1012_Homework Exercise 2

1 Description

Eighty-nine children’s scores on the reading subtest of the Peabody Individual Achievement Test (PIAT) were recorded. Each child was 6 years old in 1986, the first year** of data collection. | Column 1: Child ID
| Column 2: Wave (1 = 1986, 2 = 1988, 3 = 1990)
| Column 3: Child’s expected age on each measurement occasion
| Column 4: Age in years
| Column 5: Child’s PIAT score

2 Input Data

# Loading package
pacman::p_load(tidyverse, afex, segmented, foreign, haven, lme4, lmerTest, ggplot2, Rmisc, car)

# data input
dta2 <- read.table("C:/Users/HANK/Desktop/HOMEWORK/reading_piat.txt", header = T)

head(dta2)

##   ID Wave Age_G     Age PIAT
## 1  1    1   6.5  6.0000   18
## 2  1    2   8.5  8.3333   35
## 3  1    3  10.5 10.3333   59
## 4  2    1   6.5  6.0000   18
## 5  2    2   8.5  8.5000   25
## 6  2    3  10.5 10.5833   28

str(dta2)

## 'data.frame':    267 obs. of  5 variables:
##  $ ID   : int  1 1 1 2 2 2 3 3 3 4 ...
##  $ Wave : int  1 2 3 1 2 3 1 2 3 1 ...
##  $ Age_G: num  6.5 8.5 10.5 6.5 8.5 10.5 6.5 8.5 10.5 6.5 ...
##  $ Age  : num  6 8.33 10.33 6 8.5 ...
##  $ PIAT : int  18 35 59 18 25 28 18 23 32 18 ...

dta2 <- dta2 %>%
  mutate(Age_c = Age - 6.5)

3 plot

ggplot(dta2, aes(Age, PIAT, group = ID, color = Wave))+
  geom_point(size = rel(1))+ 
  geom_line() +
  labs(x = "Age (year)", y = "Score") +
  theme(legend.position = c(.9, .2))

4 LMER

Model:

Scoreij = b0i + b1i × (ageij - 6.5) + εij,

b0i = β0i + U0i, b1i = β1i + U1i, i = 1, 2, …, 89; j = 1, 2, 3,

where εij follows a standard normal distribution with mean zero and SD sigma and (U0i, U1i) is a bivariate normal distribution with a zero mean vector with an unknown covariance matrix.

summary(m1 <- lmer(PIAT ~ Age_c + (Age_c | ID), data = dta2))

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: PIAT ~ Age_c + (Age_c | ID)
##    Data: dta2
## 
## REML criterion at convergence: 1804.1
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.0042 -0.4893 -0.1383  0.4067  3.6892 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr
##  ID       (Intercept)  5.507   2.347        
##           Age_c        3.377   1.838    0.53
##  Residual             27.400   5.235        
## Number of obs: 267, groups:  ID, 89
## 
## Fixed effects:
##             Estimate Std. Error      df t value Pr(>|t|)    
## (Intercept)  21.0621     0.5630 75.7425   37.41   <2e-16 ***
## Age_c         4.5399     0.2622 87.7555   17.32   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##       (Intr)
## Age_c -0.288