Structural Equation Models for Comparing Dependent
Means and Proportions
Jason T. Newsom
How to Do a Paired t-test with Structural
Equation Modeling
Jason T. Newsom
Overview
• Rationale
• Structural equation model equivalents of conventional
tests
• Continuous
• Binary variables
• Multiple indicators
• Multiple time points
• Mixed factorial tests
Why Use SEM to Compare Repeated Means or
Proportions?
• Many longitudinal designs only have a few time points
(e.g., pre, post, follow-up)
• Mean or proportion change hypotheses common
• Understand underpinnings of more complex longitudinal
models and how they relate to ANOVA
• Statistical advantages
• Lots of interesting extensions
Statistical Advantages
• Robust estimation
• Same general models for binary, ordinal, other discrete
variables
• Account for measure error with multiple indicators
• Ensure measurement invariance
• Include correlated error structures
• Convenient missing data estimation
• Easily linked to random effects ANOVA
SEM
Regression
ANOVA
t-test
Several Equivalent SEMs to Compare Dependent
Means
• Single difference score
2003)
(e.g., Rovine & Liu, 2012; Rovine & Molenaar,
• Mean equality constraint (invariance) test
• Growth curve specification (Newsom, 2002; Voelkle, 2007)
• Latent difference score approach (McArdle, 2001)
Single Difference Score
• Consider test of single observed repeated measure
• Comparison of average difference to 0 is equal to paired
(dependent) t-test comparing two dependent means
y21     y2  y1   / N
t  y21 / SE y21
• Which is equivalent to repeated measures ANOVA
t2  F
Single Difference Score
• Equivalently, intercept-only regression
y2 1   0  
Single Difference Score
• Restated as SEM with mean structure
(Rovine & Molenaar, 2003)
y21  1   1
• Test of average difference in OLS (t-test) and ML (z-test)
are asymptotically equivalent as N  
• OLS model suggests simple SEM model for repeated
measures test
Single Difference Score
Equality Constraint
• Nested models compared
M 0 : 1   2
M 1 : 1   2
• Likelihood ratio test, where
 2   M2 0   M2 1
with df  df M 0  df M1
Equality Constraint
Contrast Coding Model
• ANOVA noted as special case of latent growth curve
model from the beginning (e.g.,McArdle & Epstein, 1987; Meredith &
Tisak, 1990)
• Growth specification for comparing two dependent means
or repeated measures (Newsom, 2002, 2015; Voelkle, 2007)
• Slope factor mean used to test mean difference
Contrast coding model
Contrast Coding Model
Two time points, omit i subscripts
y1  111   212
y2  121   222
If intercept loadings equal 1, slope loadings equal 0 and 1
E  y1   E 1 1  E 2  0 
E  y2   E 1 1  E 2 1
Contrast Coding Model
Simplifying,
E  y1   E 1 
E  y2   E 1   E 2 
Substitute and rearrange,
E 2   E  y2   E  y1   E  y2  y1 
or,
 2  E  y2   E  y1   E  y2  y1 
Contrast Coding Model
• Generally, growth curve model with common time codes
(0, 1, 2, ... T-1) gives baseline mean plus average
change (e.g., Bollen & Curran, 2006)
E  yti   1   2
where 1 is intercept mean and 2 is slope mean
Latent Difference Score Model
• McArdle’s (2001) latent difference score model also tests
mean difference
• Autoregressive model with constraints
• Additional factor used to capture repeated measures
difference
Latent Difference Score Model
• Autoregression model
y2   2   21 y1   2
• Set autoregression coefficient, 21, to 1, the structural
intercept, 2, to 0, the disturbance, 2, to 0, and add a new
difference score factor, 3, with loading equal to 1 and mean and
variance estimated
y2   2    21  y1   23 3   2
y2   0   1 y1  13   0 
3  y2  y1
E 3    3  E  y2  y1   E  y2   E  y1 
Latent Difference Score Model
Relation to ANOVA
• Use the contrast coding model to illustrate
Relation to ANOVA
• Because the general growth model can be stated as
E  yti   1   2
• It is the same as the repeated measures ANOVA model
(e.g., Winer, 1971)
Yti  m  p i  t t   ti
E Yti   m  t t
Here, m represents the grand mean, pi represents the average score deviation for
each case, tt represents the deviation of the mean at each time point from the grand
mean, and ti represents the error.
Relation to ANOVA
• With two time points, dfA = 1, so
MS A  SS A 
Y2  Y1 
2
2

