sem slides1
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Structural Equation Modeling (Hybrid Models)
What is it? General method for modeling = () i.e. for modeling covariance
structure
Intuitively can be thought of as the combination of confirmatory factor
analysis (CFA) with path analysis.
CFA and path analysis are each special cases of structural equation
modeling
What is it good for?
To account for measurement error in modeling relationships between
variables measured with error (i.e. latent variables)
As it uses the graphical notation of path analysis, it provides a methodfor describing the assumed causal relationships between observed vari-
ables, between observed and latent, and between latent and latent.
To take advantage of multicollinearity in a set of predictors rather than
seeing it as a hinderance.
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SEM - becoming ubiquitousSTRUCTURAL EQUATION MODELING, 10(1), 3546
Copyright 2003, Lawrence Erlbaum Associates, Inc.
The Growth of Structural EquationModeling: 19942001
Scott L. HershbergerDepartment of Psychology
California State University, Long Beach
This study examines the growth and development of structural equation modeling
(SEM) from the years 1994 to 2001. The synchronous development and growth of
the Structural Equation Modeling journal was also examined. Abstracts located on
PsycINFO were used as the primary source of data. The major results of this investi-
gation were clear: (a) The number of journal articles concerned with SEM increased;(b) the number of journals publishing these articles increased; (c) SEM acquired he-
gemony among multivariate techniques; and (d) Structural Equation Modeling be-
came the primary source of publication for technical developments in SEM.
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SEM - becoming ubiquitous
FIGURE 1 Distribution of number of articles and journals by year.
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SEM - becoming ubiquitous
Over 1100 Selected Publications that Cite Amos for Structural Equation Mod-eling March, 2004, http://www.amosdevelopment.com/
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Effect of ignoring measurement errorLet x1 = f1 + 1 and x2 = f2 + 2
where V ar(1) = 1, V ar(2) = 2, V ar(f1) = 1, V ar(f2) = 2, and Corr(f1, f2) =
. That is, the true correlation between the variables of interest f1 and f2 is .
Now, as we did not observe f1 and f2 directly we will instead have to deal with the
observed x1 and x2.
What is the Corr(x1, x2)? Is it close to the Corr(f1, f2) = ?
Corr(x1, x2) =Cov(x1, x2)
(V arx1) (V arx2)
= Cov(f1, f2)
1 + 1
2 + 2
=Corr(f1, f2)
1
2
1 + 1
2 + 2
= 1
1 + 1
2
2 + 2
= reliability of x1 reliability of x2
Correlation between x1 and x2 will be smaller than the true correlation between thevariables we are interested in f1 and f2
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SEM takes the measurement error into account
Rather than taking scales with less than perfect reliability and using them as ifthey are perfect measurements of the latent variable, SEM models incorporates
the measurement error and thus adjusts the correlations and path coefficients
appropriately. Assuming the model specification is correct (as usual).
Two nice papers discussing this:
Charles EP (2005) The Correction for Attenuation Due to Measurement
Error: Clarifying Concepts and Creating Confidence Sets, Psychological
Methods 10(2) 206-226.
DeShon, R. P. (1998). A cautionary note on measurement error corrections
in structural equation models. Psychological M ethods, 3(4), 412-423.
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Examples of correction for attenuation
observed correlation of .3
.2 .4 .6 .8 1.0
.2 - - .87 .75 .67
.4 - .75 .61 .53 .47
.6 .87 .61 .50 .43 .39
.8 .75 .53 .43 .38 .33
1.0 .67 .47 .39 .33 .30
observed correlation of .5
.2 .4 .6 .8 1.0
.2 - - - - -
.4 - - - .88 .79
.6 - - .83 .72 .65
.8 - .88 .72 .63 .56
1.0 - .79 .65 .56 .50
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A useful example from CFA
f1
x3 e1.93
x5 e2.62
x10 e3
.85
f2
x1 e4
.92
x7 e5.58
x8 e6
.58
x9 e7
.57
.54
Chi-square = 9.8 d.f. = 13, p-value = .704, Corr(f1, f2) = .54
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A useful example from CFA
A natural/practical thing to do with these 7 variables is to create two scales. One
created from X3, X5, X10, that is Scale1 = X3 + X5 + X10, and one created
from X1, X7, X8, and X9, that is Scale2 = X1 + X7 + X8 + X9
Then we can calculate the observed correlation between Scale1 and Scale2 is .45.Obviously this is smaller than the correlation found between the factors usingCFA (i.e. SEM).
