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Scientific Methods for Health Sciences - Structural Equation Modeling (SEM) and Generalized Estimating Equation (GEE) Modeling

Questions

• How to represent dependencies in linear models and examine causal effects?

• Is there a way to study population average effects of a covariate against specific individual effects?

Overview

SEM allow re-parameterization of random-effects to specify latent variables that may affect measures at different time points using structural equations. SEM show variables having predictive (possibly causal) effects on other variables (denoted by arrows) where coefficients index the strength and direction of predictive relations. SEM does not offer much more than what classical regression methods do, but it does allow simultaneous estimation of multiple equations modeling complementary relations.

GEE is a marginal longitudinal method that directly assesses the mean relations of interest (i.e., how the mean dependent variable changes over time), accounting for covariances among the observations within subjects, and getting a better estimate and valid significance tests of the relations. Thus, GEE estimates two different equations, (1) for the mean relations, and (2) for the covariance structure. An advantage of GEE over random-effect models is that it does not require the dependent variable to be normally distributed. However, a disadvantage of GEE is that it is less flexible and versatile – commonly employed algorithms for it require a small-to-moderate number of time points evenly (or approximately evenly) spaced, and similarly spaced across subjects. Nevertheless, it is a little more flexible than repeated-measure ANOVA because it permits some missing values and has an easy way to test for and model away the specific form of autocorrelation within subjects.

GEE is mostly used when the study is focused on uncovering the population average effect of a covariate vs. the individual specific effect. These two things are only equivalent for linear models, but not in non-linear models.

FIX FORMULA For instance, suppose Yi, j,   is the random effects logistic model of the jth,  observation of the ith subject, then log(p_(i,j)/(1-p_(i,j) ))=μ+ν_i, where  ν_i~N(0,σ^2) is a random effect for subject i and p_(i,j)=P(Y_(i,j)=1|ν_i).

(1) When using a random effects model on such data, the estimate of μ accounts for the fact that a mean zero normally distributed perturbation was applied to each individual, making it individual-specific.

(2) When using a GEE model on the same data, we estimate the population average log odds,

FIX FORMULA δ=log((E_ν (1/(1+e^(-μ+ν_i ) )))/(1-E_ν (1/(1+e^(-μ+ν_i ) )) )), in general μ≠δ.
If μ=1 and σ^2=1, then δ≈.83. 

empirically:

m <- 1; s <- 1; v<-rnorm(1000, 0,s); v2 <- 1/(1+exp(-m+v)); v_mean <- mean(v2)

d <- log(v_mean/(1-v_mean)); d

Note that the random effects have mean zero on the transformed, linked, scale, but their effect is not mean zero on the original scale of the data. We can also simulate data from a mixed effects logistic regression model and compare the population level average with the inverse-logit of the intercept to see that they are not equal. This leads to a difference of the interpretation of the coefficients between GEE and random effects models, or SEM.

That is, there will be a difference between the GEE population average coefficients and the individual specific coefficients (random effects models).

# theoretically, if it can be computed:

FIX FORMULAS!!
E(Y)=μ=1 (in this specific case), but the expectation of the population average log odds δ=  log[(P(Y_(i,j)=1|ν_i))/(1-P(Y_(i,j)=1|ν_i))] would be < 1  . Note that this is kind of related to the fact that a grand-total average  need not be equal to an average of partial averages. 

The mean of the ith person in the jth observation (e.g., location, time, etc.) can be expressed by:
E(Yij | Xij,α_j)= g[μ(Xij ┤|β)+Uij(α_j,Xij)],
Where μ(X_ij |β) is the average “response” of a person with the same covariates X_ij, β a set of fixed effect coefficients, and Uij(α_j,Xij) is an error term that is a function of the (time, space) random effects, α_j, and also a  function of the covariates X_ij, and g is the link function which specifies the regression type -- e.g., 
linear: 		g^(-1) (u)=u,
log:		g^(-1) (u)= log(u), 
logistic: 		g^(-1) (u)=log(u/(1-u))
E(Uij(α_j,Xij)|Xij)=0.

The link function, g(u), provides the relationship between the linear predictor and the mean of the distribution function. For practical applications there are many commonly used link functions. It makes sense to try to match the domain of the link function to the range of the distribution function's mean.

INSERT TABLE!!!!!!!!!!!

Structural Equation Modeling (SEM)

SEM is a general multivariate statistical analysis technique that can be used for causal modeling/inference, path analysis, confirmatory factor analysis (CFA), covariance structure modeling, and correlation structure modeling.

Advantages

• It allows testing models with multiple dependent variables

• Provides mechanisms for modeling mediating variables

• Enables modeling of error terms

• Facilitates modeling of challenging data (longitudinal with auto-correlated errors, multi-level data, non-normal data, incomplete data)

This method SEM allows separation of observed and latent variables. Other standard statistical procedures may be viewed as special cases of SEM, where statistical significance less important, than in other techniques, and covariances are the core of structural equation models.

