Workshop on multilevel modeling II - Université Lavalarchimede.mat.ulaval.ca/pages/duchesne/handouts.pdfWorkshop on multilevel modeling II Belkacem Abdous & Thierry Duchesne [email protected]

Workshop on multilevel modeling II

Belkacem Abdous & Thierry [email protected]

Universite Laval

Bangalore: October 17-21, 2011

Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 1 / 297

Course Aims

Recap of two level models

• Introduction• Two-level models for binary responses• Subject-specific and population-averaged inferences

Advanced MLM topics

• Higher-level models with nested random effects• Higher-level models with crossed random effects


Notes

Notes

Introduction

Multilevel Models

Also known as

random-effects models,

hierarchical models,

variance-components models,

random-coefficient models,

mixed models


Introduction

What is multilevel modeling?

Statistical models designed for data with hierarchicalstructures or multistage samples.

Examples:• take a sample of districts, then sample individuals within

each district• pupils nested within schools• patients nested in hospitals,• people in neighborhoods,• employees in firms.

Longitudinal data is a classical example where multipleobservations over time are nested within units (e.g.subjects).


Notes

Notes

Introduction

Multilevel modeling: Four Key Notions1

1-Modeling data with a complex structure:

A large range of structures that ML can handle routinely;e.g. houses nested in neighborhoods

2-Modeling heterogeneity:

standard regression models (averages), i.e. the generalrelationship, while ML additionally models variances; e.g.individual house prices vary from neighborhood toneighborhood

1From: http://www.cmm.bristol.ac.uk/Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 5 / 297

Introduction

Multilevel modeling: Four Key Notions2

3-Modeling dependent data:

potentially complex dependencies in the outcome over time,over space, over context; e.g.houses within a neighborhoodtend to have similar prices

4-Modeling contextuality: micro and macro and relations,

e.g. individual house prices depends on individual propertycharacteristics and on neighborhood characteristics

2From: http://www.cmm.bristol.ac.uk/Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 6 / 297

Notes

Notes

Introduction

Hierarchical structures


Introduction

A 3-level hierarchical structure


Notes

Notes

Introduction

Cross-classified structure


Introduction



Notes

Notes

Introduction

Multiple membership structure


Introduction

Multiple membership structure


Notes

Notes

Introduction

A mix of crossed-classifications and multiplemembership structures


Introduction

Analysis Strategies for Multilevel Data

Group-level analysis

Individual analysis

Contextual analysis

Analysis of covariance (fixed effects model)

Fit single-level model but adjust standard errors forclustering (GEE approach)

Multilevel (random effects) model


Notes

Notes

Introduction


Aggregate to level 2 and fit standard regression models 3

Example: use the regional incidence rate of coronaryheart disease (CHD) as the dependent variable andvariables including average age and income, proportionof women etc. as independent variables.

This loses a lot of information and we riskmisinterpreting the results.

3LEYLAND AND GROENEWEGEN:http://nvl002.nivel.nl/postprint/PPpp1539.pdf


Introduction


Problem : if level 2 and level 1 variables reflect differentcausal processes

Ecological or aggregation fallacy: This is the methodologicalidentification of a relationship at an area level between anoutcome and a population characteristic, and attribution ofthis relation to individuals when this relationship actuallydoes not exist at the individual level.


Notes

Notes

Introduction


Robinson (1950) : correlation between illiteracy andethnicity in the USA

Level Black illiteracy Foreign-bornilliteracy

Individual 0.20 0.11(97 million people)

State( 48 units) 0.77 -0.52


Introduction

Individual analysis

Use Level 1 variables and distribute level 2 characteristics tolevel 1 individuals then fit standard regression models 4

Example: assign the economic welfare of regions to allindividuals (i.e. identical for each individual within aregion).

Here we risk the atomistic fallacy : draw inferencesregarding the relation between group level variablesbased on individual level data when individual-levelassociations may differ of those at the group level.



Notes

Notes

Introduction

Ecological and atomistic fallacies 5

The relationship between cost and need found amongindividuals. As need increases, so does the average cost.

The fact that individuals live in different municipalities isignored



Introduction

Ecological and atomistic fallacies

The relationship between cost and need across threemunicipalities. The relationship differs little from thatfound at the individual level.


Notes

Notes

Introduction

Ecological and atomistic fallacies

This ignores the data on individuals and assumes thatthe average relationships between municipalities holdbetween individuals.The relationship is fairly consistent across the threemunicipalities. An increase in need is associated with asmaller increase in cost than in the two previous figures

The average level of spending for a fixed level of needvaries between municipalities, and the ecological andindividual analyses could not take this into account.


Introduction

Contextual analysis

Analyse individual-level data by including group-levelpredictors.

But : Assumes all group-level variance can be explainedby group-level predictors; incorrect SE’s for group-levelpredictors


Notes

Notes

Introduction

Contextual analysis

Do pupils in single-sex school experience higher exam attainment?

Structure: 4059 pupils in 65 schools

Response: Normal score across all London pupils aged 16

Predictor: Girls and Boys School compared to Mixed school

Parameter Single level Multilevel

Cons (Mixed school) -0.098 (0.021) -0.101 (0.070)Boy school 0.122 (0.049) 0.064 (0.149)Girl school 0.245 (0.034) 0.258 (0.117)

Between school 0.155 (0.030)variance(σ2

u)Between student 0.985 (0.022) 0.848 (0.019)

variance (σ2e )


Introduction

Analysis of covariance (fixed effects model)

Include dummy variables for each and every group. But

What if number of groups very large, eg households?

No single parameter assesses between group differences

Can not make inferences beyond groups in sample

Can not include group-level predictors as all degrees offreedom at the group-level have been consumed

Target of inference: individual School versus schools


Notes

Notes

Introduction

GEE approach

Fit single-level model but adjust standard errors forclustering (GEE approach) But

Treats groups as a nuisance rather than of substantiveinterest;

No estimate of between-group variance;

Not extendible to more levels and complex heterogeneity


Introduction

Multilevel (random effects) model

Partition residual variance into between- andwithin-group (level 2 and level 1) components.

Allows for un-observables at each level.

corrects standard errors.

Micro AND macro models analysed simultaneously.

Avoids ecological fallacy and atomistic fallacy.

Richer set of research questions BUT (as usual) needwell-specified model and assumptions met.


Notes

Notes

Two-level models for binary responses

Two-level models for binary responses


Two-level models for binary responses Single-level models

Logistic regression

When the distribution is binomial (e.g., binary response,such as HIV prevalence) and the link function is logit, we getthe logistic regression model:

E [yi |xi ] = P[yi = 1|xi ] = πi ;

logit(πi ) = ln(

πi1−πi

)= ln{Odds(yi = 1|xi )} = β1 +β2xi ;

Odds(yi = 1|xi ) = eβ1+β2xi ⇔ πi = eβ1+β2xi

1+eβ1+β2xi.


Notes

Notes


Interpretation of the model parameters

The exponential of the model parameters are easilyinterpreted in terms of odds ratios:

eβ1: odds of getting a response of 1 when xi = 0;

eβ2: value by which odds of getting yi = 1 are multipliedwhen xi increases by 1 (and other covariates xij ’s, ifavailabe, remain unchanged).



Example: Determinants of HIV prevalenceamong FSW

Ramesh et al. (AIDS 2008) use the IBBA1 data to exploreassociations between HIV prevalence and sociodemographic+ sex work characteristics of FSW in 23 districts of 4southern states.


Notes

Notes



Model and notation:

yi = 1 if i-th FSW is HIV positive, yi = 0 otherwise

xi = (1,xi1,xi2, . . . ,xip): sociodemographic and sex workcharacteristics of i-th FSW

Model:logit(P[Yi = 1|xi ]) = β′xi = β0 +β1xi1 +β2xi2 + · · ·+βpxip




Ramesh et al. (AIDS 2008)Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 32 / 297

Notes

Notes



Ramesh et al. (AIDS 2008)




For instance if we consider xi1 = 1 if FSW i’s sex clientvolume per week is greater than or equal to ten and xi1 = 0 ifFSW i’s sex client volume per week is less than ten.

β1 = 0.223, which corresponds to a value of 1.25 for thefollowing odds ratio:

P[HIV+|xi1 = 1, xi2 = x∗2 , . . . ,xip = x∗p ]

/P[HIV-|xi1 = 1, xi2 = x∗2 , . . . ,xip = x∗p ]

P[HIV+|xi1 = 0, xi2 = x∗2 , . . . ,xip = x∗p ]

/P[HIV-|xi1 = 0, xi2 = x∗2 , . . . ,xip = x∗p ]

.


Notes

Notes

Two-level models for binary responses Two-level random intercept model

Two-level random intercept logistic model

Suppose that we have individuals (level 1) within districts(level 2). We might need to

relax the assumption of independence among individualsin a same district;

incorporate the potential effects of omitted/unobserveddistrict-specific variables in the model;

allow the odds of having a response equal to 1 for equalxi1 to vary among districts.

This can be done by adding a district-level random interceptin the logistic regression model.



Two-level random intercept logistic model

Let

yij be the response for individual i in district j;

x1ij = (x1ij1,x1ij2, . . . ,x1ijp)′ be the individual level covariatesfor individual i in district j.

x2j = (x2j1,x2j2, . . . ,x2jq)′ be the district level covariates fordistrict j;

The random intercept logistic regression model assumes that

logit{P[yij = 1|xij , ζj ]} = β0 +β′1x1ij +β′2x2j + ζj ,

with ζj ∼N(0,Ψ) a district-specific random intercept.


Notes

Notes


Interpretation of the random intercept logisticmodel

β0: log-odds of yij = 1 when x2j = x3ij = ζj = 0;

β1k : increase in log-odds of yij = 1 when x1ijk increases byone unit, but other x1ij`’s, ζj and x2j remain unchanged⇒effect of increasing the value of x1ijk by one unitwithout changing district and holding the value of allother covariates fixed.

β2k : increase in log-odds of yij = 1 when x2jk increases byone unit, but ζj and all other covariates remainunchanged⇒effect of increasing the value of x2jk by one unitwithout changing district and holding the value of x1ij

fixed;


Two-level models for binary responses Example: HIV in FSW


Ramesh et al. (AIDS 2008) fitted a two-level randomintercept model with only individual level covariates:

yij = 1 if i-th FSW in j-th district is HIV positive, yij = 0otherwise

x1ij : sociodemographics and sex work characteristics fori-th FSW in j-th district

Model:logit(P[yij = 1|x1ij , ζj ]) = β0 +β11x1ij1 + · · ·+β1px1ijp + ζj , withζj ∼N(0,Ψ).


