Week 8, Lectures 1 & 2: Fixed-, Random-, and Mixed-Effects models

Transcription

1 Week 8, Lectures 1 & 2: Fixed-, Random-, and Mixed-Effects models 1. The repeated measures design, where each of n Ss is measured k times, is a popular one in Psych. We approach this design in 2 ways: 1. As a generalisation of the paired t-test 2. As an expansion from 1-way to 2-way designs 2. Fixed & random effects; nested & repeated measures designs 3. Using EMS tables to define appropriate F ratios for certain designs 4. Options for modeling random effects in R 1

2 Outline of Lectures 1 & 2 on Mixed- 5. kv0.csv ; skv1.r 6. Expanded HW-5: Effects models 1. range of possible models 2. Use AIC and log(likelihood) to compare models 7. Refs: Howell, Chap. 14; Repeated_Measures/repeated_measures.htm ( ucla ) ( nwu ) 2

3 Reprise of paired t-test This latter analysis aptly finesses the problem introduced by the correlation across Ss between x 0i and x 1i. Each S gives 2 scores, one in Group = 0, and the other in Group = 1. Wrong analysis wd be res1 = t.test(x0, x1, var.equal = T), because it treats the 2 groups as independent! Correct analysis is res2 = t.test(x0, x1, var.equal = T, paired = T) 3

4 Reprise of paired t-test For the i th S, compute the difference: d i = x 1i x 0i. Let s = s.d. of the {d i }. t n 1 = s / d n ; or F 2 1,n 1 = t n 1 = nd 2 s 2. d = x 1 x 0. Grand mean is x + x SS Group = n[(x 1 x) 2 + (x 0 x) 2 ] x. = n(x 1 x 0 ) 2 = nd 2, after simplifying. Thus, F 1,n 1 = nd 2 s 2 = SS Group s 2 = MS Group s 2. Because k 1 =1.

5 Reprise of paired t-test F 1,n 1 = nd 2 How best to interpret s 2? d i is also the slope of each line in the Fig. Thus variation in d i is an index of a Subject x Group interaction. That is, s 2 = var(d i ) = the Subj x Group interaction MS. F 1,n 1 = nd 2 s 2 = MS Group s 2 s 2 = MS Group MS Sub*Group This expression for F generalises to the case, k > 2. 5

6 Introduction to Mixed Models An alternative way to finesse the problem of cor(x 0i, x 1i ) is to use the package lme4 (= Linear Mixed Effects, with S4 classes), and the function, lmer(). First, arrange data in long form, d1 (as for lm()). res3 = lmer(score ~ Group + (1 suid), data = d1). NOT res3i = lm(score ~ Group, data = d1), which is incorrect, because it ignores the grouping/ correlation resulting from the repeated measures design! Of course the linear mixed model, lmer(), can do much more than paired t-tests! [Recent articles tout its efficacy: JPSP 2012, by Judd, Westfall & Kenny; Science, Oct 2013, Biology s Dry Future, by R. Service.]

7 Passage from 1- to 2- & 3-way designs Example 1: How does the Score on a memory test depend on the length of the study period, T (= 1, 2, or 3 units)? We could use a 1-way between-groups design in which, say, 24 participants (Ss) are randomly assigned, n = 8 to each level of T. T = 1 T = 2 T = Source df SS MS F Between 2 MS b MS b /MS w Within 21 MS w Total 23 Even though the data in the table are in rows, there is no Row factor because the 3 Ss in each row have nothing in common. 7

8 Example 2: Same as Ex. 1, except that you now worry that Score might also depend on Ss' verbal Ability (A). So divide Ss into 2 levels of A, lo and hi, 12 Ss at each level. At each level of ability, randomly assign n = 4 Ss to each level of T. This is a 2-way between-groups factorial design. A T = 1 T = 2 T = 3 lo hi 3,5, 4,5, 3,7, 5,1, 5,6, 7,6, Source df SS MS F T 2 MS T /MS w A 1 MS A /MS w A * T 2 MS A*T /MS w Within 18 MS w Total 23 We have a Row (A) & a Col (T) factor, so we can define the A*T interaction. The 4 independent obs in each of the 6 cells are used to estimate MS w, which is the 8 denominator in all 3 F ratios.

