201ab Quantitative methods Linear Mixed Effects ModelsWe have pre-and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section,

ED VUL | UCSD Psychology

201ab Quantitative methodsLinear Mixed Effects Models

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Our expanding model landscape

Eq. var t-tests

Multiple regression

ANCOVAOLS regressionANOVA

Gen

eral

lin

ear m

odel

Logistic regressionPoisson regression

Gen

eral

ized

linea

r mod

els

Linear mixed models

Repeated measures ANOVA

Mixed design (split-plot) ANOVA

Generalized linear mixed models

Hierarchical Bayesian models

As we saw, and in general, special cases of broader model classes are usually favored despite being less flexible because they are simpler and allow for easier estimation and inference.

A few of the methods we covered don’t fit into this scheme:- Binomial test- Pearson’s Chi-

square tests- Unequal

variance t-tests

There are other generalizations:- Multivariate

linear model- General(ized)

additive model


Motivating dataWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.

str(test.data)

'data.frame': 400 obs. of 6 variables:$ student: Factor w/ 20 levels "S.1","S.10","S.11",..: 1 1 1 1 1 1 11 ...$ sex : Factor w/ 2 levels "female","male": 2 2 2 2 2 2 2 2 2 2 ...$ test : Factor w/ 2 levels "post","pre": 2 2 2 2 2 2 2 2 2 2 ...$ part : Factor w/ 10 levels "P.1","P.10","P.2",..: 1 3 4 5 6 7 8 9 ...$ section: Factor w/ 2 levels "qualitative",..: 1 1 1 1 1 2 2 2 2 2 ...$ score : num 53 50 79 67 70 68 68 62 65 79 ...

head(test.data)

student sex test part section scoreS.1 S.1 male pre P.1 qualitative 53S.1.2 S.1 male pre P.2 qualitative 50S.1.3 S.1 male pre P.3 qualitative 79S.1.4 S.1 male pre P.4 qualitative 67S.1.5 S.1 male pre P.5 qualitative 70S.1.6 S.1 male pre P.6 quantitative 68


MotivationWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.So: 20 (students) x 20 (time x parts) = 400 measurements

What’s the correlation structure?

How do you analyze this?


MotivationWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.So: 20 (students) x 20 (time x parts) = 400 measurements1) Do students improve from pre to post?2) Do females outperform males?3) Is qualitative or quantitative easier?4) Is improvement different for males and females? For qual. vs quant.?5) Does qual./quant. improve more? easier for males/females? 6) Is male/female difference different for qual/quant? Pre/Post?7) Does learning [pre vs post] alter any qual/quant disparity between

males/females?8) Are some parts easier or harder?9) Are some parts easier for males/females? Improve more?10) Do some parts improve more for males than females?11) Do some students do better or worse?12) Are some students better at qual/quant? Improve more?13) Do some students improve more on qual/quant?


Crossed random effectsWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.So: 20 (students) x 20 (time x parts) = 400 measurements• repeated measures ANOVA worldview: two options:– Students as the unit of analysis.sex as “between student” factorsection and time as “within student” factors.

– Test part as the unit of analysis.section as a “between part” factor, sex and time as “within part” factors.


Analysis by pooling (subject)summary(aov(data=test.data,

score~sex*test*section + Error(student/(test*section))))

Error: studentDf Sum Sq Mean Sq F value Pr(>F)

sex 1 3570 3570 1.443 0.245Residuals 18 44539 2474

Error: student:testDf Sum Sq Mean Sq F value Pr(>F)

test 1 0.90 0.90 0.879 0.361 sex:test 1 33.06 33.06 32.195 2.21e-05 ***Residuals 18 18.48 1.03

Error: student:sectionDf Sum Sq Mean Sq F value Pr(>F)

section 1 5048 5048 661.6 1.20e-15 ***sex:section 1 1604 1604 210.2 2.27e-11 ***Residuals 18 137 8

Error: student:test:sectionDf Sum Sq Mean Sq F value Pr(>F)

test:section 1 1242.6 1242.6 1238.1 < 2e-16 ***sex:test:section 1 61.6 61.6 61.4 3.29e-07 ***Residuals 18 18.1 1.0

