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Teaching Styles Revisited — 30 Years of Modelling
John Hinde
Science Foundation Ireland Funded Bio-SI Project & Statistics Group,School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, GalwayIreland
john.hinde@nuigalway.ie
Murray Aitkin Meeting, RSS, London
15 April 2010
John Hinde (NUIG) 15 April 2010 1 / 24
Summary
1 Teaching Styles Study
2 Teaching Styles Study ReanalysisClustering Teachers — Latent Class ModelTeaching Style & Pupil ProgressPupil Personality Clustering
3 Teaching Styles Analyses Today
4 Murray’s Legacy
John Hinde (NUIG) 15 April 2010 2 / 24
Teaching Styles Study (Bennet, 1976)
Study of relationship between Teaching Style and Pupil Performance inprimary schools
Questionnaire on 28 items (in 6 areas) of classroom behaviour → 36binary items
sent to all 871 primary in Lancs & Cumbria for 3rd and 4th yearteachers88% response ratesubsequent analysis based 468 4th year teachers
Principal component analysis & cluster analysis
12 cluster solution chosen78 teachers not clearly classified into any clusterclusters ordered from extremely formal to extremely informal
John Hinde (NUIG) 15 April 2010 3 / 24
Teaching Styles Study (Bennet, 1976)
37 teachers selected to represent 7 clusters (2 informal, 3 mixed, 2formal)
950 Pupils in classes of selected teachers followed for one year
pre-tested on entry to class — reading, mathematics, Englishpost-tested prior to exit from schoolpersonality tests — 15 personality variables
Analysis of Covariance on post-test scores
used to test differences between teaching stylesformal/mixed/informal
adjusting for pre-testsignificant differences found
formal better than informal
John Hinde (NUIG) 15 April 2010 4 / 24
Teaching Styles Study Reanalysis (Aitkin et al, 1981)
Statistical modelling approach to each stage
model based clustering of teachers — latent class model
variance component model for pupil performance and teaching style
pupil personality — normal mixture models, factor model
Involved
software development
extensive model fitting (maximum likelihood)
model selection/comparison
model validation
John Hinde (NUIG) 15 April 2010 5 / 24
Clustering Teachers — Latent Class Model
Mixture model with conditional independence of items within each class
P(X = x) =K∑
j=1
πj
38∏l=1
P(Xl = xl |j , θjl)
gives probabilistic assignment of teachers to clusters
Estimation — EM algorithm
Number of classes? Selected 3 class modelfit sequence of models (multiple maxima)non-standard testing problem: bootstrap simulation for −2 log `Bayesian approach (Aitkin and Rubin, 1985)graphical tests based Total Item Score — normal mixtures
Conditional independence?
Discrete classes vs continuous latent variable (factor) models (Bockand Aitkin, 1981)
John Hinde (NUIG) 15 April 2010 6 / 24
Teaching Style & Pupil Progress
Variance component model — pupils nested within classes (Teachers)
Ypqr = µ+ γxpqr + αp + Tq + εpqr
xpqr pre-test score;γ common pre-test effect → γp interaction with method
αp method (teaching style) effects
Tq teacher ability: Tq ∼ N(0, σ2T )
εpqr ∼ N(0, σ2E ) independent of Tq
Method effects found not to be statistically significant — betweenclass (teacher) variation
Model estimation used probabilistic assignment of Teachers to methods(clusters)
John Hinde (NUIG) 15 April 2010 7 / 24
Teaching Style & Pupil Progress (ctd)
Could combine Teacher and method effects
Tq ∼ Mixture-N(αp, σ2T ) ≈
3∑p=1
πqpN(αp, σ2T )
where πqp are given by teacher clustering probabilistic assignments.
sort of heterogeneity model (Verbeke and Lesaffre) with fixed πqp andcommon variance; σ2
T ,p?
separation of estimation of teacher style and pupil performancejustified by conditional independence argument
joint modelling — mixture of experts heterogeneity model
models variation in pupil performance allowing for varying teacherability (variance components model)models variation in teacher ability by indicators of style (model formixing proportions)
John Hinde (NUIG) 15 April 2010 8 / 24
Pupil Personality Clustering
15 personality variables (only 8 used in original analysis)Multivariate normal mixture model with common variance matrix
K∑j=1
πj N(µj ,Σ)
EM algorithm in GENSTAT; extremely slow convergence
Σ diagonal — conditional independence; multiple maxima
Normal factor model
X|U ∼ Np(µ+ ΛU,Ψ), U ∼ Nr (0, I)
Problems with normality assumptions — marginal distributionsextremely skew
Use personality variables directly in pupil achievement model —computational limitations
John Hinde (NUIG) 15 April 2010 9 / 24
Teaching Styles Analyses Today
Statistical Modelling tools much improved
Computer power on a laptop
For example, in R . . .