Notes

Notes









Notes

Notes



For instance, the estimate of the coefficient in front of thecovariate that is 1 if the FSW has more than 10 clients perweek is β = 0.107, with a standard error of 0.06. Thecorresponding odds ratio would be exp(0.107) = 1.11.

The estimate of the variance of the random intercepts isΨ = 0.347.



Determinants of HIV prevalence among FSW

Ramesh et al fitted the model using the PQL method inMLwiN. We can fit it by maximum likelihood (more on this ina few moments) using Stata’s xtmelogit:

************

* Read data in data and declare as panel data,

* define response variable hiv_prev

************

use "F:\IBBA1.dta", clear

xtset districtnum

generate hiv_prev = abs(hiv_final-2)

********

* Null model

********

xtmelogit hiv_prev || districtnum: ///

, variance intpoints(12)


Notes

Notes


Determinants of HIV prevalence among FSW

**************

* Random intercept model

**************

xtmelogit hiv_prev ib(2).CurrentAge ///

ib(1).MaritalStatus ib(2).Literacy ///

ib(1).IncomeOtherSources ib(1).PlaceSolicit ib(1).SexVolume ///

ib(1).DurationSW ib(2).SexWorkDebut ib(2).SexDebut ///

|| districtnum: , variance intpoints(12)

* RAN IN ABOUT 2 MINUTES

estimates store RameshRI1

matrix ri1 = e(b)





Notes

Notes







Notes

Notes






Fitting the same model with Stata’s gllamm:

gllamm hiv_prev ///

AgeLess25 WidDivSepDeva Unmarried Literate ///

NoOtherIncome Brothels PubPlaces ///

Clients10plus Duration5plus ///

StartedWorkLess20 SexDebutLess15 ///

, i(districtnum) link(logit) family(binom) nip(3)

* RAN IN 2.5 MINUTES


matrix ri3 = e(b)


Notes

Notes



* Then use ri3 as starting value for more quadrature points

gllamm hiv_prev ///





, i(districtnum) link(logit) family(binom) ///

from(ri3) nip(15)

* RAN IN 7 MINUTES


matrix ri15 = e(b)





Notes

Notes




Two-level models for binary responses Latent-response formulation

Latent-response formulation

To interpret the value of the variance estimates in randomintercept models, it is useful to rewrite the logistic regressionmodel as a latent-response model.

Suppose that y ∗i is an unobserved variable (e.g., “propensity”to contract diseases), but that we observe yi defined as

yi =

{1, if y ∗i > 00, otherwise.

,

with y ∗i = β0 +β1xi + εi .

If εi follows a standard logistic distribution (mean 0, varianceθ = π2/3≈ 3.29), we get the logistic regression model for yi .


Notes

Notes



From Rabe-Hesketh & Skrondal (2005, p. 239)




Now to get the two-level random intercept model, supposethat y ∗ij is an unobserved variable, but that we observe yij

defined as

yij =

{1, if y ∗ij > 00, otherwise.

,

with y ∗ij = β0 +β′1x1ij +β′2x2j + ζj + εij .

If εij is independent of ζj and follows a standard logisticdistribution, we get the random intercept logistic regressionmodel for yij .


Notes

Notes


Latent variable formulation

This formulation yields an interesting interpretation for Ψ:

correlation(y ∗ij ,y∗i ′j |xij ,xi ′j) =

Ψ

Ψ + 3.29= VPC .

⇒ Ψ controls the conditional within-district correlation.




VPC is often referred to as the variance partition coefficient.

It is interpreted as the proportion of the residual (i.e.,unexplained by covariates) variability in the latent response(e.g., propensity to contract diseases) explained bybetween-district variations.


Notes

Notes



Because θ = Var (y ∗ij |ζj) = Var (εij) = 3.29 is fixed, omitting level1 covariates x1ijk from a random-intercept logistic modelcannot result in an increase in the unexplained level 1variability Ψ.

But because VPC is increased (there is more unexplainedvariability), there will be an increase in the estimate of Ψ.




For the “Null” random intercept model in Table 4 ofRamesh et al:

Ψ = 0.514

⇒VPC = 0.514/(0.514 + 3.29)≈ 0.135

Hence in the null model, 13.5% of the latent response’svariability is explained by unobserved between-districtcharacteristics.


Notes

Notes



In the random intercept model with covariates (column 2 ofTable 4 of Ramesh et al):

Ψ = 0.347

⇒VPC = 0.347/(0.347 + 3.29)≈ 0.095

So 9.5% of the residual variability (variability in y ∗

unexplained by the individual-level covariates) is explained bybetween-district variations.


Two-level models for binary responses Two-level random coefficient model

Two-level random coefficient (slope) model

It is possible to generalize the model so that the effect ofthe level 1 covariates is different in each district.

This can be done by adding random coefficients in front ofsome of the individual-level covariates of the model:

logit{P[yij = 1|x1ij ,x2j , ζ0j , ζ1j ]} = β0 + ζ0j + (β1 + ζ1j)x1ij1

+β12x1ij2 + · · ·+β1px1ijp

+β2x2j ,

where (ζ0j , ζ1j) are assumed to follow a bivariate normaldistribution with mean vector (0,0), Var (ζ0j) = Ψ00,Var (ζ1j) = Ψ11 and Cov (ζ0j , ζ1j) = Ψ01.


Notes

Notes


Two-level random coefficient (slope) model

Interpretation:

(β1 + ζ1j) is the increase in the log-odds of yij = 1 for anindividual in district j whose value of x1ij1 increases byone unit;

β1 is the same increase, but in an “average district”, i.e.a district for which ζ1j = 0;



Example: Determinants of HIV prevalenceamong FSW (cont’d)

Ramesh et al. (AIDS 2008) also fitted two-level randomcoefficient models to the IBBA1 data:

yij = 1 if i-th FSW in j-th district is HIV positive, yij = 0otherwise

x1ij : sociodemographic and sex work characteristics fori-th FSW in j-th district


Notes

Notes



The model in column 3 (random intercept and randomcoefficients in front of marital status indicators):

logit(P[yij = 1|x1ij , ζj ]) = (β0 + ζ0j)

+ (β11 + ζ1j)x1ij1 + (β12 + ζ2j)x1ij2

+β13x1ij3 + · · ·+β1px1ijp,

(ζ0j , ζ1j , ζ2j)′ ∼N

(0,0,0)′,Ψ =

Ψ0,0 Ψ0,1 Ψ0,2

Ψ0,1 Ψ1,1 Ψ1,2

Ψ0,2 Ψ1,2 Ψ2,2

x1ij1 = 1 if widowed/divorsed/separated/devadasi

x1ij2 = 1 if unmarried.





Notes

Notes





Determinants of HIV prevalence among FSW(cont’d)

Fitting the model with Stata’s xtmelogit

**************

* Random slope (marital status)

**************

* First, with Laplace approximation and a simple

* variance-covariance structure

xtmelogit hiv_prev ///





|| districtnum: WidDivSepDeva Unmarried, ///

variance laplace

* RAN IN 5 MINUTES

matrix rs1un1 = e(b)


Notes

Notes



* Same, with more complex variance-covariance structure

* and 5 quadrature points

matrix a1 = (rs1un1,0,0,0)







variance covariance(unstructured) ///

intpoints(5) from(a1,copy) refineopts(iterate(0))

* RAN IN 18 MINUTES





Notes

Notes






To get the values in column 3 of Table 4 of Ramesh et al:

Var (ζ0j + x1ij1ζ1j + x1ij2ζ2j) = Ψ00 + x21ij1Ψ11 + x2

1ij2Ψ22

+ 2x1ij1Ψ01 + 2x1ij2Ψ02 + 2x1ij1x1ij2Ψ12.

With maximum likelihood, 5 quadrature points, xtmelogit:

Married (x1ij1 = x1ij2 = 0): 0.55 [With PQL, from Ramesh et al., Table

4: 0.62]

Widowed/Divorced/Separated/Devadasi (x1ij1 = 1, x1ij2 = 0):0.55 + 0.13 + 2∗ (−0.20) = 0.28 [With PQL, from Ramesh et al., Table

4: 0.31]

Never married (x1ij1 = 0, x1ij2 = 1):0.55 + 0.11 + 2∗ (−0.21) = 0.24 [With PQL, from Ramesh et al., Table

4: 0.27]


Notes

Notes



The effect of marital status (odds ratios of being HIV+when divorced or never married vs married) is not the samefrom one district to the other.

If we look at the xtmelogit output, we have a coefficient of0.66 with a variance of 0.28 for Widowed, etc. vs married.

⇒ log-odds ratio of being HIV+ for Widowed, etc. vsmarried varies among districts according to a N(0.66, 0.28)distribution.




Some interesting calculations can then be made. Forexample:

1st quartile of N(0.66, 0.28) is given by0.66 + z0.25

√0.28 = 0.66 + (−0.67)(0.53) = 0.30, so 25% of

districts have odds ratio for Divorced etc. vs Married lessthan exp(0.30) = 1.35

3rd quartile N(0.66, 0.28) is given by0.66 + z0.75

√0.28 = 0.66 + (0.67)(0.53) = 1.02, so 25% of

districts have odds ratio for Divorced etc. vs Married greaterthan exp(1.02) = 2.76


Notes

Notes



Fitting the same model with Stata’s gllamm:

* Using random intercept model as starting point

matrix a2 = (ri15,0,0,0,0,0)

* Equations to define random coefficients

generate cons = 1

eq randomIntercept: cons

eq randomWid: WidDivSepDeva

eq randomUnmar: Unmarried

gllamm hiv_prev ///

AgeLess25 WidDivSepDeva Unmarried Literate NoOtherIncome ///

Brothels PubPlaces Clients10plus Duration5plus ///


, i(districtnum) nrf(3) eqs(randomIntercept randomWid randomUnmar) ///

link(logit) family(binom) from(a2) copy ip(m) nip(5)





Notes

Notes




Two-level models for binary responses Inferences in two-level logistic models

Maximum likelihood estimation

Consider the general model

logit{P[yij = 1|x1ij ,x2j , ζj ]} = β0 +β′1x1ij +β′2x2j + ζ ′jzij ,

where ζj ∼N(0,Ψ) and zij is specified so as to obtain thedesired random intercept or random coefficient model.