Note that the Cronbachs alpha for Scale 1 is 0.827 and for Scale 2 is 0.751.
Might consider fixing up the correlation between the scales by their estimated
reliabilities. That is, rewriting derivation from two pages ago we have that (also
page 197 of Kline)
=Corr(x1, x2)
reliability ofx1reliability of x2 So can calculate .45
.827.751
= .571
Notice that it overadjusted, this estimate is actually larger than the true cor-relation of .54. This may be expected since Cronbachs alpha underestimates
reliability when factor loadings are not equal.
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Using a single indicator of a latent factor to adjust forunreliability
scale1 scale2
.45
4.89
scale1
13.84
scale2
3.68
The variance of scale 1 is 4.89, the variance of scale2 is 13.84.
Notice that the simple correlation between the scales is .45 which is smaller
than .54.
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Using a single indicator of a latent factor to adjust forunreliability
To adjust for the unreliability, fix the variance of the error terms to be equal to
Variance of scale time (1-reliability). Here the Cronbachs alpha for scale 1 is
.827 and for scale 2 is .751.
f1f2
scale1
4.89*(1-.827)
e1
1
1
scale2
13.84*(1-.751)
e2
1
1
f1 f2
scale1
.84597
e1
1
1
scale2
3.44616
e2
1
1
f1f2
scale1
e1
.42
scale2
e2
.50
.57
Notice the correlation now has been adjusted for the unreliability
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Example of Structural equation modeling
From Neumark-Sztainer D, Wall MM, Story M, Perry C (2003) Correlates of
unhealthy weight-control behaviors among adolescents: Implications for pre-
vention programs, Health Psychology, 22(1), 88-98.
Figure 1. Proposed model: Correlates of unhealthy weight-control behaviors among adolescents.
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Example of SEM - Measuring the latent variables
Table 2
Results From Confirmatory Factor Analysis Including Standardized Factor Loadings and
Correlation Between the Factors
Model and factors
Factor
loading Correlation between factors
Personal measurement model 1 2 31. Weightbody concerns .73 .05
Weight concerns .68Weight importance .54
Body dissatisfaction .752. Psychological well-being .73 .22
Self-esteem .91
Depressive mood
.563. Healthnutrition attitudes .05 .22
Concern about health .77Perceived benefits of healthy eating .34
Socioenvironmental measurement model 4 5 64. Familypeer weight norms .26 .06
Parental concernsbehaviors .71Peer dieting .395. Weight teasing .26 .21
Frequency of teasing .83Source of teasing .72
6. Family connectedness .06 .21 Family communication .85Atmosphere at family meals .55
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Could create scales and do Path Analysis - Ignoring mea-surement error
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Instead, use full SEM - Incorporate CFA into the PathAnalysis - thus accounting for measurement error
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Final results of SEM
Figure 4. Final model testing among adolescent girls: Correlates of unhealthy weight-control behaviors.
BMI body mass index. * p .01. Figure 5. Final model testing among adolescent boys: Correlates of unhealthy weight-control behaviors.BMI body mass index. * p .01.
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Common to use 2-step approach to SEM
1. Develop measurement model (CFA) relating observed variables to latentvariables. Examine goodness of fit of this model on its own. Examine
correlations between all variables (usually latent variables) of interest by
looking at correlations between factors from CFA.
2. Develop full structural equation model. That is, change the spuriously
correlated relationships in the CFA to impose theoretical causal direct
effects between variables and drop relationships not assumed by theory.