Error: WithinDf Sum Sq Mean Sq F value Pr(>F)

Residuals 320 31652 98.91


Analysis by pooling (item)summary(aov(data=test.data,

score~sex*test*section + Error(part/(test*sex))))

Error: partDf Sum Sq Mean Sq F value Pr(>F)

section 1 5048 5048 1.302 0.287Residuals 8 31014 3877

Error: part:testDf Sum Sq Mean Sq F value Pr(>F)

test 1 0.9 0.9 0.046 0.836 test:section 1 1242.6 1242.6 62.891 4.65e-05 ***Residuals 8 158.1 19.8

Error: part:sexDf Sum Sq Mean Sq F value Pr(>F)

sex 1 3570 3570 205.83 5.44e-07 ***sex:section 1 1604 1604 92.48 1.14e-05 ***Residuals 8 139 17

Error: part:test:sexDf Sum Sq Mean Sq F value Pr(>F)

sex:test 1 33.06 33.06 23.53 0.001270 ** sex:test:section 1 61.62 61.62 43.86 0.000165 ***Residuals 8 11.24 1.41


Residuals 360 45042 125.1


Analysis by item/subject pooling

summary(aov(data=test.data, score~sex*test*section + Error(student/(test*section))))


sex 1 3570 3570 1.443 0.245Residuals 18 44539 2474








Residuals 320 31652 98.91

summary(aov(data=test.data, score~sex*test*section + Error(part/(test*sex))))










Residuals 360 45042 125.1

Both strategies give us the wrong answer because they neglect one source of covariation.

Subject analysis (pooling over parts)Ignores within-section among-part variation, most relevant for assessing differences between sections.

Item analysis (pooling over students)Ignores within-sex among-student variation, most relevant for assessing differences between sexes.


Analysis by explicit poolingWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.

ANOVA “by student” (disregarding part random effect)and ANOVA “by part” (disregarding subject random effect) give conflicting results.

This is because they respect some correlation structure (due to students, or due to parts) but not the other.


Crossed random effectsWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.So: 20 (students) x 20 (time x parts) = 400 measurements• repeated measures ANOVA worldview: two options:– Students as the unit of analysis.sex as “between student” factorsection and time as “within student” factors.

– Test part as the unit of analysis.section as a “between part” factor, sex and time as “within part” factors.

• Both of these omit the other source of correlated noise!


We want both

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But we can’t do this:summary(aov(data=test.data,

score~sex*test*section + Error(student/(test*section)) + Error(part/(test*sex))))

Need a more flexible way to specify correlation structure.


Random vs fixed effects: lay theoryWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.So: 20 (students) x 20 (time x parts) = 400 measurements1) Do students improve from pre to post?2) Do females outperform males?3) Is qualitative or quantitative easier?4) Is improvement different for males and females? For qual. vs quant.?5) Does qual./quant. improve more? easier for males/females? 6) Is male/female difference different for qual/quant? Pre/Post?7) Does learning [pre vs post] alter any qual/quant disparity between

males/females?8) Are some parts easier or harder?9) Are some parts easier for males/females? Improve more?10) Do some parts improve more for males than females?11) Do some students do better or worse?12) Are some students better at qual/quant? Improve more?13) Do some students improve more on qual/quant?


Fixed effects:- pre/post- male/female- qual/quantRandom effects:- student- part

Random vs fixed effects: lay theoryWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.So: 20 (students) x 20 (time x parts) = 400 measurements

Factor levels are of general relevance.We care about offsets for specific levels.

Factor levels are specific to our study. They are random samples of possible levels that might occur in the world.Maybe we care about variance across levels in general, but not the actual offsets for specific levels.


1) Do students improve from pre to post?2) Do females outperform males?3) Is qualitative or quantitative easier?4) Is improvement different for males and females? For

qual. vs quant.?5) Does qual./quant. improve more? easier for

males/females? 6) Is male/female difference different for qual/quant?

Pre/Post?7) Does learning [pre vs post] alter any qual/quant disparity

between males/females?8) Are some parts easier or harder?9) Are some parts easier for males/females? Improve more?10) Do some parts improve more for males than females?11) Do some students do better or worse?12) Are some students better at qual/quant? Improve more?13) Do some students improve more on qual/quant?

