mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Correlational Structurein the Random-Effect Structure

of Mixed Models

Harald Baayen

March 25, 2009

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Outline

introduction

Russian verbs

Reading Dutch poetry

Conclusions Poetry Experiment

Final Remarks

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

random effects in mixed-effects modeling

I goalto provide an intuitive guide to understanding the roleof correlation parameters in the random effects part of amixed model

I two examplesI example 1: variation in the realization of Russian verbs

I correlational structure involvinga fixed factorial predictor

I fixed factor levels nestedunder the random-effect factor

I example 2: self-paced reading latencies of poetryI correlational structure involving covariatesI subject covariates crossed with itemI item covariates crossed with subject

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Outline

introduction

Russian verbs

Reading Dutch poetry

Conclusions Poetry Experiment

Final Remarks

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the -a/aj alternative forms

forms with -a form with -aj

infinitive maxat′ maxat′

masculine sg past maxal maxal1sg present masu maxaju2sg present mases′ maxa(j)es′

3sg present maset maxa(j)et1pl present masem maxa(j)em2pl present masete maxa(j)ete3pl present masut maxajutimperative masi(te) maxaj(te)present active participle masuscij maxajuscijgerund masa maxaja

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

systematicities in this variation

s p f i a g

Cou

nt

010

0020

0030

0040

00

−a−aj

dental labial velar

Cou

nt

010

0020

0030

0040

00

−a−aj

Counts of -a (black) and -aj (white) realizations for sixparadigm slots (left) and place of articulation of the finalconsonant of the root (right). a: active present participle, p:third person plural, s: third person sigular, f: first/secondperson, i: infinitive, g: gerund.

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

different verbs show different patternslo

git

−404

s p f i a g

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alkat

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s p f i a g

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s p f i a g

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cherpat

s p f i a g

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s p f i a g

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s p f i a g

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glodat

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kapat

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klepat

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klikat

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kloxtat

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kolebat

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kolyxat

−404

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krapat−4

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kudaxtat

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kurlykat

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maxat

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mykat

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pleskat

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prjatat

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pryskat

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pyxat

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ryskat

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04

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schepat

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stonat

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svistat

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tykat

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xlestat

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xnykat

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●

zhazhdat

The log odds (of -a versus -aj) for each of the six paradigmslots. A log odds greater than zero indicates a preference for-a, a log odds smaller than zero a preference for -aj.

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

a model with random intercepts for verbs

I contrast coding for Paradigm and Place

I reference level (Active participle for Paradigm,dental for Place)

I contrasts (group mean differences) with respect to thereference level (e.g., Gerund versus ActiveParticiple, Labial versus Dental)

I we begin with a model with random intercepts forVerb, thereby allowing the verbs to differ in the extentto which they prefer -a over -aj (equally across all formsin the paradigm)

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

a model with random intercepts for verbs

> russian.lmer = lmer(cbind(a, aj) ~ Paradigm ++ Place + (1 | Verb), data = russian, family = "binomial")

Random effects:Groups Name Variance Std.Dev.Verb (Intercept) 6.7098 2.5903Number of obs: 222, groups: Verb, 37

Fixed effects:Estimate Std. Error z value Pr(>|z|)

(Intercept) 3.993 0.810 4.928 0.000Paradigmf -0.558 0.145 -3.847 0.000Paradigmg -3.067 0.129 -23.830 0.000Paradigmi -1.827 0.194 -9.419 0.000Paradigmp 0.250 0.112 2.236 0.025Paradigms 0.812 0.095 8.554 0.000Placelabial -3.348 1.211 -2.765 0.006Placevelar -2.071 1.027 -2.017 0.044

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

problems with this initial model

I differences between verbs are restrictedto just the intercept

I our dotplot suggests, however, that verbs may differwith respect to the paradigm slots for which they preferor disprefer -a versus -aj

