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Model-based Analytics - Growth Curve Models

Latent growth curve models may be used to analyze longitudinal or temporal data where the outcome measure is assessed on multiple occasions, and we examine its change over time, e.g., the trajectory over time can be modeled as a linear or quadratic function. Random effects are used to capture individual differences by conveniently representing (continuous) latent variables, aka growth factors. To fit a linear growth model we may specify a model with two latent variables: a random intercept, and a random slope:

#load data   05_PPMI_top_UPDRS_Integrated_LongFormat.csv ( dim(myData) 661  71), wide 
# setwd("/dir/")
myData <- read.csv("https://umich.instructure.com/files/330395/download?download_frd=1&verifier=v6jBvV4x94ka3EYcGKuXXg5BZNaOLBVp0xkJih0H",header=TRUE)
attach(myData)

# dichotomize the "ResearchGroup" variable
table(myData$\$$ResearchGroup)
 myData$\$$ResearchGroup <- ifelse(myData$\$$ResearchGroup == "Control", 1, 0)

# linear growth model with 4 timepoints
# intercept (i) and slope (s) with fixed coefficients
# i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4 (intercept/constant)
# s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4  (slope/linear term)
# ??? =~ 0*t1 + 1*t2 + 2*t3 + 3*t4  (quadratic term)

In this model, we have fixed all the coefficients of the linear growth functions:

model4 <-
' 
i =~ 1*UPDRS_Part_I_Summary_Score_Baseline + 1*UPDRS_Part_I_Summary_Score_Month_03 + 
1*UPDRS_Part_I_Summary_Score_Month_06 + 1*UPDRS_Part_I_Summary_Score_Month_09 + 
1*UPDRS_Part_I_Summary_Score_Month_12 + 1*UPDRS_Part_I_Summary_Score_Month_18 + 
1*UPDRS_Part_I_Summary_Score_Month_24 + 
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Baseline + 
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_03 + 
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_06 + 
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_09 + 
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_12 + 
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_18 +
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_24 + 
1*UPDRS_Part_III_Summary_Score_Baseline + 1*UPDRS_Part_III_Summary_Score_Month_03 + 
1*UPDRS_Part_III_Summary_Score_Month_06 + 1*UPDRS_Part_III_Summary_Score_Month_09 + 
1*UPDRS_Part_III_Summary_Score_Month_12 + 1*UPDRS_Part_III_Summary_Score_Month_18 + 
1*UPDRS_Part_III_Summary_Score_Month_24 + 
1*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Baseline + 
1*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Month_06 + 
1*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Month_12 + 
1*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Month_24 + 
1*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Baseline + 
1*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Month_06 +
1*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Month_12 + 
1*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Month_24 
s =~ 0*UPDRS_Part_I_Summary_Score_Baseline + 1*UPDRS_Part_I_Summary_Score_Month_03 + 
2*UPDRS_Part_I_Summary_Score_Month_06 + 3*UPDRS_Part_I_Summary_Score_Month_09 + 
4*UPDRS_Part_I_Summary_Score_Month_12 + 5*UPDRS_Part_I_Summary_Score_Month_18 + 
6*UPDRS_Part_I_Summary_Score_Month_24 +
0*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Baseline + 
1*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_03 + 
2*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_06 + 
3*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_09 + 
4*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_12 + 
5*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_18 +         
6*UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Month_24 + 
0*UPDRS_Part_III_Summary_Score_Baseline + 1*UPDRS_Part_III_Summary_Score_Month_03 + 
2*UPDRS_Part_III_Summary_Score_Month_06 + 3*UPDRS_Part_III_Summary_Score_Month_09 + 
4*UPDRS_Part_III_Summary_Score_Month_12 + 5*UPDRS_Part_III_Summary_Score_Month_18 + 
6*UPDRS_Part_III_Summary_Score_Month_24 + 
0*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Baseline + 
2*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Month_06 + 
4*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Month_12 +
6*X_Assessment_Non.Motor_Epworth_Sleepiness_Scale_Summary_Score_Month_24 +
0*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Baseline + 
2*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Month_06 + 
4*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Month_12 + 
6*X_Assessment_Non.Motor_Geriatric_Depression_Scale_GDS_Short_Summary_Score_Month_24
'

fit4 <- growth(model4, data=myData)
summary(fit4)
parameterEstimates(fit4)	# extracts the values of the estimated parameters, the standard errors, 
# the z-values, the standardized parameter values, and returns a data frame	
fitted(fit4)	# return the model-implied (fitted) covariance matrix (and mean vector) of a fitted model

