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Growth Modeling

Growth Modeling. The aim of growth modeling is to look at the rate of change of a variable over time. Example : attitudes, profits, working speed, life satisfaction. Potential growth patterns can be linear, quadratic, cubic, logarithmic, exponential.

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Growth Modeling

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  1. Growth Modeling

  2. The aim of growth modeling is to look at the rate of change of a variable over time. Example : attitudes, profits, working speed, life satisfaction. Potential growth patterns can be linear, quadratic, cubic, logarithmic, exponential. The hierarchy in the data is that time points are nested within people (or other entities).

  3. Possible Growth Curves : Cubic Trend Quadratic Trend Linear Trend

  4. Two important things : Growth curve can be fitted up to one less than the number of time points that we have. Example : With three time points, a linear and quadratic growth curve can be fitted. 2. A growth curve is defined by a power function. Linear (Time) Quadratic (Time*Time) Cubic (Time*Time*Time) Quartic (Time*Time*Time*Time)

  5. Life Satisfaction Over Time

  6. LS_Data

  7. Person : 115 subject • Gender : 0, 1 • Time : 4 points of time : • Satisfaction Base (0) • Satisfaction 6 Months (1) • Satisfaction 12 Months (2) • Satisfaction 18 Months (3) • LS : Life Satisfaction on a 10-point scale (0 = completely dissatisfied, 10 = completely satisfied)

  8. Data setwd) satisfactionData <- read.delim("LS_Data.dat", header=TRUE) library("reshape") LS <- melt(satisfactionData,id=c("Person","Gender"), measured=c("Satisfaction_Base", "Satisfaction_6_Months","Satisfaction_12_Months", "Satisfaction_18_Months")) Time<-c(rep(0,115),rep(1,115),rep(2,115),rep(3,115)) LS$Time<-Time rm(Time) head(LS)

  9. Data LS_Data<- data.frame(Person = LS$Person, Gender=LS$Gender, Time =LS$Time, LS=LS$value) head(LS_Data)

  10. Nonlinear Mixed Effects Models (nlme) install.packages(“nlme”) library("nlme")

  11. Model 1 Intercept-only ; no predictors Model is specified as y ~ 1 Code : intercept <- gls(LS ~ 1, data=LS_Data, method="ML", na.action=na.exclude) Pred_Y1 <- predict(intercept) # Predicted Y / Regression Line Pred_Y1 <- as.vector(Pred_Y) summary(intercept) intervals(intercept)

  12. Model 1 head(LS_D1) Person Gender Time LS Pred_Y 1 1 0 0 6 5.988584 2 1 0 1 6 5.988584 3 1 0 2 5 5.988584 4 1 0 3 2 5.988584 5 2 1 0 7 5.988584 6 2 1 1 7 5.988584

  13. Model 2 Intercept vary across people : Code : randomIntercept <- lme(LS ~ 1, data = LS_Data, random = ~1|Person, method = "ML", na.action = na.exclude, control = list(opt="optim"))

  14. Model 2 Pred_Y2 <- predict(randomIntercept) Pred_Y2 <- as.vector(Pred_Y2) summary(randomIntercept) intervals(randomIntercept)

  15. Model 2 head(LS_D2) Person Gender Time LS Pred_Y 1 1 0 0 6 5.084936 2 1 0 1 6 5.084936 3 1 0 2 5 5.084936 4 1 0 3 2 5.084936 5 2 1 0 7 6.355934 6 2 1 1 7 6.355934

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  17. Model 3 Random Intercept across people + Time as a predictor : Time as a fixed effect Code : timeRI <- update(randomIntercept, .~. + Time) timeRI <- lme(LS ~ Time, data = LS_Data, random = ~1|Person, method = "ML", na.action = na.exclude, control = list(opt="optim"))

  18. Model 3 Code : Pred_Y3 <- predict(timeRI) Pred_Y3 <- as.vector(Pred_Y3) summary(timeRI) intervals(timeRI) coef(timeRI)

  19. Model 3 head(coef(timeRI)) (Intercept) Time 1 6.269597 -0.8772182 2 7.707617 -0.8772182 3 5.242440 -0.8772182 4 6.475028 -0.8772182 5 7.502185 -0.8772182 6 6.680460 -0.8772182

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  21. Growth Modeling Rate of development /growth over time differs within entities

  22. Model 4 Effect of Time across people : timeRS <- update(timeRI, random = ~Time|Person) or timeRS <- lme(LS ~ Time, data = LS_Data, random = ~Time|Person, method = "ML", na.action = na.exclude, control = list(opt="optim"))

  23. Model 4 Pred_Y4 <- predict(timeRS) Pred_Y4 <- as.vector(Pred_Y4) summary(timeRS) intervals(timeRS) coef(timeRS)

  24. Model 4 head(coef(timeRS)) (Intercept) Time 1 6.338484 -0.9234636 2 7.705109 -0.8732796 3 5.198861 -0.8586077 4 6.725606 -1.0345095 5 7.317987 -0.7622338 6 6.814232 -0.9616653

