(PDF) Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling

MULTILEVEL ANALYSIS

An introduction to

basic and advanced multilevel modeling

Tom A. B. Snijders and Roel J. Bosker

SAGE Publications

London Thousand Oaks New Delhi

1999

2. Multilevel data and multilevel analysis 6

2. Multilevel data and multilevel analysis

Multilevel Analysis using the

hierarchical linear model :

random coeﬃcient regression analysis

for data with several nested levels.

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Each level is (potentially) a

source of unexplained variability.

2

2. Multilevel data and multilevel analysis 8

Some examples of units

at the macro and micro level:

macro-level micro-level

schools teachers

classes pupils

neighborhoods families

precincts voters

ﬁrms departments

departments employees

families children

litters animals

doctors patients

interviewers respondents

judges suspects

subjects measurements

respondents = egos alters

3

2. Multilevel data and multilevel analysis 11

Multilevel analysis is a suitable approach to take

into account the social contexts as well as the

individual actors or subjects.

The hierarchical linear model is a type of regression

analysis for multilevel data where the dependent

variable is at the lowest level.

Explanatory variables can be deﬁned at any level

(including aggregates of level-one variables).

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Z

y

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Z

y

. . . . . . . . .

-

x

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Z

y

. . . . . . . . .

-

x

Figure 2.5 The structure of macro–micro propositions.

4

2. Multilevel data and multilevel analysis 6–7

Two kinds of argument to choose for a multilevel

analysis instead of an OLS regression of

disaggregated data:

1. Dependence as a nuisance

Standard errors and tests base on OLS regression

are suspect because the assumption of

independent residuals is invalid.

2. Dependence as an interesting phenomenon

It is interesting in itself to disentangle variability

at the various levels;

moreover, this can give insight in where further

explanation may fruitfully be sought.

5

4. The random intercept model 39

4. The random intercept model

Hierarchical Linear Model:

iindicates level-one unit (e.g., individual);

jindicates level-two unit (e.g., group).

Variables for individual iin group j:

Yij dependent variable;

xij explanatory variable at level one;

for group j:

zjexplanatory variable at level two; njgroup size.

OLS regression model of Yon X:

Yij =β0+β1xij +Rij .

Group-dependent regressions:

Yij =β0j+β1jxij +Rij .

6

4. The random intercept model 41

In the random intercept model, only the intercept

varies randomly between groups:

Yij =β0j+β1xij +Rij .

where β0j= average intercept γ00

plus group-dependent deviation U0j:

β0j=γ00 +U0j.

The constant regression coeﬃcient β1now is

denoted γ10 to indicate that it is a parameter in the

overall model. Substitution yields

Yij =γ00 +γ10 xij +U0j+Rij .

These formulae may be interpreted in two ways:

the U0jcould be ﬁxed or random coeﬃcients.

In ANCOVA, they are ﬁxed; in the hierarchical linear

model, they are random.

7

4. The random intercept model 42

X

Y

β01 



β03 



β02 

regression line group 1

regression line group 3

regression line group 2

p

y12

R12{

Figure 4.1 Diﬀerent parallel regression lines.

The point y12 is indicated

with its residual R12 .

8

4. The random intercept model 43

Arguments for choosing between ﬁxed (F) and

random (R) coeﬃcient models:

1. If groups are unique entities, inference should

focus on these groups, F.

2. If groups are regarded as sample from

(hypothetical?) population and inference should

focus on this population, then R.

3. If level-two eﬀects are to be tested, then R.

4. For small group sizes, in order to avoid

overﬁtting: R.

5. If group eﬀects U0j(etc.) are not nearly

normally distributed, the Roption is risky (or

more complicated multilevel models must be

used).

Rule of thumb:

for less than 10 groups, think of F;

for more than 10 groups, think of R;

if group sizes are large (over 100), diﬀerences

between them are small.

9

4. The random intercept model 45–46

The empty model (random eﬀects ANOVA) is a

model without explanatory variables:

Yij =γ00 +U0j+Rij .

Variance decomposition:

var(Yij) = var(U0j) + var(Rij) = τ2

0+σ2.

Covariance between two individuals (i6=i0)

in the same group j:

cov(Yij, Yi0j) = var(U0j) = τ2

0,

and their correlation:

ρ(Yij, Yi0j) = ρI(Y) = τ2

0

(τ2

0+σ2).

This is the intraclass correlation coeﬃcient.

10

4. The random intercept model 47

Table 4.1 Estimates for empty model

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 40.36 0.43

Random eﬀect Var. Comp. S.E.

Level two variance:

τ2

0=var(U0j)19.42 2.92

Level one variance:

σ2=var(Rij)64.57 1.97

deviance 16253.2

11

4. The random intercept model 46–47

Intraclass correlation

ρI=19.42

19.42 + 64.57 = 0.23

Total population of individual values Yij has mean

40.36 and standard deviation

√19.42 + 64.57 = 9.16 .

Population of class means β0jhas mean 40.36

and standard deviation √19.42 = 4.4.

The model becomes more interesting,

when also ﬁxed eﬀects of explanatory variables are

included.

12

4. The random intercept model 49–51

Table 4.2 Estimates for random intercept model with

eﬀect for IQ

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 40.61 0.31

γ10 = Coeﬃcient of IQ 2.488 0.070

Random eﬀect Var. Comp. S.E.

Level two variance:

τ2

0=var(U0j)9.50 1.52

Level one variance:

σ2=var(Rij)42.23 1.29

deviance 15251.8

There are two kinds of parameters:

1. ﬁxed eﬀects: regression coeﬃcients γ;

(just like in OLS regression)

2. random eﬀects:

variance components σ2and τ2

0.

13

4. The random intercept model 49–51

Table 4.3 Estimates for ordinary least squares regression

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 40.93 0.15

γ10 = Coeﬃcient of IQ 2.654 0.072

Random eﬀect Var. Comp. S.E.

Level one variance:

σ2=var(Rij)50.90 1.51

deviance 15477.7

14

4. The random intercept model 49–51

−4−3−2−10 1 2 3 4

25

30

50

55

X= IQ

Y

.

Figure 4.2 Fifteen randomly chosen regression lines

according to the random intercept model of Table 4.2.

15

4. The random intercept model 51–54

More explanatory variables:

Yij =γ00 +γ10 x1ij +... +γp0xpij

+γ01 z1j+... +γ0qzqj

+U0j+Rij .

Especially important:

diﬀerence between within-group

and between-group regressions.

X

Y

"""""""""""""""""""""

between-group regression line



regression line within group 1

regression line within group 3

regression line

within group 2

Figure 4.3 Diﬀerent between-group and within-group

regression lines.

16

4. The random intercept model 51–54

This is obtained by having separate ﬁxed eﬀects for

the level-1 variable Xand its group mean ¯

X.

(Alternative: use the within-group deviation variable

(X−¯

X)instead of X.)

Table 4.4 Estimates for random intercept model

with diﬀerent within- and between-group regressions

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 40.74 0.28

γ10 = Coeﬃcient of IQ 2.415 0.072

γ01 = Coeﬃcient of IQ (group mean) 1.589 0.313

Random eﬀect Var. Comp. S.E.

Level two variance:

τ2

0=var(U0j)7.73 1.30

Level one variance:

σ2=var(Rij)42.15 1.28

deviance 15227.5

17

4. The random intercept model 58–59

The random eﬀects U0jare not statistical

parameters and therefore they are not estimated as

part of the estimation routine.

However, it sometimes is desirable to ‘estimate’

them. This can be done by the empirical Bayes

method; these ‘estimates’ are also called the

posterior means.

The posterior mean for group jis based on two

kinds of information:

⇒sample information: the data in group j;

⇒population information:

the value U0jwas drawn from a normal

distribution with mean 0 and variance τ2

0.

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4. The random intercept model 58–59

The empirical Bayes estimate in the case of the

empty model is a weighted average of the group

mean and the overall mean:

ˆ

βEB

0j=λjˆ

β0j+ (1 −λj) ˆγ00 ,

where the weight λj

is the reliability of the mean of group j

λj=τ2

0

τ2

0+σ2/nj

.

These ‘estimates’ are not unbiased for each speciﬁc

group, but they are more precise when the mean

squared errors are averaged over all groups.

