MULTILEVEL ANALYSIS

MULTILEVEL ANALYSIS

Kate PickettSenior Lecturer in Epidemiology

SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York

Perspective

• Health researchers:– Are interested in answering research questions

(not maths)– Want to be able to apply statistical techniques– Want to be able to interpret results– Want 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;14:32-38SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York

Levels of analysis

• Health researchers may collect and use data collected at the level of:– Individuals, patients– Families or other social groupings– Clinics or hospitals– Small areas, neighbourhoods– Large 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 side– Factors 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

050

100150200

1995 2000 2004

Year

Num

ber

• 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 70Age (years)

Tota

l cho

lest

erol

(mm

ol/l)

Total cholesterol = β0 + β1 x age + ε

Simple linear regression, adding a categorical variable

4

5

6

7

8

30 40 50 60 70Age (years)

Tota

l cho

lest

erol

(mm

ol/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 70Age (years)

Tota

l cho

lest

erol

(mm

ol/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 StataSUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York

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 0.2010 0.06 0.11 0.28

25 0.08 0.19 0.46

50 0.11 0.30 0.59100 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 high– Evidence of a contextual effect on the outcome– Evidence of differences in composition between

the clusters– Explore 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


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MULTILEVEL ANALYSIS

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