 22
2
When 2 is the contrast factor mean with loadings set to
0 and 1
Relation to ANOVA
• In repeated measures ANOVA
(Winer, 1971),
Var  y1   Var  y2 
 Cov  y1 , y2 
2
• Using path tracing rules
• If constant slope factor variance assumed, fixed at 0,
then the intercept factor variance is equal to the
covariance of observed variables
MSerror 
Cov( y1 , y2 )  1121Var 1   1222Var  2 
Cov( y1 , y2 )  1 1  Var 1    1  1   0  
Cov( y1 , y2 )  Var 1 
Relation to ANOVA
• And because the variance of the measurement residual is
a function of the observed measures, loadings, and factor
variances
Var ( ti )  Var  yti   tk2 kk
• Assuming
• constant measurement residual variance, Var(),
• and loadings for the intercept factor are equal to 1,
2
so that tk  1,
• and Cov( y1 , y2 )  Var 1  from above
• Then
Var  y1   Var  y2 
Var ( ) 
 tk2Var  k 
2
Var  y1   Var  y2 
Var ( ) 
 Cov  y1 , y2   MSerror
2
Example
• Average of five items from the positive affect measure (self-rated
feelings of "happy," "enjoying yourself," "satisfied," "joyful," and
"pleased")
• Measured twice, 6 months apart
• N = 574
• Means:
y2  2.925
y1  3.034
y2  y1  .109
• Conventional repeated measures ANOVA and three model
specifications (single-variable difference score model,
equality constraint model, contrast coding model, latent
difference score model)
Example
Conventional tests: t (573) = -3.90, (-3.90)2 = 15.21, F(1,573) =
15.24
SEM Results
Mean
difference
Est
Single-variable
difference score
model
-.109
Equality
constraint model
NA
Parameter
test (z)
-3.907
LR Test,
2(1)
15.063
15.063
Contrast coding
model
-.109
-3.907
15.063
Latent difference
score model
-.109
-3.907
15.063
Binary Variables
• Conventional test of two dependent proportions
conditional logistic (e.g., Agresti, 2013) and McNemar’s chisquare (McNemar, 1947) among others
SEM
GLIM
Regression
Logistic
ANOVA
t-test
Chi-square
Binary Variables
Time 2
Time 1 Yes No
Yes
p11
p12
p1.
No
p21
p22
p2.
p.1
p.2
where p1. and p.1 are marginal probabilities for the first
row (Time 1 response) and column (Time 2 response),
respectively, and p12 and p21 are discordant (“yes”-”no”
and “no-yes”) cell probabilities
Binary Variables
Time 2
Time 1 Yes No
Yes
p11
p12
p1.
No
p21
p22
p2.
p.1
p.2
Binary Variables
Time 2
Time 1 Yes No
Yes
p11
p12
p1.
No
p21
p22
p2.
p.1
p.2
Binary Variables
• Common SEM estimation with DWLS or full ML
• Same models can be used for any discrete variables that
can be analyzed with SEM programs (e.g., binary, ordinal,
count, categorical, zero-inflated)
• Will focus on full ML to highlight equivalence to
conventional tests
Contrast Coding with Binary Variables
Full ML, with each threshold, tt=0
Binary Variables
The subject specific form of the conditional logistic
model (Cox, 1958; Rasch, 1961) for binary variables
logit  P  yit  1   ai  bxt
Parallels the growth curve model give above
E  yti   1   2
For the contrast coding model with variance of 1 freely
estimated and full ML estimation
logit  P  yit  1   1   2
Binary Variables
Because subject-specific conditional logistic coefficient
is
p 
b  ln  21 
 p12 
It suggests
 p21 

 p12 
 2  ln 
So the difference in marginal probabilities or discordant
cell probabilities is given by
 exp 1   2    exp 1  
p.1  p1.  