Maximum likelihood estimation consists in finding the valuesof β = (β0,β′1,β

′2)′ and Ψ that maximize the probability of the

observed data.


Notes

Notes


Likelihood function

The likelihood function is the probability of the observedresponses given the observed covariates:

L(β,Ψ) = P[yij , i = 1, . . . ,nj , j = 1, . . . ,N|X]

=N∏

j=1

P[yij , i = 1, . . . ,nj |Xj ]

=N∏

j=1

∫P[yij , i = 1, . . . ,nj |Xj , ζj ]φ(ζj ;0,Ψ) dζj ,

where φ(ζj ;0,Ψ) is the density of the multivariate normaldistribution with mean vector 0 and variance matrix Ψ.

All one needs to do is find the value of β and Ψ thatmaximize L(β,Ψ), but ...



Numerical integration and maximization

... the integral in L(β,Ψ) cannot be evaluated in closed formfor two-level logistic models:

L(β,Ψ) =N∏

j=1

∫ ni∏i=1

eβ0+β′1x1ij +β

′2x2j +ζ

′j zij

1 + eβ0+β′1x1ij +β′

2x2j +ζ′j zij

× (2π||Ψ||)−d/2 exp

(−

1

2ζ′j Ψ−1ζj

)dζj ,

where d = dim(ζj) and ||Ψ|| is the absolute value of thedeterminant of Ψ.

⇒ Software combines numerical integration methods withnumerical maximization.


Notes

Notes


Stata implementation

Numerical integration in maximum likelihood is performed byxtmelogit and gllamm using quadrature methods. Thenumber of quadrature points can be specified: the higherthe number of points, the more precise the likelihoodevaluation, the better the estimates of the elements of Ψ ...but the slower the execution!

xtmelogit: Use option intpoint(#). [With #=1 we getthe Laplace approximation (good estimates of β, poorestimates of Ψ). We use Laplace to get starting pointsfor method with #≥ 3.]

gllamm: Use option nip(#). [Laplace method not allowed,so use with #≥ 3.]



Fitting random intercept and randomcoefficient models

In the random intercept model, d =dim(ζj) = 1, so integrationin one dimension: easy and quick, so we can use largenumber of quadrature points (say 12 or 20).

When fitting a model with random coefficients as well, thend =dim(ζj)≥ 2, so numerical integration now in higherdimension.

⇒Fitting such a model with maximum likelihood isnumerically quite challenging (much, much slower than whend = 1).


Notes

Notes


Getting maximum likelihood estimation toconverge

Some tricks to achieve convergence with maximumlikelihood:

Start with the Laplace method (1 quadrature point) anduse its results as starting point for quadrature with morequadrature points.

Try fitting a model with a simpler structure for Ψ (e.g.,diagonal structure, the default in xtmelogit).

If the database is not too large (unfortunately not thecase of IBBA ...), use the “data cloning” method totrick WinBUGS into giving you the maximum likelihoodestimators. (Create a new dataset that is comprised of a large number of copies of the

current dataset, then fit Bayesian model with flat priors ⇒ posterior mean is maximum likelihood

estimate and posterior variance is variance of estimate.)




Fitting the random coefficient model with Stata’s xtmelogit:

**************

* Random slope (marital status)

**************

* First, with Laplace approximation and a

* simple variance-covariance structure







variance laplace

* RAN IN 5 MINUTES

estimates store RameshRS1UN1

matrix rs1un1 = e(b)


Notes

Notes



* Same, with more complex variance-covariance structure

* and 5 quadrature points

matrix a1 = (rs1un1,0,0,0)







variance covariance(unstructured) ///

intpoints(5) from(a1,copy) refineopts(iterate(0))

* RAN IN 18 MINUTES

estimates store RameshRS5UN

matrix rs5un = e(b)





Notes

Notes






Fitting the model with Stata’s gllamm

* Using random intercept model as starting point

matrix a2 = (ri15,0,0,0,0,0)

* Equations to define random coefficients

generate cons = 1

eq randomIntercept: cons

eq randomWid: WidDivSepDeva

eq randomUnmar: Unmarried

gllamm hiv_prev ///




, i(districtnum) nrf(3) eqs(randomIntercept randomWid randomUnmar) ///

link(logit) family(binom) from(a2) copy ip(m) nip(5)


Notes

Notes


Standard errors and approximate distribution

The preceding outputs showed standard errors. These arethe square roots of the diagonal elements of the negativehessian matrix of the log of the likelihood function(automatically calculated when numerical maximization isused).

If se(βk ) is the standard error of βk , then we have that βk isapproximately normally distributed with mean βk andvariance se(βk )2. This means that a (1−α)100% confidenceinterval for βk is given by βk ±z1−α/2se(βk ), and by

exp(βk ±z1−α/2se(βk

)for the corresponding odds ratio.



Standard errors and approximate distribution

For the Ramesh et al model with random coefficients, thelast xtmelogit output gave β11 = 0.66 with associatedstandard error 0.10.

Hence a 95% confidence interval for the odds ratio of beingHIV+ for Divorced, etc. vs Married is given byexp (0.66±1.96×0.10) = (1.59, 2.35).


Notes

Notes


Test of linear combinations

As a matter of fact, the entire vector of maximum likelihoodestimates β is approximately normally distributed with meanvector given by the true value of the parameters β andvariance matrix given by the inverse of the negative of thehessian of the log of the likelihood.

This approximate normality is used to construct Wald testsof hypotheses of the form H0 : linear combination of the βcoefficients= 0 vs H1 : this combination is not equal to 0.




Say we want to test that the odds ratio in the group ofUnmarried FSW who have more than 10 clients per weekand all other covariates at the reference level is the same asthe odds ratio of the FSW who solicit in brothels and haveall other covariates at the reference level, i.e.,

Unmarried + Clients10plus = Brothels

⇒ Unmarried + Clients10plus - Brothels = 0


Notes

Notes



This can be done with the lincom function after having fittedthe model with xtmelogit:

* Wald test of Unmarried + Clients10plus = Brothels

* Output in terms of the beta’s

lincom Unmarried + Clients10plus - Brothels

* Same test, but with output in terms of the odds ratio

lincom Unmarried + Clients10plus - Brothels, or





Notes

Notes



We can see that the p-value of the test is 0.465, hence nosignificant difference between the two groups.



Test that a second level is required

In Stata, xtmelogit automatically compares the model fittedwith an ordinary (level-1 only) logistic regression model. Aconservative p-value for the test H0 : ordinary logisticregression vs H1 : two-level model is given as the last line ofthe xtmelogit output.

In the xtmelogit output for the random coefficient model,with have a p-value of 0.0000, so very small, so we have verystrong evidence against the null model and so the randomintercept and coefficients have a highly significant variability.


Notes

Notes


Estimation of the ζj

Though the ζj are not model parameters, their values mightbe useful to compare districts. Because these values areunobserved, they must be estimated.

Unfortunately, when doing maximum likelihood, we integratethem out, so we cannot estimate them.



Empirical Bayes

Software that fits GLMM by maximum likelihood can easilyfind the mode of the posterior distribution of the randomeffects given the data, i.e., the values of ζj that maximize∏

j

φ(ζj ;0,Ψ)∏

i

P[yij |xij , ζj ],

with Ψ replaced by its maximum likelihood estimate. Theseare called the empirical Bayes (modal) predictions of therandom effects.


Notes

Notes


Caterpillar plots

Plotting estimates of the ζj ’s along with their respectiveconfidence intervals can give a good idea of how significantthe between district variability may be:

If all the ζj are close to zero, then the random effects arenot significant and a two-level model is not necessary.




Stata code ran after xtmelogit that produced the randomintercept model to get a caterpillar plot

* caterpillar plot

* store random intercept estimates in u0

predict u0, reffects

* store the standard error of random effect estimates in u0se

predict u0se, reses

* u0 and u0se repeated for each FSW, we only need one per district

egen pickone = tag(districtnum)

sort u0

generate u0rank = sum(pickone)

serrbar u0 u0se u0rank if pickone==1, scale(1.96) yline(0)


Notes

Notes




Two-level models for binary responses Adding level-two explanatory variables

Random intercept vs level-two explanatoryvariables

The variability of the random intercepts in a randomintercept model can be viewed as between-district variabilityin the latent response that is due to unmodelled differencesbetween districts.


Notes

Notes


Random intercept vs level-two explanatoryvariables

Adding significant district-level explanatory variables in thelinear predictor should explain some of this variability andtherefore diminish the level of unexplained between-districtvariability.

Thus to explain the between-district variability, we can use alinear predictor of the form

logit{P[yij = 1|x1ij ,x2j , ζ0j ]} = β0 + ζ0j +β1x1ij

+β2x2j .



Example: Adding district-level variables inRamesh et al.’s study

We repeated the analysis with the random intercept modelof Table 4 of Ramesh et al, but we added district-levelvariables to the model.

Several variables were significant when added alone, but fewremained significant when added together with otherdistrict-level variables.

In the end (e.g., reverse causality, outliers, etc.), we endedup just adding the the total proportion of female (ages 15 to45) in the population, PropFem.


Notes

Notes



Random intercept model of Table 4, but with district-levelvariable PropFem added.

rename prop_fem_pop_15_49_t PropFem

* Random intercept model with MeanAgeMar added

* Start with Laplace approximation






PropFem || districtnum: ///

, variance laplace

* MeanAgeMar || districtnum: ///

* RAN IN 1 MINUTE

matrix riAge1 = e(b)




* Now with 12 integration points







, variance from(riAge1,copy) ///

refineopts(iterate(0)) intpoints(12)

* MeanAgeMar || districtnum: ///

* RAN IN 2 MINUTES

estimates store RameshRIage12

matrix riAge12 = e(b)


Notes

Notes







Notes

Notes



There is a negative coefficient in front of the variable ⇒decrease in district’s proportion of female is associatedwith increase in HIV prevalence.This effect is highly significant (p-value = 0.010)Variance of random intercepts went down from 0.34 to0.25 (see also caterpillar plot of random interceptestimates on next page).VPC = 0.25/(0.25 + 3.29) = 7.1%⇒ Instead of 9.5% ofthe variability in y ∗ due to unexplained between-districtvariability, we are down to 7.1%.Nonetheless, there is still a significant amount ofunexplained between-district variability (p-value oflikelihood ratio test of 0.0000), so the random interceptsare still required.




