Multilevel Analysis

Kate Pickett

Senior Lecturer in Epidemiology

Perspective

Health researchers:Are interested in answering research

questions (not maths)Want to be able to apply statistical

techniquesWant to be able to interpret resultsWant to be able to communicate with

consumers and statisticians

Aims for this session

Understand the rationale for multilevel analysis

Understand common terminology Interpret output from multilevel models Be able to read and critically appraise

studies using multilevel models

Context and composition

Studying populations (groups) and individuals

From Rose, G. Sick individuals and sick populations. Int J Epidemiol 1985;14:32-38

Levels of analysis

Health researchers may collect and use data collected at the level of:Individuals, patientsFamilies or other social groupingsClinics or hospitalsSmall areas, neighbourhoodsLarge populations

Population A Population B

How is Population A different from Population B?

Ecological studies

Data are aggregated and represent a group, rather than an individual incidence rate of an illness prevalence of a particular health service

We don’t know which particular individuals within the group were ill or received the service

These group-based outcome measures are analyzed by correlating them with determinants measured for the same groups

Source: Pickett KE, Kelly S, Brunner E, Lobstein T, Wilkinson RG. Wider income gaps, wider waistbands? An ecological studyof obesity and income inequality. J Epidemiol Community Health 2005;59:670–674.

The ecological fallacy

Associations at the group level may not hold at an individual level Eg, we might see that rates of obesity are correlated

internationally with per capita calorie intake But, we don’t know if it is the obese individuals who

are eating all the calories Many group-level variables are correlated so we may get

spurious correlations Eg, obesity rates may also be correlated with number

of zoos per capita or some other completely unrelated factor

The atomistic fallacy

But the ecological fallacy has a flip sideFactors that affect outcomes in individuals

may not operate in the same way at the population level

• Eg, teenage births are more common among the poor, but teenage birth rates are very high in some very wealthy countries.

Source: Pickett KE, Mookherjee S, Wilkinson RG. Adolescent Birth Rates,Total Homicides, and Income Inequality In Rich Countries, AJPH2005;95:1181-1183.

Example of teenage births

Ecological variables

Sometimes ecological studies are done because it is quick and easy

Sometimes ecological studies are the best design for the research questionBECAUSE

Some determinants are “ecological”: Population density Air quality/pollution GNP Income inequality % unemployed Ambient temperature

Context and composition

But what if we are interested in both types of variables (individual and population) simultaneously?

Eg: we might want to know about the effect of population-level unemployment on health, above and beyond the health impact of being unemployed for any given individual

Multilevel models

Introduction to multilevel models

Number of papers using multilevel

analysis: Medline

0

50

100

150

200

1995 2000 2004

Year

Nu

mb

er

Hierarchical models

Mixed effects models

Random effects models

Background

Developed in education research

Observations of students in a single class are not independent of one another

“Standard” statistical models assume that observations are independent

Two-level hierarchy Students within

classes Three-level hierarchy

Students within classes within schools

Four-level hierarchy Students within

classes within schools within local authority areas

Health research context

Patients within a medical practice Residents within neighbourhoods Subjects within trial clusters Hospitals within PCTs….

Examples for class

Some examples are drawn from Twisk JWR “Applied Multilevel Analysis” Cambridge University Press, 2006

Example data are available at: http:\www.emgo.nl\researchtools

Research question: what is the relationship between total cholesterol and age?

Statistical software: Stata but note that MLwiN is free to UK academics: http://www.cmm.bristol.ac.uk/MLwiN/download/index.shtml)

Simple linear regression

4

5

6

7

8

30 40 50 60 70

Age (years)

To

tal c

ho

les

tero

l (m

mo

l/l)

Total cholesterol = β0 + β1 x age + ε

Simple linear regression, adding a categorical variable

4

5

6

7

8

30 40 50 60 70

Age (years)

To

tal c

ho

les

tero

l (m

mo

l/l)

MalesFemales

Total cholesterol = β0 + β1 x age + β2 x gender + ε

Simple linear regression, adding another variable (doctor)

4

5

6

7

8

30 40 50 60 70

Age (years)

To

tal c

ho

les

tero

l (m

mo

l/l)

MD1MD2MD3MD4MD5MD…

Total cholesterol = β0 + β1 x age + β2 x MD1 + β3 x MD2 + β4 x MD3 + β5 x MD4 +…..+ βm x MDm-1 + ε

Multilevel analysis

Instead of estimating all those separate intercepts, we estimate the variance of them

In our example that means estimating 1 additional parameter, rather than 11

We are allowing the intercept to be random (random effects modelling)

An efficient way of correcting for a variable with many categories

Trade-off: Assumes that the different intercepts are

normally distributed

Example data

Cholesterol Dataset 441 patients Age 44-86 years Cholesterol 3.90-

8.86 mmol/l 12 doctors

Non-multilevel regression

. regress cholesterol age

Source | SS df MS Number of obs = 441-------------+------------------------------ F( 1, 439) = 142.06 Model | 99.3395851 1 99.3395851 Prob > F = 0.0000 Residual | 306.984057 439 .699280312 R-squared = 0.2445-------------+------------------------------ Adj R-squared = 0.2428 Total | 406.323642 440 .923462822 Root MSE = .83623

