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==[[SMHS| Scientific Methods for Health Sciences]] - Structural Equation Modeling (SEM) and Generalized Estimating Equation (GEE) Modeling ==
+
==[[SMHS| Scientific Methods for Health Sciences]] - Model-based Analyses ==
  
<b>Questions</b>
+
Structural Equation Modeling (SEM), Growth Curve Models (GCM), and Generalized Estimating Equation (GEE) Modeling
  
• How to represent dependencies in linear models and examine causal effects?
+
==Questions ==
  
Is there a way to study population average effects of a covariate against specific individual effects?
+
*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?
  
<b>Overview</b>
+
==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.  
 
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 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.
Line 15: Line 18:
 
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.
 
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.
  
<mark>FIX FORMULA For instance, suppose Y<sub>i</sub>, <sub>j</sub>,  is the random effects <b>logistic model</b> of the j<sup>th</sup>,  observation of the i<sup>th</sup> 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).</mark>
+
For instance, suppose $Y_{i,j}$ is the random effects <b>logistic model</b> 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  $ν_i \sim N(0,σ^2)$ is a random effect for <u>subject i</u> 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.
+
(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 <i>population average log odds</i>,
 
(2) When using a GEE model on the same data, we estimate the <i>population average log odds</i>,
  
<mark>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.
+
\begin{equation}
If μ=1 and σ^2=1, then δ≈.83. </mark>
+
δ=log\Bigg(\frac{E_v(\frac{1}{1+e^{-μ+v}i})}{1-E_v(\frac{1}{1+e^{-μ+v}i})}
 +
\Bigg),
 +
\end{equation}
 +
 
 +
in general $μ≠δ$.
 +
 
 +
If $μ=1$ and $σ^2=1$, then $δ≈.83$.  
  
 
empirically:
 
empirically:
Line 36: Line 49:
 
<b># theoretically</b>, if it can be computed:
 
<b># theoretically</b>, if it can be computed:
  
<mark>FIX FORMULAS!!
+
$E(Y)=μ=1$ (in this specific case), but the expectation of the population average log odds  
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. </mark>
+
$δ=log\Bigg[\frac{P(Y_{i,j}=1|v_i)}{1-P(Y_{i,j}=1|v_i)}\Bigg]would be $< 1$ <SUP>1</SUP>.  
 
+
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.  
<mark>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.</mark>
 
 
 
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.
 
 
 
<mark>INSERT TABLE!!!!!!!!!!!</mark>
 
 
 
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
+
The mean of the $i^{th}$ person in the $j^{th}$ observation (e.g., location, time, etc.) can be expressed by:
  
• Enables modeling of error terms
+
$E(Yij | Xij,α_j)= g[μ(Xij|β)+Uij(α_j,Xij)]$,
  
• Facilitates modeling of challenging data (longitudinal with auto-correlated errors, multi-level data, non-normal data, incomplete data)
+
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.,  
 +
 +
*<u>linear</u>:''' $g^{-1} (u)=u,$
  
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.
+
*<u>log</u>:'''         $g^{-1} (u)= log(u),$
  
Definitions
+
*<u>logistic</u>:''' $g^{-1} (u)=log(\frac{u}{1-u})$
 +
 +
*$E(Uij(α_j,Xij)|Xij)=0.$
  
* The <b>disturbance</b>, <i>D</i>, is the variance in Y unexplained by a variable X that is assumed to affect Y. 
+
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.
 
 
X    →    Y  ←    D
 
 
 
*<b>Measurement error</b>, <i>E</i>, is the variance in X unexplained by A, where X is an observed variable that is presumed to measure a latent variable, <i>A</i>.
 
 
 
A    →    X  ←    E
 
 
 
*Categorical variables in a model are <b>exogenous</b> (independent) or <b>endogenous</b> (dependent).
 
 
 
Notation
 
 
 
*In SEM <b>observed (or manifest) indicators</b> are represented by <b>squares/rectangles</b> whereas latent variables (or factors) represented by circles/ovals.
 