1

exp



1

exp





1
2 
1 


Binary Variables
• The score test can be stated in terms of difference of two
marginal probabilities (population average)
p.1  p1.
p21  p12
z

 p21  p12  / N  p21  p12  / N
• and it’s square is the McNemar test for df = 1

2
 


p.1  p1.

 p21  p12  / N 
2
where p1. and p.1 are marginal probabilities for the first
row (Time 1 response) and column (Time 2 response),
respectively, and p12 and p21 are discordant (“yes”-”no”
and “no-yes”) cell probabilities
Example
Self-report of major health event, N = 574, in national
survey of older adults
Time 1 Health Event
Yes = 1
No = 0
Yes = 1
21 (.037)
119 (.207)
140 (.244)
Time 2 Health Event
No = 0
21 (.037)
413 (.720)
434 (.756)
Total
42 (.073)
532 (.927)
Example
• Conventional tests
• p.1 – p1. = .171
• Conditional logistic = 53.707, p < .001
• McNemar’s test = 68.600, p < .001
• SEM model
• 1 = -3.250, 2 = 1.735
• Which is the log of probability ratio
p 
 .207 
ln  21   ln 
  1.735
 .037 
 p12 
• 2 /SE2= 7.329, p < .001; and (7.329)2=53.714,
same as conventional conditional logistic test
• LR test constraining (Pearson chi-square fit from
Mplus) 2 = 68.596, p < .001, same as McNemar’s
test
Example
• Conventional tests
• p.1 – p1. = .171
• Conditional logistic = 53.707, p < .001
• McNemar’s test = 68.600, p < .001
• SEM model
• 1 = -3.250, 2 = 1.735
• Which is the log of probability ratio
p 
 .207 
ln  21   ln 
  1.735
 .037 
 p12 
• 2 /SE2= 7.329, p < .001; and (7.329)2=53.714,
same as conventional conditional logistic test
• LR test constraining (Pearson chi-square fit from
Mplus) 2 = 68.596, p < .001, same as McNemar’s
test
Latent Variables with Multiple Indicators
Latent Variables with Multiple Indicators
• No advantage for estimating means, because random
measurement error does not bias means
• Identification important for mean interpretation
• referent variable
• single occasion (e.g., Byrne, 1994; Widaman & Reise,
1997), second or third
• effects coding (Little, Slegers, & Card, 2006), gives
weighted means for each time point
• With equality constraints, statistical tests are equal
across identification approaches
Latent Variables with Multiple Indicators
• No advantage for estimating means, because random
measurement error does not bias means
• Identification important for mean interpretation
• referent variable
• single occasion (e.g., Byrne, 1994; Widaman & Reise,
1997), second or third
• effects coding (Little, Slegers, & Card, 2006), gives
weighted means for each time point
• With equality constraints, statistical tests are equal
across identification approaches
Latent Variables with Multiple Indicators
• No advantage for estimating means
• Consider classical test theory, if expected value of
measurement error, E(e) = 0, then the observed mean
and true mean are equal.
E ( X )  E (T )  E (e)
E ( X )  E (T )  0
E ( X )  E (T )
Latent Variables with Multiple Indicators
• But significance tests may be improved because of
reduced error variance compared with observed variable
(Hancock, 2003)
• Standardized effect size for measured variables for
between-subjects comparison
dy 
 m1  m2 
sy
• For latent variables for between-subjects comparison
d 
 m1  m2 
s
• Where s < sy, if y has measurement error
Multiple Time Points
Multiple Time Points
• Equality constraint, contrast coding model, or latent
difference score model possible
• Contrast coding model requires T – 1 difference factors to
test T time points
• can investigate Helmert contrast (e.g., 2, -1,-1 and 0,
-1, 1), forward differencing (-1, 1, 0 and 0, -1, 1), or
trend analysis (e.g., -1, 0, 1 and 1, 0, 1), among
others
• Default specifications resemble MANOVA rather than
repeated measures ANOVA in terms of assumptions
• Assumptions can be tested, however