Ramesh et al. random intercept + proportion of females Ramesh et al. random intercept


Notes

Notes



Let us see what happens if we add a level-2 variable thatdoes not explain much between-district variability: the sexratio (sexratio).






sexratio || districtnum: ///

, variance from(riAge1,copy) ///






Notes

Notes






This variable does not have a significant effect (p-value= 0.81)

Variance of random intercepts only went down from 0.34to 0.33

VPC = 0.33/(0.33 + 3.29) = 9.1%


Notes

Notes


Random coefficient vs level-two explanatoryvariables

The variability of the random coefficients in a randomcoefficient model can be viewed as between-districtvariability in the effect of an individual-level explanatoryvariable on the latent response that is due to unmodelleddifferences between districts.



Random coefficient vs level-two explanatoryvariables

Adding a significant interaction between a district-level andthis individual-level explanatory variable in the linearpredictor should explain some of this variability and thereforediminish the level of unexplained between-district variability.

logit{P[yij = 1|x1ij ,x2j , ζ0j , ζ1j ]} = β0 + ζ0j + (β1 + ζ1j)x1ij

+β2x2j +β3x1ijx2j .


Notes

Notes


Example: Adding interactions withdistrict-level variables in Ramesh et al.’s study

Can we explain some of the between-district variability in theeffect of marital status using district level variables?

Let us try with PropFem: Does adding an interaction betweenPropFem and marital status significantly reduce the varianceof the random coefficients for marital status?




Procedure:

Fit the model with random coefficients and PropFem, butwithout the interactions.

Fit the model with random coefficients and PropFem withinteractions.

Compare the two models with a likelihood ratio test(Null: model without interactions; alternative: modelwith interactions; if p-value small, reject the null andconclude that interactions are required).


Notes

Notes



First, we fit the model with random coefficients and PropFem,but without interactions.

* Random coefficients + PropFem,

* simple variance structure, Laplace

matrix s1 = (riAge12,0,0)







WidDivSepDeva Unmarried ///

, variance from(s1,copy) refineopts(iterate(0)) laplace

* RAN IN 3 MINUTES

estimates store RameshRSage1

matrix rSAge1 = e(b)




* Samething, but with 5 pt quadrature








, variance from(rSAge1,copy) ///


* RAN IN 5 MINUTES

estimates store RameshRSage5

matrix rSAge5 = e(b)


Notes

Notes







Notes

Notes



Now we fit the model with random coefficients, PropFem, andits interactions with marital status.

generate PropDivor = WidDivSepDeva*PropFem

generate PropUnmar = Unmarried*PropFem

* First with Laplace approximation

matrix s1 = (rSAge5[1,1..12],0,0,rSAge5[1,13..16])






PropFem PropDivor PropUnmar || districtnum: ///


, variance from(s1,copy) ///


* RAN IN 5 MINUTES

estimates store RameshRSageInter1

matrix rSAgeInter1 = e(b)Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 121 / 297



* Next with full quadrature with 5 points






PropFem PropDivor PropUnmar || districtnum: ///


, variance from(rSAgeInter1,copy) ///


* RAN IN 7.5 MINUTES

estimates store RameshRSageInter5

matrix rSAgeInter5 = e(b)


Notes

Notes







Notes

Notes



We can test if the interactions are significant with alikelihood ratio test:

lrtest RameshRSageInter5 RameshRSage5 , stats





Notes

Notes



Likelihood ratio statistic:2{−4085.739− (−4086.348)} = 1.22

Degrees of freedom: model 1 has 2 parameters less(none of them variances) than model 1 ⇒ 2 degrees offreedom

Pr[chi-squared with 2 degrees of freedom> 1.22] = 0.5442

AIC (smaller is better) also suggests model 1 (BICcannot be used here, since treats all 10 093 observationsas independent)




Not surprisingly, variances of random coefficients not muchdifferent between the two models:

With interactions Without interactionsσ2

WidDiv 0.0440 (0.03) 0.0436 (0.03)σ2

Unmar 0.0180 (0.05) 0.0175 (0.05)σ2

cons 0.288 (0.10) 0.287 (0.10)

⇒ Between-district differences in the effect of marital statusnot explained by between-district differences in proportion offemales in population.


Notes

Notes

Subject-specific and population-averaged inferences

Subject-specific and population-averagedinferences


Subject-specific and population-averaged inferences Differences between SS and PA inferences

Differences between subject-specific andpopulation-averaged inferences


Notes

Notes


Subject-specific vs population-averagedinferences

In multi-level models inferences can generally be categorizedinto two main types:

subject-specific (or conditional) inferences;

population-averaged (or marginal) inferences.

Warning:Differences between these two types of inferences are subtle,both conceptually and numerically!




Subject-specific effect

The subject-specific effect of a covariate is the effect of a changeof its value on the individual (subject-specific, level 2) probabilities.

The term “subject-specific” arises from the longitudinal dataanalysis literature and can be somewhat misleading. “Subjects”really denote the level 2 units.

For instance the fixed-effects (β′xij part) in our previous two-level

models estimate district-specific effects, as they give changes in

log-odds of conditional probabilities P[Yij = 1|xij , ζj ] for a “typical

district” with ζj = 0.


Notes

Notes



Population-averaged effect

The population-averaged effect of a covariate is the effect ofa change of its value on the average probability of Y = 1 inthe entire population.

The fixed-effects (β′xij part) in a single level model estimatesuch population-averaged effects, as they give the change inlog-odds of marginal (unconditional, population-wide)probabilities P[Yij = 1|xij ].




ExampleIf we look at the Ramesh et al study, we can think of effectsthat we would more likely want at the district-specific level:

Suppose that interventions to reduce the number ofclients per week will be applied to certain districts. Thenthe district-specific effect of the number of clients perweek should be more interesting than itspopulation-averaged effect.

Perhaps a district-specific interpretation of the effect ofthe place of solicitation would make more sense than apopulation-averaged interpretation.


Notes

Notes



Example (cont’d)We can also think of effects that we would more likely wantto estimate at the population-averaged level:

We would like to compare the prevalence betweenliterate and illiterate FSWs. In this case we areinterested in a population-averaged effect.

Since district-level variables are usually difficult tochange for a given district, the population-averagedeffect of these variables is often easier to interpret (e.g.,prevalence in districts with a high proportion of womenvs districts with a low proportion of women).




In a typical longitudinal study where, say, patients are thelevel 2 units and several measures (level 1) are taken oneach patient.

The effect of a variable that cannot change for a givenpatient (e.g., gender) makes more sense at thepopulation-averaged level.

We usually want to assess the patient-specific effect ofvariables that can be changed at the patient level, forinstance the effect of treatment.


Notes

Notes



Mathematically, we can get from a two-level (hencesubject-specific, or conditional) logistic regression model toa marginal (population-averaged) regression model by“integrating the random effects out” of the model:

P[yij = 1|xij ] =

∫ exp(β′xij + ζ′j zij

)1 + exp

(β′xij + ζ′j zij

)φ(ζj ;0,Ψ) dζj .

Unfortunately, there is no formula to go from thepopulation-averaged to the subject-specific probabilities ingeneral ...




The equation from the previous slide implies that in practice,parameter estimates of a marginal (e.g., single-level) model,say, βSL

k , are attenuated values of parameter estimates of thecorresponding two-level model (e.g., random-intercept modelwith βRI

k ).

As a matter of fact, for the random intercept model, it ispossible to show that

βSLk =

√3.29

3.29 +σ2ζ

βRIk .


Notes

Notes



From Rabe-Hesketh & Skrondal (2005, p. 255). Bold line: population-averaged probability. Dashed lines:

district-specific probabilities from a random-intercept model.Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 139 / 297


Example: Ramesh et al study

Effect of AgeLess25 and PropFem in the Ramesh et al randomintercept model, with proportion of females in the 15-49 agegroup added.

District-specific Population-averagedestimate std. err. p-value estimate std. err. p-value

AgeLess25 -.295 .093 0.002 -.292 .123 0.018PropFem -11.3 4.35 0.010 -9.97 4.60 0.030

We will see how these estimates were obtained in a fewmoments ...


Notes

Notes

Subject-specific and population-averaged inferences Subject-specific inferences

Subject-specific and population-averaged inferences



Types of subject specific inferences

All inferences based on multi-level models seen so far todayhave been subject-specific inferences.

We will now see how to get predicted individual-levelprobabilities, which are another type of subject-specificinferences that can be carried out with a multi-levelregression model


Notes

Notes


Predicted individual-level probabilities

Let ζj denote the empirical Bayes estimate of the randomeffects for district j. We may want various types ofprediction of the probability of yij = 1:

1. FSW i of district j:

P[yij = 1|xij , ζj ] = exp(β′xij + ζ ′jzij

)/{1 + exp

(β′xij + ζ ′jzij

)}.

After xtmelogit, you simply need to typepredict PredictedProb, mu

and PredictedProb will contain the desired probability foreach individual.



Subject-specific probabilities with xtmelogit





PropFem ///


* RAN IN 1 MINUTE

estimates store RameshRI1b

matrix ri1b = e(b)





PropFem ///



matrix ri12 = e(b)

* predicted individual-level probabilities

predict PredictedProb, mu


Notes

Notes


Subject-specific model fit

Ramesh et al, random intercept, with PropFem




2. New FSW in district j: We will have covariateinformation for this new FSW, say x0j . Thus we compute(by hand)

P[yij = 1|x0j , ζj ] = exp(β′x0j + ζ′j z0j

)/{1 + exp

(β′x0j + ζ′j z0j

)}.

You can get the random effects estimates ζj using the optionreffects in predict.


Notes

Notes


Subject-specific probabilities with xtmelogit





PropFem ///


* RAN IN 1 MINUTE

estimates store RameshRI1b

matrix ri1b = e(b)





PropFem ///



matrix ri12 = e(b)

* estimates of the random effects

predict Rinter, reffects





Predicted probabilities (PredictedProb) and random interceptestimate (Rinter) for the first few FSW in the Bangaloredistrict (using Data Editor)


Notes

Notes



3. New individual in new district: We will have covariateinformation for this new individual, say x00. For the newdistrict, we must assume a value for ζ, say ζ0. For a“typical” district, this would be ζ0 = 0.

Thus we compute (by hand)

P[yij = 1|x00, ζ0] = exp(β′x00 + ζ′0z00

)/{1 + exp

(β′x00 + ζ′0z00

)}.




