SMHS GEE - Revision history

Dinov: /* Example 2 */

2016-07-06T14:44:04Z

‎Example 2

Dinov: /* Problems */

2016-07-06T14:43:45Z

‎Problems

Dinov: /* Problems */

2016-07-06T14:43:01Z

‎Problems

Show changes

Dinov: /* Problems */

2016-07-06T14:37:44Z

‎Problems

Dinov: /* References */

2016-07-06T14:37:06Z

‎References

Dinov: /* Motivation */

2016-07-06T14:35:31Z

‎Motivation

Dinov: /* Motivation */

2016-07-06T14:34:17Z

‎Motivation

Dinov: /* Motivation */

2016-07-06T14:32:41Z

‎Motivation

Dinov: /* Motivation */

2016-07-06T13:26:01Z

‎Motivation

Dinov: /* Motivation */

2016-07-06T13:21:59Z

‎Motivation

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   Estimated effects may be unbalanced
-===Example 2===
+====Example 2====
 Use [[SOCR_Data_Oct2009_ID_NI| Normal and Schizophrenia Neuroimaging study of Children Data]]
 ▪	Subject: Subject identifier

@@ Line 107: / Line 107: @@
 ===Problems===
-Example 1:
+====Example 1====
-library(geepack)   # install this package first
+ library(geepack)   # install this package first
-data(dietox)
+ data(dietox)
-summary(dietox)
+ summary(dietox)
       Weight            Feed             Time             Pig
   Min.   : 15.00   Min.   :  3.30   Min.   : 1.000   Min.   :4601

@@ Line 213: / Line 213: @@
 Terms:
-                      Sex       Age       TBV       GMV       WMV       CSF Background   Sex:Age Residuals
+  Sex       Age       TBV       GMV       WMV       CSF Background   Sex:Age Residuals
 Sum of Squares    497.829   740.825  1040.712    25.337  3131.065   205.501    384.453   366.533 15459.175
 Deg. of Freedom         1         1         1         1         1         1          1         1        54

← Older revision		Revision as of 14:37, 6 July 2016
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	[https://onlinecourses.science.psu.edu/stat504/node/180 Introductions to GEE]		[https://onlinecourses.science.psu.edu/stat504/node/180 Introductions to GEE]
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@@ Line 57: / Line 57: @@
 :: $0=U(\hat{\beta})=\sum_{i=1}^N{\underbrace{D_i^T(\hat{\beta})}_{\text{scale}} \underbrace{V_i^{-1}(\hat{\beta},\alpha)}_{\text{variance weight}} \underbrace{\{Y_i-\mu_i(\hat{\beta})\}}_{\text{model mean}}}$
 ::: '''Scale''': a change of scale term transforming the scale of the mean, $\mu_i$, to the scale of the regression coefficients (covariates)
-::: '''Variance weight''': the inverse of hte variance-covariance matrix is used to weight in the data for subject $i$, i.e., giving more weight to differences between observed and expected values for subjects that contribute more information
+::: '''Variance weight''': the inverse of the variance-covariance matrix is used to weight in the data for subject $i$, i.e., giving more weight to differences between observed and expected values for subjects that contribute more information
 ::: '''Model Mean''': specifies the mean model, $\mu_i(\beta)$, compared to the observed data, $Y_i$. This fidelity term minimizes the difference between actually-observed and mean-expected (within the $i^{th}$ cluster/subject).

@@ Line 52: / Line 52: @@
 * Without an explicit likelihood function, to ''estimate the parameter'' vector $\beta$ (and perhaps the covariance parameter matrix $R_i(\alpha)$) and perform valid ''statistical inference'' that takes the dependence into consideration, we need to construct an unbiased estimating function:
 : $D_i(\beta) = \frac{\partial \mu_i}{\partial \beta}$, the partial derivative, w.r.t. $\beta$, of the mean-model for subject $i$
-: $D_i(j,k) = \frac{\partial \mu_{i,j}}{\partial \beta_k}$, the partial derivative, w.r.t. $k^{th}$ regression coefficient ($\beta_k$), of the mean-model for subject $i$ and measurement (e.g., time-point) $j$.
+: $D_i(j,k) = \frac{\partial \mu_{i,j}}{\partial \beta_k}$, the partial derivative, w.r.t. the $k^{th}$ regression coefficient ($\beta_k$), of the mean-model for subject $i$ and measurement (e.g., time-point) $j$.
 : '''Estimating (cost) function''': $U(\beta)=\sum_{i=1}^N{D_i^T(\beta)V_i^{-1}(\beta,\alpha)\{Y_i-\mu_i(\beta)\}}$.
-: Solving the Estimating Equations leads to parameter estimating solutions: $0=U(\hat{\beta})=\sum_{i=1}^N{D_i^T(\hat{\beta})V_i^{-1}(\hat{\beta},\alpha)\{Y_i-\mu_i(\hat{\beta})\}}$.
+: Solving the Estimating Equations leads to parameter estimating solutions:
 :: $0=U(\hat{\beta})=\sum_{i=1}^N{\underbrace{D_i^T(\hat{\beta})}_{\text{scale}} \underbrace{V_i^{-1}(\hat{\beta},\alpha)}_{\text{variance weight}} \underbrace{\{Y_i-\mu_i(\hat{\beta})\}}_{\text{model mean}}}$
 ::: '''Scale''': a change of scale term transforming the scale of the mean, $\mu_i$, to the scale of the regression coefficients (covariates)