• Sphericity and compound symmetry can be assessed
through equality constraints on difference factor
variances and covariances
Mixed ANOVA Designs
• MIMIC or multigroup approach
• For MIMIC approach, predication of difference factor,
tests the between by within interaction
• For multigroup approach, equality constraints on
difference factor tests interaction
Mixed ANOVA Designs
• Simple difference score
Mixed ANOVA Designs
• Equality constraint approach
Mixed ANOVA Designs
• Contrast coding model
Mixed ANOVA Designs
• Latent difference score model
SEM Extensions
• Mixed between and within ANOVA models including
covariates, with advantage of latent variable covariates
• Mixed between and within factorial designs with binary or
ordinal variables not presently conveniently tested with
conventional tests
• Incorporate into larger models with differences as
predictors, outcomes, mediators
• Readily extends to random effects interpretations
Limitations/Cautions
• Alternative reference distribution for small N (e.g., <
120)
• critical ratio correct, p-values incorrect
• Sample size requirements larger for:
• robust estimates, bootstrapping, estimation for
noncontinuous, or missing data used
• Sometimes a structural equation model is just a paired ttest
Why consider structural equation modeling
approaches to comparing dependent means or
proportions?
• Understand underpinnings of more complex longitudinal
models and how they relate to ANOVA
• Estimate measure error with multiple indicators
• Ensure measurement invariance
• Include correlated error structures
• Generalize to binary, ordinal, other discrete variables
• Mixed between and within by including covariates, with
advantage of latent variable covariates
• Incorporate into larger models with differences as
predictors, outcomes, mediators
• Convenient missing data estimation
References
Hancock, G. R. (2003). Fortune cookies, measurement error, and experimental design. Journal of Modern
Applied Statistical Methods, 2, 293–305.
McArdle, J. J. (2001). A latent difference score approach to longitudinal dynamic structural analysis. In R.
Cudeck, S. du Toit, & D. Sorbom (Eds.), Structural equation modeling: Present and future (pp. 342–
380). Lincolnwood, IL: Scientific Software International.
McArdle, J. J., & Epstein, D. (1987). Latent growth curves within developmental structural equation
models. Child Development, 58, 110–133
McNemar, Q. (June 18, 1947). Note on the sampling error of the difference between correlated
proportions or percentages". Psychometrika 12, 153–157.
Meredith, W., & Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107–122.
Newsom, J.T. (2002). A multilevel structural equation model for dyadic data. Structural Equation
Modeling, 9, 431-447.
Newsom, J.T. (2015). Longitudinal Structural Equation Modeling: A Comprehensive Introduction. New
York: Routledge.
Rovine, M. J., & Molenaar, P. C. M. (2003). Estimating analysis of variance models as structural equation
models. In B. Pugesek, A. Tomer, & A. von Eye (Eds.), Structural equation modeling: Applications in
ecological and evolutionary biology research (pp. 235–280). New York: Cambridge.
Rovine, M. J., & Liu, S. (2012). Structural equation modeling approaches to longitudinal data. In J.T.
Newsom, R. N. Jones, & S. M. Hofer (Eds.). Longitudinal data analysis: A practical guide for
researchers in aging, health, and social science (pp. 243–270). New York: Routledge.
.Voelkle, M. C. (2007). Latent growth curve modeling as an integrative approach to the analysis of
change. Psychology Science, 49, 375.
Thank you
[email protected]
Computer code for Mplus and lavaan (R package) available
from the author or download from
www.longitudinalSEM.com (coming soon)