Today's data: Sleep Study

• Reaction: RT in ms
• Days: day 0 is normal sleep baseline (interval da -> Numeric)
• Subject: numbered (categorical, non ordinal -> Factor)

In R

> # example data providedy by lme4
> library(lme4)
> data(sleepstudy)
> # for more info, try ?sleepstudy
> library(lattice)  # provides some nice plotting functions

> str(sleepstudy)

## 'data.frame':    180 obs. of  3 variables:
##  $ Reaction: num  250 259 251 321 357 ...
##  $ Days    : num  0 1 2 3 4 5 6 7 8 9 ...
##  $ Subject : Factor w/ 18 levels "308","309","310",..: 1 1 1 1 1 1 1 1 1 1 ...

Make a linear model

• basic line, no error term: y = mx + b
• dep = slope*indep + baseline.offset
• outcome = (model) + error

Fit a line

• Fit a line to observed data with magic and matrices:
  • Y = β₁X + β₀ + ε
  • Y = β₂X + β₁X + β₀ + ε
  • Y = β₃X + β₂X + β₁X + β₀ + ε
  • ...

• R has this built in:

  > sleep.lm <- lm(Reaction ~ Days, data = sleepstudy)

• additional predictors with + (no interaction) or * (interaction)

Add a regression line with lattice graphics

> # update() effectively performs the previous call for us PLUS other
> # options we add now
> sleep.xy <- update(sleep.xy, type = c("p", "r"))  # p for points, r for regression
> sleep.xy

Models for single subjects

> sleep.xy.bysubj <- xyplot(Reaction ~ Days|Subject,
+                           data=sleepstudy,
+                           xlab = "Days of sleep deprivation",
+                           ylab = "Average reaction time (ms)")
> sleep.xy.bysubj

With regression lines

> sleep.xy.bysubj <- update(sleep.xy.bysubj, type = c("p", "r"))
> sleep.xy.bysubj

Fixed vs Random Effects

• Model subjects as fixed effect?
  • only when we want to make intrasample predictions
  • i.e. sample==population
  • fixed means known variance / manipulation
  • fixed-effects: directed, preferably "exhaustive" manipulation

• Model subjects as random effects?
  • "random" means unknown variance
  • error term is a random effect
  • correction for the error resulting from our particular choice of sample
  • correction per grouping for slope and intercept possible
  • error term per grouping!

Random effect structure

• combine by-subject and by-item analyses in one step
• cf. Clark (1973)

Random effect structure

• Early idea: build up from minimal structure until improvements don't bring you anything on ANOVA (Baayen et al. 2008 )
• New idea: Use the most complicated random effects structure possible (Barr et al. 2013 )

Random effect structure

• Possible random effect structures for ONE fixed factor:
  1. Intercepts only by random factor:
     (1 | random.factor)
  2. Slopes only by random factor:
     (0 + fixed.factor | random.factor)
  3. Intercepts and slopes by random factor:
     (1 + fixed.factor | random.factor)
  4. Intercept and slope, separately, by random factor:
     (1 | random.factor) + (0 + fixed.factor | random.factor)

Relationship to ANOVA

• ezANOVA() depends on aov() which depends on lm()
• anova() can be used to compare existing lm()s
• linear models compared with F and t tests
• no continuous predictors with ANOVA
• ANOVA works on per-subject item averages and examines variance over subjects for each condition
• MEMs work at an individual trial level and can accomodate empty cells and unbalanced designs!

ezANOVA()

> # hide startup output
> suppressPackageStartupMessages(library(ez))
> sleep.ez <- ezANOVA(sleepstudy,
+                     dv=.(Reaction),
+                     wid=.(Subject),
+                     within=.(Days),
+                     detailed=TRUE,
+             # we'll take a look at the aov() in a sec
+                     return_aov=TRUE) 

## Warning: "Days" will be treated as numeric.

## Warning: There is at least one numeric within variable, therefore aov()
## will be used for computation and no assumption checks will be obtained.

> sleep.ez$ANOVA

##   Effect DFn DFd    SSn   SSd     F         p p<.05    ges
## 1   Days   1  17 162703 60322 45.85 3.264e-06     * 0.7295

aov() from ezANOVA()

> sleep.ez$aov

## 
## Call:
## aov(formula = formula(aov_formula), data = data)
## 
## Grand Mean: 298.5
## 
## Stratum 1: Subject
## 
## Terms:
##                 Residuals
## Sum of Squares     250618
## Deg. of Freedom        17
## 
## Residual standard error: 121.4
## 
## Stratum 2: Subject:Days
## 
## Terms:
##                   Days Residuals
## Sum of Squares  162703     60322
## Deg. of Freedom      1        17
## 
## Residual standard error: 59.57
## Estimated effects are balanced
## 
## Stratum 3: Within
## 
## Terms:
##                 Residuals
## Sum of Squares      94312
## Deg. of Freedom       144
## 
## Residual standard error: 25.59

formula from aov() from ezANOVA()

> aov.formula <- formula(attr(sleep.ez$aov, "terms"))
> print(aov.formula, showEnv = FALSE)

## Reaction ~ Days + Error(Subject/(Days))

Extensions of linear models to non-linear data

• traditional linear models can be extended to model other types of data such as binary (e.g. yes/no responses)
• basically works by strapping a transformation (link function) onto the front and back ends -- R does this for you!
• fixed effects: glm()
• mixed effects: glmer()

Family types

• binomial: (aka logistic regression) binary ~ continuous
• Gaussian: (normal linear regression) continuous ~ continuous and continuous ~ categegorial
• Gamma: continuous ~ exp(continuous) (exponential response)
• Poisson: count ~ continuous
• (inverse.gaussian, quasi, quasibinomial, quasipoisson)

Binomial models

• casuality of grouping
  • traditional t-test vs. detection prediction
  • Johannes' data
• existence / evidence for a priori categories
  • connecting theory and empiry
  • difficult vs non difficult violations
• behavioral ~ eeg
  • performance (cf. VanRullen (2011)
  • anomaly detection
  • Sarah's data

(More) References

• Stack Exchange and Cross Validated
  • lmer Tag
  • mixed model Tag
  • lme4 Tag
• GLM Families
• Especially relevant questions
  • Residuals for Binomial GLM
  • Comparing non-nested models

• Florian Jaeger's blog
• Jonathan Harrington (phonetics prof. in Munich):
  • Mixed Models (in German)
  • Generalized Linear Mixed Models (in English)
• Tutorial for sociolinguists

• lmer, p-values and all that
• Understanding glmer()output
• Package nlme (older, more specialized MEM implementation), see also here

• Tutorial from lme4 coauthor: (Bolker et al. 2009 )
• Modelling interindividual differences via mixed models
  • Kliegl et al. (2010)
  • Roehm et al. (2012)
• In corpus linguistics / dialectology: Wieling et al. (2011)