Correlational Structure in the Random-Effect Structure of Mixed Models

Transcription

1 Correlational Structure in the Random-Effect Structure of Mixed Models March 25, 2009

2 Outline

3 random effects in mixed-effects modeling goal to provide an intuitive guide to understanding the role of correlation parameters in the random effects part of a mixed model two examples example 1: variation in the realization of correlational structure involving a fixed factorial predictor fixed factor levels nested under the random-effect factor example 2: self-paced reading latencies of correlational structure involving covariates subject covariates crossed with item item covariates crossed with subject

4 Outline

5 the -a/aj alternative forms forms with -a form with -aj infinitive maxat maxat masculine sg past maxal maxal 1sg present mašu maxaju 2sg present mašeš maxa(j)eš 3sg present mašet maxa(j)et 1pl present mašem maxa(j)em 2pl present mašete maxa(j)ete 3pl present mašut maxajut imperative maši(te) maxaj(te) present active participle mašuščij maxajuščij gerund maša maxaja

6 systematicities in this variation Count a aj Count a aj s p f i a g dental labial velar Counts of -a (black) and -aj (white) realizations for six paradigm slots (left) and place of articulation of the final consonant of the root (right). a: active present participle, p: third person plural, s: third person sigular, f: first/second person, i: infinitive, g: gerund.

7 different verbs show different patterns logit xnykat zhazhdat schepat schipat stonat pleskat poloskat prjatat pryskat kudaxtat kurlykat maxat kapat klepat klikat svistat metat kloxtat tykat pyxat murlykat mykat kolebat vnimat ryskat schekotat kolyxat xlestat paxat krapat alkat blistat bryzgat cherpat dremat dvigat glodat s p f i a g s p f i a g s p f i a g s p f i a g s p f i a g s p f i a g s p f i a g The log odds (of -a versus -aj) for each of the six paradigm slots. A log odds greater than zero indicates a preference for -a, a log odds smaller than zero a preference for -aj.

8 a model with random intercepts for verbs contrast coding for Paradigm and Place reference level (Active participle for Paradigm, dental for Place) contrasts (group mean differences) with respect to the reference level (e.g., Gerund versus Active Participle, Labial versus Dental) we begin with a model with random intercepts for Verb, thereby allowing the verbs to differ in the extent to which they prefer -a over -aj (equally across all forms in the paradigm)

9 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) Number of obs: 222, groups: Verb, 37 Fixed effects: Estimate Std. Error z value Pr(> z ) (Intercept) Paradigmf Paradigmg Paradigmi Paradigmp Paradigms Placelabial Placevelar

10 problems with this initial model differences between verbs are restricted to just the intercept our dotplot suggests, however, that verbs may differ with respect to the paradigm slots for which they prefer or disprefer -a versus -aj furthermore, we have assumed that the likelihood of a given variant for a given verb in one paradigm slot is independent of the likelihood of a given variant for that same verb in another paradigm slot, which seems unlikely

11 the observations across paradigm cells for a given verb are not independent (dots represent verbs) 2 2 a r = 0.63 rs = 0.6 p = 1e f r = rs = r = 0.55 p = 4e 04 rs = 0.52 p = 9e 04 g r = rs = r = 0.71 rs = 0.72 r = 0.64 rs = 0.65 r = 0.73 rs = 0.73 r = 0.88 rs = 0.87 r = 0.8 rs = 0.75 r = 0.63 rs = 0.52 p = 9e 04 r = 0.55 p = 4e 04 rs = r = rs = i r = 0.66 rs = 0.65 r = 0.62 rs = 0.6 p = 1e 04 p r = 0.83 rs = 0.82 s

12 anticipating the consequences of contrast coding for the random effects structure we are modeling Paradigm with contrast coding bringing flexibility into the model for verb specific preferences across the paradigm will therefore be implemented in terms of adjustments to the intercept and adjustments to contrast coefficients to anticipate the correlational structure of these adjustments, we redo our previous plot, retaining the log odds for the reference level, Active participle but for all other levels of Paradigm, we replace the observed log odds by its difference with the corresponding value for the reference level

13 anticipating the consequences of contrast coding for the random effects structure the correlations involving the reference level change sign, the other correlations remain positive this pattern is strongest for weakly correlated random variables