I furthermore, we have assumed that the likelihood of agiven variant for a given verb in one paradigm slot isindependent of the likelihood of a given variant for thatsame verb in another paradigm slot, which seemsunlikely

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the observations across paradigm cellsfor a given verb are not independent(dots represent verbs)

a

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r = 0.63p = 0

rs = 0.6p = 1e−04

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r = 0.24p = 0.1533rs = 0.22

p = 0.1826

r = 0.55p = 4e−04rs = 0.52

p = 9e−04

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p = 0.0088

r = 0.73p = 0

rs = 0.73p = 0

r = 0.63p = 0

rs = 0.52p = 9e−04

i●

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r = 0.71p = 0

rs = 0.72p = 0

r = 0.88p = 0

rs = 0.87p = 0

r = 0.55p = 4e−04

rs = 0.5p = 0.0018

r = 0.66p = 0

rs = 0.65p = 0

p−

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−4

04 r = 0.64p = 0

rs = 0.65p = 0

r = 0.8p = 0

rs = 0.75p = 0

−4 0 4

r = 0.46p = 0.0046rs = 0.34

p = 0.0412

r = 0.62p = 0

rs = 0.6p = 1e−04

−2 2

r = 0.83p = 0

rs = 0.82p = 0

s

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

anticipating the consequences of contrastcoding for the random effects structure

I we are modeling Paradigm with contrast coding

I bringing flexibility into the model for verb specificpreferences across the paradigm will therefore beimplemented in terms of adjustments to the interceptand adjustments to contrast coefficients

I to anticipate the correlational structure of theseadjustments, we redo our previous plot,

I retaining the log odds for the reference level, Activeparticiple

I but for all other levels of Paradigm, we replace theobserved log odds by its difference with thecorresponding value for the reference level

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

anticipating the consequences of contrastcoding for the random effects structure

I the correlations involving the reference levelchange sign, the other correlations remain positive

I this pattern is strongest for weakly correlatedrandom variables

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

simulated data before contrasts

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mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

simulated data after contrasts

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mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

anticipating the consequences of contrastcoding for the random effects structure(dots represent verbs)

a

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r = −0.62p = 0

rs = −0.5p = 0.0017

r = 0.69p = 0

rs = 0.74p = 0

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r = 0.78p = 0

rs = 0.63p = 0

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r = −0.44p = 0.0063rs = −0.41p = 0.0117

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r = 0.66p = 0

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r = 0.7p = 0

rs = 0.69p = 0

p−

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r = −0.37p = 0.0242rs = −0.33p = 0.0431

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−2 2 4

r = 0.74p = 0

rs = 0.82p = 0

s

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

an improved model

> russian.lmer1 = lmer(cbind(a, aj) ~ Paradigm ++ Place + (1 + Paradigm | Verb), data = russian,+ family = "binomial")

Random effects:Name Variance Std.Dev. Corr(Intercept) 11.3268 3.3655Paradigmf 4.9671 2.2287 -0.654Paradigmg 12.3863 3.5194 -0.398 0.766Paradigmi 11.0430 3.3231 -0.757 0.879 0.638Paradigmp 2.6471 1.6270 -0.472 0.919 0.893 0.777Paradigms 4.1690 2.0418 -0.609 0.838 0.535 0.950 0.676

> pairscor.fnc(ranef(russian.lmer1)$Verb)

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

visualization of the BLUPs

Intercept

−4 0 2 4

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r = −0.33p = 0.0432rs = −0.24p = 0.1514

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rs = 0.56p = 4e−04

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r = −0.39p = 0.0162rs = −0.36p = 0.0314

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r = 0.94p = 0

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r = 0.77p = 0

rs = 0.72p = 0

p

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rs = 0.94p = 0

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r = 0.69p = 0

rs = 0.74p = 0

s

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

are 15 additional parameters justified?

