Introduction to mixed-effects modelling
Christophe Pallier
2017-05-29
Plan
Meaning of "X has an effect on Y" mean
Example
Sampling
Inference
Classical Simple Regression
Graphical Representation
Example
Ordinary Least Squares solution to Regression
Prediction
The equation \( \alpha.x+\beta \) can be used to compute the expected value of \( Y \) for a given value \( x \) of \( X \)
plot(y ~ x, bty='l', pch=16)
abline(fit, col=2, lwd=2)
Hyothesis testing in simple regression
Testing the hypothesis of a non-null slope
Estimates and standard errors of coefficients
Remark 1: X can be binary
The regression regression of \( Y \) on \( X \) is strictly equivalent to a Student T test.
x = rep(1:2, 20)
y = x + rnorm(40)
m1 <- lm(y ~ x)
plot(y ~ x, pch=16, xlim=c(.5, 2.5))
abline(m1)
Case of binary X
Remark 2: testing for non linear effect of X
Remark 2: testing for non linear effect of X
plot(x, y)
lines(x, fitted(m1), col=2) # linear
lines(x, fitted(m2), col=3) # polynomial deg 2
Remark 3: Non-parametric analyses as an alternative
plot(y ~ x, bty='l', pch=16)
lines(lowess(x, y), col='blue', lwd=2)
Remark 4: nicer plot with ggplot2
Multiple regression
Interpretation of coefficients
Recommended readings for linear modelling
An issue with repeated measurements
An issue with repeated measurements
Justifications for averaging over repeated measurements
One issue with averaging
Fixed and Random factors
The score model
Mixed-effects
Advantages of mixed-effect modelling over ANOVA and rANOVA
Drawback of mixed effect modelling
First example
First example
First example
Classical Anova
bysub <- aggregate(y ~ cond + sub, data=data1, mean)
with(bysub, { print(tapply(y, cond, mean)); interaction.plot(cond, sub, y)})
1 2
88.45269 106.35811
Classical anova
Mixed-effects model
Estimated subject's' effects
$sub
We assumed that the effect of condition was the same across all subjects, but we can fit a more flexible model allowing for a random effect of cond within subject
Fitting a more flexible model
Second example: simple regressions within subjects
To test if a continuous predictor \( X \) has an effect on a continuous variable \( Y \), the classic linear regression framework assumes that measurements are independent.
x <- rnorm(100)
y <- 5 + x + rnorm(100)
plot(y ~ x, pch=16)
summary(lm1 <- lm(y ~ x))
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-3.04035 -0.52200 -0.02043 0.72914 1.91953
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.88012 0.09464 51.57 <2e-16 ***
x 1.03755 0.09589 10.82 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.944 on 98 degrees of freedom
Multiple R-squared: 0.5443, Adjusted R-squared: 0.5397
F-statistic: 117.1 on 1 and 98 DF, p-value: < 2.2e-16
abline(lm1)
Clustered measurements
Let us suppose that the measures are “clustered”, that is, come from different participants. The independence assumption is violated.
Classical approaches: group as a fixed factor
Classical approach II: random regressions
More principled approach: fit a multilevel regression
Price of houses as a function of latitude (Galwey; 2006)
Models with several random factors are not a novelty !
Expected Mean Squares
Expected Mean squares for SxW<T> design
Notice that there is no simple ratio between sources of variance to test the hypothesis that \( \sigma_t > 0 \).
For example, EMS\( _T \) exceeeds EMS\( _{T{\rm x}S} \) by \( \sigma_w^2 + qr\sigma_t^2 \).
MinF'
Example of a lexical decision experiment
Filtering out mistakes and long RTs
nrow(lexdec)
[1] 1659
plot(density(lexdec$RT))
lexdec2 <- subset(lexdec, RT < 7)
nrow(lexdec2)
[1] 1618
lexdec3 <- subset(lexdec2, Correct=="correct")
nrow(lexdec3)
[1] 1557
Inspect subject data to see if there is an effect of Trial