9 Example 3: Same as Ex. 2, except that you now decide that the best way to control for Ability (A) is to use each S as her or his own control, and to measure each S s score at all 3 levels of T. Suppose we have 8 Ss. This is a 2-way within-group design, with S and T as the 2 factors. S T = 1 T = 2 T = Source df SS MS F T 2 MS T /MS res S 7 MS S /MS res Residual 14 MS res Total 23 Because n = 1 obs per cell, we cannot estimate the within-cell variance, MS w, separately from the interaction MS. Hence we lump the two sources of variation into MS res, as use this as the denominator in the F ratios for the 2 main effects. 9

10 Example 3 (cont d): One way to finesse the problem that the interaction MS and MS w are confounded is to assume that the interaction MS is 0 and, therefore, that MS res = MS w. Assuming that the interaction MS is 0 is equivalent to assuming that S and T have additive effects. The lm() model would be: rs3 = lm(score ~ subid + time, data=d0), rather than rs3a = lm(score ~ subid * time, data=d0), which wd give an error message! S T = 1 T = 2 T = Source df SS MS F T 2 MS T /MS res S 7 MS S /MS res Residual 14 MS res Total 23 10

11 ANOVA Table for the additive model Source df MS F df of F Row, e.g., gift Column, e.g., income r-1 MS r MS r / MS error c-1 MS c MS c / MS error r-1, df error c-1, df error Within Cell, or Error, or Residual df error = rc(n-1)+ (r-1)(c-1) Total N - 1 = rcn - 1 MS error or MS resid The formulae for the expected MS (EMS) tell us how to define F for testing each effect. n obs per cell. 11

12 ANOVA Table for the interactive model: **Note reduction in df error when we test for RxC interaction; and that df error = 0 if n = 1 Source df MS F df of F Row r-1 MS r MS r / MS error Column c-1 MS c MS c / MS error r-1, df error c-1, df error RxC interaction (r-1)(c-1) MS rc MS rc / MS error (r-1)(c-1), df error Within cell df error = rc(n-1) ** MS error Total N - 1 = rcn

13 CAVEAT: When n = 1, one cannot test for interaction! cat('interactive Model ) rs3a = lm(score ~ subid * time, data=d0) print(anova(rs3a)) Response: score Df Sum Sq Mean Sq F value Pr(>F) ability time ability:time Residuals #Note Residuals: ALL 12 residuals are 0: no residual degrees of freedom! Warning message: In anova.lm(rs3a) : ANOVA F-tests on an essentially perfect fit are unreliable 13

14 Design Differences between Ex. 2 & 3 A T = 1 T = 2 T = 3 lo 3,5, 4,5, 3,7, hi 5,1, 5,6, 7,6, S T = 1 T = 2 T = Both are 2-way factorial designs: A by T, and S by T. However, because we are rarely interested in S per se as an explanatory factor (Why?), but are interested in A, we refer to the S by T design as a 1-factor, within-s design! The number of levels of A is small ; that of S is large (more a symptom than a principled diff). 14

15 Design Differences between Ex. 2 & 3 A T = 1 T = 2 T = 3 lo 3,5, 4,5, 3,7, hi 5,1, 5,6, 7,6, S T = 1 T = 2 T = We are rarely interested in the levels S per se (i.e., in john vs mary vs ); these are merely random selections (or effects ) from a large popn of possible values. We wish to make inferences about this popn of S levels. We are interested in the levels of A ( lo vs hi ). These are fixed effects to be interpreted. 15

16 Design Differences between Ex. 2 & 3 A T = 1 T = 2 T = 3 S T = 1 T = 2 T = 3 lo hi 3,5, 4,5, 3,7, 5,1, 5,6, 7,6, In the A by T design, all obs at each level of A come from different Ss and, therefore, are statistically independent (i.e., uncorrelated) In the S by T design, all obs at each level of S come from the same S and, therefore, are correlated. That is, the correlation between scores when, e.g., T=1 and T=2 shd be positive. 16

17 Design Differences between Ex. 2 & 3 A T = 1 T = 2 T = 3 lo 3,5, 4,5, 3,7, hi 5,1, 5,6, 7,6, S T = 1 T = 2 T = The S by T design is also called a repeated measures design. The within-s correl between scores introduces complexities into the calculation of F ratios for this design To solve these complexities, we assume that the within-s correls satisfy certain simplifying conditions, e.g., Compound Symmetry. 17