Definitions

• The disturbance, D, is the variance in Y unexplained by a variable X that is assumed to affect Y.

X → Y ← D

• Measurement error, E, is the variance in X unexplained by A, where X is an observed variable that is presumed to measure a latent variable, A.

A → X ← E

• Categorical variables in a model are exogenous (independent) or endogenous (dependent).

Notation

• In SEM observed (or manifest) indicators are represented by squares/rectangles whereas latent variables (or factors) represented by circles/ovals.

PLEASE FIX ARROWS *Relations: Direct effects (→), Reciprocal effects (<--> or ), and Correlation or covariance ( ) all have different appearance in SEM models.

Model Components

The measurement part of SEM model deals with the latent variables and their indicators. A pure measurement model is a confirmatory factor analysis (CFA) model with unmeasured covariance (bidirectional arrows) between each possible pair of latent variables. There are straight arrows from the latent variables to their respective indicators and straight arrows from the error and disturbance terms to their respective variables, but no direct effects (straight arrows) connecting the latent variables. The measurement model is evaluated using goodness of fit measures (Chi-Square test, BIC, AIC, etc.) Validation of the measurement model is always first.

Then we proceed to the structural model (including a set of exogenous and endogenous variables together with the direct effects (straight arrows) connecting them along with the disturbance and error terms for these variables that reflect the effects of unmeasured variables not in the model).

Notes

• Sample-size considerations: mostly same as for regression - more is always better

• Model assessment strategies: Chi-square test, Comparative Fit Index, Root Mean Square Error, Tucker Lewis Index, Goodness of Fit Index, AIC, and BIC.

• Choice for number of Indicator variables: depends on pilot data analyses, a priori concerns, fewer is better.

Hands-on Example 1 (School Kids Mental Abilities)

These data (Holzinger & Swineford 1939) include mental ability test scores of 7 & 8 grade children from two schools (Pasteur and Grant-White). This version of the dataset includes only 9 (out of the 26) tests. We can build and test a confirmatory factor analysis (CFA) SEM model for 3 correlated latent variables (or factors), each with three indicators:

o visual factor measured by 3 variables: x1, x2 and x3,

o textual factor measured by 3 variables: x4, x5 and x6,

o speed factor measured by 3 variables: x7, x8 and x9.

ID	lhs	op	rhs	user	free	ustart
1	Visual	=~	x1	1	0	1
2	Visual	=~	x2	1	1	NA
3	Visual	=~	x3	1	2	NA
4	Textual	=~	x4	1	0	1
5	Textual	=~	x5	1	3	NA
6	Textual	=~	x6	1	4	NA
7	Speed	=~	x7	1	0	1
8	Speed	=~	x8	1	5	NA
9	Speed	=~	x9	1	6	NA
10	x1	~~	x1	0	7	NA
11	x2	~~	x2	0	8	NA
12	x3	~~	x3	0	9	NA
13	x4	~~	x4	0	10	NA
14	x5	~~	x5	0	11	NA
15	x6	~~	x6	0	12	NA
16	x7	~~	x7	0	13	NA
17	x8	~~	x8	0	14	NA
18	x9	~~	x9	0	15	47.8
19	Visual	~~	Visual	0	16	NA
20	Textual	~~	Textual	0	17	NA
21	Speed	~~	Speed	boy	18	NA
22	Visual	~~	Textual	girl	19	NA
23	Visual	~~	Speed	girl	20	NA
24	Textual	~~	Speed	boy	21	NA

There are 3 latent variables (factors) in this model, each with 3 indicators, resulting in 9 factor loadings that need to be estimated. There are also 3 covariances among the latent variables {another three parameters}.

These 12 parameters are represented in the path diagram as single-headed and double-headed arrows, respectively. We also need to estimate the residual variances of the 9 observed variables and the variances of the 3 latent variables, resulting in 12 additional free parameters. In total, we have 24 parameters.

To fully identify the model we need to set the metric of the latent variables. There are 2 ways to do this:

o for each latent variable, fix the factor loading of one of the indicators (typically the first) to a constant (e.g., 1.0), or

o standardize the variances of the 3 latent variables.

Either way, we fix 3 of these 24 parameters, and 21 parameters remain free.

The parTable(fit) method, generates this table output.

The `rhs', `op' and `lhs' columns define the parameters of the model. All parameters with the `=~' operator are factor loadings, whereas all parameters with the `~~' operator are variances or covariances. Nonzero elements in the `free' column are the free parameters of the model. Zero elements in the `free' column correspond to fixed parameters, whose value is found in the `start' column.