For models with explanatory variables,

the same principle can be applied:

the values that would be obtained

as OLS estimates per group

are “shrunk towards the mean”.

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4. The random intercept model 58–59

There are two kinds of standard errors

for empirical Bayes estimates:

comparative standard errors

S.E.comp ˆ

UEB

hj!=S.E. ˆ

UEB

hj −Uhj!

for comparing the random eﬀects of diﬀerent level-2

units

(use with caution

– E.B. estimates are not unbiased!);

and diagnostic standard errors

S.E.diag ˆ

UEB

hj!=S.E. ˆ

UEB

hj!

used for model checking

(e.g., checking normality of the level-two residuals).

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4. The random intercept model 62

−10

−5

0

5

10

U0j

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Figure 4.4 The added value scores for 131 schools with

comparative posterior conﬁdence intervals.

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5. The hierarchical linear model 62

5. The hierarchical linear model

It is possible that not only the group average, but

also the eﬀect of Xon Yis randomly dependent

on the group.

In other words, in the equation

Yij =β0j+β1jxij +Rij ,

also the regression coeﬃcient β1jhas a random

part:

β0j=γ00 +U0j

β1j=γ10 +U1j.

Substitution leads to

Yij =γ00 +γ10 xij

+U0j+U1jxij +Rij .

Variable Xnow has a random slope.

22

5. The hierarchical linear model 62

There are more parameters:

var(U0j) = τ00 =τ2

0;

var(U1j) = τ11 =τ2

1;

cov(U0j, U1j) = τ01 .

23

5. The hierarchical linear model 71

Table 5.1 Estimates for random slope model

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 40.75 0.29

γ10 = Coeﬃcient of IQ 2.459 0.083

γ01 = Coeﬃcient of IQ (group mean) 1.405 0.322

Random eﬀect Var. Comp. S.E.

Level two random eﬀects:

τ2

0=var(U0j)7.92 1.32

τ2

1=var(U1j)0.200 0.098

τ01 =cov(U0j, U1j)-0.820 0.267

Level one variance:

σ2=var(Rij)41.35 1.29

deviance 15213.5

The equation for this table is

Yij = 40.75 + 2.459 IQij + 1.405 IQ.j

+U0j+U1jIQij +Rij .

The slope β1jhas average 2.459 and standard

deviation √0.200 = 0.45.

24

5. The hierarchical linear model 71

−4−3−2−101234

25

30

50

55

X= IQ

Y

.

Figure 5.2 Fifteen random regression lines

according to the model of Table 5.1.

Note the heteroscedasticity:

the variance is larger for low Xthan for high X.

The intercept variance and intercept-slope

covariance

depend on the position of the X= 0 value,

because the intercept is deﬁned by the X= 0 axis.

25

5. The hierarchical linear model 73

The next step is to explain the random slopes:

β0j=γ00 +γ01 zj+U0j

β1j=γ10 +γ11 zj+U1j.

Substitution then yields

Yij = (γ00 +γ01 zj+U0j)

+ (γ10 +γ11 zj+U1j)xij +Rij

=γ00 +γ01 zj+γ10 xij +γ11 zjxij

+U0j+U1jxij +Rij .

The term γ11 zjxij is called the

cross-level interaction eﬀect.

26

5. The hierarchical linear model 75

Table 5.2 Estimates for model with random slope

and cross-level interaction

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 40.89 0.29

γ10 = Coeﬃcient of IQ 2.443 0.082

γ01 = Coeﬃcient of IQ 1.246 0.326

γ02 = Coeﬃcient of Z20.057 0.037

γ12 = Coeﬃcient of Z2×IQ -0.022 0.011

Random eﬀect Var. Comp. S.E.

Level two random eﬀects:

τ2

0=var(U0j)7.67 1.29

τ2

1=var(U1j)0.178 0.095

τ01 =cov(U0j, U1j)-0.769 0.260

Level one variance:

σ2=var(Rij)41.36 1.29

deviance 15208.4

27

5. The hierarchical linear model 77

Table 5.3 Estimates for model with random slopes

and many eﬀects

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 42.28 1.31

γ10 = Coeﬃcient of IQ 2.110 0.383

γ20 = Coeﬃcient of SES 0.178 0.053

γ01 = Coeﬃcient of IQ 0.908 0.328

γ02 = Coeﬃcient of GS -0.037 0.050

γ03 = Coeﬃcient of COMB -1.602 0.798

γ11 = Coeﬃcient of IQ ×IQ -0.057 0.083

γ12 = Coeﬃcient of IQ ×GS -0.002 0.015

γ13 = Coeﬃcient of IQ ×COMB 0.368 0.232

γ21 = Coeﬃcient of SES ×IQ -0.020 0.019

γ22 = Coeﬃcient of SES ×GS 0.001 0.003

γ23 = Coeﬃcient of SES ×COMB 0.057 0.046

Random eﬀect Var. Comp. S.E.

Level two random eﬀects:

τ2

0=var(U0j)7.73 1.27

τ2

1=var(U1j)0.127 0.084

τ01 =cov(U0j, U1j)-0.666 0.242

τ2

2=var(U2j)0.0 0.0

τ02 =cov(U0j, U2j)0.0 0.0

Level one variance:

σ2=var(Rij)39.23 1.22

deviance 15087.8

28

5. The hierarchical linear model 78

Table 5.4 Estimates for a more parsimonious model

with a random slope and many eﬀects

Fixed Eﬀect Coeﬃcient S.E.

γ00 = Intercept 41.28 0.35

γ10 = Coeﬃcient of IQ 2.103 0.094

γ20 = Coeﬃcient of SES 0.156 0.015

γ01 = Coeﬃcient of IQ 0.947 0.323

γ03 = Coeﬃcient of COMB -1.297 0.589

γ13 = Coeﬃcient of IQ ×COMB 0.485 0.171

Random eﬀect Var. Comp. S.E.

Level two random eﬀects:

τ2

0=var(U0j)7.76 1.28

τ2

1=var(U1j)0.143 0.087

τ01 =cov(U0j, U1j)-0.598 0.245

Level one variance:

σ2=var(Rij)39.24 1.22

deviance 15093.0

29

6. Testing 86–89

6. Testing

To test ﬁxed eﬀects, use the t-test with test statistic

T(γh) = ˆγh

S.E.(ˆγh).

(Or the Wald test for testing

several parameters simultaneously.)

For parameters in the random part,

do not use t-tests.

Simplest test for random part parameters

is the deviance (or likelihood ratio) test:

subtract deviances, use chi-squared test.

(d.f. = number of parameters tested).

However, one special circumstance:

variance parameters are necessarily positive.

Therefore, they may be tested one-sided.

This works as follows:

if deviance diﬀerence = 0, then no signiﬁcance;

if deviance diﬀerence >0,

calculate p-value from chi-squared distribution

and divide by 2.

30

6. Testing 91, 123

Other test for random part:

Rao’s eﬃcient score test

(the same as Lagrange multiplier test).

First estimate null hypothesis;

from there, make one step of the (R)IGLS or

Fisher scoring algorithm and deﬁne

one-step estimate = ˜τ2.

Test

˜

τ2

S.E.(˜τ2),

in standard normal (or t) distribution.

Note: t-test may not be applied to

ML estimate of variance parameter.

Advantage of this test: cheap to compute.

Therefore useful in model exploration.

Also good power properties.

31

6. Testing 91, 123

Table 6.1 Estimates for two models

with diﬀerent between- and within-group regressions

Model 1 Model 2

Fixed Eﬀects Coeﬃcient S.E. Coeﬃcient S.E.

γ00 = Intercept 40.78 0.29 40.78 0.29

γ10 = Coeﬃcient of IQ 2.249 0.082

γ20 = Coeﬃcient of ˜

IQ 2.249 0.082

γ30 = Coeﬃcient of SES 0.156 0.015 0.156 0.015

γ01 = Coeﬃcient of IQ 1.084 0.325 3.333 0.320

Random eﬀects Var. Comp. S.E. Var. Comp. S.E.

level two random eﬀects:

τ2

0=var(U0j)8.19 1.33 8.19 1.33

τ2

1=var(U1j)0.170 0.091 0.170 0.091

τ01 =cov(U0j, U1j)-0.722 0.258 -0.722 0.258

level one variance:

σ2=var(Rij)39.29 1.22 39.29 1.22

deviance 15103.7 15103.7

Test for equality of within- and between-group

regressions is t-test for IQ in Model 1:

t= 1.084/0.325 = 3.23, p < 0.01.