18 Defn of Compound Symmetry (CS) We assume that the correlation across Ss between the scores at T = 1 and T = 2, cor(x 1i, x 2i ) = cor(x 1i, x 3i ) = cor(x 3i, x 2i ). If there is a between-ss factor, A, we also assume that the correls, cor(x 1i, x 2i ), cor(x 1i, x 3i ) and cor(x 3i, x 2i ), are the same when A = 1 as when A = 2. An almost equivalent set of conditions, known as sphericity, is that the variance of the differences between scores at T = i and T = j is the same for all i and j. 18

19 Mixed Models Fixed effect factors, e.g., gender, A (lo/ med/hi). For quant, B, fixed effects are the linear ( slope ) and quadratic effects of B Random effect factors, e.g., subject (suid), word, school, classroom Mixed models (MM) contain fixed and random effects. F ratios have to be defined carefully Subject is often present in a mixed model; subjects usually give > 1 obs (i.e., there is a within-s factor), and are often nested within a between-s factor. Hence MM analysis is often complicated by nesting and within-s correls 19

20 Example 4: 1 between-s factor, A, 1 within- S factor, T; Ss are thus nested within A The research questions are the same as before. Ss are classified as Low or High ability (A); n Ss at each A level, and each S is tested at all levels of T. A varies between Ss; and T varies within Ss - thus we have a repeated measures design. In addition to the fixed effects factors, A and T, there is the (unmentioned) random effects factor, S. But S 1 at Low A is not the same person as S 1 at High Ability. Thus, S is not crossed with A. We say that Subject is nested within Ability. 20

21 We need to be able to recognise a nested design (as opposed to a factorial design). Some R and SPSS functions require that we specify the nested factors; other functions do not. The data for n = 3 might be: Ability T Subject Low High

22 Structural (or formal) models are useful for calculating the EMS The structural model for the 2-way additive, fixed effects model is: X ijk = µ + a i + b j + e ijk, where a i is the fixed effect of Row i, b j is the fixed effect of Column j, Σ i a i = 0, Σ j b j = 0, etc. var(e ijk ) is labeled σ e2. The overall Row and Column effects are defined by parameters, θ a and θ b, that are quasivariances: θ a = a i=1 a i 2 a 1, θ b = b j =1 b j 2 b 1 22

23 Under H 0, each a i = 0, implying θ a = 0, and each b j = 0, implying θ b = 0. Under H 1, θ a > 0. Indeed θ a is the key determinant of E(MS r ), the Expected Mean Square for Rows. It can be shown that: E(MS r ) = σ e 2 + nbθ a, and E(MS c ) = σ e 2 + naθ b. Whether H 0 is true or not, E(MS error ) = σ e2. Good estimates of θ a and θ b are: θ a = (MS r - MS error )/nb, and θ b = (MS c - MS error )/na. In random effects models, a i is a random variable and θ a is replaced by var(a i ) = σ 2 a, etc. 23

24 Mixed-Effects Modeling in R Various functions in R can be used to analyse mixed models aov() in the base package; ideal for balanced designs with no missing data lme() and gls() in nlme, with p-values! lmer() in lme4, with the latest tricks, no p s! Use pvals.fnc {languager} or {lmertest} for p-values fastrml.lmm() in lmm, with improved est of the random effects Each function has its own syntax for distinguishing fixed from random effects See R script, srepmeas0.r 24

25 Drug Responses 1-1, 0, 1, 2, 1, -2, 3, 0 2 2, 3, 2, 0, 6, 1, 1, 5 3 4, 6, 3, 5, 3, 2, 5, 6 Drug may be a fixed- or randomeffects factor, as discussed earlier. 25

26 srepmeas0.r library(nlme) library(lme4) library(languager) # see also {lmertest} plot(d0) cat( 1. Fixed Effects model ) rs.fix = aov(score ~ drug, d0) print(summary(rs.fix)) print(model.tables(rs.fix, means,digits = 3) cat( 2. Random Effects model with aov() ) rs.ran = aov(score ~ drug + Error(drug), d0) #Drug random print(summary(rs.ran)) 26

27 cat( 3. Random Effects model with lme() ) rs.lme1 = lme(score ~ 1, random = ~ 1 drug, d0) #Package nlme print(summary(rs.lme1)) cat( 4. Random Effects model with lmer() ) rs.lmer1 = lmer(score ~ (1 drug), d0) #Package lme4 print(summary(rs.lmer1)) #To get p-values rs.mcmc = pvals.fnc(rs.lmer1, nsim = 10000, addplot = T) print(rs.mcmc) 27