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mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

model comparison with likelihood ratio test

russian.lmer = lmer(cbind(a, aj) ~ Paradigm + Place +(1|Verb), data=russian, family="binomial")russian.lmer1 = lmer(cbind(a, aj) ~ Paradigm + Place +(1+Paradigm|Verb), data=russian, family="binomial")anova(russian.lmer, russian.lmer1)

...Df AIC BIC logLik Chisq Chi Df Pr(>Chisq)

russian.lmer 9 1397.04 1427.67 -689.52russian.lmer1 29 525.08 623.76 -233.54 911.96 20 < 2.2e-16

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Summary

I models become more precise if you takenon-independence seriously

I the coding used for factor levels determines yourinterpretation of the random effects correlationalstructure

I for contrast coding, the default in R, pairwisecorrelations change sign for pairs involving the referencelevel

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Outline

introduction

Russian verbs

Reading Dutch poetry

Conclusions Poetry Experiment

Final Remarks

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Reading Dutch poetry

I self-paced reading experimentI 87 poems, in all 2315 different word formsI 326 subjectsI 275996 self-paced reading latencies

I three random-effect factorsI PoemI WordI Subject

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Words

I random interceptsI possibly, additional random slopes/contrasts for

properties of the subjectsI the subject’s age

I reading latencies for a given word might dependspecifically on whether you are a younger or an oldersubject)

I RT in questionaire (the subject’s response latency inan on-line questionaire requesting from the subject anestimate of the number of poems read annually)

I reading latencies for a given word might depend onwhether you are a slow, careful evaluator or a fast,superficial responder

I the subject’s sexI a given word might be read more quickly by females

(or males) (female words versus male words)

I note: it is important to center predictors

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

spoken British English (BNC)

I femalesshe, her, said, n’t, I, and, to, cos, oh, Christmas,thought, lovely, nice, mm, had, did, going, yes, really

I malesfucking, er, the, yeah, aye, right, hundred, fuck, is, of,two, three, a, four, ah, no

I http://www.comp.lancs.ac.uk/ucrel/papers/rlh97.html

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

spoken British English (BNC)

I femalesshe, her, said, n’t, I, and, to, cos, oh, Christmas,thought, lovely, nice, mm, had, did, going, yes, really

I malesfucking, er, the, yeah, aye, right, hundred, fuck, is, of,two, three, a, four, ah, no

I http://www.comp.lancs.ac.uk/ucrel/papers/rlh97.html

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

spoken British English (BNC)

I femalesshe, her, said, n’t, I, and, to, cos, oh, Christmas,thought, lovely, nice, mm, had, did, going, yes, really

I malesfucking, er, the, yeah, aye, right, hundred, fuck, is, of,two, three, a, four, ah, no

I http://www.comp.lancs.ac.uk/ucrel/papers/rlh97.html

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

spoken British English (BNC)

I femalesshe, her, said, n’t, I, and, to, cos, oh, Christmas,thought, lovely, nice, mm, had, did, going, yes, really

I malesfucking, er, the, yeah, aye, right, hundred, fuck, is, of,two, three, a, four, ah, no

I http://www.comp.lancs.ac.uk/ucrel/papers/rlh97.html

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

spoken British English (BNC)

I femalesshe, her, said, n’t, I, and, to, cos, oh, Christmas,thought, lovely, nice, mm, had, did, going, yes, really

I malesfucking, er, the, yeah, aye, right, hundred, fuck, is, of,two, three, a, four, ah, no

I http://www.comp.lancs.ac.uk/ucrel/papers/rlh97.html

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Word: exploration with lmList

> items.lmList = lmList(ReadingTime ~ Age + RTquestionaire ++ Sex | Word, data = dat)> items = data.frame(coef(items.lmList))> pairscor.fnc(items, cex = 0.5)

I for each word, we fit a separate model to the readingtimes of the subjects reading that word with aspredictors the age, questionaire RT, and sex of thosesubjects

I for each word, we thus obtain an intercept, slopes forAge and RTquestionaire, and a contrast coefficient forSex

I we plot these coefficients using a pairwise scatterplotmatrix

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Word: exploration with lmList

Intercept

−0.6 −0.2 0.2

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p = 0.1504

rs = 0

p = 0.9262

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r = 0.08

p = 2e−04

rs = 0.08

p = 1e−04

r = 0.05

p = 0.0268

rs = −0.02

p = 0.2601

RTquestionaire

−0.