One possible use of predicted district-level probabilities is theestimation of the potential number of cases averted by anintervention (suppose that there is an intervention variable xin the model that is 1 if there is an intervention in thedistrict and that is 0 otherwise). Suppose that there is anintervention in district j.

Count the number of observed cases (yij = 1) in district j;

Compute the predicted prevalence if no intervention forthat district using ζj , β, x = 0 for the interventionvariable and the district average for the value of theother variables;

Compare the observed number of cases to the number ofcases expected when there is no intervention.


Notes

Notes

Subject-specific and population-averaged inferences Population averaged-inference based on GEE

Population-averaged inferences



Estimation of population-averaged effects

For factors that cannot be modified within level 2 (e.g.,gender in a longitudinal study following individuals), it makesmore sense to infer about the average change in response inthe population when these factors are modified.

(So it makes more sense to talk about the differencebetween men and women in a population than to talk aboutthe effect of changing one’s gender from man to woman.)

We have seen how to obtain subject-specific inferences frommulti-level models. How can we obtain population-averagedinferences?


Notes

Notes


Estimation of population-averaged effects

Some possible avenues:

Fitting an ordinary one-level regression model:Fitting such a model to multilevel data would yield validpopulation-averaged estimates, but invalid standarderrors, confidence intervals or p-values because of thewithin-district correlation. We will not consider thisavenue any further ...

Computing population-averaged estimates from amulti-level model: As we have seen, this is possible,but seems to be numerically challenging (the formulainvolved a complicated integral). (This can be done withgllamm.)

Using generalized estimating equations: Thisapproach readily yields valid population-averagedinferences. Subject-specific inferences cannot be derived.



Predicted population-averaged probabilitiesfrom a multi-level model

Population-averaged probability for given xij: Supposethat we want to know the marginal probability P[yij = 1|xij ]from a two-level model. This is estimated by

P[yij = 1|xij ] =

∫ exp(β′xij + ζ′j zij

)1 + exp

(β′xij + ζ′j zij

)φ(ζj ;0,Ψ) dζj .

(We simply replace unknown quantities by their maximum

likelihood estimates in the equation seen a few slides ago ...)


Notes

Notes


Predicted population-averaged probabilitiesfrom a multi-level model

In Stata, this can only be done with gllamm:

1. Run gllamm to get the random intercept model

2. Type gllapred PredictProbPopAve, mu marginal

You can compare gllapred PredictProbSubSpec, mu andPredictProbPopAve using the Data Editor!!!



Population-averaged probabilities using gllamm

Ramesh et al random intercept model with PropFem

gllamm hiv_prev ///




PropFem ///

, i(districtnum) link(logit) family(binom) from(ri1b) copy nip(3)

gllapred PredictProbPopAve, mu marginal

gllapred PredictedProb2, mu


Notes

Notes




Subject-specific predicted probabilities using xtmelogit

(PredictedProb), gllamm (PredictedProb2), estimates ofrandom intercepts using xtmelogit (Rinter) andpopulation-averaged probabilities using gllamm

(PredictProbPopAve) for the first few FSW of the Bangaloredistrict.



Generalized estimating equations (GEE)

We can directly estimate population-averaged effects andmake population-averaged inferences using GeneralizedEstimating Equations (GEE):

No numerical integration required, no model for randomeffects required, but no level-1 estimation or predictionpossible.


Notes

Notes



GEE use an iterative algorithm to find the values of theregression parameters that solve∑

j

Wj(Yj −P[Yj = yj ;β]) = 0,

where Wj is a weight matrix associated with the j-th level 2unit and depends on our guess of the correlation structurebetween the level 1 observations within the level 2 units.

A nice property of GEE estimates is that even if we guess thecorrelation structure wrong, inferences about β based on arobust variance estimate remain valid; however guessing thecorrelation structure right yields more powerful inferences.




Procedure:

Specify a model for the mean of the observations (i.e.,an ordinary one-level regression model)

Try to guess the form of the within-district correlation(independent, exchangeable, ar #, unstructured). If youget it wrong, you lose a bit of efficiency, but inferencesusing GEE remain valid.

Use coefficient estimates and robust standard errors tomake inferences.


Notes

Notes


GEE in Stata

For multilevel logistic regression, we can fit a model with theGEE approach using xtlogit or xtgee. Actually xtgee iscomparable to gllamm and can perform GEE inference forany generalized linear model.

* Population-averaged inference for the random intercept model

* With xtgee ... requires more space in memory for large matrices

set matsize 1000

xtgee hiv_prev ///




PropFem ///

, i(districtnum) link(logit) family(binomial) ///

corr(exchangeable) vce(robust)



Population-averaged inference with xtgee

E.g., to get valid 95% confidence interval for odds ratio

corresponding to Unmarried, simply compute

exp(0.431±1.96×0.098) = (1.27, 1.86) or ...


Notes

Notes


GEE in Stata

... add eform option in xtgee statement:

* Population-averaged inference for the random intercept model

* With xtgee ... requires more space in memory for large matrices

set matsize 1000

xtgee hiv_prev ///




PropFem ///


corr(exchangeable) vce(robust) eform



Population-averaged inference with xtgee


Notes

Notes


GEE in Stata using xtlogit

Or, equivalently, using xtlogit:

* We could have obtained identical results

* with xtlogit:

xtlogit hiv_prev ///




PropFem ///

, i(districtnum) pa corr(exchangeable) vce(robust)

The output is exactly the same as with xtgee.



Getting estimates of the working correlationmatrix parameters

When fitting a model with GEE, we have to “guess” thecorrelation structure between the level 1 observations (e.g.,FSW) within a same level 2 unit (e.g., district).

With Stata, it is possible to output the value of theparameter estimates in the working correlation structure, aslong as the xtgee command was used to fit the model.


Notes

Notes



set matsize 1000

xtgee hiv_prev ///




PropFem ///



* FOLLOWING LINES MUST BE RUN FOLLOWING xtgee

* WILL NOT WORK AFTER xtlogit!!!!

* to get an idea of the working correlation matrix

estat wcorrelation, compact





Notes

Notes


Getting population-averaged estimatedprobabilities

If we compute the inverse logit of β′xij when β is obtainedwith the GEE method,

e β′xij

1 + e β′xij,

we obtain an estimate of the proportion of individuals withY = 1 in a population where everyone has covariate value xij .

Again, it is easy to get these values with Stata after havingrun xtgee.



Getting population-averaged estimatedprobabilities

set matsize 1000

xtgee hiv_prev ///




PropFem ///



* FOLLOWING LINES MUST BE RUN FOLLOWING xtgee

* WILL NOT WORK AFTER xtlogit!!!!!

* to get population averaged predicted probabilities

predict marginprob


Notes

Notes



Predicted population-averaged probabilities with covariate values of the

first few FSW in the Bangalore district


Advanced MLM topics Variance-components three-level model

Three-level models with nested randomeffects

Chapter 10, MLMUS2


Notes

Notes


Three-level models: Introduction

A hierarchical structure:Units −→ Clusters −→ Superclusters

For example, we might have repeated measurementoccasions (units) for patients (clusters) who areclustered in hospitals (superclusters).




What if the response were measured on each woman atseveral visits at a clinic?

District

FSW 1 FSW 2 FSW 3

Level 3

Level 2Woman

Visit 1 Visit 2 Visit 1Visit 3Visit 2 Visit 3 Level 1Visit

District


Notes

Notes



We could think of other examples with 3 levels using thecurrent IBBA round 1 data. For example:

Level 1: FSW

Level 2: District

Level 3: State

Warning: To estimate level 3 variance parameters, we needto have several level 3 units (e.g., several states)!



Three-level models : continuousresponses


Notes

Notes


Variance-components model 6

Example: Do peak-expiratory-flow measurements varybetween method?Peak expiratory flow is measured using two methods, thestandard Wright peak flow and the Mini Wright meter, eachon two occasions on 17 subjects.

Level 1: Occasion (i)

Level 2: Method (j)

Level 3: Subjects (k)

6From: MLMUS2, Chapter 2Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 177 / 297


Variance-components model

See MLMUS2, p.53Abdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 178 / 297

Notes

Notes



Let Yijk be the response of subject k for method j atoccasion i. A model with random intercept for subjects :

Model 1 : Yijk = β+ ζ(3)k + εijk

Within subjects residual :

εijk |ζ(3)k ∼N(0,θ)

Random effect of subjects (between) :

ζ(3)k ∼N(0,ψ(3))

We ignore the fact that different methods were used



Variance-components model : Stata code

use ./mlmus2/pefr, clear

reshape long wm wp, i(id) j(occasion)

generate i = nreshape long w, i(i) j(meth) string

sort id meth occasion

list id meth occasion w in 1/8, clean noobs

encode meth, gen(method)

recode method 2=0


Notes

Notes


Variance-components model : Stata code

Data are reshaped as follows



Variance-components model : Model (1)

xtmixed w ‖ id:, mleestimates store model1


Notes

Notes



To allow for a systematic difference between the 2 methods,we might add a binary variable for estimating the methods?effect

Model 2 : Yijk = β1 +β2xj + ζ(3)k + εijk

with xj a dummy variable and

εijk |xj , ζ(3)k ∼ N(0,θ)

ζ(3)k |xj ∼ N(0,ψ(3))




xtmixed w method ‖ id:, mleestimates store model2


Notes

Notes



The intraclass correlation coefficient is the correlationbetween the 4 repeated measures on the same individual( the method used for the measurement is ignored)

ρ =ψ(3)

ψ(3) +θ=

109.222

109.222 + 23.822= 0.95

This is the % of the total variance of the measurementsthat is explained by the variance of the individualmeasurments.




In addition to subjects and occasions variations,measurements obtained with the same method might bemore similar to each other than measurements obtained withtwo different methods


Notes

Notes


Variance-components of a three-level model:

The between subjects variation modeled by ζ(3)k

it reflects the fact that measurements on the samesubjects are more similar than measurements on differentsubjects.




The conditional independence assumption of model (1)

Yijk = β+ ζ(3)k + εijk is violated

For a given subject, the measurements using the samemethod tend to be more similar to each other.


Notes

Notes



Model 2: Yijk = β1 +β2xj + ζ(3)k + εijk is unsatisfactory

The shift of the measurements using one method relativeto the other is not constant, it varies between subjects.



