Overview of Experiment

I am conducting a meta-analysis on some task switching data to see if there is an age-related increase in an effect called the “congruency effect”. A simplified explanation of the experiments is that the task requires participants to switch between two tasks on single-digit stimuli: 1) Judge whether the number is lower/higher than 5; 2) Judge whether the number is odd/even. There are two response keys (left & right). Participants press the left key for an odd and a lower than 5 response; they press the right key for an even and a higher than 5 response.

Congruency Effect

Due to the way the stimuli are mapped to response keys, some stimuli are called congruent and some are incongruent: * congruent stimuli require the same response regardless of the currently-relevant task. For example, if the stimulus is the number 1, it is both odd and lower than 5, so it doesn’t matter what the correct task is, the correct response is always left. * incongruent stimuli require different responses for each task. For example, the number 7 is odd (left response) and higher than 5 (right response). So, the participant must ensure they attend to the relevant task so they can make the correct response.

Response times are faster to congruent stimuli than to incongruent stimuli. This is thought to reflect facilitated response selection during task switching performance.

Age Effects

I am interested in whether this congruency effect is influenced by healthy ageing. As such, I have conducted a meta-analysis of all papers that have reported congruency data.

# the data is in my dropbox
data <- read.csv("https://dl.dropboxusercontent.com/u/33530204/data_for_ray.csv")

head(data)

##             study congruency rt_younger rt_older
## 1   Capeda et al.  congruent        595    896.0
## 2   Capeda et al.  congruent        595   1021.0
## 3     Eich et al.  congruent        943   1053.0
## 4     Eich et al.  congruent       1015   1097.0
## 5 Eppinger et al.  congruent        594    859.5
## 6 Eppinger et al.  congruent        605    910.0

str(data)

## 'data.frame':    42 obs. of  4 variables:
##  $ study     : Factor w/ 11 levels "Capeda et al.",..: 1 1 2 2 3 3 3 3 4 5 ...
##  $ congruency: Factor w/ 2 levels "congruent","incongruent": 1 1 1 1 1 1 1 1 1 1 ...
##  $ rt_younger: num  595 595 943 1015 594 ...
##  $ rt_older  : num  896 1021 1053 1097 860 ...

Some studies had multiple experiments, so sometimes I have a few data sets from the same study.

Brinley Plots

In ageing research, people use Brinley Plots in meta-analyses to assess age-related increases in response time effects. This controls for the fact that generally older adults are slower at responding, so response time differences between two experimental conditions could just be due to being slower overall (i.e., it is therefore not an interesting difference; it’s just general slowing).

A Brinly plot is basically a regression plot, with one data point for each study and condition, predicting RT for older adults from RT for younger adults. The theoretical question of interest is whether the data are best described with one regression line (i.e., the same fit for “congruent” and “incongruent”) or whether two are required (i.e., one regression line for “congruent” and one for “incongruent”).

Below is a Brinley plot with fake data to show what I mean. Here, two regression lines are required for the data. This is evidence that there is an age-related decline that is above-and-beyond general slowing.

# get the fake data
fake_data <- read.csv("https://dl.dropboxusercontent.com/u/33530204/agingData.csv")


# make the Brinley Plot
library(ggplot2)
min_rt <- min(c(fake_data$rtYounger, fake_data$rtOlder))
max_rt <- max(c(fake_data$rtYounger, fake_data$rtOlder))

p <- ggplot(fake_data, aes(y = rtOlder, x = rtYounger, colour = condition))
p <- p + geom_point(aes(colour = condition))
p <- p + geom_smooth(method = lm, se = FALSE)
p <- p + coord_cartesian(xlim = c(0, max_rt), ylim = c(0, max_rt))
p <- p + theme(panel.background = element_rect(fill = "grey86"))
p <- p + scale_x_continuous("Response Time (Younger)") + 
  scale_y_continuous("Response Time (Older)")
p

Each data point represents the average response time for older adults plotted against the average response time for younger adults, for that condition, for that study.

Brinley Plot for My Data & LMMs

Here is the Brinley plot for the data I have from the meta-analysis.

min_rt <- min(c(data$rt_younger, data$rt_older))
max_rt <- max(c(data$rt_younger, data$rt_older))

p <- ggplot(data, aes(y = rt_older, x = rt_younger, colour = congruency))
p <- p + geom_point(aes(colour = congruency))
p <- p + geom_smooth(method = lm, se = FALSE)
# p <- p + geom_abline(intercept = 0, slope = 0.9, aes(colour = "blue"))
p <- p + coord_cartesian(xlim = c(0, max_rt), ylim = c(0, max_rt))
p <- p + theme(panel.background = element_rect(fill = "grey86"))
p <- p + scale_x_continuous("Response Time (Younger)") + 
  scale_y_continuous("Response Time (Older)")
p

It seems that only one line is required, but this uses standard linear modelling, not accounting for the fact that sometimes multiple data points come from the same study.

My Question

Basically, I guess I need an LMM with study as a random effect. What I need to do is construct a model comparison test to see whether one regression line is required or whether two is required (so, a main effect model). How do I do this in R?