHere we need to determine our first level equation.
We have already determined that we need a Quadratic so our basic model will be a random coefficients model.
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The ModelThe Model
20 1 2( ) ( )ti i i ti i ti tiY a L a L eπ π π= + − + − +
ii r0000 += βπAgain, we want to think about what these parameters mean
ii r1101 += βπ
these parameters mean.
AlsoAlso,
•What does it mean if we find that the level-2 error terms are correlated
ii r2202 += βπ ii 2202 β
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HLM Model ParametersHLM Model ParametersThe following parameters are estimated:
Fixed intercept: β00Fixed intercept: β00 Fixed linear slope: β10 Fixed quadratic slope: β20 V i f d i Variance of random intercept: τ00Variance of random linear slope: τ11Variance of random quadratic slope: τ11Covariance of random intercept and linear slope: τ01Covariance of random intercept and quadratic slope: τ02Covariance of random linear slope and random quadratic slope: τ12Covariance of random linear slope and random quadratic slope: τ12Level-1 error variance: σ2
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SimplificationSimplificationThings to look at:
Fixed Coefficients.
Random Coefficients.
Associations.
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HLM InputHLM Input
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Variance Component EstimatesVariance Component Estimates
Level-1 error variance: σ2
Random Effect Covariance Matrix
Random Effect Correlation MatrixMatrix
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Fixed Effect EstimatesFixed Effect Estimates
Fixed intercept: β00 Fixed linear slope: β10ed te cept β00 ed ea s ope β10
Fixed quadratic slope: β20
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Other Growth CurvesOther Growth CurvesWe can always fit more terms:
We could have a polynomial with up to P terms as long as We could have a polynomial with up to P terms as long as we have (P+1) observations with in several observations.
W l f d d i blWe can always transform our dependent variable.
The average level 1 equation can look different from The average level-1 equation can look different from the model at level-1.
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Complex Level-1 ErrorsComplex Level 1 ErrorsAlso, if need be, we may be interested in:
Specifying level-1 variance as a function of person variables.
Specifying level-1 variance as a function of time variables.
We can also allow the error terms within a person to We can also allow the error terms within a person to correlate (AR(1)).
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Piecewise Linear Growth ModelsPiecewise Linear Growth ModelsWhile a quadratic growth curve may work just fine to describe the data, there is a alternative.
There may be instances when you do not like assuming that the growth rate is constantly changingthe growth rate is constantly changing.
This is also useful when we want to compare growth rates between two different periods.
As an alternative we can fit a nonlinear trend with two linesAs an alternative, we can fit a nonlinear trend with two lines.
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Piecewise Linear Growth ModelsPiecewise Linear Growth ModelsTo do this we get to define a few new variables.
Depending on how we define the variables we will have two different interpretationshave two different interpretations.
To do this I first present the table on page 179To do this I first present the table on page 179.
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RecodingRecoding
These are for two
differentdifferent slopes
This is for the
incremental h ichange in
the slope between the two periods
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ModelsModels
Once we have our new variables we use the model:
raaY +π+π+π= ijtiitiiiti raaY +π+π+π= 22110
Of course, we can model each level-1 coefficient as a function of level-2 coefficients.
How are these parameters interpreted?
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Time-Varying CovariatesTime Varying CovariatesOne other thing to notice is that by modeling responses using an HLM we can easily have time responses using an HLM we can easily have time varying covariates.
For example:Weather conditions.Amount of sleep on the night prior to a test.
These could also be fixed or random effects.
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CenteringCenteringNext we see how centering can have an effect on growth curves.curves.
Specifically, we will look at:p y,Centering in linear growth curves.Centering in quadratic growth curves.Bias of studying time-varying predictors.Variance growth parameters.
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Centering (Linear Growth)Centering (Linear Growth)As you may expect centering is all the same here.
How we interpret the intercept is determined by where we center.
Here, we are more likely to center based on a ytheoretically interesting point.
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Centering (Quadratic Growth)Centering (Quadratic Growth)Now in a quad growth curve things are going to be slightly differentslightly different.
Remember that the intercept π is equal to that Remember that the intercept π0i is equal to that point at which our variable equals 0.
Also π1i is the instantaneous growth rate at that same time.
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Centering (Quadratic Growth)Centering (Quadratic Growth)
This means that if we use the start time at 0 then:
The intercept is the starting value.
π1i is the growth rate at the initial time individuals start.
f There could be little information at that point.
This may also cause high correlations between This may also cause high correlations between these two values.
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Centering (Quadratic Growth)Centering (Quadratic Growth)Centering at “the middle”
Intercept is ability at the middle of study.
π1i is the growth rate at the middle.This is also the average growth rate.
More information about growth rate.
Minimizes correlation between random effects.
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Bias for Time-Varying CovariatesBias for Time Varying Covariates
In addition Time-varying covariates may have biased In addition, Time-varying covariates may have biased slopes because of compositional effects (covered in detail in Chapter 5).
The aggregate of our variable may mean something different than the individual observationsthan the individual observations.
Think about thing like a variable “has a job.”
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Variance of Growth ParmsVariance of Growth ParmsCentering can have an effect on variance, because of the change in interpretation of each parameter.
If have equal time points across people centering will not change variancenot change variance.