14 simulated data before contrasts y x r= z x r= y z r= y x r= z x r= y z r= y x r= z x r= y z r=0.97

15 simulated data after contrasts y x x r= z x x r= y x z x r= y x x r= z x x r= y x z x r= y x x r= z x x r= y x z x r=0.97

16 anticipating the consequences of contrast coding for the random effects structure (dots represent verbs) a r = 0.56 p = 3e 04 rs = 0.55 p = 5e 04 r = 0.62 rs = f r = 0.69 rs = 0.74 g r = 0.8 rs = 0.79 r = rs = r = rs = r = 0.8 rs = 0.81 r = 0.84 rs = 0.87 r = 0.72 rs = 0.75 r = 0.78 rs = 0.63 r = 0.66 rs = 0.64 r = 0.53 p = 8e 04 rs = 0.56 p = 3e 04 i r = 0.7 rs = 0.69 r = 0.61 p = 1e 04 rs = 0.6 p = 1e 04 p r = 0.74 rs = 0.82 s

17 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) Paradigmf Paradigmg Paradigmi Paradigmp Paradigms > pairscor.fnc(ranef(russian.lmer1)$verb)

18 visualization of the BLUPs Intercept r = 0.61 p = 1e 04 rs = 0.63 r = rs = f r = 0.81 rs = 0.72 g r = 0.72 rs = 0.75 r = 0.92 rs = 0.91 r = 0.64 rs = 0.56 p = 4e 04 i r = rs = r = 0.93 rs = 0.9 r = 0.94 rs = 0.89 r = 0.77 rs = 0.72 p r = 0.54 p = 5e 04 rs = 0.56 p = 4e 04 r = 0.86 rs = 0.91 r = 0.54 p = 6e 04 rs = 0.55 p = 6e 04 r = 0.96 rs = 0.94 r = 0.69 rs = 0.74 s

19 are 15 additional parameters justified? observed proportion expected proportion model observed proportion expected proportion model 2

20 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 russian.lmer < 2.2e-16

21 Summary models become more precise if you take non-independence seriously the coding used for factor levels determines your interpretation of the random effects correlational structure for contrast coding, the default in R, pairwise correlations change sign for pairs involving the reference level

22 Outline

23 self-paced reading experiment 87 poems, in all 2315 different word forms 326 subjects self-paced reading latencies three random-effect factors Poem Word Subject

24 Words random intercepts possibly, additional random slopes/contrasts for properties of the subjects the subject s age reading latencies for a given word might depend specifically on whether you are a younger or an older subject) RT in questionaire (the subject s response latency in an on-line questionaire requesting from the subject an estimate of the number of poems read annually) reading latencies for a given word might depend on whether you are a slow, careful evaluator or a fast, superficial responder the subject s sex a given word might be read more quickly by females (or males) (female words versus male words) note: it is important to center predictors

25 spoken British English (BNC) females she, her, said, n t, I, and, to, cos, oh, Christmas, thought, lovely, nice, mm, had, did, going, yes, really males fucking, er, the, yeah, aye, right, hundred, fuck, is, of, two, three, a, four, ah, no rlh97.html

26 spoken British English (BNC) females she, her, said, n t, I, and, to, cos, oh, Christmas, thought, lovely, nice, mm, had, did, going, yes, really males fucking, er, the, yeah, aye, right, hundred, fuck, is, of, two, three, a, four, ah, no rlh97.html

27 spoken British English (BNC) females she, her, said, n t, I, and, to, cos, oh, Christmas, thought, lovely, nice, mm, had, did, going, yes, really males fucking, er, the, yeah, aye, right, hundred, fuck, is, of, two, three, a, four, ah, no rlh97.html

28 spoken British English (BNC) females she, her, said, n t, I, and, to, cos, oh, Christmas, thought, lovely, nice, mm, had, did, going, yes, really males fucking, er, the, yeah, aye, right, hundred, fuck, is, of, two, three, a, four, ah, no rlh97.html

29 spoken British English (BNC) females she, her, said, n t, I, and, to, cos, oh, Christmas, thought, lovely, nice, mm, had, did, going, yes, really males fucking, er, the, yeah, aye, right, hundred, fuck, is, of, two, three, a, four, ah, no rlh97.html