28 Compare outputs from different functions in mixed-models analysis Our preferred function for analysing mixedmodels is lmer {lme4}. However, it is helpful to have some knowledge of others, esp lme {nlme}, because its syntax is sometimes used in yet other functions. A broad survey shows that the fixed effects are characterised by estimates and p-values, and the random effects by variances and covariances/correlations. 28

29 1. Drug as a fixed effects factor, using aov Df Sum Sq Mean Sq F value Pr(>F) drug ** Residuals Redo aov with Drug as random effects factor Error: drug Df Sum Sq Mean Sq drug Error: Within Df Sum Sq Mean Sq F value Pr(>F) Residuals Notes: MS drug = and MS resid = in both analyses. In 1-way designs, the F ratio is the same for the 2 models. 29

30 3. Redo random effects model with lme() Linear mixed-effects model fit by REML (Default: method=reml ) Data: d0 AIC BIC loglik Random effects: Formula: ~1 drug (Intercept) Residual StdDev: Fixed effects: score ~ 1 Value Std.Error DF t-value p-value (Intercept) Notes: MS resid = = 3.024, as found previously. Recall that E(MS b ) = σ e 2 + n σ b 2 = * = 28.2, as found previously. 30

31 4. Output from lmer() (default is REML=T ) Formula: score ~ 1 + (1 drug) Data: d0 [deviance = -2*logLik] AIC BIC loglik deviance REMLdev Random effects: Groups Name Variance Std.Dev. drug (Intercept) Residual Number of obs: 24, groups: drug, 3 Fixed effects: Estimate Std. Error t value (Intercept) These results are consistent with earlier ones. 31

32 $fixed Estimate MCMCmean HPD95lower HPD95upper pmcmc Pr(> t ) $random Groups Name StDev MCMCmedn MCMCmean HPD95lower HPD95upr drug (Int) Residual [Comments on loglik, deviance, REMLdev, Highest Posterior density (HPD); AIC = deviance + 2(p + 1)] Output from lme() AIC BIC loglik Output from lmer() AIC BIC loglik deviance REMLdev

33 Week 8, Lec 2: Issues in Mixed Models Review of various issues, preliminary to the introduction of lmer(), that arose in Lec. 1. Describe kv0.csv in HW-5: DV = score; attn is a between-s factor (2 levels); nsol is a within-s factor (3 levels); 10 Ss nested in each level of attn. What is the set of plausible (mixed) models for use in analysing the data? What is the theoretical rationale for these models? How to implement and test/compare models in R? 33

34 Preliminary Issues in Mixed Models S T = 1 T = 2 T = Source df SS MS F T 2 MS T /MS res S 7 MS S /MS res Residual 14 MS res Total 23 Subject (suid) is usually a random effects factor (unless we re interested in the roles of the specific Ss); is the variance, σ a2, of these random effects (e.g., of the overall mean or intercept of each S) significantly different from 0? σ 2 a is a parameter to be est. We are interested in the effect associated with each level of T; thus T is a fixed effects factor. If T is categorical with 2 df, there is a fixed effect associated with each df; this is a parameter to be est. If T is quantitative, the linear effect of T (i.e., the slope ) is a fixed effect with 1 df; and the quadratic effect of T (i.e., the non-linearity ) is also a fixed effect with 1 df; ; these are parameters to be est. 34

35 Preliminary Issues in Mixed Models When we have repeated obs for each S, S T = 1 T = 2 T = 3 this introduces a correl across Ss between, e.g., score at T=1 and score at T=2. This correl complicates the definition of test statistics for the different null hypotheses lmer() is designed to take account of these correls in defining appropriate testing procedures. Models with only fixed effects are analysed using lm(). Models with both fixed and random effects are called mixed models and are analysed using lmer() (or lme() or aov(), ). Subject (or suid) is sometimes referred to as the grouping factor. In linguistic studies, stimuli (e.g., words) can also be a random effect if each speaker responds to each word. Subjects can vary randomly in their overall average score (or intercept ); but they can also vary in their sensitivity to T (or slope ). How to specify and test these models with lmer()? 35

36 Preliminary Issues in Mixed Models The issue of p-values. We will discuss this issue more fully later on. Note that lme() gives p-values for fixed and random effects parameters, whereas lmer() gives t statistics, but no p-values, and only for fixed effects parameters. This is sometimes inconvenient (but never more than that!) For intercept only mixed models (the simplest type), pvals.fnc() {languager} does give p-values. See also {lmertest}. The recommendation is that we use, e.g., anova(rs1, rs2), to compare nested models, one containing the variance parameter of interest, and the other omitting this parameter. This comparison yields a p-value. Or the 36 rule-of-thumb: Reject if t > 2!