Examine goodness of fit of this model as a whole.Common reference advocating this approach is
Anderson, J.C. and Gerbing, D.W. (1988) Psychological Bulletin
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Comparing Multiple Regression and SEMSTATISTICS IN MEDICINE
Statist. Med. 2003;22
:36713685 (DOI: 10.1002/sim.1588)
TUTORIAL IN BIOSTATISTICS
Comparison of multiple regression to two latent variabletechniques for estimation and prediction
Melanie M. Wall1;;; and Ruifeng Li2;
1 Division of Biostatistics; School of Public Health; University of Minnesota; Minneapolis; MN 55455; U.S.A.2 Department of Epidemiology; School of Public Health; Harvard University; U.S.A.
SUMMARY
In the areas of epidemiology, psychology, sociology, and other social and behavioural sciences, re-searchers often encounter situations where there are not only many variables contributing to a particular
phenomenon, but there are also strong relationships among many of the predictor variables of interest.By using the traditional multiple regression on all the predictor variables, it is possible to have problemswith interpretation and multicollinearity. As an alternative to multiple regression, we explore the useof a latent variable model that can address the relationship among the predictor variables. We considertwo dierent methods for estimation and prediction for this model: one that uses multiple regression onfactor score estimates and the other that uses structural equation modelling. The rst method uses mul-tiple regression but on a set of predicted underlying factors (i.e. factor scores), and the second methodis a full-information maximum-likelihood technique that incorporates the complete covariance structure
of the data. In this tutorial, we will explain the model and each estimation method, including how tocarry out prediction. A data example will be used for demonstration, where respiratory disease deathrates by county in Minnesota are predicted by ve county-level census variables. A simulation studyis performed to evaluate the eciency of prediction using the two latent variable modelling techniquescompared to multiple regression. Copyright ? 2003 John Wiley & Sons, Ltd.
KEY WORDS: multiple regression; factor analysis; structural equation modelling; respiratory disease
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Data Source - MN county example
Minnesota county-level census death record data from 1990 to 1998
Outcome: Log of age-adjusted respiratory disease death rate
Observed Predictors: Five census variables on the county-level
Goal: establish the relation of predictors with outcome for interpretationand prediction
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FIVE PREDICTORS (all on the county-level)- MN countyexample
eduhs: percent with high school education
medhhin: median households income (in dollars)
percapit: per capita income (in dollars)
pubwater: percent of households with access to public water
wood: percent of households using wood to heat the home
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Multiple Regression -MN county example
.00
educhs
.33
m edhhinc
.04
percapin
.04
pubwater
.01
wood
respm ort
1.29
.11
-.02
.07
.53
.03
e11
.03
.01
.00
.00
.11
.05
-.02.02
-.01
-.02
resp = 0 + 1eduhs + 2medhhin + 3percapit + 4pubwater + 5wood +
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Tool in AMOS to draw many covariance arrows
Click on each variable in the set which will be correlated with each other
Click on Tools then Macros then Draw Covariance
This will then draw all the desired double headed arrows.
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Multiple Regression-MN county example
Examining unstandardized estimates. The coefficients are scaled up by 10 (for
the percents) and 1000 for the dollar amounts compared to numbers in the
original paper because units of raw data are scaled down.
Regression W eights
Estimate S.E. C.R. P
respm ort
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Multiple Regression-MN county example
educhs
m edhhinc
percapin
pubwater
wood
.23
respm ort
.58
.10
-.02
.20
.14
e1
.84
.86
.36
-.30
.93
.43
-.43.53
-.47
-.87
Interpretation Problem: highly correlated - multicollinearity
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Multiple Regression-MN county example
Sample Correlations - Estimates
educhs medhhinc percapin pubwater wood respm ort
educhs 1.000000 0.841414 0.859896 0.355381 -0.304913 0.157416
medhhinc 0.841414 1.000000 0.925435 0.433244 -0.425844 0.102968
percapin 0.859896 0.925435 1.000000 0.534107 -0.469779 0.074351pubwater 0.355381 0.433244 0.534107 1.000000 -0.874654 -0.276009
wood -0.304913 -0.425844 -0.469779 -0.874654 1.000000 0.369008respm ort 0.157416 0.102968 0.074351 -0.276009 0.369008 1.000000
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Consider latent variables as explanation for correlation-MN county example
Hence we consider the following latent variable model that takes into account the existenceof the two latent factors ruralness and SES:
eduhs= 10 + 11SES + u1
medhhin = 20 + 21SES + u2
percapit = SES + u3
pubwater = 30 + 32ruralness + u4
wood = ruralness + u5
RESP= 0 + 1SES + 2ruralness +
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Structural equation model-MN county example
respm ort
.03
e11
.04
ses
percapin.00
e21.001
m edhhinc
.03
e32.761
.03
access toutilities
wood
.00
e4-.49
1
pubwater
.01
e5
1.001
educhs.00
e6 .261
.02-.63
.40
Chi-square = 21.1 d.f = 7, ratio = 3.0
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Structural equation model-MN county example
Regression W eightsEstimate S.E. C.R. P
percapin
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Structural equation model-MN county example
.23
respm ort
e1
ses
.96percapine2
.98
.89
m edhhince3.95
access toutilities
.90
woode4-.95
.85
pubwatere5.92
.77educhse6 .88
.51-.54
.37
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Results from paper where 2nd latent variable codeddifferently
Table IV. Estimated coecients incorporating the latent variable model.