Model 2 gives within-group coeﬃcient 2.249

and between-group coeﬃcient

3.333 = 2.249 + 1.084.

32

7. Explained variance 100

7. Explained variance

The individual variance parameters may go up

when eﬀects are added to the model.

Table 7.1 Estimated residual variance parameters ˆσ2and

ˆτ2

0for models with within-group and between-group

predictor variables

ˆσ2ˆτ2

0

I. BALANCED DESIGN

A. Yij =β0+U0j+Eij 8.694 2.271

B. Yij =β0+β1X.j +U0j+Eij 8.694 0.819

C. Yij =β0+β2(Xij −X.j) + U0j+Eij 6.973 2.443

II. UNBALANCED DESIGN

A. Yij =β0+U0j+Eij 7.653 2.798

B. Yij =β0+β1X.j +U0j+Eij 7.685 2.038

C. Yij =β0+β2(Xij −X.j) + U0j+Eij 6.668 2.891

33

7. Explained variance 102

The best way to deﬁne R2, the proportion of

variance explained, is the

proportional reduction in total variance,

which for the random intercept model is (σ2+τ2

0).

Table 7.2 Estimating the level-1 explained variance

(balanced data)

ˆσ2ˆτ2

0

A. Yij =β0+U0j+Eij 8.694 2.271

D. Yij =β0+β1(Xij −X.j) + β2X.j +U0j+Eij 6.973 0.991

Explained variance at level 1:

R2

1= 1 −6.973 + 0.991

8.694 + 2.271 = 0.27.

34

8. Heteroscedasticity 114

8. Heteroscedasticity

In MLwiN, possibility to specify

heteroscedastic models where residual variance

depends on observed variables.

Complex variation in MLwiN jargon.

E.g.,

random part at level one =R0ij +R1ij x1ij .

Then the level-1 variance

is quadratic function of X:

var(R0ij +R1ij xij) = σ2

0+ 2 σ01 x1ij +σ2

1x2

1ij .

For σ2

1= 0, this is a linear function:

var(R0ij +R1ij xij) = σ2

0+ 2 σ01 x1ij .

Possible as a variance function,

without random eﬀects interpretation.

34

8. Heteroscedasticity 112–113

Level-1 variance function for Model 3:

37.83 −4.02 IQ

Maybe further diﬀerentiation possible

between low-IQ pupils?

Model 4 uses

IQ2

−=









IQ2if IQ <0

0if IQ ≥0,

IQ2

+=









0if IQ <0

IQ2if IQ ≥0.

Y

IQ

−4

−2

2

4

−4−22 4

.

Figure 8.1 Eﬀect of IQ on language test.

35

8. Heteroscedasticity 112–113

Table 8.2 Heteroscedastic models

depending on IQ.

Model 3 Model 4

Fixed Eﬀect Coeﬃcient S.E. Coeﬃcient S.E.

Intercept 39.61 0.31 39.73 0.31

IQ 2.223 0.077 3.236 0.157

IQ2

−0.246 0.046

IQ2

+−0.306 0.039

SES 0.146 0.014 0.144 0.014

Gender 2.51 0.26 2.35 0.25

IQ 1.02 0.32 1.21 0.31

Random Eﬀect Parameter S.E. Parameter S.E.

Level-two random eﬀects:

Intercept variance 8.06 1.31 7.19 1.17

IQ slope variance 0.133 0.078 0.0 0.0

Intercept - IQ slope covariance −0.51 0.24 0.0 0.0

Level-one variance parameters:

σ2

0constant term 37.83 1.20 37.88 1.18

σ01 IQ eﬀect −2.01 0.26 −2.37 0.22

Deviance 14960.0 14908.1

36

8. Heteroscedasticity 117–119

Heteroscedasticity can be very important

for the researcher

(although mostly (s)he doesn’t know it yet).

Bryk & Raudenbush: Correlates of diversity.

Explain not only means, but also variances!

Heteroscedasticity also possible

for level-2 random eﬀects:

give a random slope at level 2

to a level-2 variable.

37

9. Assumptions of the hierarchical linear model 120–123

9. Assumptions of the

Hierarchical Linear Model

Yij =γ0+r

X

h=1 γhxhij

+U0j+p

X

h=1 Uhj xhij +Rij .

Questions:

1. Does the ﬁxed part contain

the right variables (now X1to Xr)?

2. Does the random part contain

the right variables (now X1to Xp)?

3. Are the level-one residuals

normally distributed?

4. Do the level-one residuals

have constant variance?

5. Are the level-two random coeﬃcients

normally distributed?

6. Do the level-two random coeﬃcients

have a constant covariance matrix?

38

9. Assumptions of the hierarchical linear model 121–123; also 54

Follow the logic of the HLM

1. Include contextual eﬀects

For every level-1 variable Xh,

check the ﬁxed eﬀect of the group mean ¯

Xh.

Econometricians’ wisdom:

the U0jmay not be correlated with the Xhij.

Therefore test this correlation by testing

the eﬀect of ¯

Xh(’Hausman test’)

Use a ﬁxed eﬀects model if this eﬀect is signiﬁcant.

Diﬀerent approach to the same assumption:

Include the ﬁxed eﬀect of ¯

Xhif it is signiﬁcant,

and continue to use a random eﬀects model.

(Also include the variables ¯

Xh.j Zj

for all cross-level interactions involving Xh!)

Also the random slopes Uhj must not be

correlated with the Xkij.

This can be checked by testing

the ﬁxed eﬀect of ¯

Xk.j Xhij .

This procedure widens the scope of random

coeﬃcient models beyond what is allowed by the

conventional rules of econometricians.

39

9. Assumptions of the hierarchical linear model 121–123; also 54

2. Check random eﬀects of level-1

variables.

Think of the score test for the possibility of testing

many random slopes without a lot of work.

(Berkhof and Snijders, J. Ed. Beh. Stats. 2001)

3. Check heteroscedasticity.

Implemented in MLwiN by random eﬀects at level 1.

See Chapter 8.

40

9. Assumptions of the hierarchical linear model 128–133

OLS level-1 residuals

Test the ﬁxed part of the level-1 model

using OLS residuals, calculated per group separately.

Test the random part of the level-1 model

using squared standardized OLS residuals.

These can be calculated by macro res1.obe.

Empirical Bayes level-2 residuals

Test the ﬁxed part of the level-2 model

using EB residuals.

Test the random part of the level-1 model

using squared EB residuals

standardized by diagnostic variance.

41

9. Assumptions of the hierarchical linear model 134–139

Inﬂuence of level-two units

Inﬂuence of higher-level units can be assessed

by Cook-type statistics

and their ﬁt by standardized multivariate residuals.

Ref.: Snijders & Berkhof, 2003 (see website).

42

10. Designing Multilevel Studies 140–142

10. Designing Multilevel Studies

Note: each level corresponds to

a sample from a population.

For each level, the total sample size counts.

E.g., 3-level design:

15 municipalities;

in each municipality, 200 households;

in each household, 2-4 individuals.

Total sample sizes are 15 (level 3),

3000 (level 2), ≈9000 (level 1).

Much information about individuals and households,

but little about municipalities.

43

10. Designing Multilevel Studies 140–142

Power and standard errors

Consider testing a parameter β, based on a t-ratio

ˆ

β

s.e.(ˆ

β).

For signiﬁcance level αand power γ

β

s.e.(ˆ

β)≈(z1−α+zγ)=(z1−α−z1−γ),

where z1−α, zγand z1−γ

are values for standard normal distribution.

E.g., for α=.05, γ =.80, eﬀect size β=.20,

sample size must be such that

standard error ≤.20

1.64 + 0.84 = 0.081 .

Following discussion mainly in terms of standard

errors, always for two-level designs,

two-stage samples of Nclusters each with nunits.

44

10. Designing Multilevel Studies 142–143

Design eﬀect for estimation of a mean

Empty model:

Yij =µ+Uj+Rij .

with var(Uj) = τ2, var(Rij) = σ2.

Parameter to be estimated is µ.