  25. Model 5 First Order Autoregressive ARModel <- update(timeRS, correlation = corAR1(0, form = ~Time|Person)) ARModel <- lme(LS ~ Time, data = LS_Data, random = ~Time|Person, method = "ML", na.action = na.exclude, control = list(opt="optim"), correlation = corAR1(0, form = ~Time|Person))

  26. Model 5 Pred_Y5 <- predict(ARModel) Pred_Y5 <- as.vector(Pred_Y5) summary(ARModel) intervals(ARModel) coef(ARModel)

  27. Model 5 head(coef(ARModel)) (Intercept) Time 1 6.296443 -0.8711929 2 7.528736 -0.8702692 3 5.418097 -0.8692794 4 6.414589 -0.8733059 5 7.409368 -0.8674247 6 6.569258 -0.8714521

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  29. Comparing 5 models : anova(intercept, randomIntercept, timeRI, timeRS, ARModel)

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  31. Quadratic TrendModel 6 If time is the predictor variable, a quadratic is tested by including a predictor timeQuadratic <- update(ARModel, .~. + I(Time^2)) Or : timeQuadratic <- lme(LS ~ Time+I(Time^2), data = LS_Data, random = ~Time|Person, method = "ML", na.action= na.exclude, control = list(opt="optim"), correlation = corAR1(0, form = ~Time|Person))

  32. Cubic TrendModel 7 timeCubic<-update(timeQuadratic, .~. + I(Time^3)) timeCubic<- lme(LS ~ Time+I(Time^2)+I(Time^3), data = LS_Data, random = ~Time|Person, method = "ML", na.action= na.exclude, control = list(opt="optim"), correlation = corAR1(0, form = ~Time|Person))

  33. Comparing 3 models : anova(ARModel, timeQuadratic, timeCubic)

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  35. Quadratic TrendModel 8 Update Model 3 : Mod8 <- update(timeRI, .~. + I(Time^2)) Mod8 <- lme(LS ~ Time+I(Time^2), data = LS_Data, random = ~1|Person, method = "ML", na.action= na.exclude, control = list(opt="optim"))

  36. Model 8 summary(Mod8) intervals(Mod8) Pred_Y8 <- predict(Mod8) Pred_Y8 <- as.vector(Pred_Y8)

  37. Model 8 coef(Mod8) head(coef(Mod8)) (Intercept) Time I(Time^2) 1 5.739092 0.7352972 -0.5512887 2 7.232865 0.7352972 -0.5512887 3 4.672111 0.7352972 -0.5512887 4 5.952488 0.7352972 -0.5512887 5 7.019469 0.7352972 -0.5512887 • 6.165884 0.7352972 -0.5512887

  38. Model 8 # Data frame LS_D8 <- data.frame(Person = LS_Data$Person[c(1,116,231,346,2,117,232,347,3,118,233,348,4,119,234,349,5,120,235,350,6,121,236,351)], Gender=LS_Data$Gender[c(1,116,231,346,2,117,232,347,3,118,233,348,4,119,234,349,5,120,235,350,6,121,236,351)], Time =LS_Data$Time[c(1,116,231,346,2,117,232,347,3,118,233,348,4,119,234,349,5,120,235,350,6,121,236,351)], LS=LS_Data$LS[c(1,116,231,346,2,117,232,347,3,118,233,348,4,119,234,349,5,120,235,350,6,121,236,351)], Pred_Y=Pred_Y8[c(1,116,231,346,2,117,232,347,3,118,233,348,4,119,234,349,5,120,235,350,6,121,236,351)])

  39. Model 8 head(LS_D8) Person Gender Time LS Pred_Y 1 1 0 0 6 5.739092 2 1 0 1 6 5.923100 3 1 0 2 5 5.004531 4 1 0 3 2 2.983385 5 2 1 0 7 7.232865 6 2 1 1 7 7.416873

  40. Model 8 ## GGPLOT library(ggplot2) plot <- ggplot(LS_D8, aes(x=LS_D8[,3], y=LS_D8[,5],colour=factor(LS_D8[,1]))) plot + layer(geom="point") + layer(geom="line")+ xlab("Time")+ylab("LS")+ ggtitle("Regression Line / Predicted Y")+labs(colour= "Person")

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  42. Cubic TrendModel 9 Mod9 <-update(Mod8, .~. + I(Time^3)) Mod9 <- lme(LS ~ Time+I(Time^2) +I(Time^3), data = LS_Data, random = ~1|Person, method = "ML", na.action= na.exclude, control = list(opt="optim"))

  43. Model 9 Code : summary(Mod9) intervals(Mod9) Pred_Y9 <- predict(Mod9) Pred_Y9 <- as.vector(Pred_Y9)

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