Regression on factor scores SEM-FIML
Parameter Estimate (s.e.) P-value Estimate (s.e.) P-value
0: intercept
7:68 (0.14) 0:0001
7:85 (0.16) 0:00011: SES 0.00003 (0.00001) 0.0161 0.00004 (0.00001) 0.00102: ruralness 0.010 (0.003) 0.0006 0.013 (0.003) 0.0001
R2 0.14 0.21
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f S
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Explanation of how SEM might predict better than Mul-tiple regression
How is it possible for the SEM-FIML technique to beat ordinary least squares in terms ofprediction? It is well known that E(Y|X) is the best mean square predictor of Y.Although in general the form of E(Y|X) is unknown, when (Y;X) is jointly normal, withE(Y;X) = (Y; \X), Var(X) =XX and Cov(Y;X) =YX, then E(Y|X) =Y+YX
1XX
(X\X).
The best predictor given a particular data set is then equal to E(Y|X
), with the maximum-likelihood estimates plugged in for \Y, \X, YX and XX. When nothing is assumed aboutthe p(p + 1)=2 unique elements of the symmetric matrix XX, the maximum-likelihood esti-mator forXX is simply the sample covariance matrix ofX, i.e. every element is estimatedindependently, and the E(Y|X) with maximum-likelihood estimators plugged in yields the
OLS predictor. However, if we have some model for the elements ofXX
as is the casein the latent variable model where XX is a function of fewer parameters than p(p + 1)=2,and these parameters also appear in YX, then maximum-likelihood estimators based on themodelled XX and YX plugged into E(Y|X) such as (11) should be best with respect tomean squared prediction error. Simply put, if the SEM model () is a good model for,
then
YX()1
XX() is more ecient than the OLS estimator
SxyS1
xx for estimating
YX1
XX.Furthermore, we point out that like the ordinary least-squares regression predictor, thepredictor using factor score estimates (9) is also a linear predictor (i.e. a linear functionof Y). On the other hand, the SEM-FIML predictor (11) is not linear since the parameterestimators are non-linear functions of both the Y and X variables. This may help to further
explain how it performs more eciently than the other methods.
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S l i d l i ifi d
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Structural equation model - misspecified measurementmodel
.19
respm ort
e1ses
1.05
percapine21.03
.81
m edhhince3
.90
access toutilities
.84
woode4
-.92
.90
pubwatere5
.95
.15
educhse6
.56
-.51
.35
.39
Chi-square = 125.2 d.f = 7, ratio = 17.9
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C d f SEM i M l
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Code for SEM in MplusHere is code for fitting the MN county data SEM in Mplus
data: file is mncountycensus.txt;
variable: names are eduhs medhhin percapit pubwater wood resp;
usevariables are eduhs medhhin percapit pubwater resp wood1;
define: wood1 = 1-wood;
analysis: Type = general;
model:
ses by eduhs medhhin percapit;
ruralness by pubwater wood1;
resp on ses ruralness;
output: standardized sampstat;
As before the by command is used to describe the indicators of new latent variables (in this case ses and ruralness).