Random vs fixed effects: lay theoryWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.So: 20 (students) x 20 (time x parts) = 400 measurements

We can classify these kinds of questions as:Fixed main effects.Fixed 2-way interactions.Fixed 3-way interactions.Random main effects.Random 2-way interactions.Random 3-way interactions.


Complete / No / Partial Pooling• Complete pooling:

• No pooling:

• Partial pooling:

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yi ~α +βXi + ei

yi =α j[i] +βXi + eiα j = anything

yi =α j[i] +βXi + eiα j ~ Normal(γ,σα )

We estimate an overall mean starting weight, and an overall weight/time slope. These are completely pooled over all people, disregarding individual differences.

We estimate an overall weight/time slope (complete pooling for slope). And individual-specific starting weights with no constraints on their relationship (no pooling for intercept).

We estimate an overall weight/time slope (complete pooling for slope). And individual-specific starting weights with the understanding that they are somewhat similar, and thus are modeled as Normally distributed around some common mean (partial pooling for intercept).


Complete / No / Partial Pooling• Complete pooling:

Consequently: we get one intercept

• No pooling: Consequently: we get lots of intercepts that may differ a lotand be noisy.

• Partial pooling:Consequently: we get lots of intercepts, but they will be drawn toward the overall meanby an amount that scales with our uncertainty about them.

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yi ~α +βXi + ei

yi =α j[i] +βXi + eiα j = anything

yi =α j[i] +βXi + eiα j ~ Normal(γ,σα )


Complete / No / Partial Pooling

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Gelman’s example: Our estimates of coefficients for counties with little data are noisy. In the no pooling analysis these counties look extreme because of the noise. In the partial pooling analysis, this noisiness yields uncertainty and coefficients are adjusted to look more like the overall mean.

No pooling Partial pooling

Sample size per county Sample size per county


Random vs fixed effects: formalRandom effect coefficients are thought to come from some distribution. Their variation is treated as noise. They are ‘partially pooled’ so that they ‘shrink’ toward the average (0).

Fixed effects are parameters.There is no distribution for them.They are not pooled at all. They do not “shrink”.


Back to the design we couldn’t handleWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.m = lmer(data=test.data, score~sex*test*section + (1|student), REML = F)

Linear mixed model fit by maximum likelihood ['lmerMod']Formula: score ~ sex * test * section + (1 | student)

Data: test.dataAIC BIC logLik deviance df.resid

2991.907 3031.822 -1485.953 2971.907 390 Random effects:Groups Name Std.Dev.student (Intercept) 10.352 Residual 9.152 Number of obs: 400, groups: student, 20Fixed Effects:

(Intercept) sexmale69.18 -2.18

testpre sectionquantitative-3.64 6.80

sexmale:testpre sexmale:sectionquantitative0.42 -6.44

testpre:sectionquantitative sexmale:testpre:sectionquantitative8.62 -3.14

Here we allow the intercept to vary with student: +(1|student)


Back to the design we couldn’t handleWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.m = lmer(data=test.data, score~sex*test*section +(1|student)+(1|student:test)+(1|student:section)+(1|student:test:section))

This is the full “student” random effect structure.

m = lmer(data=test.data, score~sex*test*section +(1|part)+(1|part:test)+(1|part:sex)+(1|part:test:sex))

This is the full “part” random effect structure.

m = lmer(data=test.data, score~sex*test*section +(1|student)+(1|student:test)+(1|student:section)+(1|student:test:section)+(1|part)+(1|part:test)+(1|part:sex)+(1|part:test:sex)+(1|student:part))

This is the full random effect structure of the design.


Linear Mixed Effects AnalysisWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.m0 = lmer(data=test.data, score~1

+ (1|student) + (1|part) + (1|student:test) + (1|student:section) + (1|part:test) + (1|part:sex) + (1|part:sex:test) + (1|student:test:section)+ (1|student:part), REML = FALSE)

This is the exhaustive list of random effects that might exist in our data.

This exhaustive strategy makes sense for such a factorial design with random intercepts capturing sources of correlated error in our data – to get the by-item and by-subject ANOVAs together, but it’s not something you would do with complex real-world data with variable slopes, etc. In those cases the usually advised procedure is to start with the fixed effects, and add random effects so long as they improve the model.