# resid() function return (unstandardized) residuals of a fitted model including the difference between 
# the observed and implied covariance matrix and mean vector
resid(fit4)

Measures of model quality (Comparative Fit Index (CFI), Root Mean Square Error of Approximation (RMSEA))

# report the fit measures as a signature vector: Comparative Fit Index (CFI), Root Mean Square Error of 
# Approximation (RMSEA)
fitMeasures(fit4, c("cfi", "rmsea", "srmr"))

Comparative Fit Index (CFI) is an incremental measure directly based on the non-centrality measure. If d = χ2(df) where df are the degrees of freedom of the model, the Comparative Fit Index is:

FIX THIS!!!!!!!!!!!!!!!!(d(Null Model) - d(Proposed Model))/(d(Null Model)).

0≤CFI≤1 (by definition). It is interpreted as:

• CFI<0.9 - model fitting is poor.

• 0.9≤CFI≤0.95 is considered marginal,

• CFI>0.95 is good.

CFI is a relative index of model fit – it compare the fit of your model to the fit of (the worst) fitting null model.

Root Mean Square Error of Approximation (RMSEA) - “Ramsey”

An absolute measure of fit based on the non-centrality parameter: FIX EQUATION!!!!>√((χ2 - df)/(df×(N - 1))) ,

where N the sample size and df the degrees of freedom of the model. If χ² < df, then the RMSEA∶=0. It has a penalty for complexity via the chi square to df ratio. The RMSEA is a popular measure of model fit.

• RMSEA < 0.01, excellent,

• RMSEA < 0.05, good

• RMSEA > 0.10 cutoff for poor fitting models

Standardized Root Mean Square Residual (SRMR) is an absolute measure of fit defined as the standardized difference between the observed correlation and the predicted correlation. A value of zero indicates perfect fit. The SRMR has no penalty for model complexity. SRMR <0.08 is considered a good fit.

# inspect the model results (report parameter table)
inspect(fit4)

#install.packages("semTools")
# library("semTools")

A Simpler Model (fit5)

model5 <- '
 # intercept and slope with fixed coefficients
i =~ UPDRS_Part_I_Summary_Score_Baseline + UPDRS_Part_I_Summary_Score_Month_03 + UPDRS_Part_I_Summary_Score_Month_24
s =~ 0*UPDRS_Part_I_Summary_Score_Baseline + 1*UPDRS_Part_I_Summary_Score_Month_03 + 6*UPDRS_Part_I_Summary_Score_Month_24
 # regressions
i ~  R_fusiform_gyrus_Volume + Weight + ResearchGroup + Age + chr12_rs34637584_GT                                                              
s ~ R_fusiform_gyrus_Volume + Weight + ResearchGroup + Age + chr12_rs34637584_GT
 # time-varying covariates
   UPDRS_Part_I_Summary_Score_Baseline ~ Weight
   UPDRS_Part_I_Summary_Score_Month_03  ~ ResearchGroup 
    UPDRS_Part_I_Summary_Score_Month_24 ~ Age
'

fit5 <- growth(model5, data=myData)
summary(fit5); fitMeasures(fit5, c("cfi", "rmsea", "srmr"))
parameterEstimates(fit5)	# extracts the values of the estimated parameters, the standard errors, 
# the z-values, the standardized parameter values, and returns a data frame

lavaan (0.5-18) converged normally after  99 iterations
 Number of observations                           661
 Estimator                                         ML
 Minimum Function Test Statistic                3.703
 Degrees of freedom                                 1
 P-value (Chi-square)                           0.054
Parameter estimates:
 Information                                 Expected
 Standard Errors                             Standard
                  Estimate  Std.err  Z-value  P(>|z|)
Latent variables:
 i =~
   UPDRS_P_I_S_S     1.000
   UPDRS_P_I_S_S     1.074
   UPDRS_P_I_S_S     1.172
 s =~
   UPDRS_P_I_S_S     0.000
   UPDRS_P_I_S_S     1.000
   UPDRS_P_I_S_S     6.000

