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Scientific Methods for Health Sciences - Model-based Analyses

Structural Equation Modeling (SEM), Growth Curve Models (GCM), 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.

Growth Curve (or latent growth) modeling is a statistical technique employed in SEM for estimating growth trajectories for longitudinal data (over time). It represent repeated measures of dependent variables as functions of time and other covariates. When subjects or units are observed repeatedly over known time points latent growth curve models reveal the trend of an individual as a function of an underlying growth process where the growth curve parameters can be estimated for each subject/unit.

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.

DOUBLE CHECK FORMULA

For instance, suppose $Y_{i,j}$ is the random effects logistic model of the $j^{th}$, observation of the $i^{th}$ subject, then $log\Bigg(\frac{p_{i,j}}{1-p_{i,j}} \Bigg)=μ+ν_i,$ 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,

DOUBLE CHECK FORMULA $$δ=log\Bigg(\frac{E_v(\frac{1}{1+e^{-μ+v}i})}{1-E_v(\frac{1}{1+e^{-μ+v}i})} \Bigg),$$

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:

CHECK FORMULAS!!

$E(Y)=μ=1$ (in this specific case), but the expectation of the population average log odds $δ=log\Bigg[\frac{P(Y_{i,j}=1|v_i)}{1-P(Y_{i,j}=1|v_i)}\Bigg]$ would be $< 1^1$. 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 $i^{th}$ person in the $j^{th}$ 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: $g^{-1} (u)=log(\frac{u}{1-u})$
• $E(Uij(α_j,Xij)|Xij)=0.$

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!!!!!!!!!!!

 Distribution Support of distribution Typical uses Link name Link function Mean function Normal real: $(-∞, +∞)$ Linear-response data Identity $X\beta=\mu$ $\mu=X\beta$ Exponential, Gamma real:$(0, +∞)$ Exponential-response data, scale parameters Inverse $X\beta=-\mu^{-1}$ $\mu=-(X\beta)^{-1}$ Inverse Gaussian real:$(0, +∞)$ Inverse squared $X\beta=-\mu^{-2}$ $\mu=(-X\beta)^{-1/2}$