ˆµ=N

X

j=1

n

X

i=1 Yij ,

var(ˆµ) = τ2

N+σ2

Nn .

The sample mean of a simple random sample of

Nn elements from this population has variance

τ2+σ2

Nn .

45

10. Designing Multilevel Studies 142–143

Relative eﬃciency of simple random sample w.r.t.

two-stage sample is the design eﬀect

(cf. Kish, Cochran)

nτ 2+σ2

τ2+σ2= 1 + (n−1)ρI,(1)

where

ρI=τ2

τ2+σ2.

46

10. Designing Multilevel Studies 142–143

Eﬀect of a level-two variable

Two-level regression with random intercept model

Yij =β0+β1xj+Uj+Eij .

When var(X) = s2

X,

var(ˆ

β1) = τ2+ (σ2/n)

Ns2

X

.

For a simple random sample from the same

population,

var(ˆ

βdisaggregated

1) = τ2+σ2

Nns2

X

.

Relative eﬃciency again equal to (1).

47

10. Designing Multilevel Studies –

Eﬀect of a level-one variable

Now suppose Xis a pure level-one variable,

i.e., ¯

Xthe same in each cluster,

ρI=−1/(n−1).

Assume var(X) = s2

Xwithin each cluster.

E.g.: time eﬀect in longitudinal design;

within-cluster randomization.

Again

Yij =β0+β1xij +Uj+Eij .

Now

ˆ

β1=1

Nns2

X

N

X

j=1

n

X

i=1 xij Yij (2)

=β1+1

Nns2

X

N

X

j=1

n

X

i=1 xij Eij

with variance

var(ˆ

β1) = σ2

Nns2

X

.

48

10. Designing Multilevel Studies –

For a simple random sample from the same

population,

var(ˆ

βdisaggregated

1) = σ2+τ2

Nns2

X

.

Design eﬀect now is

σ2

τ2+σ2= 1 −ρI.

Eﬃciency due to blocking on clusters!

49

10. Designing Multilevel Studies –

For random intercept model,

design eﬀect <1for level-one variables,

and >1for level-two variables.

Conclusion: for a comparison of randomly assigned

treatments with costs depending on N n,

randomising within clusters more eﬃcient

than between clusters

(Moerbeek, van Breukelen, and Berger, JEBS 2000.)

But not necessarily for variables

with random slope!

50

10. Designing Multilevel Studies –

Level-one variable with random slope

Assume a random slope for X:

Yij =β0+β1xij

+U0j+U1jxij +Eij .

Variance of (2) now is

var(ˆ

β1) = nτ 2

1s2

X+σ2

Nns2

X

.

Marginal residual variance of Yis

σ2+τ2

0+τ2

1s2

X

so design eﬀect now is

nτ 2

1s2

X+σ2

τ2

0+τ2

1s2

X+σ2.

51

10. Designing Multilevel Studies –

Two-stage sample

(”within-subject design” in psychology)

‘neutralizes’ variability due to random intercept,

not random slope of X.

In practice:

if there is a random slope for X

then ﬁxed eﬀect does not tell the whole story

and it is relevant to choose a design

in which the slope variance can be estimated.

52

10. Designing Multilevel Studies –

Optimal sample size for estimating a

regression coeﬃcient

Estimation variance of regression parameters ≈

σ2

1

N+σ2

2

Nn

for suitable σ2

1, σ2

2.

Total cost is usually not a function of total sample

size Nn, but of the form

c1N+c2Nn .

Minimizing the variance under the constraint of a

given total budget leads to the optimum

nopt =v

u

t

c1σ2

2

c2σ2

1

(rounded).

Optimal Ndepends on available budget.

53

10. Designing Multilevel Studies –

For level-one variables with constant cluster means,

σ2

1= 0

so that nopt =∞: single-level design.

For level-two variables,

σ2

1=τ2

s2

X

σ2

2=σ2

s2

X

so that

nopt =v

u

t

c1σ2

c2τ2.

Cf. Cochran (1977),

Snijders and Bosker (JEBS 1993),

Raudenbush (Psych. Meth. 1997, p. 177),

Chapter Snijders in Leyland & Goldstein (eds.,

2001),

Moerbeek, van Breukelen, and Berger (JEBS 2000),

Moerbeek, van Breukelen, and Berger (The

Statistician, 2001).

54

10. Designing Multilevel Studies –

How much power is gained by using covariates?

In single-level regression:

covariate reduces unexplained variance

by factor 1−ρ2.

Assume random intercept models:

Yij =β0+β1xij +Uj+Rij

Yij =˜

β0+β1xij +β2zij

+˜

Uj+˜

Rij

Zij =γ0+UZj +RZij .

Also assume that Zis uncorrelated with X

within and between groups

(regression coeﬃcient β1not aﬀected

by control for Z).

Denote population residual within-group correlation

between Yand Zby

ρW=ρ(Rij, RZij ),

55

10. Designing Multilevel Studies –

population residual between-group correlation by

ρB=ρ(Uj, UZj).

Reduction in variance parameters given by

˜σ2= (1 −ρ2

W)σ2,

˜τ2= (1 −ρ2

B)τ2.

In formulae for standard errors, control for Z

leads to replacement of σ2and τ2

by ˜σ2and ˜τ2.

Therefore:

for pure within-group variables,

only within-group correlation counts;

for level-two variables, both correlations count,

but between-group correlations play the major role

unless nis rather small.

56

10. Designing Multilevel Studies 144–150

Estimating ﬁxed eﬀects in general

Assumptions in preceding treatment very stringent.

Approximate expressions for standard errors

given by Snijders and Bosker (JEBS 1993)

and implemented in computer program PinT

(‘Power in Two-level designs’) available from

http://stat.gamma.rug.nl/snijders/multilevel.htm

Raudenbush (Psych. Meth. 1997) gives more

detailed formulae for some special cases.

Main diﬃculty in the practical application is the

necessity to specify parameter values:

means,

covariance matrices of explanatory variables,

random part parameters.

57

10. Designing Multilevel Studies 144–150

Example:

sample sizes for therapy eﬀectiveness study.

Outcome variable Y, unit variance, depends on

X1(0–1) course for therapists: study variable,

X2therapists’ professional training,

X3pretest.

Means:µ1= 0.4, µ2=µ3= 0.

Variances between groups:

var(X1)=0.24 (because µ1= 0.4)

var(X2) = var(X3)=1(standardization)

ρI(X3)=0.19 (prior knowledge)

ρ(X1, X2) = −0.4(conditional randomization)

ρ(X1,¯

X3|X2)=0 (randomization)

⇒ρ(X1,¯

X3)=0.2

ρ(X2,¯

X3)=0.5(prior knowledge).

This yields σ2

X3(W)= 1 −0.19 = 0.81 and

ΣX(B)=





0.24 −0.20 0.04

−0.20 1.0 0.22

0.04 0.22 0.19







.

58

10. Designing Multilevel Studies 144–150

Parameters of the random part:

var(Yij) = β2

1σ2

X(W)+β0ΣX(B)β

+τ2

0+σ2

(cf. Section 7.2 of Snijders and Bosker, 1999).

Therefore total level-one variance of Yis

β2

1σX(W)+σ2

and total level-two variance is

β0ΣX(B)β+τ2

0.

Suppose that ρI(Y)= 0.2 and that the available

explanatory variables together explain 0.25 of the

level-one variance and 0.5 of the level-two variance.

Then σ2= 0.6 and τ2

0= 0.10.

Budget structure:

assume c1= 20, c2= 1, k= 1000.

59

10. Designing Multilevel Studies 144–150

With this information, PinT can run. Results:

5 10 20 30 40

0.14

0.17

0.20

0.23

0.26

n

S.E.(ˆ

β1)

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

∗∗

◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦

Figure 1 Standard errors for estimating β1,

for 20N+Nn ≤1,000;

∗for σ2= 0.6, τ 2

0= 0.1;

◦for σ2= 0.5, τ 2

0= 0.2.

For these parameters, 7≤n≤17 acceptable.

Sensitivity analysis: also calculate for

σ2= 0.5, τ 2

0= 0.2.

(Residual intraclass correlation twice as big.)

Now 5≤n≤12 acceptable.