If thinking of other covariates or time points that g pvary across people then it will matter.
Grand mean will shrink standard errors.Group mean will remove effect of the covariate differencesGroup mean will remove effect of the covariate differences.
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Growth Model Extensions
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HLM, MRM, and SEMHLM, MRM, and SEMNow we provide a discussion about the comparisons of three different methods to deal with repeated measures:p
Hierarchical Linear Models
Multivariate-Repeated Measures
Structural Equation Models
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HLMHLMWe begin by the concept of HLM.
Repeated measures are conceptualized as a person’s trajectory that changes as a function of person specific j y g p pvariables.
Level-2 variables describe some of the variation across people.
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MRMMRMIn MRM the repeated measures are treated as a response vector from a person.
We define our groups:These can be defined by a combination of main effects and interactionsThese can be defined by a combination of main effects and interactions.
Then we have to define the variation across time.
Then we compare across groups:This is just like a MANOVA with contrasts or profile analysis.This is just like a MANOVA with contrasts or profile analysis.
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Limitations of MRMLimitations of MRMFor the most part MRM requires that all time periods have been observed for all people:p p
This is not required by HLM.
MRM d t t d t it ti h th l MRM does not extend to a situations where the people (level-2) are nested inside of a level-3.
HLM does this naturally.
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SEMSEMIn general, there are a number of growth curves that can be phrased as a structural equation modelcan be phrased as a structural equation model.
In these cases:In these cases:The SEM measurement part is the level-1 analysis.The latent variables are the growth parameters.g pThe structural part then corresponds to the between-person part of the model.
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SEMSEMOne major advantage of SEM is that:
It can provide numerous correlation structures at either level.
You will have any number of fit indices that generally are not supplied for HLM.
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SEMSEMThe disadvantage is that:
It requires that each examinee has the same spacing between time points.
If l h h ld d If we can conceptualize a situation where this would occur and we simply have missing data we can still use SEM.So there can be some variation in time points.
It requires that level-1 predictors have the same distributions across all participants in the same subgroup.
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HLM and Covariance Structures HLM and Covariance Structures So just quickly…
What happens whenever we do not assume that our ppobservations within a level-2 unit are independent?
While something like this would be easy in SEMWhile something like this would be easy in SEM…In HLM you would need to do something with a MHLM (which I will not talk about now)
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Just a little BackgroundJust a little Background
Notice that HLM already models dependency among our b ( k h f h h k observations (to take this a step farther we are thinking
about how HLM will model the variance and covariance structure of our data)
Think of a model with a random intercept (homogeneous variance):Uses 2 parameters.
Think of a model with a random intercept and slope (homogeneous variance):
Uses 4 parameters.
Think of a model with a random intercept and slope (heterogeneous variances)
Uses more parameters depends on the number of time pointsUses more parameters…depends on the number of time points
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Just a little BackgroundJust a little BackgroundThere are times when this is not enough.
Or the parameterization is not rightOr the parameterization is not right.
Think in terms of…what happens if within a level-2 unit the observations have a certain correlation structure.
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HLM with level-1 Cov StructuresHLM with level 1 Cov StructuresProbably the most common is an autoregressive correlation matrixcorrelation matrix.
This assumes that an observed value at time t actually This assumes that an observed value at time t actually depends on the value at time (t-1).
In the end this means that the correlation between observations given group is something other than 0.
And can be characterized with a single parameterAnd can be characterized with a single parameter.
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HLM and CovarianceHLM and CovarianceSo what we can see is that HLM is generally thought as having independent errors within the level-2 units.g p
We are able to incorporate some structure.p
Not as well as SEM, so there is a trade off.SEM has fit indices and more options with structure.HLM handles “missing data” better in certain situations.
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One Other PointOne Other Point
Again we get back to predicting future slopes for a g g p g pperson:
OLSWorks fine in situations where the coefficients are highly reliablereliable.
EBWill give better estimates in data with a lot of “noise”.We have shared information.
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Power ConsiderationsPower ConsiderationsThe same hold as before:
That is if we can add more peopleThat is, if we can, add more people.If we can’t, add more observations.
With observations we can either:Lengthen the study.Add more time points within a study.
Software such as Optimal Design or PINT could be used Software such as Optimal Design or PINT could be used to add in Power computations
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Chapter 6 SummaryChapter 6 SummaryLinear and quadratic growth curves.
These are the same as any other 2-level HLM.
Also, talked about interpretation on the parameters.
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Chapter 6 SummaryChapter 6 SummaryPiecewise linear growth models:
Said by recoding we can explore the difference between two Said by recoding we can explore the difference between two time periods.
Time-Varying CovariatesCan always add in time varying covariates.
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Chapter 6 SummaryChapter 6 SummaryCentering Effects
Has the same effects as beforeHas the same effects as before.Covered effects of center on quadratic models.
Comparison of HLM to MRM and SEMModels differ only slightly.SEM has added advantage of modeling association.HLM has added advantage of being flexible with time points.
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Chapter 6 SummaryChapter 6 SummaryFuture Prediction:
In predicting future growth curves we can use OLS if reliableIn predicting future growth curves we can use OLS if reliable.Otherwise use EB for reduced standard errors.
Power:Same basic idea as before only now we had to consider length f t dof study.
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