John Hinde (NUIG) 15 April 2010 10 / 24
Teaching Styles: Latent Class Models
class3x<-poLCA(f,styles,nclass=3,nrep=10) ### try a number of startsModel 1: llik = -9789.05 ... best llik = -9789.05Model 2: llik = -9790.939 ... best llik = -9789.05Model 3: llik = -9789.05 ... best llik = -9789.05Model 4: llik = -9789.05 ... best llik = -9789.05Model 5: llik = -9797.303 ... best llik = -9789.05Model 6: llik = -9789.05 ... best llik = -9789.05Model 7: llik = -9789.05 ... best llik = -9789.05Model 8: llik = -9799.223 ... best llik = -9789.05Model 9: llik = -9790.939 ... best llik = -9789.05Model 10: llik = -9798.57 ... best llik = -9789.05.......
Estimated class population shares0.3975 0.3012 0.301
Teaching Styles paper:
Estimated class population shares0.366 0.312 0.322
Variance Components Model: English
Linear mixed model fit by maximum likelihoodFormula: eng.s.p ~ eng.scen + (1 | teacher)
Data: pupilfAIC BIC logLik deviance REMLdev
6224 6243 -3108 6216 6222Random effects:Groups Name Variance Std.Dev.teacher (Intercept) 10.746 3.2781Residual 46.730 6.8360Number of obs: 920, groups: teacher, 36
Fixed effects:Estimate Std. Error t value
(Intercept) 106.45824 0.59410 179.19eng.scen 0.74348 0.01777 41.84
NPMLE Variance Components Model: EnglishFinite mass point distribution for Teacher effect Tq
No. points −2 log `
1 6325.7312 6212.7123 6206.9424 6206.6495 6206.656
Call: allvc(formula = eng.s.p ~ eng.scen, random = ~1 | teacher,data = pupilf, k = 3, random.distribution = "np")
Coefficients:eng.scen MASS1 MASS2 MASS3
0.7448 102.1820 105.2298 110.3676
Component distribution - MLE of sigma: 6.828Random effect distribution - standard deviation: 3.21772
Mixture proportions:MASS1 MASS2 MASS3
0.2230806 0.4065043 0.3704151-2 log L: 6206.9
Variance Components Model with Methods: English
Linear mixed model fit by maximum likelihoodFormula: eng.s.p ~ eng.scen + z2 + z3 + (1 | teacher)
Data: pupilfAIC BIC logLik deviance REMLdev
6225 6254 -3107 6213 6212Random effects:Groups Name Variance Std.Dev.teacher (Intercept) 9.5494 3.0902Residual 46.7512 6.8375Number of obs: 920, groups: teacher, 36
Fixed effects:Estimate Std. Error t value
(Intercept) 105.42793 0.86798 121.46eng.scen 0.74287 0.01773 41.89z2 -0.77508 3.48716 -0.22z3 2.19786 1.22388 1.80
Variance Components Model with Methods: Mathematics
Linear mixed model fit by maximum likelihoodFormula: math.s.p ~ math.scen + z2 + z3 + (1 | teacher)
Data: pupilfAIC BIC logLik deviance REMLdev
6437 6466 -3212 6425 6421Random effects:Groups Name Variance Std.Dev.teacher (Intercept) 20.122 4.4858Residual 57.364 7.5739Number of obs: 921, groups: teacher, 36
Fixed effects:Estimate Std. Error t value
(Intercept) 102.63308 1.21366 84.56math.scen 0.81115 0.02192 37.01z2 -3.60902 4.89355 -0.74z3 0.69271 1.71107 0.40
> anova(pupil.mathz,pupil.math)Models:pupil.math: math.s.p ~ math.scen + (1 | teacher)pupil.mathz: math.s.p ~ math.scen + z2 + z3 + (1 | teacher)
Df AIC BIC logLik Chisq Chi Df Pr(>Chisq)pupil.math 4 6433.7 6453.0 -3212.9pupil.mathz 6 6437.0 6465.9 -3212.5 0.739 2 0.691
English Scores
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Reading Scores
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John Hinde (NUIG) 15 April 2010 18 / 24
Variance Components Model: All Three Subjects
Linear mixed model fit by maximum likelihoodFormula: post ~ subject + subject:pre - 1 + (subject - 1 | teacher/pupil)
Data: pupil.allAIC BIC logLik deviance REMLdev
19395 19509 -9679 19357 19374Random effects:Groups Name Variance Std.Dev. Corrpupil:teacher subjecteng 34.580 5.8805