Linear Mixed Effects AnalysisWe have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.

m0 = lmer(data=test.data, score~1+ (1|student) + (1|part) + (1|student:test) + (1|student:section) + (1|part:test) + (1|part:sex) + (1|part:sex:test) + (1|student:test:section) + (1|student:part), REML = FALSE)

mF = lmer(data=test.data, score~sex*test*section+ (1|student) + (1|part) + (1|student:test) + (1|student:section) + (1|part:test) + (1|part:sex) + (1|part:sex:test) + (1|student:test:section) + (1|student:part), REML = FALSE)

This is a null model that includes only the random effects, no fixed effects of interest.

This is the full model including all the fixed effects.

anova(m0,mF)

Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq) m0 11 1494.4 1538.3 -736.21 1472.4 mF 18 1440.4 1512.3 -702.22 1404.4 67.977 7 3.783e-12 ***

The full model is significantly better than null, so something is going on with our fixed effects.


Interpreting random effect SDs.We have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.

summary(m0)

Random effects: Groups Name Std.Dev.student:part (Int) 0.88364student:test:section (Int) 0.36681part:sex:test (Int) 0.95617student:section (Int) 0.71859student:test (Int) 0.01777part:sex (Int) 2.94028part:test (Int) 2.54545student (Int) 11.08337part (Int) 9.27407Residual 0.60322

Let’s look at the random effects (errors) to see what happened.

These are partitioning all the variation of our data into independent sources that add up to yield the full covariance. The SDs tell us how much variation is attributable to each source.

Can I test if these reflect significantly non-zero var?- Why do it? They reflect the structure of your

experiment, not really something worth testing.- Better to just compare the amount of variation in

one source to another (e.g. students vs residuals)But I really want to!- You could drop one you are interested in from

the model, and see if a model that includes it is better (AIC, or LRT)

- You could extract coefficients with ranef(), their standard errors with se.ranef() then calculate X=sum((ranef/se.ranef)^2) and compare to chi-squared distribution… Not really common.


Interpreting random effect SDs.

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We have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.

summary(m0) summary(mF)

Random effects: Groups Name Std.Dev.student:part (Int) 0.88364student:test:section (Int) 0.36681part:sex:test (Int) 0.95617student:section (Int) 0.71859student:test (Int) 0.01777part:sex (Int) 2.94028part:test (Int) 2.54545student (Int) 11.08337part (Int) 9.27407Residual 0.60322

Random effects:Groups Name Std.Dev.student:part (Int) 0.88375student:test:section (Int) 0.34807part:sex:test (Int) 0.28981student:section (Int) 0.70360student:test (Int) 0.04714part:sex (Int) 0.80917part:test (Int) 0.85788student (Int) 10.69397part (Int) 8.95206Residual 0.60474

Let’s look at the random effects (error) SDs to see what happened.

SDs reduced by a large proportion tell us that fixed effects explained that variation:part:test, part:sex, part:sex:test

Which fixed effects could account for this reduction?


Random effect SDs and residuals



sex 1 3570 3570 1.443 0.245Residuals 18 44539 2474








Residuals 320 31652 98.91











Residuals 360 45042 125.1

These give us estimates roughly comparable to the random effects:SD(students) = 10.7 ~= sqrt(2474/20)=11.18 ~= sqrt(125)=11.12SD(parts) = 8.95 ~= sqrt(3877/40)=9.84 ~= sqrt(98.91)=9.94However, even these are higher in aov, because we have more error terms to partition the variance in the full mixed model, and because we have partial pooling to draw estimates closer together. This is even more true for the other residual terms.

Subject analysis (pooling over parts)Ignores within-section among-part variation, most relevant for assessing differences between sections.

Item analysis (pooling over students)Ignores within-sex among-student variation, most relevant for assessing differences between sexes.


Interpreting fixed effects.We have pre- and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section, with 5 parts in each section.

Caution! It’s contentious whether it is sensible to get p values from those t values!

