Demonstration of Clustered Data

library(lme4) # MLM

library(ggplot2) # beautiful graphs

library(gganimate) # animated ggplots

library(plotly) # animated graphs

# library(broom)

library(pander) # nice tables

library(sjPlot) # nice tables for MLM

1 Grouped and Individual Data

“The data were generated from random numbers, and there is no relation between X and Y at all. Firstly, values of X and Y were generated for each ‘subject,’ then a further random number was added to make the individual observation.”

From Bland and Altman, BMJ, 1994, 308, 896.

So… we follow their procedure.

set.seed(3846) # random seed

g <- seq(1, 5) # number of groups

x <- rnorm(5, 100, 10) # group x

y <- rnorm(5, 100, 10) # group y

# 5 obs / group

id <- seq(1, 25) 

group_num <- rep(g, 1, each=5)

x_group <- rep(x, each = 5) # x for the group

y_group <- rep(y, each = 5) # y for the group

x_noise <- rnorm(25, 0, 1) # random noise

y_noise <- rnorm(25, 0, 1) # random noise

x_individual <- x_group + x_noise # individual = group + noise

y_individual <- y_group + y_noise # individual = group + noise

mydata <- data.frame(id, 
                     group_num,
                     x_group, 
                     y_group, 
                     x_individual, 
                     y_individual)

The animation below illustrates the process of simulating the data.

ggplot(mydata,
       aes(color = factor(group_num),
           label = group_num)) +
  geom_point(aes(x = x_group, 
                 y = y_group),
             size = 10,
             show.legend = FALSE) + 
  geom_text(aes(x = x_group,
                y = y_group),
            color = "white",
            show.legend = FALSE) +
  geom_point(aes(x = x_individual, 
                y = y_individual),
             show.legend = FALSE) + 
  transition_layers(layer_length = 1,
                    transition_length = 7) +
  enter_fade() +
  enter_grow() +
  exit_fade() +
  exit_shrink() +
  labs(title = "Illustrating the Process of Simulating the Data",
       x = "x",
       y = "y") +
  theme_minimal() +
  scale_color_viridis_d(name = "group")

2 Analyses

2.1 OLS

myOLS <- lm(y_individual ~ x_individual, data = mydata)

# summary(myOLS)

sjPlot::tab_model(myOLS,
                  show.se = TRUE,
                  show.ci = FALSE,
                  show.stat = TRUE)

	y_individual
Predictors	Estimates	std. Error	Statistic	p
(Intercept)	4.49	12.44	0.36	0.722
x individual	1.05	0.14	7.75	<0.001
Observations	25
R2 / R2 adjusted	0.723 / 0.711

# pander(tidy(myOLS))

2.2 MLM

myMLM <- lmer(y_individual ~ x_individual + (1 | group_num), 
              data = mydata)

# summary(myMLM)

sjPlot::tab_model(myMLM,
                  show.se = TRUE,
                  show.ci = FALSE,
                  show.stat = TRUE)

	y_individual
Predictors	Estimates	std. Error	Statistic	p
(Intercept)	98.71	15.74	6.27	<0.001
x individual	0.02	0.16	0.12	0.903
Random Effects
σ2	0.62
τ00 group_num	97.39
ICC	0.99
N group_num	5
Observations	25
Marginal R2 / Conditional R2	0.000 / 0.994

# pander(tidy(myMLM))

2.3 Compare OLS and MLM

tab_model(myOLS, myMLM,
          dv.labels = c("OLS", "MLM"), 
          show.se = TRUE,
          show.ci = FALSE,
          show.stat = TRUE)

	OLS				MLM
Predictors	Estimates	std. Error	Statistic	p	Estimates	std. Error	Statistic	p
(Intercept)	4.49	12.44	0.36	0.722	98.71	15.74	6.27	<0.001
x individual	1.05	0.14	7.75	<0.001	0.02	0.16	0.12	0.903
Random Effects
σ2					0.62
τ00					97.39 group_num
ICC					0.99
N					5 group_num
Observations	25				25
R2 / R2 adjusted	0.723 / 0.711				0.000 / 0.994