60

10. Designing Multilevel Studies 151–153

Estimating intraclass correlation

Donner (ISR 1986):

S.E.(ˆρI) = (1 −ρI)

×(1 + (n−1)ρI)v

u

t

2

n(n−1)(N−1) .

Budget constraint: substitute N=k/(c1+c2n)

and plot as a function of nfor various ρI.

E.g., suppose 20 N+N n ≤1000

and 0.1 ≤ρI≤0.2.

61

10. Designing Multilevel Studies 151–153

2 10 20 30 40

0.025

0.05

0.075

0.10

0.125

0.15

n

S.E.(ˆρI)

ρI= 0.10

∗

∗∗

∗

ρI= 0.20

◦

Figure 2 Standard error for estimating

intraclass correlation coeﬃcient for

budget constraint 20N+Nn ≤1000

with ρI= 0.1 and 0.2.

Clusters sizes between 16 and 27 are acceptable.

62

10. Designing Multilevel Studies 154

Variance parameters

Approximate formulae for random intercept model

(Longford, 1993)

S.E.(ˆσ2)≈σ2v

u

t

2

N(n−1)

and

S.E.(ˆτ2

0)≈σ22

N n

v

u

t

1

n−1+2τ2

0

σ2+n τ 4

0

σ4.

Standard errors for estimated standard deviations:

S.E.(ˆσ)≈S.E.(ˆσ2)

2σ,

and similarly for S.E.(ˆτ0).

Same procedure:

substitute N=k/(c1+c2n)and plot as

function of n.

Example in Cohen (J. Oﬀ. Stat., 1998).

63

11. Crossed random coeﬃcients 156–157

11. Crossed random coeﬃcients

In the usual statistical theory of mixed models

(models with random and ﬁxed eﬀects),

crossed random coeﬃcients are nothing special.

The advantage of the hierarchical linear model

is that the estimation algorithms and methodology

take advantage of the nested structure.

The combination of this with crossed random eﬀects

is possible but a bit complicated.

Crossed eﬀects in a two-level model

Example: school jcrossed with neighborhood f.

Pupil iin school jis in neighborhood G(i, j).

Yij =γ0+q

X

h=1 γhxhij

+U0j+p

X

h=1 Uhj xhij

+WG(i,j)+Rij .

Usual assumptions: U,W,Rall independent.

64

11. Crossed random coeﬃcients 156–157

Trick: deﬁne dummy variables bf(f= 1,...,F)

bfij =









1G(i, j) = f

0G(i, j)6=f .

Random neighborhood eﬀect rewritten as

WG(i,j)=F

X

f=1 Wfbfij

and the random slopes W1to WFmust be

uncorrelated with equal variances

var(W1) = var(W2) = ... =var(WF).

Crossed random eﬀects are represented by

uncorrelated equal-variance slopes

of dummy variables.

65

11. Crossed random coeﬃcients 158

Table 11.1 Two cross-classiﬁed models

for maths achievement.

Model 1 Model 2

Fixed Eﬀect Coeﬀ. S.E. Coeﬀ. S.E.

γ0Intercept 7.99 0.23 8.62 0.14

γ1IQ 0.054 0.005

γ2Pretest 0.170 0.008

γ3Motivation 0.037 0.008

γ4Father’s education 0.057 0.015

γ5Gender 0.237 0.112

Random Eﬀect Var. C. S.E. Var. C. S.E.

Crossed random eﬀect:

τ2

W=var(Wf)secondary school 2.590 0.567 0.727 0.179

Level-two random eﬀect:

τ2

0=var(U0j)primary school 0.179 0.076 0.174 0.059

Level-one variance:

σ2=var(Rij)9.102 0.217 6.370 0.152

Deviance 19334.4 17958.7

66

11. Crossed random coeﬃcients 160–161

Correlated crossed random coeﬃcients

Contribution of neighborhood fto random part:

W0f+W1fxij .

Rewrite this as

W0,f(i,j)+W1,f(i,j)xij

=F

X

f=1(W0fbfij +W1fbf ij xij ).

In other words, we need random slopes

of bfij and also bfij xij .

Restrictions:

Independence for diﬀerent values of f;

var(W01) = var(W02) = ... =var(W0F),

var(W11) = var(W12) = ... =var(W1F),

cov(W01, W11) = cov(W02, W12)

=... =cov(W0F, W1F).

67

11. Crossed random coeﬃcients 162–163

Social networks

Relations between individuals fand g:

(e.g., friendship) (f, g = 1, ..., F )

Yfg =µ+Af+Bg+Ufg +Rf g ,

where

Af=outgoingness eﬀect

Bg=popularity eﬀect

Ufg =reciprocity eﬀect, with Ufg =Ugf

Rfg =residual.

How to model this with

crossed and nested random coeﬃcients?

Nesting structure: relations within dyads.

A dyad is a pair (Yfg , Ygf ).

There are F(F−1)/2dyads, labeled j,

each dyad includes 2 relations i= 1,2.

68

11. Crossed random coeﬃcients 162–163

Zij =relation iin dyad j,

sfij = 1 if fis sender for relation Zij ,

rfij = 1 if fis receiver for relation Zij .

Indicate Ufg by Ujand Rfg by Rij .

Then the model is equivalent to

Zij =µ+F

X

f=1(Afsfij +Bfrf ij )

+Uj+Rij

with restrictions

var(A1) = var(A2) = ... =var(AF),

var(B1) = var(B2) = ... =var(BF),

cov(A1, B1) = cov(A2, B2) = ... =cov(AF, BF).

69

11. Crossed random coeﬃcients 164

Table 11.2 Estimates for two social network models

Model 1 Model 2

Fixed Eﬀects Coeﬃcient S.E. Coeﬃcient S.E.

Intercept 2.32 0.08 2.20 0.17

Sender’s gender (F) -0.22 0.12

Receiver’s gender (F) -0.06 0.07

Similar gender (F-F) 0.74 0.16

Random eﬀects Var. C. S.E. Var. C. S.E.

level three random eﬀects:

school intercept 0.01 0.03 0.03 0.03

sender variance 0.47 0.05 0.42 0.05

receiver variance 0.18 0.02 0.13 0.02

sender-receiver covariance 0.12 0.03 0.09 0.02

level two random eﬀect:

reciprocity 0.19 0.02 0.17 0.01

level one variance:

residual 0.33 0.01 0.33 0.01

deviance 7419.0 7237.66

70

12. Longitudinal data 158

12. Longitudinal data

1. Fixed occasions,

2. Variable occasions.

Table 12.1 Estimates for random intercept models

Model 1 Model 2

Fixed Eﬀect Coeﬃcient S.E. Coeﬃcient S.E.

µ0Mean at time 0 0.648 0.053 0.585 0.063

µ1Mean at time 1 0.648 0.053 0.718 0.067

µ2Mean at time 2 0.648 0.053 0.672 0.067

µ3Mean at time 3 0.648 0.053 0.639 0.070

Random eﬀect Parameter S.E. Parameter S.E.

Level two (i.e., individual) variance:

τ2

0=var(U0i)0.121 0.029 0.121 0.029

Level one (i.e., occasion) variance:

σ2=var(Rti)0.074 0.010 0.072 0.010

deviance 123.39 118.35

71

12. Longitudinal data 173

Table 12.2 Estimates for random slope models

Model 3 Model 4

Fixed Eﬀect Coeﬀ. S.E. Coeﬀ. S.E.

µ0Mean at time 0 0.574 0.076 0.610 0.090

µ1Mean at time 1 0.721 0.066 0.733 0.075

µ2Mean at time 2 0.665 0.057 0.655 0.065

µ3Mean at time 3 0.633 0.058 0.601 0.065

αMain eﬀect gender -0.133 0.169

γInt.act. gender ×experience 0.089 0.057

Random eﬀect Par. S.E. Par. S.E.

Level two variation:

τ2

0Intercept variance 0.233 0.056 0.239 0.057

τ2

1Slope variance 0.017 0.006 0.017 0.007

τ01 Intercept-slope covariance -0.052 0.017 -0.053 0.017

Level one (i.e., occasion) variance:

σ2Residual variance 0.048 0.009 0.047 0.009

deviance 102.66 99.91

72

12. Longitudinal data 176

Table 12.4 The empty multivariate model

for the teachers’ evaluations

Fixed Eﬀect Coeﬃcient S.E.