37 A simple example: 1-way design rs.lmer1 = lmer(score ~ (1 drug), d0) print(summary(rs.lmer1)) #To get p-values rs.mcmc = pvals.fnc(rs.lmer1, nsim = 10000, addplot = T) print(rs.mcmc) In this simple, 1-way design, the results are the same whether we regard drug as a fixed- or a random-effects factor. This is far from true for more complex models. Let us discuss the terms in the lmer() output. 37

38 Formula: score ~ 1 + (1 drug) Data: d0 AIC BIC loglik deviance REMLdev Random effects: Groups Name Variance Std.Dev. drug (Intercept) Residual Number of obs: 24, groups: drug, 3 Fixed effects: Estimate Std. Error t value (Intercept)

39 rs.mcmc = pvals.fnc(rs.lmer1, nsim = 10000, addplot = T) $fixed Estimate MCMCmean HPD95lower HPD95upper pmcmc Pr(> t ) $random Groups Name StDev MCMCmedn MCMCmean HPD95lower HPD95upr drug (Int) Residual [Comments on loglik, deviance, REMLdev, Highest Posterior density (HPD)] 39

40 Assessing an Intervention 50 patients enrolled in a weight loss program note their weights at the start of an intervention, at the end of Week 1, and at the end of Week 2. Let W 0, W 1 and W 2 be the initial, end-wk1, and end-wk2 weights. Data in wtloss1.csv. R script in srepmeas3.r Weights at the 3 time-points are correlated, because each person gives 3 weights, i.e., 3 repeated measures. So it wd be incorrect to treat this design as a 1-way design with time as group. 40

41 id w1 w2 w How to test H 0 : E(W 1 ) = E(W 2 ) = E(W 2 ), taking account of cor(w i, W j ), i j? Ans. Reshape the data into long form, and use lmer().

42 melt(), lm(), lmer() d1 = melt(d0, id.vars = c('id'), measure.vars = c('w1', 'w2', 'w3'), variable.name = 'time', value.name = 'weight') d1$id = factor(d1$id) # The random effects factors have to be defined as factors d1$time = as.numeric(d1$time) # Change 'time' from factor to quantitative variable, 1:3; for use in ggplot(). But CHECK that this line of code gives the correct coding for 'time'! 42

43 id w1 w2 w3 Data in long form, using melt() id time weight res2a = lmer(weight ~ time + (1 id), d1)

44 Plot data for each Subject to guage how Ss vary: Is there appreciable variation in intercept, as well as in slope? Data has to be in long form for ggplot2() g2 = (qplot(time, weight, facets = ~ id, geom = c('point', 'smooth'), method = lm, data = d1) + ) theme_bw() + ylim(40, 90) ggsave('wtloss1a.pdf', plot = g2) 44

45 weight time 45

46 Mixed models analyses with lmer() There appears to be appreciable variation in both intercept and slope across Ss. We start with intercept-only mixed models, then consider an intercept + slope model. ( slope = linear effect of time for each S. intercept is approx the mean wt of each S.) res3 = lmer(weight ~ time + (1 id), d1) print(summary(res3)) res4 = lmer(weight ~ poly(time, 2) + (1 id), d1) print(summary(res4)) print(anova(res3, res4))

47 Discuss results! res3 = lmer(weight ~ time + (1 id), d1) print(summary(res3)) Random effects: Groups Name Variance Std.Dev. id (Intercept) Residual Number of obs: 150, groups: id, 50 Fixed effects: Estimate Std. Error t value (Intercept) time

48 Discuss results! res4 = lmer(weight ~ poly(time, 2) + (1 id), d1) print(summary(res4)) Random effects: Groups Name Variance Std.Dev. id (Intercept) Residual Number of obs: 150, groups: id, 50 Fixed effects: Estimate Std. Error t value (Intercept) poly(time, 2) poly(time, 2)