Lavaan’s user-friendly model-specification approach is implemented in the fitting functions: cfa() and sem().

Since these data contain 3 latent variables, and no regressions, the minimalist syntax is:

data.model <- 'visual 	=~ x1 + x2 + x3
textual 	                =~ x4 + x5 + x6
speed 	                        =~ x7 + x8 + x9'

Fit the CFA model:

fit.1 <- cfa(data.model, data = HolzingerSwineford1939)

The `user' column (parTabale) shows which parameters were explicitly contained in the user-specified model syntax (= 1), and which parameters were added by the cfa() function (= 0).

parTable(fit.1)

If we prefer not to fix the factor loadings of the first indicator, but instead want to fix the variances of the latent variances, the model syntax would be changed to:

fit.2 <- 'visual =~ NA*x1 + x2 + x3
textual =~ NA*x4 + x5 + x6
speed =~ NA*x7 + x8 + x9
visual ~~ 1*visual
textual ~~ 1*textual
speed ~~ 1*speed'

More complex model specifications can be made using the full lavaan model syntax:

fit.full <- '			# latent variables
visual 	=~ 1*x1 + x2 + x3
textual =~ 1*x4 + x5 + x6
speed 	=~ 1*x7 + x8 + x9
# residual variances observed variables
x1 	~~ x1
x2 	~~ x2
x3 	~~ x3
x4 	~~ x4
x5 	~~ x5
x6 	~~ x6
x7 	~~ x7
x8 	~~ x8
x9 	~~ x9
# factor variances
visual 	~~ visual
textual ~~ textual
speed 	~~ speed
# factor covariances
visual 	~~ textual + speed
textual ~~ speed'
fit.3 <- lavaan(fit.full, data = HolzingerSwineford1939)

We can specify the model where the first factor loadings are explicitly fixed to one, and the covariances among the factors are added manually.

fit.mixed <- '		# latent variables
visual =~ 1*x1 + x2 + x3
textual =~ 1*x4 + x5 + x6
speed =~ 1*x7 + x8 + x9
# factor covariances
visual ~~ textual + speed
textual ~~ speed'
fit <- lavaan(fit.mixed, data = HolzingerSwineford1939, auto.var = TRUE)

The best method to view results from a SEM fitted with lavaan is summary(), which can be called with optional arguments like fit.measures, standardized, and rsquare.

Core Lavaan Methods

summary() print a long summary of the model results

show() print a short summary of the model results

coef() returns the estimates of the free parameters in the model as a named numeric vector

fitted() returns the implied moments (covariance matrix and mean vector) of the model

resid() returns the raw, normalized or standardized residuals (difference between implied and observed moments)

vcov() returns the covariance matrix of the estimated parameters

predict() compute factor scores

logLik() returns the log-likelihood of the fitted model (if maximum likelihood estimation was used)

AIC(), BIC() compute information criteria (if maximum likelihood estimation is used)

update() update a fitted lavaan object

inspect() peek into the internal representation of the model; by default, it returns a list of model matrices counting the free parameters in the model; can also be used to extract starting values, gradient values, and much more

If these args are set to TRUE, the output includes additional fit measures, standardized estimates, and R2 values for the dependent variables, respectively

fit.model <- 'visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
 speed =~ x7 + x8 + x9'

fit <- cfa(fit.model, data = HolzingerSwineford1939)

summary(fit, fit.measures = TRUE)

fit <- cfa(fit.model, data=HolzingerSwineford1939, estimator="GLS", group="sex")
fit.4 <- cfa(fit.model, data=HolzingerSwineford1939, estimator="GLD", group="sex", group.equal="regressions") 
anova(fit, fit.4)

The output consists of three sections.

• The first section (first 6 lines) contains the package version number, an indication whether the model has converged (and in how many iterations), and the effective number of observations used in the analysis.

• The second section contains the model χ^2 test statistic, degrees of freedom, and a p value are printed. If fit.measures = TRUE, it also prints the test statistic of the baseline model (where all observed variables are assumed to be uncorrelated) and several popular fit indices. If maximum likelihood estimation is used, this section will also contain information about the log-likelihood, the AIC, and the BIC.

• The third section provides an overview of the parameter estimates, including the type of standard errors used and whether the observed or expected information matrix was used to compute standard errors. Then, for each model parameter, the estimate and the standard error are displayed, and if appropriate, a z value based on the Wald test and a corresponding two-sided p value are also shown. To ease the reading of the parameter estimates, they are grouped into three blocks:

o factor loadings,

o factor covariances, and

o residual variances of both observed variables and factors.

The summary() method provides a nice summary of the model results for visualization purposes. The parameterEstimates() method returns the actual parameter estimates as a data.frame, which can be processed further