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MULTILEVEL ANALYSIS. Kate Pickett Senior Lecturer in Epidemiology. SUMBER: www-users.york.ac.uk/.../Multilevel%20 Analysis . ppt University of York. Perspective. Health researchers: Are interested in answering research questions (not maths) Want to be able to apply statistical techniques - PowerPoint PPT Presentation
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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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
Population A Population B
How is Population A different from Population B?
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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.
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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.
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
Example data
Cholesterol Dataset• 441 patients• Age 44-86 years• Cholesterol 3.90-8.86
mmol/l• 12 doctors
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
Back to unemployment example
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
Population A
Population B
Red = unemployed
Data Structure
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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)
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
Population A
Population B
Aside from unemployment, subjects in A are different fromB in other ways: composition (shape, size), context (density)
Data Structure
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
A multi-level regression model
i = individual, j=context:
yij = bxij + BXi + Ej + eij
Health = b (unemployedij) + B(% unemployedi) +Ej + eij
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York
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
SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.pptUniversity of York