“External” individual differences can be added as fixed effects

Example: National Youth Survey longitudinal data on tolerance for deviant behavior, exposure to deviant behavior, and gender. Subset of 16 subjects printed in Willett (1997, Table 11.1). (DeviantBehavior)

Tolerance was measured by asking whether it is wrong for someone their age to: cheat on tests, purposely destroy property of others, use marijuana, steal something worth less than $5, hit or threaten someone without reason, use alcohol, break into a building or vehicle to steal, sell hard drugs, or steal something worth more than $50.

##      SubjID      Gender      Exposure          Age       Tolerance   
##  S0009  : 5   Female:45   Min.   :0.810   Min.   :11   Min.   :1.00  
##  S0045  : 5   Male  :35   1st Qu.:0.922   1st Qu.:12   1st Qu.:1.22  
##  S0268  : 5               Median :1.145   Median :13   Median :1.50  
##  S0314  : 5               Mean   :1.191   Mean   :13   Mean   :1.62  
##  S0442  : 5               3rd Qu.:1.395   3rd Qu.:14   3rd Qu.:1.99  
##  S0514  : 5               Max.   :1.990   Max.   :15   Max.   :3.46  
##  (Other):50

“Internal” individual differences

• No “external” measure that can be entered as a fixed effect
• Individual differences needed for a different analysis (e.g., VLSM)

For such situations, random effects provide a way to quantify individual effect sizes in the context of a model of overall group performance.

A simple example:

Participant A: \(\zeta_{A1} - \zeta_{A0} = 1 - (-1) = 2\)

Participant B: \(\zeta_{B1} - \zeta_{B0} = (-1) - 1 = -2\)

“Internal” individual differences: Example

Data (Made-up): Effect of school mental health services on educational achievement (EducMH)

• Condition = Treatment (students who received mental health services) vs. Control (academically matched group of students who did not receive services)
• SDQ = Strengths and Difficulties Questionnaire: a brief behavioral screening for mental health, only available for Treatment group. Lower scores are better (Total difficulties).
• Math = Score on standardized math test

summary(EducMH)

##        ID           Condition        Year           SDQ            Math      
##  101    :   6   Control  :540   Min.   :2009   Min.   : 6.6   Min.   : 81.7  
##  102    :   6   Treatment:540   1st Qu.:2010   1st Qu.:16.2   1st Qu.:147.5  
##  103    :   6                   Median :2012   Median :17.9   Median :165.1  
##  104    :   6                   Mean   :2012   Mean   :17.9   Mean   :164.8  
##  105    :   6                   3rd Qu.:2013   3rd Qu.:19.6   3rd Qu.:181.2  
##  106    :   6                   Max.   :2014   Max.   :33.8   Max.   :259.3  
##  (Other):1044                                  NA's   :540

“Internal” individual differences: Example

Data (Made-up): Effect of school mental health services on educational achievement (EducMH)

Question 1: Did the school mental health services improve academic achievement? That is, did the two groups differ on math achievement at baseline and over the 6 years of the study?

Question 2: For the treatment group, was individual-level improvement in mental health associated with improvement in math scores?

“Internal” individual differences: Example

Question 1: Did the school mental health services improve academic achievement? That is, did the two groups differ on math achievement at baseline and over the 6 years of the study?

# adjust time variable to have a sensible intercept
EducMH$Time <- EducMH$Year - 2009
# fit the models
m.base <- lmer(Math ~ Time + (Time | ID), data=EducMH, REML=F)
m.0 <- lmer(Math ~ Time + Condition + (Time | ID), data=EducMH, REML=F)
m.1 <- lmer(Math ~ Time*Condition + (Time | ID), data=EducMH, REML=F)
# compare the models
anova(m.base, m.0, m.1)

## Data: EducMH
## Models:
## m.base: Math ~ Time + (Time | ID)
## m.0: Math ~ Time + Condition + (Time | ID)
## m.1: Math ~ Time * Condition + (Time | ID)
##        npar  AIC  BIC logLik deviance Chisq Df Pr(>Chisq)   
## m.base    6 7847 7876  -3917     7835                       
## m.0       7 7848 7883  -3917     7834  0.15  1     0.6947   
## m.1       8 7843 7883  -3914     7827  7.02  1     0.0081 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#summary(m.1)
coef(summary(m.1))

##                         Estimate Std. Error  df t value   Pr(>|t|)
## (Intercept)              149.297     2.0295 180 73.5636 3.006e-136
## Time                       5.216     0.3737 180 13.9560  1.776e-30
## ConditionTreatment         1.337     2.8701 180  0.4659  6.419e-01
## Time:ConditionTreatment    1.414     0.5285 180  2.6755  8.149e-03

There was no group difference at baseline, but there was a group difference on slope. That is, math achievement in the two groups started out the same, but increased more quickly in the Treatment group.