µ0Mean at time 0 0.574 0.079

µ1Mean at time 1 0.717 0.066

µ2Mean at time 2 0.650 0.052

µ3Mean at time 3 0.657 0.057

deviance 91.10

Estimated covariance matrix of the complete data vector:

c

Σ(Yc) =







0.303 0.173 0.111 0.112

0.173 0.196 0.098 0.118

0.111 0.098 0.120 0.109

0.112 0.118 0.109 0.141







73

12. Longitudinal data 177

Table 12.5 The multivariate model

with the teachers’ self-evaluation.

Fixed Eﬀect Coeﬃcient S.E.

µ0Eﬀect time 0 0.212 0.087

µ1Eﬀect time 1 0.375 0.073

µ2Eﬀect time 2 0.257 0.075

µ3Eﬀect time 3 0.288 0.070

αSelf-evaluation 0.469 0.070

Deviance 56.42

Estimated residual covariance matrix of the complete data

vector:

c

Σ(Yc) =







0.228 0.115 0.070 0.044

0.115 0.120 0.049 0.057

0.070 0.049 0.092 0.061

0.044 0.057 0.061 0.084







.

74

12. Longitudinal data 180

Table 12.6 Estimates for models

with self-evaluation.

Fixed Eﬀect Coeﬃcient S.E.

µ0Eﬀect time 0 0.188 0.082

µ1Eﬀect time 1 0.345 0.082

µ2Eﬀect time 2 0.232 0.090

µ3Eﬀect time 3 0.233 0.088

αEﬀect of self-evaluation 0.515 0.081

Random Eﬀect Parameter S.E.

Level-two (i.e., individual) variance:

τ2

0=var(U0i)0.069 0.019

Level-one (i.e., occasion) variance:

σ2=var(Rti)0.066 0.009

Deviance 84.26

75

12. Longitudinal data 183

Variable occasions:

Populations of curves.

Table 12.7 Linear growth model

for 5–10-year-old children with retarded growth.

Fixed Eﬀect Coeﬃcient S.E.

γ00 Intercept 96.32 0.285

γ10 Age 5.53 0.08

Random Eﬀect Parameter S.E.

Level-two (i.e., individual) random eﬀects:

τ2

0Intercept variance 19.79 1.91

τ2

1Slope variance for age 1.65 0.16

τ01 Intercept-slope covariance −3.26 0.46

Level-one (i.e., occasion) variance:

σ2Residual variance 0.82 0.03

Deviance 7099.87

76

12. Longitudinal data 185

Table 12.8 Cubic growth model

for 5–10-year-old children with retarded growth.

Fixed Eﬀect Coeﬃcient S.E.

γ00 Intercept 110.40 0.22

γ10 t−7.55.23 0.12

γ20 (t−7.5)2−0.007 0.038

γ30 (t−7.5)30.009 0.020

Random Eﬀect Variance S.E.

Level-two (i.e., individual) random eﬀects:

τ2

0Intercept variance 13.80 1.19

τ2

1Slope variance t−t02.97 0.32

τ2

2Slope variance (t−t0)20.255 0.032

τ2

3Slope variance (t−t0)30.066 0.009

Level-one (i.e., occasion) variance:

σ2Residual variance 0.37 0.02

Deviance 6603.75

77

12. Longitudinal data 185

Estimated correlation matrix for the level-two random slopes

(U0i, U1i, U2i, U3i):

d

RU=







1.0 0.17 −0.27 0.04

0.17 1.0 0.11 −0.84

−0.27 0.11 1.0−0.38

0.04 −0.84 −0.38 1.0







.

78

12. Longitudinal data 188

Table 12.9 Piecewise linear growth model

for 5–10-year-old children with retarded growth.

Fixed Eﬀect Coeﬃcient S.E.

γ00 Intercept 110.40 0.22

γ10 f1(5–6 years) 5.79 0.24

γ20 f2(6–7 years) 5.59 0.18

γ30 f3(7–8 years) 5.25 0.16

γ40 f4(8–9 years) 5.16 0.15

γ50 f5(9–10 years) 5.50 0.16

Random Eﬀect Variance S.E.

Level-two (i.e., individual) random eﬀects:

τ2

0Intercept variance 13.91 1.20

τ2

1Slope variance f13.97 0.82

τ2

2Slope variance f23.80 0.57

τ2

3Slope variance f33.64 0.50

τ2

4Slope variance f43.42 0.45

τ2

5Slope variance f53.77 0.53

Level-one (i.e., occasion) variance:

σ2Residual variance 0.302 0.015

Deviance 6481.87

79

12. Longitudinal data 188

Estimated correlation matrix

of the level-two random eﬀects (U0i, ..., U5i):

c

RU=







1.0 0.22 0.31 0.14 −0.05 0.09

0.22 1.0 0.23 0.01 0.18 0.33

0.31 0.01 1.0 0.12 −0.16 0.48

0.14 0.01 0.12 1.0 0.47 −0.23

−0.05 0.18 −0.16 0.47 1.0 0.03

0.09 0.33 0.48 −0.23 0.03 1.0







.

80

12. Longitudinal data 191

Table 12.10 Cubic spline growth model

for 12–17-year-old children with retarded growth.

Fixed Eﬀect Coeﬃcient S.E.

γ00 Intercept 150.00 0.42

γ10 f1(linear) 6.43 0.19

γ20 f2(quadratic) 0.25 0.13

γ30 f3(cubic left of 15) −0.038 0.030

γ40 f4(cubic right of 15) −0.529 0.096

Random Eﬀect Variance S.E.

Level-two (i.e., individual) random eﬀects:

τ2

0Intercept variance 52.07 4.46

τ2

1Slope variance f16.23 0.71

τ2

2Slope variance f22.59 0.34

τ2

3Slope variance f30.136 0.020

τ2

4Slope variance f40.824 0.159

Level-one (i.e., occasion) variance:

σ2Residual variance 0.288 0.014

Deviance 6999.06

81

12. Longitudinal data 191

Estimated correlation matrix of the level-two random eﬀects

(U0i, ..., U4i):

d

RU=







1.0 0.26 −0.31 0.32 0.01

0.26 1.0 0.45 −0.08 −0.82

−0.31 0.45 1.0−0.89 −0.71

0.32 −0.08 −0.89 1.0 0.40

0.01 −0.82 −0.71 0.40 1.0







.

82

12. Longitudinal data 193

t(age in years)

length

120

130

140

150

160

170

12 13 14 15 16 17

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

.

...

.

....

.

..

.

Figure 12.1 Average growth curve (∗) and 15 random

growth curves for 12–17-year-olds for cubic spline model.

83

12. Longitudinal data 194

Table 12.11 Growth variability

explained by gender and parents’ length.

Fixed Eﬀect Coeﬃcient S.E.

γ00 Intercept 150.20 0.47

γ10 f1(linear) 5.85 0.18

γ20 f2(quadratic) 0.053 0.124

γ30 f3(cubic left of 15) −0.029 0.030

γ40 f4(cubic right of 15) −0.553 0.094

γ01 Gender −0.385 0.426

γ11 f1×gender −1.266 0.116

γ21 f2×gender −0.362 0.037

γ02 Parents’ length 0.263 0.071

γ12 f1×parents’ length 0.0307 0.0152

Random Eﬀect Variance S.E.

Level-two (i.e., individual) random eﬀects:

τ2

0Intercept variance 49.71 4.31

τ2

1Slope variance f14.52 0.52

τ2

2Slope variance f22.37 0.33

τ2

3Slope variance f30.132 0.020

τ2

4Slope variance f40.860 0.156

Level-one (i.e., occasion) variance:

σ2Residual variance 0.288 0.013

Deviance 6885.18

84

12. Longitudinal data 194

Estimated correlation matrix of the level-two random eﬀects

(U0i, ..., U4i):

d

RU=







1.0 0.22 −0.38 0.35 0.07

0.22 1.0 0.38 −0.05 −0.81

−0.38 0.38 1.0−0.91 −0.75

0.35 −0.05 −0.91 1.0 0.48

0.07 −0.81 −0.75 0.48 1.0







.

85

12. Longitudinal data 197

Table 12.12 Estimates for two models

for cortisol data.