Regressions:
 i ~
   R_fsfrm_gyr_V     0.000
   Weight            0.003
   ResearchGroup    -0.880
   Age              -0.009
   c12_34637584_    -0.907
 s ~
   R_fsfrm_gyr_V    -0.000
   Weight           -0.000
   ResearchGroup    -0.084
   Age               0.002
   c12_34637584_    -0.047
 UPDRS_Part_I_Summary_Score_Baseline ~
   Weight           -0.000
 UPDRS_Part_I_Summary_Score_Month_03 ~
   ResearchGroup     0.693
 UPDRS_Part_I_Summary_Score_Month_24 ~
   Age              -0.002

Covariances:
 i ~~
   s                 0.074

Intercepts:
   UPDRS_P_I_S_S     0.000
   UPDRS_P_I_S_S     0.000
   UPDRS_P_I_S_S     0.000
   i                 1.633
   s                -0.023

Variances:
   UPDRS_P_I_S_S     1.017
   UPDRS_P_I_S_S     1.093
   UPDRS_P_I_S_S     2.993
   i                 1.019
   s                -0.025

 cfi rmsea  srmr
0.996 0.064 0.008

fitted(fit5)	# return the model-implied (fitted) covariance matrix (and mean vector) of a fitted model
# write.table(fitted(fit5), file="C:\\Users\\Dinov\\Desktop\\test1.txt")

# resid() function return (unstandardized) residuals of a fitted model including the difference between 
# the observed and implied covariance matrix and mean vector
resid(fit5)

# report the fit measures as a signature vector
fitMeasures(fit5, c("cfi", "rmsea", "srmr"))   # comparative fit index (CFI)

# inspect the model results (report parameter table)
inspect(fit5)

Note: See discussion of SEM modeling pros/cons.

Generalized Estimating Equation (GEE) Modeling

Generalized Estimating Equations (GEE) modeling is used for analyzing data with the following characteristics: (1) the observations within a group may be correlated, (2) observations in separate clusters are independent, (3) a monotone transformation of the expectation is linearly related to the explanatory variables, and (4) the variance is a function of the expectation. The expectation (#3) and the variance (# 4) are conditional given group-level or individual-level covariates.

GEE is applied to handle correlated discrete and continuous outcome variables. For the outcome variables, it only requires specification of the first 2 moments and correlation among them. The goal is to estimate fixed parameters without specifying their joint distribution. The correlation is specified by one of these 4 alternatives (which is specified in the R call: geeglm(outcome ~ center + treat + sex + baseline + age, data = respiratory, family = "binomial", id = id, corstr = " exchangeable", scale.fix = TRUE):

Respiratory Illness GEE R example

This example is based on a data set on respiratory illness and the geepack package. The data is from a clinical study of the treatment effects on patients with respiratory illness. N=111 patients from 2 clinical centers randomized to receive either placebo or active treatments. 4 temporal examinations assessed the respiratory state of patients as good (=1) or poor (=0). Explanatory variables characterizing a patient were: center (1,2), treatment (A=active, P=placebo), sex (M=male, F=female), age (in years) at baseline. The values of the covariates were constant for the repeated elementary observations on each patient.

Table 1 shows the number of patients for the response patterns across the 4 visits split by baseline-status and treatment. Baseline respiratory status = 0 appear to have either low or high number of positive responses. Baseline respiratory status = 1 tend to respond positively. Table 2 describes the distribution of the number of positive responses per patient for sex and center.

See also

• Back to Model-based Analytics
• Structural Equation Modeling (SEM)
• Next Section: Generalized Estimating Equation (GEE) Modeling

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• SOCR Home page: http://www.socr.umich.edu

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