“Internal” individual differences: Example

Question 2: For the treatment group, was individual-level improvement in mental health associated with improvement in math scores?

First step: For the treatment group, plot change in the SDQ over time showing both group mean and individual variability

ggplot(subset(EducMH, Condition == "Treatment"), aes(Year, SDQ)) + 
  geom_line(aes(group=ID), color="gray") +
  stat_summary(fun=mean, geom="line") +
  stat_summary(fun.data=mean_se, geom="errorbar", width=0.3) +
  theme_bw(base_size=12)

Within the treatment group, it looks like there is lots of variability in response to the mental health services. Some people responded really well (big decreases in difficulties on SDQ), some people didn’t respond well (increased difficulties according to SDQ).

We want to know whether this variability is associated with variability in improved math achievement.

“Internal” individual differences: Example

Question 2: For the treatment group, was individual-level improvement in mental health associated with improvement in math scores?

Analysis strategy:

• Build separate models for change in SDQ and change in Math scores over time
• Use random effects to quantify individual differences in change over time for the two scores
• Test the correlation between change in SDQ and in Math achievement (and make a scatterplot showing this).

“Internal” individual differences: Example

Question 2: For the treatment group, was individual-level improvement in mental health associated with improvement in math scores?

Analysis strategy:

• Build separate models for change in SDQ and change in Math scores over time

m.math <- lmer(Math ~ Time + (Time | ID), 
               data=subset(EducMH, Condition == "Treatment"), REML=F)
m.sdq <- lmer(SDQ ~ Time + (Time | ID), 
              data=subset(EducMH, Condition == "Treatment"), REML=F)

“Internal” individual differences: Example

Question 2: For the treatment group, was individual-level improvement in mental health associated with improvement in math scores?

Analysis strategy:

• Build separate models for change in SDQ and change in Math scores over time
• Use random effects to quantify individual differences in change over time for the two scores

#psy811::get_ranef() will extract the named random effect and clean them up a bit
re.math <- get_ranef(m.math, "ID")
re.sdq <- get_ranef(m.sdq, "ID")
#merge() will combine those into one data frame, but needs some help because the variable names are all the same
re <- merge(re.math, re.sdq, by="ID", suffixes = c(".math", ".sdq"))
summary(re)

##        ID      Intercept.math     Time.math      Intercept.sdq    
##  Min.   :201   Min.   :-59.50   Min.   :-3.372   Min.   :-2.0326  
##  1st Qu.:223   1st Qu.: -9.85   1st Qu.:-0.733   1st Qu.:-0.7895  
##  Median :246   Median : -0.01   Median : 0.118   Median : 0.0341  
##  Mean   :246   Mean   :  0.00   Mean   : 0.000   Mean   : 0.0000  
##  3rd Qu.:268   3rd Qu.: 11.48   3rd Qu.: 0.694   3rd Qu.: 0.6024  
##  Max.   :290   Max.   : 57.64   Max.   : 2.747   Max.   : 3.0828  
##     Time.sdq      
##  Min.   :-2.4951  
##  1st Qu.:-0.7468  
##  Median : 0.0989  
##  Mean   : 0.0000  
##  3rd Qu.: 0.5482  
##  Max.   : 3.0944

head(re)

##    ID Intercept.math Time.math Intercept.sdq Time.sdq
## 1 201         0.8412 -1.542050       -0.9793   0.8381
## 2 202         9.6071  0.029529        0.5728   0.3952
## 3 203       -14.6657 -0.573849        0.5872   1.1176
## 4 204        -3.5781 -0.290299       -0.8056   0.4423
## 5 205         2.1411 -0.852670        0.4991   0.8463
## 6 206        31.4592  0.006922        0.2029   0.6119

“Internal” individual differences: Example

Question 2: For the treatment group, was individual-level improvement in mental health associated with improvement in math scores?

Analysis strategy:

• Build separate models for change in SDQ and change in Math scores over time
• Use random effects to quantify individual differences in change over time for the two scores
• Test the correlation between change in SDQ and in Math achievement (and make a scatterplot showing this).

cor.test(re$Time.math, re$Time.sdq)

## 
##  Pearson's product-moment correlation
## 
## data:  re$Time.math and re$Time.sdq
## t = -11, df = 88, p-value <2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.8455 -0.6749
## sample estimates:
##     cor 
## -0.7739

ggplot(re, aes(Time.math, Time.sdq)) + geom_point() + stat_smooth(method="lm") + 
  labs(x="Relative Rate of Increase in Math Score", 
       y="Relative Rate of Decrease in SDQ Total Difficulties Score") + 
  theme_bw(base_size=12)

## `geom_smooth()` using formula 'y ~ x'

Strong correlation (\(r = -0.77\), \(p < 0.0001\)) indicating that response to mental health intervention (decreased difficulties) was associated with larger increases in math achievement. Note that the key quantities here are slopes. That is, the rate of decreased mental health difficulties is associated with a higher rate of math achievement.