30 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) for each word, we fit a separate model to the reading times of the subjects reading that word with as predictors the age, questionaire RT, and sex of those subjects for each word, we thus obtain an intercept, slopes for Age and RTquestionaire, and a contrast coefficient for Sex we plot these coefficients using a pairwise scatterplot matrix

31 the Word: exploration with lmlist Intercept r = rs = Age r = 0.08 p = 2e 04 rs = 0.08 p = 1e 04 r = rs = RTquestionaire r = 0.28 rs = 0.23 r = 0.09 rs = r = rs = 0.08 p = 1e 04 Sex each point represents a verb type lines are nonparametric scatterplot smoothers

32 the Word: Age and Intercept items$age items$intercept to the left in the graph: here we see words that are read faster by older subjects (a regression of Reading Time on Age has negative slope for these words) to the right in the graph: here we see words that are read slower by older subjects (a regression of Reading Time on Age has positive slope for these words)

33 the Word: Age and Intercept items$age items$intercept in the center of the graph: zero slope, so no Age effect: here we see words that are processed the same irrespective of Age these words also have the smallest intercepts, so overall, these words elicit the shortest mean reading latencies

34 the Word: Sex and Intercept the left panel shows the intercepts for females (vertical) and the contrast for males (horizontal) there are relatively few words to the right of X=0 (words for which the intercept for males has to be adjusted upwards compared to the intercept for females) as we move to the left, we meet words for which the intercept (appropriate for females) has to be adjusted downward for males the right panel shows the intercepts for females (vertical) and the reconstructed intercepts for males (horizontal) items$sex items$intercept items$sex + items$intercept items$intercept

35 the Word: Sex and Intercept what makes words more or less easy to process by males or females? to answer this question, we consider Subject as random-effect factor

36 the Subject random intercepts possibly, random slopes for properties of the Words, e.g., the word s frequency the word s number of constituent morphemes background: Ullman s hypothesis that females have superior verbal memory and hence have a stronger frequency effect than males

37 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 = , df = , p-value = Intercept Nmorphs SurfFreq

38 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)

39 joint analysis: random effects Random effects: Groups Name Std.Dev. Corr Word (Intercept) RTquestionaire Age Subject (Intercept) Nmorphs SurfFreq Poem (Intercept) Residual Number of obs: , groups: Word, 2315; Subject, 326; Gedicht, 87

40 visualization subject BLUPs (Intercept) r = 0.69 rs = 0.67 Nmorphs r = 0.67 rs = 0.72 r = 0.51 rs = SurfFreq

41 visualization word BLUPs (Intercept) r = 0.98 rs = 0.97 ChoiceRT r = 0.94 rs = 0.92 r = 0.86 rs = Leeftijd

42 joint analysis: random effects likelihood ratio tests support each additional parameter in the model for instance, comparing a model with only random intercepts for subject with a model with additional structure for Nmorphs and SurfFreq: anova(dat.lmer0, dat.lmer1) Df AIC BIC loglik Chisq Chi Df Pr(>Chisq) dat.lmer dat.lmer < 2.2e-16

43 modeling strategy explore with visualization where random slopes and correlations might be required add additional parameters incrementally: complex random effects structure can be difficult to fit

44 joint analysis: fixed effects Fixed effects: Estimate Std. Error t value... Nmorphs RTquestionaire Sexm Age SurfFreq I(SurfFreq^2) SurfFreq:Sexm Sexm:Nmorphs

45 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() quantile(resid(dat.lmer2), seq(0.001, 0.999, length = 20)) qnorm(p = seq(0.001, 0.999, length = 20))

46 model criticism 2 Residual Frequency

47 Outline

48 we have validated the Sex by Frequency interaction in the fixed-effect part of the model by bringing into the model all potential other sources that might explain this interaction: a potential confound with other available subject-specific properties (1+Age+RTquestionaire Word) a potential confound with individual differences in sensitivity to frequency (1+Frequency+Nmorphs Subject)

49 Outline

50 we obtain better models when we pay careful attention to the modeling of the correlational structure for the random effect factors we have a better tool for understanding subject (item) variability than traditional methods such as median splits with separate subanalyses