49 Compare res3 & res4: Is the quadratic effect of time sig? No, because the coeff of poly(time, 2)2 in the lmer() output is not sig (t < 1). For another answer: print(anova(res3, res4)) Models: res3: weight ~ time + (1 id) res4: weight ~ poly(time, 2) + (1 id) Df AIC BIC loglik Chisq ChiDf Pr(>Chisq) res res res4 is not sig better than res3, so keep simpler model, res3. 49

50 Intercept + slope model res5 = lmer(weight ~ time + (1 + time id), d1) print(summary(res5)) print(anova(res3, res5)) Models: res3: weight ~ time + (1 id) res5: weight ~ time + (1 + time id) Df AIC BIC loglik Chisq ChiDf Pr(>Chisq) res res e-05 *** The variation in slope is sig, and this parameter shd be included in mixed models analyses of these data. 50

51 Summary of mixed models analyses The individual-s plots show that the time effect is approx linear for all Ss. A formal test of the relevant coefficient, and a formal model comparison using anova() show that the quadratic effect of time is not sig. The individual-s plots show appreciable variation in both intercept and slope across Ss. A formal model comparison using anova() shows that the variation in slope is sig, and shd be included in any mixed models analysis of these data. 51

52 HW-5 using kv0.csv The fictitious data set, kv0.csv, is based on Dr. Katerina Velanova s FYP data that she shared with us when she was a Psych 252 student. I found this acknowledgement in an old handout: Data and analyses courtesy of Katerina Velanova, to whom Psy 252 owes a debt of gratitude [11/8/96].! Relevant R script: skv1.r 52

53 Ten subjects were run in a divided attention condition (attn = 1), and 10 different subjects were run in a focused attention condition (attn = 2). Each subject was then tested on a word task (anagrams) that had a unique solution (numsol = 1), two solutions (numsol = 2), or multiple solutions (numsol = 3). The dependent measure was a memory score (higher numbers reflect better performance). This design involves both between- and withinfactors and is probably the most useful design for psychologists. 53

54 Attention SubjID Numsol=1 Numsol=2 Numsol= Attn = Attn =

55 55

56 #Scripts to analyse kv0.csv; in skv1.r source('makerm.r') library(lme4) library(nlme) d0 = read.csv('kv0.csv', header = TRUE) con1 = cbind(c(-2,1,1), c(0,-1,1)) #Convert d0 into long form with make.rm() #Now we know how to use melt() {ggplot2}! dl1 = make.rm(constant = 1:2, repeated = 3:5, data = d0) colnames(dl1) = c('suid','atten','score','nsol') 56

57 dl1$nsol = factor(dl1$nsol, labels=c('one','two','many')) dl1$atten = factor(dl1$atten, labels=c('div','foc')) dl1$suid = factor(dl1$suid) contrasts(dl1$atten) = c(-1,1) contrasts(dl1$nsol) = cbind(lin=c(-1,0,1), quad=c(-1,2,-1)) 57

58 rs.aov = aov(score ~ atten*nsol + Error(suid), dl1) print(summary(rs.aov)) Error: suid Df Sum Sq Mean Sq F value Pr(>F) atten *** Residuals Error: Within Df Sum Sq Mean Sq F value Pr(>F) nsol ** atten:nsol ** Residuals

59 rs.lme1 = lmer(score ~ atten*nsol + (1 suid), dl1) print(summary(rs.lme1)) Linear mixed model fit by REML Formula: score ~ atten * nsol + (1 suid) Data: dl1 AIC BIC loglik deviance REMLdev Random effects: Groups Name Variance Std.Dev. suid (Intercept) Residual Number of obs: 60, groups: suid, 20 Fixed effects: Estimate Std. Error t value (Intercept) atten nsollin nsolquad atten1:nsollin atten1:nsolquad

60 rs.lme2a = lmer(score ~ atten * nsol + (1 + nsol suid), dl1) print(summary(rs.lme2a)) Linear mixed model fit by REML Formula: score ~ atten * nsol + (1 + nsol suid) Data: dl1 AIC BIC loglik deviance REMLdev Random effects: Groups Name Variance Std.Dev. Corr suid (Intercept) nsollin nsolquad Residual Number of obs: 60, groups: suid, 20 60

61 Fixed effects: Estimate Std. Error t value (Intercept) atten nsollin nsolquad atten1:nsollin atten1:nsolquad Describe the data! 61