subjectmath 45.085 6.7145 0.341subjectread 36.064 6.0053 0.384 0.444
teacher subjecteng 11.907 3.4506subjectmath 23.867 4.8854 0.755subjectread 23.394 4.8368 0.666 0.701
Residual 13.908 3.7293Number of obs: 2849, groups: pupil:teacher, 950; teacher, 37
Fixed effects:Estimate Std. Error t value
subjecteng 106.39793 0.61328 173.49subjectmath 102.76120 0.84360 121.81subjectread 105.12937 0.82992 126.67subjecteng:pre 0.66580 0.01712 38.88subjectmath:pre 0.73812 0.02093 35.27subjectread:pre 0.66661 0.01595 41.81
General Observations
Computing issues — time; space; software
EM algorithm — powerful general framework, but slow and nostandard errors
Random effects — normal distribution
Model based approach — preferable to ad hoc data reduction,clustering, etc
Conditional independence — a fundamental principle in modelbuilding
Model validation — limited by compute power
Constraints on modelling software capability force thinking over fitting
Murray’s modelling legacy . . .
John Hinde (NUIG) 15 April 2010 20 / 24
EM Algorithm
Mixture applications of the EM algorithm in GLIM.
NPML estimation of the mixing distribution in general statistical models
Likelihood and Bayesian analysis of mixtures
A hybrid EM/Gauss-Newton algorithm for maximum likelihood in mixturedistributions
A general maximum likelihood analysis of measurement error in generalizedlinear models
Estimation and hypothesis testing in finite mixture models
Mixture models, outliers, and the EM algorithm
Marginal maximum likelihood estimation of item parameters: Application ofan EM algorithm
John Hinde (NUIG) 15 April 2010 21 / 24
Random Effects
Regression models for repeated measurements
A general maximum likelihood analysis of overdispersion in generalized linearmodels
Random effect extensions of generalized linear models.
A general maximum likelihood analysis of variance components ingeneralized linear models
Regression models for binary longitudinal responses
Variance component models for longitudinal count data with baselineinformation: Epilepsy data revisited.
Random coefficient models for binary longitudinal responses with attrition.
Variance component models with binary response: Interviewer variability.
John Hinde (NUIG) 15 April 2010 22 / 24
Modelling
Statistical modelling issues in school effectiveness studies
A note on the regression analysis of censored data
Statistical modelling: The likelihood approach.
Modelling variance heterogeneity in normal regression using GLIM.
Model choice in contingency table analysis using the posterior Bayes factor
Meta-analysis by random effect modelling in generalized linear models.
The fitting of exponential, Weibull and extreme value distributions tocomplex censored survival data using GLIM
Fitting the multinomial logit model with continuous covariates in GLIM
An analysis of models for the dilution and adulteration of fruit juice
Statistical modelling of unemployment rates from EEC labour force survey
A reanalysis of the Stanford heart transplant data
Bayesian model comparison and model averaging for small-area estimation
Still-births among the offspring of male radiation workers at the Sellafieldnuclear reprocessing plantJohn Hinde (NUIG) 15 April 2010 23 / 24
References
Aitkin, M, Francis, B, Hinde, J, Darnell, R (2009)Statistical Modelling in R, Oxford.
Also . . .
Aitkin, M, Francis, B, Hinde, J (2005)Statistical Modelling in Glim4, 2nd Edition, Oxford.
John Hinde (NUIG) 15 April 2010 24 / 24
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