Variance-components of a three-level model

To accommodate the between-method within-subjectheterogeneity we might add a random intercept for eachcombination of method and subject

Model 3 : Yijk = β1 + ζ(2)jk + ζ

(3)k + εijk

The random effect of method is nested within subjects: ittakes a different value for each combination method-subject


Notes

Notes



Interpretation of the parameters

Yijk = β1 + ζ(2)jk + ζ

(3)k + εijk

where

β1 = Population average

(across occasions, methods, and subjects)

β1 + ζ(3)k = Average for subject k

(across occasions and methods)

β1 + ζ(2)jk + ζ

(3)k = Average for method j and for subject k

(across occasions)




Assumptions for Model 3

Yijk = β1 + ζ(2)jk + ζ

(3)k + εijk

where

ζ(3)k |xj ∼ N(0,ψ(3))

ζ(2)jk |xj , ζ

(3)k ∼ N(0,ψ(2))

εijk |xj , ζ(2)jk , ζ

(3)k ∼ N(0,θ)


Notes

Notes



xtmixed w ‖ id: ‖ method:, mleestimates store model3




We might add a systematic variation between methods

Model 4 : Yijk = β1 +β2xj + ζ(2)jk + ζ

(3)k + εijk

Two-way mixed-effects ANOVA model:

Two factors: method (fixed) and subject (random)

Replicates: occasion

Main effects: Method (β2), Subject (ζ(3)k )

Interaction : Method by Subject (ζ(2)jk )


Notes

Notes



xtmixed w method ‖ id: ‖ method:, mleestimates store model4



Variance-components three-level model

Intraclass correlations for pairs of responses

Same subject k but different methods j and j ′:

ρ(subject)≡Cor (Yijk ,Yi ′j ′k |xj ,xj ′) =ψ(3)

ψ(2) +ψ(3) +θ

Same subject k and same method j :

ρ(method,subject)≡Cor (Yijk ,Yi ′jk )|xj) =ψ(2) +ψ(3)

ψ(2) +ψ(3) +θ


Notes

Notes



Note that ρ(method,subject) > ρ(subject)Measurements for the same subject are more correlated ifthey use the same method than if they use different methods




Three-stage formulation

Stage 1: Level-1 model

Yijk = η1jk + εijk

where η1jk varies between methods j and subjects k.


η1jk = π11k +π12xj + ζ(2)jk

where the intercept π11k is in turn modeled using thelevel-3 model.


π11k = γ111 + ζ(3)k


Notes

Notes



Upon substituting π11k into level-2 model, and η1jk intoLevel-1 model we end up with

Yijk = γ111︸︷︷︸β1

+ π12︸︷︷︸β2

xj + ζ(2)jk + ζ

(3)k + εijk



Variance-components three-level model :summary

Two-Level models Three-Level modelsModel 1 Model 2 Model 3 Model 4Est (SE) Est (SE) Est (SE) Est (SE)

Fixed partβ1 450.9(26.6) 447.9(26.8) 450.9(26.6) 447.9(26.9)β2 6.0(5.7) 6.0(7.8)Random part√ψ(2) 19.5 19.0√ψ(3) 109.2 109.2 108.6 108.6√θ 23.8 23.6 17.8 17.8

Log Likelihood -349.89 -349.34 -345.29 -345.00


Notes

Notes


Variance-components three-level model :Comparison of Model 2 and Model 4

Thus, a random effect for methods is requiredHowever, the fixed effect β2 is not significant at 5%



Variance-components three-level model : Thefinal Model

xtmixed w ‖ id: ‖ method:, mle


Notes

Notes



Intraclass correlations for pairs of responses

Same subject k but different methods j and j ′:

ρ(subject) =ψ(3)

ψ(2) +ψ(3) +θ= 0.94

Same subject k and same method j :

ρ(method,subject) =ψ(2) +ψ(3)

ψ(2) +ψ(3) +θ= 0.97




In summary

No evidence of systematic bias between methods

Subject by method interaction bias

Subject-specific bias

Methods have good test-retest reliability(ρ(method,subject))


Notes

Notes

Advanced MLM topics Three-Level logistic model

Three-level models : binary responses



Three-level random intercept logistic model

The dataset (guatemala.dat) refers to completeimmunization among Guatemalan children receiving anyimmunization. It has 2159 observations on 19 variables.


Notes

Notes



Level 1: child i- immun: indicator (yijk ) is 1 if child has received full

immunization, 0 otherwise- kid2p: dummy variable (x2ijk ) is the indicator that the

child is at least 2 years old and hence eligible for full setof immunizations

Level 2: mother j, several mother-level covariates(x3ij–x9ij)

- mom: identifier for mothers- indNoSpa: mother is indigenous, not Spanish speaking

(x3jk )- indSpa: mother is indigenous, Spanish speaking (x4jk )- . . .

Level 3: community, k, two community-level covariates,rural (x10k), pcInd81 (x11k)




Children i nested in mothers j who are nested incommunities k

logit{P(yijk = 1|xijk , ζ(2)jk , ζ

(3)k )} =

(β1 + ζ(2)jk + ζ

(3)k ) +β2x2ijk + · · ·+β11x11k ,

where

xijk = (x2ijk , . . . ,x11k )′ is a vector of all covariates

ζ(3)k |xijk ∼N(0,ψ(3)) a random-intercept varying over

communities

ζ(2)jk |ζ

(3)k ,xijk ∼N(0,ψ(2)) a random-intercept varying over

mothers

random effects are assumed independent.


Notes

Notes



As usual, there is a latent-response representation of thismodel:

y ∗ijk = β0 + ζ(2)jk + ζ

(3)k

+β1x1ijk + · · ·+β11x11k + εijk ,

where εijk |ζ(3)k , ζ

(2)jk ,xijk follows a standard logistic distribution

(with variance =π2/3 = 3.29), and

yijk =

{1 if y ∗ijk > 0 ;

0 if y ∗ijk ≤ 0 ;



Intra-class correlations

There are a couple of ways to define and compute intra-classcorrelation in a three-level random intercept logistic model:

Children from same community, different mothers:

corr(y ∗ijk ,y∗i ′j ′k |xijk ,xij ′k ′) =

ψ(3)

ψ(2) +ψ(3) + 3.29

Children from same community and mother:

corr(y ∗ijk ,y∗i ′jk |xijk ,xijk ′) =

ψ(2) +ψ(3)

ψ(2) +ψ(3) + 3.29

In these models ψ(2),ψ(3) > 0⇒ children from a same motherare more correlated than children from a same communitybut different mothers.


Notes

Notes



Stage 1: Level-1 : Child-Level : x2ijk


(3)k )} = η1jk + β2x2ijk︸︷︷︸

Level−1var .

where the intercept η1jk varies between mothers j andcommunities k.




Stage 2: Level-2 : mother-level: 7 variables w2jk , . . . ,w8jk

η1jk = π11k +π12w2jk + · · ·+π18w8jk︸︷︷︸Level−2var .

+ζ(2)jk

the intercept π11k is in turn modeled using thecommunity-level (level-3) model.


Notes

Notes



Stage 3: Level-3 : community-level model : v2k = rural,v3k = pcInd81

π11k = γ111 +γ112v2k +γ113v3k︸︷︷︸Level−3var .

+ζ(3)k

Substitute step-3 in step-2 then step-2 in step-1



Fitting three-level random intercept logisticmodel in Stata

This can be done using either xtmelogit or gllamm. We willillustrate using the Guatemalan immunization study.

First we must download the data

* get the data from Stata website

use http://www.stata-press.com/data/mlmus2/guatemala, clear

* save to local file for future use

save "F:\Data-Stata\guatemala.dta", replace


Notes

Notes


three-level random interceptlogistic model with gllamm



Fitting three-level random intercept logisticmodel with gllamm

Download gllamm fromhttp://www.gllamm.org/install.html or simply usessc install gllamm

Use gllamm with 5 quadrature points (...otherwise, timeconsuming...)

gllamm immun kid2p indNoSpa indSpa momEdPri momEdSec husEdPri ///

husEdSec husEdDK rural pcInd81, ///

family(binomial) link(logit) i(mom cluster) nip(5)


Notes

Notes






Store the previous estimates in order to use them asstarting valuesmatrix a=e(b)

Increase the number of quadrature points to 8 (default)per dimension + adaptive quadraturegllamm immun kid2p indNoSpa indSpa momEdPri momEdSec husEdPri ///

husEdSec husEdDK rural pcInd81, ///

family(binomial) link(logit) i(mom cluster) from(a) adapt


Notes

Notes






Same results for ”odds ratios”: gllam, eform


Notes

Notes


Three-level random intercept logistic model,Guatemalan immunization

The variable of main interest x2ijk =kid2p has a large OR5.55> 0 and is highly significant.

The variance of the mother-level random intercepts isψ(2) = 5.19.

The variance of the community-level random intercepts isψ(3) = 1.03.



Three-level random intercept logistic model,Guatemalan immunization

The correlation between the latent-responses of two childrenfrom a same community but from different mothers is

ψ(3)/(ψ(2) + ψ(3) + 3.29) = 1.03/(5.19 + 1.03 + 3.29) = 0.11.

The correlation between the latent-responses of two childrenfrom a same mother is

(ψ(2) +ψ(3))/(ψ(2) +ψ(3) +3.29) = (5.19+1.03)/(5.19+1.03+3.29) = 0.65.


Notes

Notes


three-level random interceptlogistic model with xtmelogit



Fitting three-level random intercept logisticmodel with xtmelogit

Start with Laplace method obtained with the optionintpoints(1) or laplace which is computationallyefficient.** 10.7.2 Using xtmelogit

* Laplace

xtmelogit immun kid2p indNoSpa indSpa momEdPri momEdSec ///

husEdPri husEdSec husEdDK rural pcInd81 ///

|| cluster: || mom:, intpoints(1)

Save these Laplace estimates in order to use them asstarting values for MLE with adaptive quadrature with 3or 4 or 5 integration points....TIME consuming...

matrix a = e(b)


Notes

Notes



Laplace estimates




Adaptive quadrature: 5 points

matrix a = e(b)

xtmelogit immun kid2p indNoSpa indSpa momEdPri momEdSec husEdPri husEdSec ///

husEdDK rural pcInd81 || cluster: || mom:, intpoints(5) ///

from(a) refineopts(iterate(0))


Notes

Notes









Notes

Notes





Three-level random coefficientlogistic model


Notes

Notes


Three-level random coefficient logistic model

It is possible to let the coefficients in front of level 1(Child) variables vary from level 2 to level 2 (Mother )and/or from level 3 to level 3 unit (Community).

