library(tidyverse)
library(lme4)
library(lmerTest)
library(gazer) # helper functions and example data sets
Mirman, D., Yee, E., Blumstein, S., and Magnuson, J.S. (2011). Theories of spoken word recognition deficits in aphasia: Evidence from eye-tracking and computational modeling. Brain and Language, 117, 53-68. pdf
These are cohort and rhyme competition data from 5 participants with Broca’s aphasia, 3 participants with Wernicke’s aphasia, and 12 age-matched controls (Mirman et al., 2011). The group means are shown below, with cohort competition in the top row and rhyme competition in the bottom row. Visually, it seems that the participants with Wernicke’s aphasia exhibited more cohort competition and less rhyme competition, whereas participants with Broca’s aphasia exhibited the opposite pattern.
The starting data frame is shown below.
summary(CohortRhyme)
## subjID Group Time timeBin Object
## 101 : 80 Control :960 Min. : 50 Min. : 1.00 Competitor:800
## 102 : 80 Broca :400 1st Qu.: 525 1st Qu.: 5.75 Unrelated :800
## 103 : 80 Wernicke:240 Median :1000 Median :10.50 Target : 0
## 104 : 80 Mean :1000 Mean :10.50
## 105 : 80 3rd Qu.:1475 3rd Qu.:15.25
## 106 : 80 Max. :1950 Max. :20.00
## (Other):1120
## Type FixProp
## Cohort:800 Min. :0.0000
## Rhyme :800 1st Qu.:0.0000
## Median :0.0417
## Mean :0.0665
## 3rd Qu.:0.1000
## Max. :0.5833
##
The first step is to create a fourth-order polynomial based on the timeBin values to capture the rise and fall of the competition curves:
When the individual differences are “experiment external”, such as aphasia subtype (or continuous variables like age), the best option is to add the individual difference variable as a fixed effect to the model. So we start with a base model of the overall competition effect (Object fixed effect) and add a fixed effect of Group. It would be a good idea to build up the Group effects gradually, but we’ll skip to the full model to keep this example more brief:
From the critical Object:Group parameter estimates below, we can see that this overall Group difference is due to the Wernicke’s aphasia group differing from the Control (baseline reference) group. We’re just going to look at the effects up to the quadratic term because the higher order terms are just capturing differences in the tails. Note that the parameter estimates are for the Unrelated object relative to the Related object, so a negative estimate for the Intercept term (e.g., ObjectUnrelated:GroupWernicke) corresponds to a larger competition effect (i.e., for the Wernicke’s group compared to the Control group, the Unrelated object has an even lower intercept than the Competitor does).
So far, we have established a double dissociation – one group has larger cohort competition, the other group has larger rhyme competition – but we can also ask whether there is an association between the effects. That is, do participants with larger cohort effects tend to show smaller rhyme effects? To test this kind of “experiment internal” individual differences, we can use the random effects to estimate each participants effect size and then test the correlation between the effect sizes.
Individual differences
The random effects are the systematic deviations from the “mean” pattern predicted by the fixed effects. So we’ll use the random effects from the base model, which did not distinguish between participant groups, so we can get individual effect sizes relative to the overall mean (otherwise we’d be looking at individual effect sizes relative to the sub-group mean). You can extract random effects from a model using the lme4::ranef function, but it’s somewhat more convenient to use gazer::get_ranef, which allows you to specify which random effects you want and makes them a little more convenient to work with:
To estimate individual participant effect sizes, we need to compute the differences between random effect estimates (Intercept, poly1, poly2) for Competitor vs. Unrelated objects for each participant:
ES.coh <- re.cohort %>% group_by(subjID) %>%
summarise(Coh.Intercept = Intercept[Object=="Competitor"] -
Intercept[Object=="Unrelated"],
Coh.Quadratic = poly2[Object=="Competitor"] -
poly2[Object=="Unrelated"])
ES.rhy <- re.rhyme %>% group_by(subjID) %>%
summarise(Rhy.Intercept = Intercept[Object=="Competitor"] -
Intercept[Object=="Unrelated"],
Rhy.Quadratic = poly2[Object=="Competitor"] -
poly2[Object=="Unrelated"])
ES <- merge(ES.coh, ES.rhy, by="subjID")
group <- unique(subset(CohortRhyme, select=c(subjID, Group))) #get group assignments from original data frame
ES <- merge(ES, group, by="subjID")
summary(ES)
## subjID Coh.Intercept Coh.Quadratic Rhy.Intercept
## Length:20 Min. :-0.049232 Min. :-0.30990 Min. :-0.046640
## Class :character 1st Qu.:-0.017123 1st Qu.:-0.08317 1st Qu.:-0.017738
## Mode :character Median :-0.009845 Median : 0.02630 Median :-0.009355
## Mean : 0.000000 Mean : 0.00000 Mean : 0.000000
## 3rd Qu.: 0.021994 3rd Qu.: 0.08961 3rd Qu.: 0.016755
## Max. : 0.068690 Max. : 0.14444 Max. : 0.067391
## Rhy.Quadratic Group
## Min. :-0.12373 Control :12
## 1st Qu.:-0.05222 Broca : 5
## Median : 0.01677 Wernicke: 3
## Mean : 0.00000
## 3rd Qu.: 0.05412
## Max. : 0.11931
Now we can examine a scatterplot of the individual cohort and rhyme competition effect sizes (effects on the intercept, shown on the right) and compute correlations across the complete sample and just for the participants with aphasia:
with(ES, cor.test(Coh.Intercept, Rhy.Intercept))
##
## Pearson's product-moment correlation
##
## data: Coh.Intercept and Rhy.Intercept
## t = -1.7573, df = 18, p-value = 0.09586
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.70569266 0.07204616
## sample estimates:
## cor
## -0.382675
with(subset(ES, Group != "Control"), cor.test(Coh.Intercept, Rhy.Intercept))
##
## Pearson's product-moment correlation
##
## data: Coh.Intercept and Rhy.Intercept
## t = -4.138, df = 6, p-value = 0.006092
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.9743601 -0.3959868
## sample estimates:
## cor
## -0.860535
with(subset(ES, Group == "Control"), cor.test(Coh.Intercept, Rhy.Intercept))
##
## Pearson's product-moment correlation
##
## data: Coh.Intercept and Rhy.Intercept
## t = 1.1352, df = 10, p-value = 0.2828
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.2928018 0.7636921
## sample estimates:
## cor
## 0.3378764
The correlation is significant for participants with aphasia (r = -0.86, p = 0.006), but not for control participants (r = 0.34, p > 0.25). This suggests that there may be a single mechanism behind this pattern, in contrast to the standard separable components interpretation of a double dissociation.