20.

2

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5.8 6.2 6.6

−0.

50.

00.

5 r = −0.28

p = 0

rs = −0.23

p = 0

r = −0.09

p = 0

rs = −0.02

p = 0.397

−0.2 0.2

r = 0.03

p = 0.1904

rs = 0.08

p = 1e−04

Sex

I each point represents a verb type

I lines are nonparametric scatterplot smoothers

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Word: Age and Intercept

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−0.6 −0.4 −0.2 0.0 0.2

5.8

6.2

6.6

items$Age

item

s$In

terc

ept

I to the left in the graph: here we see words that areread faster by older subjects (a regression of ReadingTime on Age has negative slope for these words)

I to the right in the graph: here we see words that areread slower by older subjects (a regression of ReadingTime on Age has positive slope for these words)

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Word: Age and Intercept

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−0.6 −0.4 −0.2 0.0 0.2

5.8

6.2

6.6

items$Age

item

s$In

terc

ept

I in the center of the graph:I zero slope, so no Age effect: here we see words that are

processed the same irrespective of AgeI these words also have the smallest intercepts, soI overall, these words elicit the shortest mean reading

latencies

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Word: Sex and InterceptI the left panel shows the intercepts for females

(vertical) and the contrast for males (horizontal)I there are relatively few words to the right of X=0

(words for which the intercept for males has to beadjusted upwards compared to the intercept for females)

I as we move to the left, we meet words for which theintercept (appropriate for females) has to be adjusteddownward for males

I the right panel shows the intercepts for females(vertical) and the reconstructed intercepts for males(horizontal)

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−0.5 0.0 0.5

5.8

6.2

6.6

items$Sex

item

s$In

terc

ept

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5.6 5.8 6.0 6.2 6.4 6.6 6.8

5.8

6.2

6.6

items$Sex + items$Intercept

item

s$In

terc

ept

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Word: Sex and Intercept

I what makes words more or less easy to process by malesor females?

I to answer this question, we consider Subject asrandom-effect factor

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Subject

I random interceptsI possibly, random slopes for properties of the Words,

e.g.,I the word’s frequencyI the word’s number of constituent morphemes

I background: Ullman’s hypothesis that females havesuperior verbal memory and hence have a strongerfrequency effect than males

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

the Subject: Nmorphs and Frequency> subjects.lmList = lmList(Leestijd ~ Nmorphs ++ SurfFreq | Subject, data = dat)> subjects = data.frame(coef(subjects.lmList))> pairs(subjects[, 1:3])> t.test(SurfFreq ~ Sex, data = subjects)

t = -2.6782, df = 313.152, p-value = 0.007792

Intercept

−0.04 0.00 0.04

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−0.20 −0.10 0.00

−0.

20−

0.10

0.00

SurfFreq

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

joint analysis: the model specification

dat.lmer = lmer(ReadingTime ~ Trial + NumberOfWordsIntoLine +SentenceLength + ... + Sex*SurfFreq + I(SurfFreq^2) +RTquestionaire + Nmorphs*Sex + Age +(1|Poem)+(1+Nmorphs+SurfFreq|Subject)+(1+RTquestionaire+Age|Word),data=dat)

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

joint analysis: random effects

Random effects:Groups Name Std.Dev. CorrWord (Intercept) 0.075

RTquestionaire 0.008 0.716Age 0.010 -0.596 -0.171

Subject (Intercept) 0.214Nmorphs 0.006 0.387SurfFreq 0.035 -0.642 -0.251

Poem (Intercept) 0.043Residual 0.314Number of obs: 275996, groups: Word, 2315; Subject, 326; Gedicht, 87

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

visualization subject BLUPs

(Intercept)

−0.010 0.000 0.010

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p = 0

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p = 0

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r = −0.67

p = 0

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p = 0

r = −0.51

p = 0

rs = −0.54

p = 0

−0.10 0.00 0.05

−0.