Model 1 Model 2

Fixed Eﬀect Coeﬃcient S.E. Coeﬃcient S.E.

γ00 Intercept 3.21 0.33 3.16 0.34

γ10 Gestational age −0.201 0.117 −0.185 0.118

γ01 X−2.80 0.358 0.045 0.532 0.096

γ02 Z0.009 0.117

γ03 Z×(X−2.80) −0.224 0.108

Random Eﬀect Parameter S.E. Parameter S.E.

Level-two (i.e., individual) random eﬀects:

τ2

0Interc. var. 1.25 0.66 1.32 0.68

τ2

1Slope var. 0.151 0.082 0.155 0.082

τ01 Interc.-sl. cov. −0.43 0.23 −0.45 0.23

Level-one (i.e., occasion) variance:

σ20.175 0.018 0.172 0.018

Deviance 280.57 276.42

86

12. Longitudinal data 197

Table 12.13 Estimates for a model

controlling for having been asleep.

Model 3

Fixed Eﬀect Coeﬃcient S.E.

γ00 Intercept 3.04 0.31

γ10 Gestational age −0.142 0.099

γ01 X−2.80 0.548 0.084

γ02 Z−0.058 0.118

γ03 Z×(X−2.80) −0.136 0.096

γ04 Sleeping 0.358 0.117

Random Eﬀect Parameter S.E.

Level-two (i.e., individual) random eﬀects:

τ2

0Intercept variance 0.869 0.496

τ2

1Slope variance 0.092 0.057

τ01 Intercept-slope covariance −0.28 0.17

Level-one (i.e., occasion) variance parameters:

σ2

0Basic residual variance 0.130 0.015

σ01 Sleeping eﬀect 0.116 0.045

Deviance 251.31

87

13. Multivariate multilevel models 175

13. Multivariate Multilevel Models

Incomplete paired data

The simplest multilevel example:

combination of a paired and an unpaired t-test.

Yhi is measurement hfor subject i;

for some subjects, only 1 measurement available,

for others 2.

The model is

Yhi =γ0+γ1d1hi +Uhi

where d1hi = 1 for h= 1, 0 otherwise.

Null hypothesis is ‘γ1= 0’.

88

13. Multivariate multilevel models 175

Table 12.3 Estimates for incomplete paired data.

Fixed Eﬀect Coeﬃcient S.E.

γ0Constant term 0.577 0.081

γ1Eﬀect time 1 0.151 0.064

Deviance 95.05

Estimated covariance matrix of complete data vector:

c

Σ(Yc) = 





0.309 0.166

0.166 0.184 



.

t-test for γ1:t= 0.151/0.064 = 2.36, p < 0.05.

This is an example where a 2-level model is used

for multivariate 1-level data.

Now continue with multivariate 2-level data.

89

13. Multivariate multilevel models 199–202

Why the extra complexity of multivariate models?

1. Inference about correlations:

partition covariance between levels.

2. More powerful tests (use all data).

Gain mostly is small; larger in case of high correlations and

incomplete data.

3. Test hypotheses about several dependent variables

simultaneously.

4. One multivariate test instead of several ’univariate’ tests.

Dependent variable is

Yhij measurement of variable h

for individual iin group j ,

complete data vector

Yc=







Y1ij

.

Ymij







.

90

13. Multivariate multilevel models 199–202

Multivariate random intercept model

Yhij =γ0h+γ1hx1ij +γ2hx2ij +...

+γph xpij +Uhj +Rhij

where level-1 and level-2 residuals are

components of vectors

Rij =







R1ij

.

Rmij







,Uj=







U1j

.

Umj







with residual covariance matrices

Σ = cov(Rij),T = cov(Uj).

Total residual covariance matrix is

var(Yc) = Σ + T .

91

13. Multivariate multilevel models 199–202

Random intercept model is represented as a

3-level model using dummy variables

dshij =









1h=s ,

0h6=s

with observations at level 1,

’individuals’ at level 2, ’groups’ at level 3:

Yhij =m

X

s=1 γ0sdshij +p

X

k=1

m

X

s=1 γks dshij xkij +

m

X

s=1 Usj dshij +m

X

s=1 Rsij dshij .

Now Rsij are random slopes at level 2

and Usj are random slopes at level 3.

Random part at level 1 is empty.

Multivariate empty model: no Xvariables.

The dependent variables Yhmay have

diﬀerent explanatory variables.

92

13. Multivariate multilevel models 204–205

Example: Language and arithmetic scores.

m= 2 dependent variables.

Table 13.1 Parameter estimates for multivariate empty

model.

Language Arithmetic (Covariance)

h= 1 h= 2

Fixed Eﬀect Par. S.E. Par. S.E.

γ0h= Intercept 40.34 0.42 18.93 0.34

Random Eﬀect Par. S.E. Par. S.E. Par. S.E.

Between-schools covariance matrix:

τ2

h=var(Uhj)19.00 2.86 12.76 1.84

τ12 =cov(U1j, U2j)14.54 2.17

Within-schools covariance matrix:

σ2

h=var(Rhij)64.64 1.97 32.25 0.98

σ12 =cov(R1ij, R2ij )28.62 1.16

Deviance 29674.1

93

13. Multivariate multilevel models 204–205

Population correlation coeﬃcients

ρ(U1j, U2j) = 14.54

√12.76 ×19.00 = 0.93 ,

ρ(R1ij, R2ij ) = 28.62

√32.25 ×64.64 = 0.63 .

Correlations between observed variables:

ˆρ(Y1ij, Y2ij ) = 14.54 + 28.62

r(19.00 + 64.64)(12.76 + 32.25)

= 0.70

and between group means (for n= 30)

ˆρ(¯

Y1.j,¯

Y2.j) = 14.54 + 28.62/30

r(19.00 + 64.64/30)(12.76 + 32.25/30)

= 0.90 .

94

13. Multivariate multilevel models 204–205

Table 13.2 Parameter estimates

for multivariate model

for language and arithmetic tests.

Language Arithmetic (Covariance)

h= 1 h= 2

Fixed Eﬀect Par. S.E. Par. S.E.

γ0hIntercept 40.84 0.29 19.33 0.25

γ1hIQ 2.422 0.072 1.464 0.054

γ2hIQ (Group mean) 1.33 0.32 1.21 0.27

γ3hGroup size 0.046 0.036 0.044 0.031

γ4hIQ ×Group size −0.021 0.010 −0.016 0.007

Random Eﬀect Par. S.E. Par. S.E. Par. S.E.

Residual between-schools covariance matrix:

τ2

h=var(Uhj)7.36 1.24 6.06 0.94

τ12 =cov(U1j, U2j)5.70 0.97

Residual within-schools covariance matrix:

σ2

h=var(Rhij)42.12 1.28 24.07 0.73

σ12 =cov(R1ij, R2ij )15.04 0.76

Deviance 28540.7

ρ(U1j, U2j)= 0.85, ρ(R1ij, R2ij )= 0.47.

95

13. Multivariate multilevel models 206

Multivariate random slope model

Yhij =γ0h+γ1hx1ij +... +γph xpij

+U0hj +U1hj x1hj +Rhij .

Three-level formulation:

Yhij =m

X

s=1 γ0sdshij +p

X

k=1

m

X

s=1 γks dshij xkij +

m

X

s=1 U0sj dshij +m

X

s=1 U1sj dshij x1ij

+m

X

s=1 Rsij dshij .

Same principle as above.

Lots of random slopes...

96

14. Discrete dependent variables 211–212

14. Discrete dependent variables

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 14.1 Proportion of experience with cohabitation.

Diﬀerence between proportions:

X2= 35.40, d.f. = 18, p < 0.01.

ˆτ=√0.0022 = 0.047.

0.050.12 0.27 0.50 0.73 0.880.95

p

−4−3−2−101234

logit(p)

.

Figure 14.3 Correspondence between pand logit(p).

97

14. Discrete dependent variables 213–215

Empty multilevel logistic regression model:

logit(Pj) = γ0+U0j.

Table 14.1 Estimates for empty multilevel logistic model

Fixed Eﬀect Coeﬃcient S.E.

γ0= Intercept -0.276 0.062

Random eﬀect Var. Comp. S.E.

Level two variance:

τ2

0=var(U0j)0.032 0.023

−2−10 1 2

logit(p)

.