Similarly, it is possible to let the coefficients in front oflevel 2 variables vary from level 3 to level 3 unit.

In the Guatemalan immunization study, does the effectof eligibility (level 1 variable x2ijk) vary from communityto community (level 3 units)?




We can refit the model, but with a random coefficient thatvaries across level 3 units in front of x2ijk :


(3)k )} = β1 + ζ

(2)jk + ζ

(3)1k

+ (β2 + ζ(3)2k )x2ijk

+β3x3jk · · ·+β11x11k ,

where (ζ(3)1k , ζ

(3)2k )|xijk ∼N[(0,0),Ψ(3)]

with

Ψ(3) =

[ψ

(3)11 ) ψ

(3)12 )

ψ(3)21 ) ψ

(3)22 )

]

and ζ(2)jk |ζ

(3)0k , ζ

(3)2k ,xijk ∼N(0,ψ(2)).


Notes

Notes



�

Normally if a random coefficient is included at acertain level, corresponding random coefficients should alsobe included at all lower levels.

In the Guatemalan immunization example, this has beendone for the intercepts, but not for the coefficient of x2ijk

because the treatment was applied at the community leveland x2ijk does not vary much at the mother-level (seeMLMUS2 p. 454 for detailed discussion).



Fitting three-level random coefficient logisticmodel with gllamm

First, fit a random-intercept model


(3)k )} =

β1 +β2x2ijk +β10x10k +β11x11kζ(2)jk + ζ

(3)k

Use the previous model to retrieve the estimations of thefull model : estimates restore glri8 and the skip optionsince some covariates have been dropped

matrix a = e(b)

gllamm immun kid2p rural pcInd81, ///

family(binomial) link(logit) ///

i(mom cluster) from(a) skip adapt eform

estimates store glri0


Notes

Notes






Specify equation for intercept (one for community leveland one for mother level) and a slope for kid2p atcommunity levelgenerate cons = 1

eq inter: cons

eq slope: kid2p

Use previous estimates as starting values + two morevalues (for ψ(3)

22 and ψ(3)21 )

matrix a = e(b)

matrix a = (a,.2,0)


Notes

Notes



gllamm immun kid2p rural pcInd81, ///

family(binomial) link(logit) i(mom cluster) ///

nrf(1 2) eqs(inter inter slope) nip(8 4 4) ///

from(a) copy adapt eform

estimates store glrc

The nrf() option specifies the number of random levelsin i() statement: nrf(1 2) means 1 for mom and 2 forcluster

The eqs() option specifies one equation for each randomeffect

The nip() option specifies the number of quadraturepoints for each random effect





Notes

Notes


Three-level random coefficient model,Guatemalan immunization

The random-intercept variance at level 3 is ψ(3)11 = 2.42, it

might be interpreted as the residual between-communityvariance for too young children (kid2p=0)

The random-slope variance at level 3 is ψ(3)22 = 1.80, it

might be interpreted as the residual variability in theeffectiveness of the campaign across communities.



Three-level random coefficient model,Guatemalan immunization

The negative covariance (ψ(3)12 =−1.51) or correlation

(ψ(3)12 /

√ψ

(3)11 ψ

(3)22 =−1.51/

√2.41×1.79 =−0.73) between

the random intercept and coefficient at level 3 suggeststhat effect of kid2p is stronger in communities where theimmunization rate is low.

Interactions between kid2p and level 3 covariates may beable to explain part of the variability in the randomcoefficients.


Notes

Notes



lrtest glrc glri0 =⇒ LRchi2(2) = 8.43 andProb > chi2 = 0.0148




xtmelogit immun kid2p rural pcInd81 ///

|| cluster:kid2p, cov(unstructured) || mom: ///

,intpoints(4 8) or

estimates store xtrc

The level 3 specification: ‖ cluster:kid2p,cov(unstructured). A random slope for kid2p, anintercept is included by default.

No restriction for the covariance matrix


Notes

Notes


Fitting three-level random coefficient logisticmodel with xtmelogit


Advanced MLM topics Crossed random effects: continuous response

Multilevel models with crossed randomeffects


Notes

Notes


Hierarchical structures



Nested vs crossed random-effects

Nested design: When observations from level L aresampled (e.g., state), then the observations from levelL−1 (e.g., district) are drawn from units sampled atlevel L.


Notes

Notes


Cross-classified structure7

7www.bristol.ac.uk/cmm/team/cross-classified-review.pdAbdous & Duchesne (Laval) MLM-Workshop October 17-21, 2010 247 / 297




Notes

Notes


Nested vs crossed random-effects

Crossed design: The fact that units from level L aresampled does not determine whether units at level L−1(e.g., Round) will be observed or not. For instanceresults from Round 1 and Round 2 will be sampledregardless of which FSW are sampled.



Typical situations with crossed random-effects

See MLMUS2, p. 500

Panel (longitudinal) data with random time effects: theinvestment by each of 10 firms is recorded each yearfrom 1935 to 1965 (see pp. 474-480 of MLMUS2)

Observational data on individuals classified in two ways:an attainment score at age 16 is obtained from pupilswho each attended one of 148 primary schools and oneof 19 secondary schools (see pp. 481-492 of MLMUS2)


Notes

Notes


Typical situations with crossed random-effects

Data from an experimental design with crossed blockingfactors (see pp. 493-499 of MLMUS2)

Several raters rate each of several objects (see exercises11.4 and 11.7 of MLMUS2)

Social network data (e.g., individuals rate how muchthey like every other individual)



Example: How does investment depend onexpected profit and capital stock?

Data on 10 large American corporations : grunfeld.dta

fn: firm identifier i

firmname: firm name

yr: year j

I: Annual gross investment (in $ 1, 000, 000) = amountspent on plant and equipment, . . . (yij)

F: market value of firm (in $ 1, 000, 000) = value of allshares plus book value of all debts . . . (x2ij)

C: real vaue of capital stock (in $ 1, 000, 000) =deviation of stock of plant and equipment from stock in1933 . . . (x3ij)


Notes

Notes


A basic cross-classified model

The gross investment yij for firm i in year j could beexpressed in terms of the market value and capital stockx2ij and x3ij :

yij = β1 +β2x2ij +β3x3ij

The investment behavior of corporations is surely notdeterministic. Allow the effect of both firms and yearson gross investment to vary:

yij = β1 +β2x2ij +β3x3ij + ζ1i + ζ2j + εij



A basic cross-classified model

The two random intercepts ζ1i and ζ2i represent thefactors firm (i) and year (j) that are crossed instead ofnested, while εij is a residual error with

ζ1i ∼N(0,ψ1), ζ2i ∼N(0,ψ2), εij ∼N(0,θ).

The random intercept ζ1i is shared across all years for agiven firm i.

The random intercept ζ2j is shared across all firms for agiven year j.

The residual error εij = interaction between year andfirm and any other effect specific to firm i in year j


Notes

Notes


Example: How does investment depend onexpected profit and capital stock?

�

The STATA xtmixed is designed for nested randomeffects. Use the following trick to fit crossed effects

Consider the entire dataset as a level-3 unit a (say)within which both firms and years are nested.

Choose the factor with more levels (i.e. years) as level-2

units with random intercepts u(2)ja .

Specify a level-3 random intercept for each unit of theother factor ( firm): u(3)

pa for p = 1, . . . ,10. To this end,construct the dummy variable dpij as follows

dpij =

{1 if p = i0 otherwise



Example: investment ?

The previous model becomes

yija = β1 +β2x2ij +β3x3ij + u(2)ja +

∑p

u(3)pa dpij + εija


Notes

Notes



Residual intra-class correlations

ρ(firm): Correlation of observations on the same firmover time

ρ(firm) = cor (yij ,yij ′ |x2ij ,x3ij ,x2ij ′ ,x3ij ′) =ψ1

ψ1 +ψ2 +θ

ρ(year ): Correlation for the same year across firms

ρ(year ) = cor (yij ,yi ′j |x2ij ,x3ij ,x2i ′j ,x3i ′j) =ψ2

ψ1 +ψ2 +θ.




Stata code

To specify the random intercept u(3)pa for each firm use

the syntax R.fn.

R.fn sets a covariance matrix proportional to the identity(same variances and correlations=0)

The artificial level-3 identifier is created by all

use http://www.stata-press.com/data/mlmus2/grunfeld, clear

xtmixed I F C || all: R.fn || yr:, mle


Notes

Notes






High correlation over years within firms

ρ(firm) =ψ1

ψ1 +ψ2 +θ= 0.7

Negligible correlation over firms within years

ρ(year ) =ψ2

ψ1 +ψ2 +θ= 0.002

The syntax xtmixed I F C || all: R.yr || fn:, mle

produces the same results as above, BUT it has 20random effects at level 3


Notes

Notes



Empirical Bayes (BLUP) preditions of the random effectsof firms and year using predict and reffects option:

** 11.3.4 Prediction

predict firm year, reffects

sort fn yr

list fn firmname yr firm year if yr<1938&fn<5, clean noobs



Example: Primary and secondary school vsattainment at age 16

Dataset fife.dta : students cross-classified by 148primary schools and 19 secondary schools.

Pupils from a given primary school can go to several ofthe secondary schools and vice-versa =⇒ Primary andsecondary schools are not nested

Not every combination of primary and secondary schoolexists

Many combinations occur multiple times.


Notes

Notes


Example: Primary and secondary school

attain: attainment score at age 16 of pupil i who wentto primary school k and secondary school j (yijk)

pid: identifier of primary school (up to age 12) (k)

sid: identifier of secondary school (from age 12) (j)

vrq: verbal-reasoning score (last year of primary school)

sex: 1: female, 0: male



A look at the data structure

Create dummy variables for each combination of primaryand secondary schools

egen pick_comb = tag(pid sid)

Compute the number of such combinations by primaryschools and list them

egen numsid = total(pick_comb), by(pid)

sort pid sid

list pid sid numsid if pick_comb & pid<10, sepby(pid) noobs


Notes

Notes


A look at the data structure



Additive crossed random-effects model

A simple model with random intercepts for primary andsecondary schools is

yijk = β1 + ζ1j + ζ2k + εijk ,

where εijk , ζ1j , ζ2k are independent and normally distributedwith mean 0 and respective variances θ, ψ1 and ψ2.