 
 
[[Image:SMHS_BigDataBigSci1.png|500px]]
 
 
 
<mark> PLEASE FIX ARROWS *Relations: Direct effects (→), Reciprocal effects (<--> or  ), and Correlation or covariance ( ) all have different appearance in SEM models.</mark>
 
 
 
Model Components
 
 
 
The <b>measurement part</b> 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 <u>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</u>. The <b>measurement model</b> is evaluated using goodness of fit measures (Chi-Square test, BIC, AIC, etc.) <b>Validation of the measurement model is always first.</b>
 
 
 
<b>Then we proceed to the structural model</b> (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.
 
  
 +
<center>Common distributions with typical uses and canonical link functions</center>
 
<center>
 
<center>
 
{| class="wikitable" style="text-align:center; " border="1"
 
{| class="wikitable" style="text-align:center; " border="1"
 
|-
 
|-
|ID||lhs||op||rhs||user||free||ustart
+
|<b>Distribution</b> ||<b>Support of distribution</b>||<b>Typical uses</b>||<b>Link name</b>||<b>Link function</b>||<b>Mean function</b>
|-
 
|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
+
|Normal||real: $(-&#8734;, +&#8734;)$||Linear-response data||Identity||$X\beta=\mu$||$\mu=X\beta$
 
|-
 
|-
|15||x6||~~||x6||0||12||NA
+
|Exponential, Gamma||real:$(0, +&#8734;)$||Exponential-response data, scale parameters||Inverse||$X\beta=-\mu^{-1}$||$\mu=-(X\beta)^{-1}$
 
|-
 
|-
|16||x7||~~||x7||0||13||NA
+
|Inverse Gaussian||real:$(0, +&#8734;)$|| ||Inverse squared||$X\beta=-\mu^{-2}$||$\mu=(-X\beta)^{-1/2}$
|-
 
|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
 
 
 
 
|}
 
|}
 
</center>
 
</center>
  
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}.  
+
===Footnotes===
 +
 
 +
*<sup>1</sup> http://www.researchgate.net/publication/41895248
  
These <b>12 parameters</b> 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 <b>12 additional free parameters</b>. In total, we have <b>24 parameters.</b>
+
==Model-based Analytics==
  
[[Image:SMHS_BigDataBigSci2.png|500px]]
+
===[[SMHS_BigDataBigSci_SEM| Structural Equation Modeling (SEM)]]===
  
To fully identify the model we need to set the metric of the latent variables. There are 2 ways to do this:
+
===[[SMHS_BigDataBigSci_GCM| Growth Curve Modeling (GCM)]]===
  
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
+
===[[SMHS_BigDataBigSci_GCM| Generalized Estimating Equation (GEE) Modeling]]===
  
o standardize the variances of the 3 latent variables.
+
===[[SMHS_BigDataBigSci_CrossVal|Internal Validation - Statistical n-fold cross-validaiton]]===
  
Either way, we fix 3 of these 24 parameters, and 21 parameters remain free.  
+
<hr>
 +
* SOCR Home page: http://www.socr.umich.edu
  
The <b>parTable(fit)</b> method, generates this table output.
+
{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=SMHS_BigDataBigSci}}

Latest revision as of 09:57, 24 May 2016

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.

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 $ν_i \sim N(0,σ^2)$ 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,

\begin{equation} δ=log\Bigg(\frac{E_v(\frac{1}{1+e^{-μ+v}i})}{1-E_v(\frac{1}{1+e^{-μ+v}i})} \Bigg), \end{equation}

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:

$E(Y)=μ=1$ (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.

Common distributions with typical uses and canonical link functions
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 $X\beta=-\mu^{-2}$ $\mu=(-X\beta)^{-1/2}$

Footnotes

Model-based Analytics

Structural Equation Modeling (SEM)

Growth Curve Modeling (GCM)

Generalized Estimating Equation (GEE) Modeling

Internal Validation - Statistical n-fold cross-validaiton




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