@@ Line 10: / Line 10: @@
 * Covariate vector: $\{X_{i,1}, X_{i,2}, ..., X_{i,n_t}\}$.
-The primary focus of GEE is the estimation of the '''mean model''': $E(Y_{i,j}|X_{i,j})=\mu_{i,j}$, where $g(\mu_{i,j})=\beta_o+\beta_1 X_{i,j}(1) +\beta_2 X_{i,j}(2)+\beta_3 X_{i,j}(3)+...+\beta_p X_{i,j}(p)=X_{i,j}\times \beta$.
+The primary focus of GEE is the estimation of the '''mean model''':
-This '''mean model''' can be any generalized linear model, e.g.,
+This '''mean model''' can be any generalized linear model. For example:
 : $P(Y_{i,j}=1|X_{i,j})=\pi_{i,j}$ (marginal probability, as we don't condition on any other variables)
 : $g(\mu_{i,j})= \log ( \frac{\pi_{i,j}}{1-\pi_{i,j}}) = X_{i,j}\times \beta$.
-Since the data could be clustered (e.g., within subject, or within unit), we need to choose a corr'''elation model''':
+Since the data could be clustered (e.g., within subject, or within unit), we need to choose a '''correlation model''':
 : $var(Y_{i,j}|X_{i}) = V_{i,j}$
 : $A_{i}= diag(V_{i,j})$
@@ Line 50: / Line 52: @@
 * Without an explicit likelihood function, to ''estimate the parameter'' vector $\beta$ (and perhaps the covariance parameter matrix $R_i(\alpha)$) and perform valid ''statistical inference'' that takes the dependence into consideration, we need to construct an unbiased estimating function:
 : $D_i(\beta) = \frac{\partial \mu_i}{\partial \beta}$, the partial derivative, w.r.t. $\beta$, of the mean-model for subject $i$
-: $D_i(j,k) = \frac{\partial \mu_{i,j}}{\partial \beta_k}$
+: $D_i(j,k) = \frac{\partial \mu_{i,j}}{\partial \beta_k}$, the partial derivative, w.r.t. $k^{th}$ regression coefficient ($\beta_k$), of the mean-model for subject $i$ and measurement (e.g., time-point) $j$.
 : '''Estimating (cost) function''': $U(\beta)=\sum_{i=1}^N{D_i^T(\beta)V_i^{-1}(\beta,\alpha)\{Y_i-\mu_i(\beta)\}}$.
 : Solving the Estimating Equations leads to parameter estimating solutions: $0=U(\hat{\beta})=\sum_{i=1}^N{D_i^T(\hat{\beta})V_i^{-1}(\hat{\beta},\alpha)\{Y_i-\mu_i(\hat{\beta})\}}$.

@@ Line 7: / Line 7: @@
 GEE is applicable when (1) $\beta$, a generalized linear model regression parameter, characterizes systematic variation across covariate levels, (2) the date represents repeated measurements, clustered data, multivariate response, and (3) the correlation structure is a nuisance feature of the data. Denote:
-* Response variables: $\{Y_{i,1}, Y_{i,2}, ..., Y_{i,n_t}\}$, where $i \in [1,N]$ is the index for clusters or subjects, and $j \in [1, n_t]$ is the index of the measurement within cluster/subject.
+* Response variables: $\{Y_{i,1}, Y_{i,2}, ..., Y_{i,n_t}\}$, where $i \in [1,N]$ is the ''index for clusters or subjects'', and $j \in [1, n_t]$ is the ''index of the measurement within cluster/subject''.
 * Covariate vector: $\{X_{i,1}, X_{i,2}, ..., X_{i,n_t}\}$.
@@ Line 21: / Line 21: @@
 : paired correlations: $\rho_{i,j,k} = corr(Y_{i,j},Y_{i,k}|X_{i})$
 : correlation matrix: $R_i= \{\rho_{i,j,k}\}$, for all $j$ and $k$
-: paired predictor-response covariances: $V_i=cov(Y_i|X_i)=A_i^{1/2} R_iA_i^{1/2}$.
+: paired predictor-response covariances: $V_i=cov(Y_i|X_i)=A_i^{1/2} R_i A_i^{1/2}$.
 Different correlation structures assumed in the data lead to alternative models. For example:
@@ Line 44: / Line 44: @@
 * and so on.
-Notes:
+'''Notes''':
 * GEE is a semi-parametric technique because:
-: The specification of a mean model, $\mu_{i,j}(\beta)$, and a correlation
+: The specification of a mean model, $\mu_{i,j}(\beta)$, and a correlation model, $R_i(\alpha)$, ''does not identify'' a complete probability model for $Y_i$
 : The model $\{ \mu_{i,j}(\beta), R_i(\alpha)\}$ is semi-parametric since it only specifies the first two multivariate moments (mean and covariance) of $Y_i$. Higher order moments are not specified.
 * Without an explicit likelihood function, to ''estimate the parameter'' vector $\beta$ (and perhaps the covariance parameter matrix $R_i(\alpha)$) and perform valid ''statistical inference'' that takes the dependence into consideration, we need to construct an unbiased estimating function:
-: $D_i(\beta) = \frac{\partial \mu_i}{\partial \beta}$
+: $D_i(\beta) = \frac{\partial \mu_i}{\partial \beta}$, the partial derivative, w.r.t. $\beta$, of the mean-model for subject $i$
 : $D_i(j,k) = \frac{\partial \mu_{i,j}}{\partial \beta_k}$
 : '''Estimating (cost) function''': $U(\beta)=\sum_{i=1}^N{D_i^T(\beta)V_i^{-1}(\beta,\alpha)\{Y_i-\mu_i(\beta)\}}$.