In general, we assess significance by either:- Dropping a coefficient from the full

model, and comparing it to full via LRT (makes sense in cases that are not a factorial experiment)

- Adopting a sequential ANOVA-like procedure of adding coefficients in sequence, and assessing whether they improve the model.

summary(mF)

Fixed effects:Estimate Std. Error t value

(Intercept) 69.1800 5.2767 13.110sexmale -2.1800 4.8309 -0.451testpre -3.6400 0.6060 -6.006sectionquantitative 6.8000 5.7284 1.187sexmale:testpre 0.4200 0.3818 1.100sexmale:sectionquantitative -6.4400 0.9640 -6.681testpre:sectionquantitative 8.6200 0.8565 10.064sexmale:testpre:sectionquantitative -3.1400 0.5384 -5.833

These are the various fixed effect coefficients. In this case, because we have a 2x2x2 fixed design, we have one coefficient per main-effect/interaction.

What do these coefficients mean?


Mixed model vs subject/itemsummary(aov(data=test.data, score~sex*test*section

+ Error(student/(test*section))))


sex 1 3570 3570 1.443 0.245Residuals 18 44539 2474








Residuals 320 31652 98.91











Residuals 360 45042 125.1

Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq) m0 11 1494.4 1538.3 -736.21 1472.4 m0.S 12 1496.4 1544.3 -736.20 1472.4 0.0065 1 0.93594 m0.ST 13 1497.0 1548.9 -735.51 1471.0 1.3915 1 0.23815 m0.STS 14 1497.6 1553.5 -734.78 1469.6 1.4457 1 0.22922 m0.STS.I1 15 1477.9 1537.7 -723.93 1447.9 21.7012 1 3.186e-06 ***m0.STS.I2 16 1476.0 1539.8 -721.98 1444.0 3.9105 1 0.04799 * m0.STS.I3 17 1455.6 1523.5 -710.82 1421.6 22.3145 1 2.315e-06 ***mF 18 1440.4 1512.3 -702.22 1404.4 17.2073 1 3.351e-05 ***

Mixed model gives us the worst of both by-subject and by-item analyses, by appropriately recognizing that here, part-variance is relevant for section, and subject-variance relevant for sex.

0.+ sex

. + test.+ section

.+ test:section.+ sex:test

.+ sex:section.+ sex:test:section


Crossed random effects.

• Taking into account only one random effect (separating subject and item analyses) gives us incorrect answers.

• We need to take both random effects into account jointly.• lmer() allows us to add many random effects and their

interactions.• For such factorial designs, I favor a sequential model-building

comparison:– Include all random effects from the design.– Add explanatory variables sequentially, and do nested model

comparison via LRT in anova() – like regular Type I ANOVA


What kind of random effect structure?

M = lmer(data=test.data, score~sex*test*section+ (test*section|student) + (sex*test|part) + (1|student:part)) )

M = lmer(data=test.data, score~sex*test*section+(1|student)+(1|part)+(1|student:test)+(1|student:section)+(1|part:test)+(1|part:sex)+(1|part:sex:test)+(1|student:test:section)+(1|student:part) )

This is a dummy coded random effect error structure.It will only work for categorical factors. It makes a few assumptions that may be violated. It is most similar to what the ‘repeated measures’ aov() command does. It will be anti-conservative in a few particular cases with odd covariances for the subject:X interactions.

This is the full “varying intercept and varying slope” random effect structure. Folks who think deeply about these things believe this to be the most generally effective specification. It uses quite a few more parameters to capture the (more flexible) random effect covariance structure. It estimates more parameters and is sometimes harder to fit.

In practice: you may run into estimation problems before you’ve managed to add the full random effect structure. You probably should avoid such highly factorial crossed, structured designs. (If you keep the design simple, the latter specification is better).


Other factors calling for LMM• Repeated measures designs that are unbalanced

(e.g., due to missing data).

• Need to:– Account for lower-level variation when estimating higher-level

coefficients.– Model variation of lower-level coefficients between groups.– Estimating coefficients for specific groups that might have little

data by partially pooling across all groups.– Make predictions at different grouping levels.