100.

000.

05SurfFreq

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

visualization word BLUPs

(Intercept)

−0.02 0.00 0.02

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−0.

020.

000.

02 r = 0.98

p = 0

rs = 0.97

p = 0

ChoiceRT

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−0.2 0.0 0.2 0.4

r = −0.94

p = 0

rs = −0.92

p = 0

r = −0.86

p = 0

rs = −0.82

p = 0

−0.03 −0.01 0.01

−0.

03−

0.01

0.01Leeftijd

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

joint analysis: random effects

I likelihood ratio tests support each additional parameterin the model

I for instance, comparing a model with only randomintercepts for subject with a model with additionalstructure for Nmorphs and SurfFreq:

anova(dat.lmer0, dat.lmer1)

Df AIC BIC logLik Chisq Chi Df Pr(>Chisq)dat.lmer0 37 153425 153814 -76675dat.lmer1 42 153197 153639 -76557 237.59 5 < 2.2e-16

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

modeling strategy

I explore with visualization where random slopes andcorrelations might be required

I add additional parameters incrementally: complexrandom effects structure can be difficult to fit

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

joint analysis: fixed effects

Fixed effects:Estimate Std. Error t value

...Nmorphs -0.0002 0.0015 -0.13RTquestionaire 0.0588 0.0095 6.13Sexm -0.0937 0.0243 -3.86Age 0.0615 0.0096 6.35SurfFreq -0.0213 0.0066 -3.20I(SurfFreq^2) 0.0059 0.0021 2.76SurfFreq:Sexm 0.0097 0.0041 2.35Sexm:Nmorphs -0.0028 0.0014 -1.95

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

model criticism 1> pdf("qqplot.pdf", he = 5, wi = 5)> plot(qnorm(p = seq(0.001, 0.999, length = 20)),+ quantile(resid(dat.lmer2), seq(0.001, 0.999,+ length = 20)))> dev.off()

●

●

●●

●●●●●●●●●●●

●●

●

●

●

−3 −2 −1 0 1 2 3

−1.

0−

0.5

0.0

0.5

1.0

qnorm(p = seq(0.001, 0.999, length = 20))

quan

tile(

resi

d(da

t.lm

er2)

, seq

(0.0

01, 0

.999

, len

gth

= 2

0))

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

model criticism 2

−3 −2 −1 0 1

−1.

5−

0.5

0.0

0.5

1.0

1.5

Frequency

Res

idua

l

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Outline

introduction

Russian verbs

Reading Dutch poetry

Conclusions Poetry Experiment

Final Remarks

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Final Remarks

I we have validated the Sex by Frequency interaction inthe fixed-effect part of the model by bringing into themodel all potential other sources that might explain thisinteraction:

I a potential confound with other availablesubject-specific properties(1+Age+RTquestionaire|Word)

I a potential confound with individual differences insensitivity to frequency(1+Frequency+Nmorphs|Subject)

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Outline

introduction

Russian verbs

Reading Dutch poetry

Conclusions Poetry Experiment

Final Remarks

mixed models

Harald Baayen

introduction

Russian verbs

Reading Dutchpoetry

ConclusionsPoetryExperiment

Final Remarks

Final Remarks

I we obtain better models when we pay careful attentionto the modeling of the correlational structure for therandom effect factors

I we have a better tool for understanding subject (item)variability than traditional methods such as mediansplits with separate subanalyses