Figure 14.5 Observed log-odds and estimated normal

distribution of population log-odds of cohabitation experience.

98

14. Discrete dependent variables 216–217

The random intercept model for binary data:

logit(Pij) = γ0+r

X

h=1

γhxhij +U0j.

Table 14.1’ Estimates for multilevel logistic model with

eﬀect of religion

Model 1 Model 2

Fixed Eﬀect Coeﬀ. S.E. Coeﬀ. S.E.

γ0= Intercept -0.276 0.062 -0.149 0.060

γ1= Eﬀect Religion -1.791 0.223

Random eﬀect Var. Comp. S.E. Var. Comp. S.E.

Level two variance:

τ2

0=var(U0j)0.032 0.023 0.024 0.021

99

14. Discrete dependent variables 216–217

Table 14.1” Estimates for single level logistic model with

eﬀect of religion

Model 1 Model 2

Fixed Eﬀect Coeﬀ. S.E. Coeﬀ. S.E.

γ0= Intercept -0.266 0.044 -0.139 0.046

γ1= Eﬀect Religion -1.807 0.223

Table 14.2” Estimates for single level logistic model for

cohabitation data

Fixed Eﬀects Coeﬃcient S.E.

γ0Intercept -1.111 0.144

γ1Age - 20 0.546 0.052

γ2(Age - 20)2-0.0292 0.0038

γ3(Age - 30)2for Age >30 0.0239 0.0055

γ4(Age - 40)2for Age >40 0.0073 0.0025

γ5Religion -1.89 0.24

100

14. Discrete dependent variables 217

Table 14.2 Estimates for two models for cohabitation data

Model 1 Model 2

Fixed Eﬀects Coeﬀ. S.E. Coeﬀ. S.E.

γ0Intercept -1.200 0.154 -1.107 0.153

γ1Age - 20 0.539 0.051 0.544 0.052

γ2(Age - 20)2-0.0290 0.0037 -0.0289 0.0038

γ3(Age - 30)2

+0.0241 0.0054 0.0235 0.0055

γ4(Age - 40)2

+0.0068 0.0025 0.0075 0.0025

γ5Religion -1.85 0.24

Random eﬀects Var. Comp. S.E. Var. Comp. S.E.

random intercept:

τ2

0=var(U0j)0.061 0.037 0.049 0.034

101

14. Discrete dependent variables 223–227

Estimated level-two intercept variance may go up

when level-1 variables are added

and always does

when these have no between-group variance.

This can be understood by threshold model

which is equivalent to logistic regression:

Y=









0if ˘

Y≤0

1if ˘

Y > 0,

where ˘

Yis a latent continuous variable

˘

Yij =γ0+r

X

h=1

γhxhij +U0j+Rij

and Rij has a logistic distribution,

with variance π2/3.

The fact that the latent level-1 variance is ﬁxed

implies that explanation of level-1 variation

by a new variable Xr+1

will be reﬂected by increase of γh(0≤h≤r)

and of var(U0j).

102

14. Discrete dependent variables 223–227

Measure of explained variance (‘R2’)

for multilevel logistic regression can be based

on this threshold representation, as the

proportion of explained variance in the latent variable.

Because of the arbitrary ﬁxation of σ2

Rto π2/3,

these calculations must be based on one single model ﬁt.

Let

ˆ

Yij =γ0+r

X

h=1

γhxhij

be the latent linear predictor; then

˘

Yij =ˆ

Yij +U0j+Rij .

Calculate ˆ

Yij (using estimated coeﬃcients) and then

σ2

F=var ˆ

Yij

in the standard way from the data; then

var ˘

Yij=σ2

F+τ2

0+σ2

R

where σ2

R=π2/3=3.29.

103

14. Discrete dependent variables 223–227

The proportion of explained variance now is

R2

dicho =σ2

F

σ2

F+τ2

0+σ2

R

.

Of the unexplained variance, the fraction

τ2

0

σ2

F+τ2

0+σ2

R

is at level 2, and the fraction

σ2

R

σ2

F+τ2

0+σ2

R

is at level one.

104

14. Discrete dependent variables 223–227

Table 14.4 Estimates for probability to take a science

subject.

Model 1

Fixed Eﬀect Coeﬃcient S.E.

γ0Intercept 2.487 0.110

γ1Gender –1.515 0.102

γ2Minority status –0.727 0.195

Random Eﬀect Var. Comp. S.E.

Level-two variance:

τ2

0=var(U0j)0.481 0.082

Deviance 3238.27

Linear predictor

ˆ

Yij = 2.487 −1.515 genderij −0.727 minorityij

has variance

ˆσ2

F= 0.582 .

Therefore

R2

dicho =0.582

0.582 + 0.481 + 3.29 = 0.13 .

105

14. Discrete dependent variables 221–222

Random slopes can be added in the usual way

to the multilevel logistic model.

The following is a model for the logit with one random slope:

logit(Pij) = γ0+r

X

h=1

γhxhij +U0j+U1jx1ij .

Testing based on deviances cannot be done from MLwiN,

because it does not give good approximations to the deviance

for this type of non-linear model.

MIXOR does provide such a deviance,

and was used for the following example.

106

14. Discrete dependent variables 221–222

Table 14.3 Parameter estimates for two models

for distrust relations.

Model 1 Model 2

Fixed Eﬀect Par. S.E. Par. S.E.

γ0Intercept −1.92 0.13 −2.96 0.24

Political variables:

γ1Party member ego 0.30 0.24 0.26 0.25

γ2Political function alter −0.40 0.93 −0.15 0.78

γ3Party member ×pol. function −1.09 1.07

Role relations:

γ4Colleague 1.19 0.21

γ5Superior 1.34 0.24

γ6Subordinate −0.22 0.90

γ7Neighbor 2.31 0.31

Random Eﬀect Par. S.E. Par. S.E.

Random intercept:

τ2

0=var(U0j)intercept variance 1.55 0.37 1.68 0.44

Random slope:

τ2

2=var(U2j)slope variance pol. function 5.76 7.43 5.54 6.27

τ04 =cov(U0j, U2j)intercept-slope cov. −0.06 1.02 0.22 1.04

Deviance 1528.29 1519.95

107

14. Discrete dependent variables 231

Multicategory ordinal logistic regression

‘Measurement model’:

Y=











0if ˘

Y≤θ0

1if θ0<˘

Y≤θ1,

kif θk−1<˘

Y≤θk(k= 2, ..., c −2),

c−1if θc−2<˘

Y .

‘Structural model’:

˘

Yij =γ0+r

X

h=1 γhxhij +U0j+Rij .

108

14. Discrete dependent variables 233

Table 14.6 Multilevel 4-category logistic regression model

number of science subjects

Model 1 Model 2

Threshold parameters Threshold S.E. Threshold S.E.

θ1Threshold 1 - 2 1.541 0.041 1.763 0.045

θ2Threshold 2 - 3 2.784 0.046 3.211 0.054

Fixed Eﬀects Coeﬃcient S.E. Coeﬃcient S.E.

γ0Intercept 1.370 0.057 2.591 0.079

γ1Gender girls -1.680 0.066

γ2SES 0.117 0.037

γ3Minority status -0.514 0.156

Level two random eﬀect Parameter S.E. Parameter S.E.

τ2

0Intercept variance 0.243 0.034 0.293 0.040

deviance 9308.8 8658.2

109

14. Discrete dependent variables 234–237

Multilevel Poisson regression

ln(Lij) = γ0+r

X

h=1

γhxhij +U0j.

Table 14.5 Two Poisson models for

number of calls outside oﬃce hours.

Model 1 Model 2

Fixed Eﬀect Coeﬃcient S.E. Coeﬃcient S.E.

γ0Intercept 0.062 0.041 0.049 0.048

γ1Weekend 1.297 0.056

γ2Saturdays 1.381 0.065

γ3Sundays 1.199 0.068

γ4June −0.197 0.104

γ5July −0.170 0.104

γ6August 0.147 0.087

γ7September 0.174 0.089

γ8December 0.215 0.088

Level-two Random Eﬀect Variance S.E. Variance S.E.

τ2

0Intercept variance 0.028 0.007 0.012 0.004

Deviance 2293.87 2272.64

110

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