The on command is used to create the path analysis (structural) part of the model in theis case resp on ses
ruralness, note that either observed or latent variables can be included in an on command.
Note, in this code the variable wood has been recoded (and renamed wood1) so that it is represents the percent
of households that do NOT use wood to heat their home. There were optimization problems in Mplus when this
variable was coded the other direction. Note that results in a change in sign for the loading of wood as compared
to the previous results.
The Define: command is used in Mplus to create new variable. It is necessary to put the new variable name on
the usevariables are command, and it is necessary that this new varname comes at the end of the list.
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R lt f SEM i M l
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Results for SEM in MplusTHE MODEL ESTIMATION TERMINATED NORMALLY
MODEL RESULTS
Estimates S.E. Est./S.E. Std StdYX
SES BY
EDUHS 1.000 0.000 0.000 0.050 0.879 R-SQUARE
MEDHHIN 10.787 0.759 14.217 0.541 0.945
PERCAPIT 3.902 0.255 15.312 0.196 0.980 Observed
Variable R-squareRURALNESS BY
PUBWATER 1.000 0.000 0.000 0.181 0.923 EDUHS .772
WOOD1 0.489 0.044 11.244 0.089 0.947 MEDHHIN .893
PERCAPIT .961
RESP ON PUBWATER .853
SES 1.554 0.488 3.182 0.078 0.373 RESP .227
UTILITY -0.629 0.138 -4.549 -0.114 -0.545 WOOD1 .897
RURALNESS WITH
SES 0.005 0.001 3.905 0.513 0.513
Variances
SES 0.003 0.000 5.192 1.000 1.000
UTILITY 0.033 0.006 5.288 1.000 1.000
Residual Variances
EDUHS 0.001 0.000 5.822 0.001 0.228
MEDHHIN 0.035 0.009 4.083 0.035 0.107
PERCAPIT 0.002 0.001 1.728 0.002 0.039
PUBWATER 0.006 0.002 2.324 0.006 0.147
RESP 0.034 0.005 6.388 0.034 0.773
WOOD1 0.001 0.001 1.600 0.001 0.103
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E i i M d t (I t ti ) ff t
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Examining Moderator (Interaction) effectsThere are basically two general methods for examining moderator effects:
1. Stratify the data into different levels of the moderator and then examine the
relationship between the predictor and the outcome in each of the strata. If the
relationship between the predictor and outcome is different across the different strata,
then it can be said there is a moderator effect, if the relationships are not significantlydifferent, then there is not a moderator effect.
2. Create a new variable which is the cross-product between the predictor and the
moderator. Include this interaction term directly into the path model.
If the moderator and predictor variable are observed, then method 1 or 2 is straightforward to implement. For
Method 1, if the moderator is continuous, some decision would be necessary for how to stratify the moderator
(maybe split at the median, or else create several equally spaced cut-offs). For Method 2, a cross-product is formed
(not if one of the variables is categorical, then separate cross products with dummy variables representing the
different groups is necessary) and included. The Define: command can be used to create new cross-product
variables.
If the moderator is observed and the predictor is latent, method 1 can be implemented in AMOS and other basic
SEM software (LISREL, Proc CALIS). Method 2 can be implemented in Mplus 4 and beyond using the special
xwith command.
If either the moderator is latent or both the moderator and predictor are latent then method 1 could not actually be
done, since it would not be possible to stratify the data on the latent variable. Method 2 can be can be implemented
in Mplus 4 and beyond using the special xwith command.
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E i i M d t (I t ti ) ff t
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Examining Moderator (Interaction) effects
NOTE: When considering a moderator of the relationship between a predictor
and an outcome, it is the case that the predictor also moderates the relationship
between the moderator and the outcome. The two variables moderate each
others relationships with the outcome.
For more on conceptualizing moderators (and mediators), see e.g., Petrosino(2000) Mediators and moderators in the evaluation of programs for children.
Current Practice and Agenda for Improvement. Evaluation Review, 24(1) 47-
72.
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Interaction between SES and ruralness?
