In R

> <span class="co"># example data providedy by lme4</span>
> <span class="kw">library</span>(lme4)
> <span class="kw">data</span>(sleepstudy)
> <span class="co"># for more info, try ?sleepstudy</span>
> <span class="kw">library</span>(lattice)  <span class="co"># provides some nice plotting functions</span>

> <span class="kw">str</span>(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 ...

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 <- <span class="kw">lm</span>(Reaction ~ Days, <span class="dt">data =</span> sleepstudy)

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

Add a regression line with lattice graphics

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

With regression lines

> sleep.xy.bysubj <- <span class="kw">update</span>(sleep.xy.bysubj, <span class="dt">type =</span> <span class="kw">c</span>(<span class="st">"p"</span>, <span class="st">"r"</span>))
> sleep.xy.bysubj

• 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

• 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)

ezANOVA()

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

## 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 <- <span class="kw">formula</span>(<span class="kw">attr</span>(sleep.ez$aov, <span class="st">"terms"</span>))
> <span class="kw">print</span>(aov.formula, <span class="dt">showEnv =</span> <span class="ot">FALSE</span>)

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

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

• 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)