Notes

Notes



We can rewrite the model as a three-level model with nestedrandom effects:

Level 3: the entire dataset is the only level 3 unit

Level 2: treat one of the two crossed factors as level 2(usually the factor with more levels) and put a level 2

random intercept in the model, say u(2)j

Level 1: Define a dummy variable dpjk that will be 1 ifp = k, 0 otherwise and put a level-3 random coefficientu(3)

p in front of dpjk in the model




The model:

yijk = β1 + u(2)j +

∑p

u(3)p dpjk + εijk

We have 19 SS and 148 PS =⇒ Level-2 Units: Primaryschool, Level-3: artificial level (dataset) + 19 randomeffects for the SS

Stata code

** 11.5.2 Estimation using xtmixed

xtmixed attain || _all: R.sid || pid:, mle variance

estimates store model1


Notes

Notes






Within primary school:ψ2/(ψ1 +ψ2 +θ) = 1.12/(0.35 + 1.12 + 8.11) = 0.12

Within secondary school:ψ1/(ψ1 +ψ2 +θ) = 0.35/(0.35 + 1.12 + 8.11) = 0.037

θ captures variation unexplained by primary school orsecondary school effects (e.g., pupil-level effects, covariates,interaction between primary and secondary school).


Notes

Notes


Additive crossed random-effects model withrandom interaction

Since we have more than one pupil in mostprimary-secondary school combination, we can fit a modelwith a random interaction:

yijk = β1 + ζ1j + ζ2k + ζ3jk + εijk ,

with εijk , ζ1j , ζ2k , ζ3jk independent and normally distributedwith mean 0 and respective variances θ, ψ1, ψ2 and ψ3.

⇒ The effect of a secondary school is now allowed todepend on which primary school the pupil attended (whereasit was the same regardless of the primary school attended inthe previous model).




Severall residual intra-class correlations :Residual correlation between pupils from different PSsbut from the same SS

ρ(j) = ρ(SS) = cor (yijk ,yi ′jk ′) =ψ1

ψ1 +ψ2 +ψ3 +θ

Residual correlation between pupils from different SSsbut from the same PS

ρ(k) = ρ(PS) = cor (yijk ,yi ′j ′k ) =ψ2

ψ1 +ψ2 +ψ3 +θ

Residual correlation between pupils from the sameprimary and SS

ρ(jk) = ρ(SS,PS) = cor (yijk ,yi ′jk ) =ψ1 +ψ2 +ψ3

ψ1 +ψ2 +ψ3 +θ


Notes

Notes



If we consider all pupils in SS j, what is the residualcorrelation between pupils from the same PS?

ρ(k |j) = ρ(PS|SS) = cor (yijk ,yi ′jk |j) =ψ2 +ψ3

ψ2 +ψ3 +θ

If we consider all pupils in PS k, what is the residualcorrelation between pupils from the same SS?

ρ(j|k) = ρ(SS|PS) = cor (yijk ,yi ′jk |k) =ψ1 +ψ3

ψ1 +ψ3 +θ



Fitting the model with random interactionwith xtmixed

Two ways to fit the model

Create an identifier variable for SS and SP combinations

egen comb=group(sid pid)

xtmixed attain || _all: R.sid || pid: || comb: ///

, mle variance

Since sid is nested within pid, we do not need to createcomb

egen comb=group(sid pid)

xtmixed attain || _all: R.sid || pid: || sid: ///

, mle variance

estimates store model2


Notes

Notes


Model with random interaction




Within primary school:ψ2/(ψ1 +ψ2 +ψ3 +θ) = 0.90/(0.31+0.90+0.24+8.09) = 0.09

Within secondary school:ψ1/(ψ1 +ψ2 +θ) = 0.31/(0.31 + 0.90 + 0.24 + 8.09) = 0.032

Same primary & secondary:(ψ1 +ψ2 +ψ3)/(ψ1 +ψ2 +θ) =(0.31 + 0.90 + 0.24)/(0.31 + 0.90 + 0.24 + 8.09) = 0.15

primary|secondary: (ψ2 +ψ3)/(ψ2 +ψ3 +θ) =(0.90 + 0.24)/(0.90 + 0.24 + 8.09) = 0.12

secondary|primary: (ψ1 +ψ3)/(ψ1 +ψ3 +θ) =(0.31 + 0.24)/(0.31 + 0.24 + 8.09) = 0.06


Notes

Notes


Is the interaction needed?

A likelihood ratio test can be used to see whether theinteraction is significant (but since this tests if ψ3 = 0, thep-value is conservative and should be divided by 2.

The stata code lrtest model1 model2 gives a p-value of 0.28(0.14 when divided by 2), so the interaction is notsignificant.


Advanced MLM topics Crossed random effects: logistic regression

Crossed random-effects: logisticregression


Notes

Notes


Crossed random-effects: logistic regression

We can fit a similar model to the data when the response isbinary. In this case, the model becomes

logit{P[yijk = 1|xijk , ζ1j , ζ2k ]} = β1 +β′2xijk + ζ1j + ζ2k ,

with ζ1j , ζ2k independent and normally distributed withmean 0 and respective variances ψ1 and ψ2.




Once again, we can obtain the model from the precedingpage when we observe yijk = 1 if y ∗ijk > 0 and yijk = 0 ify ∗ijk ≤ 0, with

y ∗ijk = β1 +β′2xijk + ζ1j + ζ2k + εijk ,

with ζ1j , ζ2k independent and normally distributed withmean 0 and respective variances ψ1 and ψ2 and εijk followinga standard logistic distribution.


Notes

Notes



Residual intra-class correlations between latent-responses aregiven by

ρ(j) =ψ1

ψ1 +ψ2 + 3.29

ρ(k) =ψ2

ψ1 +ψ2 + 3.29.




Do salamanders from different populations matesuccessfully? (MLMUS2, p. 493)

Two salamander populations: roughbutt (RB) andwhiteside (WS) which had been geographically isolatedfrom each other.

Scope: Investigate whether salamanders wouldcross-breed.

Three experiment have been conducted : 1 in 1986 and2 in 1987.


Notes

Notes


Salamanders data: design

Experiment 1 in 1986: two groups of 20 salamanders

Each group = 5 RB male (RBM), 5 RB female (RBF),5 WS male (WSM) and 5 WS female (WSF).

Within each group: 60 male-female pairs. Eachsalamander has 3 partners from the same population and3 from the other population.

Experiment 2 in 1987: The same salamanders as inExperiment 1

Experiment 3 in 1987: new set of salamanders





Notes

Notes


Salamanders data: variables

y: indicator for successful mating (1: successful, 0:unsuccessful)

female: male identifier 1-60

male : female identifier 1-60

group: identifier of 6 groups of 20 salamanders (seetable 11-4)

experiment: experimnet number (1-3)

rbm rbf wsm wsf: dummy variables for salamander type





Notes

Notes






Treat salamanders from experiment 1 and 2 as independentand fit a logistic random-effects regression model

logit{P[yij = 1|x2i ,x3j , ζ1i , ζ2j ]} =

β1 +β2x2i +β3x3j +β4x2ix3j + ζ1i + ζ2j ,

where

ζ1i and ζ2j are independent random intercepts for males iand females j with variances ψ1 and ψ2 (givencovariates).

The covariates wsm (x2i ) and wsf (x3j) and theirinteraction are included


Notes

Notes


Fitting the model with xtmelogit

This is done in exactly the same manner as with xtmixed.xtmelogit y wsm wsf ww || all: R.male || female:,

orxtmelogit y wsm wsf ww || all: R.female || male:,

Unfortunately, it is computationally extremelydemanding. In both cases we need 60 randomcoefficients.




How to set up the crossed-random effects model for ahierarchical software package?

See Harvey Goldstein Chapter 8,Multilevel StatisticalModels Second Edition. (London: Edward Arnold,1995), esp. pp. 116-17 and pp. 123-24.)

Choose one of the crossed random effects as the secondlevel

Take the other crossed effect and create dummy(indicator) variables of it as random effects at a thirdlevel.

Constraint the variance-covariance matrix of this secondrandom effect to be proportional to the identity(diagonal elements are equal and off-diagonal elementsare zero)


Notes

Notes



In chapter 11 of MLMUS2, interested readers can find atrick to partition the data so that numerical integrationis performed in lower dimension.

It requires the use of the Stata command supclust.

This trick cannot always be applied: it can only be usedwhen the data can be partitioned into clusters withinwhich other factors are nested.

split the data on primary/secondary schools into 3“regions” so that all primary and secondary schools (i)appear in a single region and (ii) a secondary school withwhich a primary school is paired must be in the sameregion.




Since Salamanders are nested within groups with nomatings occurring across groups, we can use group asthe highest level.

We only need 10 random coefficients for males

The first random coefficient is for a dummy variablecorresponding to the male salamanders number: 1, 11,21, 31,41 and 51.

The second random coefficient is for a dummy variablecorresponding to the male salamanders number: 2, 12,22, 32, 42 and 52.

and so forth.

Groups of 6 salamanders with different randomcoefficients (each group has its own coefficient)


Notes

Notes



Relabel the males : salamanders within the same groupwill have the same labelgenerate m = male - (group-1)*10

generate f = female - (group-1)*10

Use xtmelogit with laplace optiongenerate ww = wsf*wsm

xtmelogit y wsm wsf ww || ///

group: R.m || f:, laplace



Salamanders data: xtmelogit-Laplace


Notes

Notes


Fitting the model with xtmelogit-intpoints(2)

The previous trick enabled us to reduce thedimensionality of integration to 10 at level 3 and 1 atlevel 2 (instead of 60 at level 3 and 1 at level 2)

Use xtmelogit with 2 integration points + Laplaceapproximation as starting valuesmatrix a = e(b)

xtmelogit y wsm wsf ww || group: R.m || f:,///

intpoints(2) from(a) refineopts(iterate(0))



Salamanders data: xtmelogit-intpoints(2)


Notes

Notes


Fitting the model with xtmelogit-intpoints(3)

Increase the integration points to 3matrix a = e(b)

xtmelogit y wsm wsf ww || group: R.m || f:,///

intpoints(3) from(a) refineopts(iterate(0))

The option intpoints(3) took a night to run!!!!


Notes

Notes

Documents

Workshop on multilevel modeling II - Université Lavalarchimede.mat.ulaval.ca/pages/duchesne/handouts.pdfWorkshop on multilevel modeling II Belkacem Abdous & Thierry Duchesne [email protected]