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Interaction between SES and ruralness?data: file is mncountycensus.txt;
variable: names are eduhs medhhin percapit pubwater wood resp;usevariables are eduhs medhhin percapit pubwater resp wood1;
define: wood1 = 1-wood;
analysis: Type = random;
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Interaction between SES and ruralness?*** WARNING in Output command
STANDARDIZED option is not available for analysis with
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Latent interaction models - nonlinear latent variablesOnce the model includes a nonlinear function of a latent variables, the traditional
methods for estimating SEM (which are based on modeling the observed covariancematrix S) are not useful. The traditional methods only apply to linear structural
models, NOTE, the well-known term LISREL stands for Linear structural relations.
During the past decade, much work has been done to develop methods for estimating
nonlinear structural relations. Mplus implements one such method which allows the
direct fitting of products of latent variables in the structural part of the model. Note,
quadratic terms can also be created by taking a latent variable xwithed with itself,
e.g. sesquad
|ses xwith ses; would create a latent quadratic ses term.
In Mplus, the estimation method is directly fitting the latent interaction using full
maximum likelihood (via the EM algorithm) with the nonlinear structural model di-
rectly included. Full maximum likelihood can also now be done using SAS PROC
NLMIXED and can similarly be implemented in Winbugs (within a Bayesian frame-work).
See Wall M.M. Maximum likelihood and Bayesian estimation for nonlinear structural
equation models using SAS, Mplus, and Winbugs Research Report 2007-021, Division
of Biostatistics, University of Minnesota, 2007.
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Latent interaction models - nonlinear latent variables
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Latent interaction models - nonlinear latent variablesThe one drawback of the full maximum likelihood or fully Bayesian method is that
it make distributional assumptions about the latent variables. Other methods aredeveloped (although not implemented easily in existing software) that do not require
strong distributional assumptions on the latent variables, see
Wall, M.M. and Amemiya, Y, (2000) Estimation for polynomial structural equation models. JASA, 95,
929-940.
Wall, M.M. and Amemiya, Y, (2001) Generalized appended product indicator procedure for nonlinear struc-
tural equation analysis. Journal of Educational and Behavioral Statistics, 26, 1-29.
Wall, M.M. and Amemiya, Y, (2003) A method of moments technique for fitting interaction effects in struc-
tural equation models, British Journal of Mathematical and Statistical Psychology, 56, 47-64.
Wall M.M. and Amemiya, Y. (2007) A review of nonlinear factor analysis and nonlinear structural equation
modeling In Factor Analysis at 100: Historical Developments and Future Directions, eds. Robert Cudeck and RobertC. MacCallum, Chapter 16 pp 337-362, Lawrence Erlbaum Associates.
Wall M.M. and Amemiya, Y. (2007) Nonlinear structural equation modeling as a statistical method In
Handbook of Latent Variable and related Models, ed Sik-Yum Lee, Chapter 15, 321-344, Elsevier, The Netherlands.
and references therein.
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Multilevel modeling
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Multilevel modeling
Data collection involves: patients within clinics, students within classrooms,
employees within units, repeated measures within patient (i.e. longitudinal
data). In each case the grouping or clustering variable is: clinics, classrooms,
units, patient.from Heck (2001) Multilevel Modeling in SEM, Chapter 4, New Developments and Techniques in SEM, eds Marcoulides
and Schumacker, 89-127
Ignoring the presence of substantial similarities among individuals within groups
can result in substantially biased estimates of the models parameters, standard
errors, and fit indexes.
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Multilevel modeling - Intraclass correlation
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Multilevel modeling Intraclass correlationThe intraclass correlation describes the degree of correspondence within clusters or
groups and can be expressed as:
=2b
2b
+ 2w
where 2b is the variability between groups and 2
w is the within-group varibility.
Thus indicates the proportion of the total variability that can be attributed to
variability between the groups. The should be zero when the data are independent
- thus, its magnitude depends on characteristics of the variable measured and theattributes of the groups. The larger the intraclass correlation, the larger the distortion
in parameter estimation that results from ignoring this similarity.
Note it is typically assumed that different groups are independent of one another.
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