------------------------------------------------------------------------------ cholesterol | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | .0512619 .0043009 11.92 0.000 .042809 .0597148 _cons | 2.798691 .268571 10.42 0.000 2.270847 3.326536------------------------------------------------------------------------------

Example using Stata

MultilevelModel inStata

. xtmixed cholesterol age ||doctor:, ml var

Performing EM optimization:

Performing gradient-based optimization:

Iteration 0: log likelihood = -404.68939 Iteration 1: log likelihood = -404.68939

Computing standard errors:

Mixed-effects ML regression Number of obs = 441Group variable: doctor Number of groups = 12

Obs per group: min = 36 avg = 36.8 max = 39

Wald chi2(1) = 262.76Log likelihood = -404.68939 Prob > chi2 = 0.0000

------------------------------------------------------------------------------ cholesterol | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | .0495866 .003059 16.21 0.000 .0435911 .0555822 _cons | 2.905812 .259134 11.21 0.000 2.397919 3.413705------------------------------------------------------------------------------

------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]-----------------------------+------------------------------------------------doctor: Identity | var(_cons) | .3685781 .1541985 .1623381 .8368327-----------------------------+------------------------------------------------ var(Residual) | .3314923 .0226341 .2899706 .3789597------------------------------------------------------------------------------LR test vs. linear regression: chibar2(01) = 282.37 Prob >= chibar2 = 0.0000

Do we need the multilevel model?

Likelihood ratio test:Compare -2 log likelihood of model

with random intercept to -2 log likelihood of ordinary linear model

Difference has a Chi-square distribution with df = difference in number of parameters estimated

Difference = 284.73, highly significant

Model parameters

Effects of age in each model: Coefficient in ordinary model = 0.0513 Coefficient in multilevel model = 0.0496

95% CI in ordinary model (0.0428, 0.0597) 95% CI in multilevel model (0.0435,0.0556)

Age is significant in both models

Intraclass correlation coefficient This measures how dependent the

observations are within clusters Eg, how correlated are the observations of

patients belonging to the same doctor? Defined as:

Variance between clusters/Total variance The smaller the variance within clusters, the

greater the ICC

ICC (a)

Distribution of an outcome variable

Assume that the total variance = 10

ICC (b)

ICC is low because:

Variance within groups is high (9)

Variance between groups is low (1)

Numerator is small, relative to denominatorICC = 1/10=0.1

ICC (c)

The groups are now more spread out, more different, and:

ICC is bigger because:

Variance within groups is lower (5)

Variance between groups is higher (5)

ICC=5/10 = 0.5

ICC (d)

The groups are now completely different, and:

ICC is maximised because:

Variance within groups is minimal (1)

Variance between groups is maximal (9)

Numerator is large, relative to denominator

ICC=9/10 = 0.9

MUCH MORE DEPENDENCE WITHIN CLUSTER – each observation provides less unique information

Impact on significance tests

Table of alpha values under different conditions of sample size and ICC

 Intraclass Correlation Coefficient

Sample size 0.01 0.05 0.20

10 0.06 0.11 0.28

25 0.08 0.19 0.46

50 0.11 0.30 0.59

100 0.17 0.43 0.70

ICC in our example

ICC = between doctor variance/total variance

ICC = 0.3686/(0.3686+0.3315) = 0.3686/0.7001 = 0.52652.6% of the total individual

differences in cholesterol are at the doctor level

ICC

When ICC is highEvidence of a contextual effect on the

outcomeEvidence of differences in composition

between the clustersExplore by including explanatory

variables at each level When ICC is low

No need for a multilevel analysis

Back to unemployment example

Population A

Population B

Red = unemployed

Data Structure

An ordinary regression model

Health =b0 + b1 (unemployed) + b2 (% unemployed) + e

e represents the effect of all omitted variables and measurement error and is assumed to have a random effect (so it gets ignored)

Population A

Population B

Aside from unemployment, subjects in A are different fromB in other ways: composition (shape, size), context (density)

Data Structure

A multi-level regression model

i = individual, j=context:

yij = bxij + BXi + Ej + eij

Health = b (unemployedij) + B(% unemployedi) +Ej

+ eij

What does this mean for critical appraisal of the health literature?

When data are hierarchical or multi-level by nature, they should be analysed appropriately

The coefficients or odds ratios from the models can be interpreted as usual

The ICC shows how much variance in the outcome occurs between the higher-level contexts

If appropriate methods are not used, standard errors and significance tests may be wrong and coefficients biased

A summary

Ecological studies Appropriate when the research question concerns

only ecological effects Ecological fallacy may be a problem

Individual-level studies Appropriate when the research question concerns

only individual-level effects Atomistic fallacy may be a problem

Multi-level studies Appropriate when the research question concerns

both context and composition of populations