SMHS BigDataBigSci - Revision history

Pineaumi: /* Generalized Estimating Equation (GEE) Modeling */

2016-05-24T13:57:39Z

‎Generalized Estimating Equation (GEE) Modeling

Pineaumi: /* Overview */

2016-05-23T19:09:25Z

‎Overview

Imoubara at 13:36, 23 May 2016

2016-05-23T13:36:25Z

Dinov: /* Overview */

2016-05-21T16:24:17Z

‎Overview

Pineaumi: /* Overview */

2016-05-20T13:20:02Z

‎Overview

Pineaumi: /* Overview */

2016-05-20T13:19:30Z

‎Overview

Imoubara at 19:43, 18 May 2016

2016-05-18T19:43:37Z

Imoubara: /* Overview */

2016-05-18T19:43:17Z

‎Overview

Imoubara at 19:32, 18 May 2016

2016-05-18T19:32:10Z

Imoubara at 15:46, 16 May 2016

2016-05-16T15:46:21Z

@@ Line 93: / Line 93: @@
 ===[[SMHS_BigDataBigSci_GCM| Growth Curve Modeling (GCM)]]===
-===[[SMHS_BigDataBigSci_GEE| Generalized Estimating Equation (GEE) Modeling]]===
+===[[SMHS_BigDataBigSci_GCM| Generalized Estimating Equation (GEE) Modeling]]===
 ===[[SMHS_BigDataBigSci_CrossVal|Internal Validation - Statistical n-fold cross-validaiton]]===

@@ Line 82: / Line 82: @@
 |}
 </center>
 ==Model-based Analytics==

@@ Line 5: / Line 5: @@
 ==Questions ==
-<li>How to represent dependencies in linear models and examine causal effects?</li>
+*How to represent dependencies in linear models and examine causal effects?
-<li>Is there a way to study population average effects of a covariate against specific individual effects?</li>
+*Is there a way to study population average effects of a covariate against specific individual effects?
 ==Overview==
@@ Line 59: / Line 59: @@
 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.,
-<li> '''<u>linear</u>:''' 		$g^{-1} (u)=u,$</li>
+*<u>linear</u>:''' 		$g^{-1} (u)=u,$
-<li>'''<u>log</u>:'''		        $g^{-1} (u)= log(u),$ </li>
+*<u>log</u>:'''		        $g^{-1} (u)= log(u),$
-<li> '''<u>logistic</u>:''' 		$g^{-1} (u)=log(\frac{u}{1-u})$</li>
+*<u>logistic</u>:''' 		$g^{-1} (u)=log(\frac{u}{1-u})$
-<li>$E(Uij(α_j,Xij)|Xij)=0.$</LI>
+*$E(Uij(α_j,Xij)|Xij)=0.$
 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.

@@ Line 17: / Line 17: @@
 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 <b>logistic model</b> of the $j^{th}$,  observation of the $i^{th}$ subject, then
@@ Line 24: / Line 22: @@
 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 <u>subject i</u> and $p_{i,j}=P(Y_{i,j}=1|ν_i).$
+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''.

@@ Line 50: / Line 50: @@
 <b># theoretically</b>, 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 $< 1$ <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>
+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:

@@ Line 18: / 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.
-<mark>DOUBLE CHECK FORMULA
+<mark> PLEASE ADD TILDA~ TO FORUMLA BELOW, BETWEEN $ν_i$ AND $~N(0,σ^2)$!!</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
@@ Line 26: / Line 26: @@
 where  $ν_i~N(0,σ^2)$ is a random effect for <u>subject i</u> and $p_{i,j}=P(Y_{i,j}=1|ν_i).$
-</mark>
+(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>,
 \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})}
@@ Line 41: / Line 38: @@
 If $μ=1$ and $σ^2=1$, then $δ≈.83$.
 empirically:
@@ Line 58: / Line 54: @@
 $E(Y)=μ=1$ (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 $< 1^1$.
+$δ=log\Bigg[\frac{P(Y_{i,j}=1|v_i)}{1-P(Y_{i,j}=1|v_i)}\Bigg]$  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>
 $E(Yij | Xij,α_j)= g[μ(Xij|β)+Uij(α_j,Xij)]$,
-</blockquote>
 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.,
@@ Line 74: / Line 69: @@
 <li> '''<u>logistic</u>:''' 		$g^{-1} (u)=log(\frac{u}{1-u})$</li>
-<li>$E(Uij(α_j,Xij)|Xij)=0.$
+<li>$E(Uij(α_j,Xij)|Xij)=0.$</LI>
 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.

@@ Line 78: / Line 78: @@
 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.
-<center><b>Common distributions with typical uses and canonical link functions</b></center>
+<center>Common distributions with typical uses and canonical link functions</center>
 <center>
 {| class="wikitable" style="text-align:center; " border="1"

@@ Line 4: / Line 4: @@
 == Questions ==
-* How to represent dependencies in linear models and examine causal effects?
+<li>How to represent dependencies in linear models and examine causal effects?</li>
-* Is there a way to study population average effects of a covariate against specific individual effects?
+<li>Is there a way to study population average effects of a covariate against specific individual effects?</li>
 ==Overview==