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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.