ED VUL | UCSD Psychology 34


Data structure decomposition

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s.idA[s.id] B[s.id] ... C1[s.id]... Y[s.id]

s.id SEX[s.id] MAJOR[s.id] IQ[s.id]SCORE[s.id]JohnDoe m psychology 110 60JaneSmith f business 125 75SueChang f psychology 105 40...

eA[A] eB[B] eAB[A,B]... eC ...

eSEX[]m -1f +1

eMAJOR[]psychology -1business +3cogsci -0.5linguistics -1.5

eIQ0.13


Single table decomposition: LMs.idA[s.id] B[s.id] ... C1[s.id]... Y[s.id]

eA[A] eB[B] eAB[A,B]... eC ...

• Every t-test/ANOVA/ANCOVA (LM/GLM) data structure can be reduced to this single-table “3rd normal form”.

• Each measurement (Y) is uniquely associated with a particular subject/unit.

• Each subject may have infinitely many other properties (like factor level category, covariate values, etc.)


Repeated and mixed-design ANOVAs.id sA[s.id]sB[s.id]... sC1[s.id] ...

eSA[A] eSB[B] eSAB[A,B] ... eSC1...

m.id s.id[m.id] Q[m.id] R[m.id] ... W[m.id] Y[m.id]

eMQ[Q] eMR[R] eQR[Q,R]... eSW ...

• Data structures for repeated measures and mixed-design ANOVA can be reduced to such a 2-table form*

• Every measurement is associated with a particular subject, and might have some measurement categorization scheme (within-subject factors).

• In the course of factoring into error strata, we effectively estimate lots of “random effects” Es[s.id] EQs[s.id,Q]


Predictors at many levels

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Private Public Private PublicEast Coast West Coast

SchoolsSchoolA,Private, East, $1000/studentSchoolB, Public, East, $2500/student...

Classes (in Schools)

SchoolB

Textbook1 Textbook2 Textbook1 Textbook2Written Exam Oral Exam

SchoolB,Class.B.1, Textbook1, Written, 24 studentsSchoolB, Class.B.2, Textbook1, Oral, 32 students...

Class.B.1Students(in Classes)


Predictors at many levelsWe can manage this design by partitioning variance into strata and proceeding with a mixed-design ANOVA.

However, repeated measures ANOVA can’t handle all the possible structures we may need to deal with.

Mixed-design ANOVA will fail us if we…- add explanatory factors to students- consider crossed random effects (substitute teachers)- add other levels (districts; …, teachers within schools, classes within

teachers,…; etc.)


Something we can’t do…

s.Id d.id[s.id] sG[s.id]sH[s.id]...

eSG[G] eSH[H] eSGH[G,H] ...

m.id s.id[m.id] Q[m.id] R[m.id] ... W[m.id] Y[m.id]


More than two explanatory levels.• E.g., students (with explanatory variables), in schools (with

explanatory variables), in districts (with explanatory variables)

d.iddA[s.id]dB[d.id]...

eDA[A] eDB[B] eDAB[A,B] ...


Something we can’t do…s.id sG[s.id]sH[s.id]...

eSG[G] eSH[H] eSGH[G,H] ...

m.Id s.id[m.id] i.id[m.id] Q[m.id] ... Y[m.id]


Crossed “random effects”• E.g., each measurement is associated with a particular

student (with explanatory variables), and a particular item (with explanatory variables)

i.id iA[s.id]iB[d.id]...

eIA[A] eIB[B] eIAB[A,B] ...


Data structures in need of LMMNormal LM/GLM structure: each unit is uniquely associated with a measurement.

Repeated measures structure:Explanatory variables at only two levels. (e.g., “between subject” and “within subject”).

More levels!Explanatory variables at more than two levels. (e.g., classes in schools in districts)

Crossed random effects!(e.g., subjects with explanatory variables crossed with items with explanatory variables)

Stuff we can’t do appropriately without mixed-effect models.



s.idSEX[s.id]

m.idt.id[m.id] p.id[m.id] SCORE[m.id]

p.idSECTION[p.id]

t.ids.id[t.id] TIME[t.id]



female malepr

e

post

qualitativequantitative

parts

stud

ents

201ab Quantitative methods Linear Mixed Effects ModelsWe have pre-and post-class exam scores for 10 males and 10 females. The exam is divided into a qualitative and quantitative section,

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