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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

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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