Clustered or Multilevel Data

1
Clustered or Multilevel Data

• What are clustered or multilevel data?
• Why are multilevel data common in outcomes
  research?
• What methods of analysis are available?
• What are random versus fixed effects?
• How does the N at each level affect model choice?
  
• How does the study question affect model choice?

2
What are clustered data?

• Gathering individual observations into larger
  groups does not create clustered data
• Individual observations from a simple, random
  sample are never clustered
• Clustering is a result of sampling/design
• Usually from stages/levels in obtaining the
  individual units of observation

3
Examples of Clustered Data

• Litters of puppies
• Pieces of leaves (several per leaf)
• Intervention on institutions (eg, schools)
• TB cases and their contacts
• Survey stratified by county and census tract
• A sample of physicians and their patients
• Repeated measurements on individuals

4
Clustered or Multilevel Data
Level 2 (cluster)
Physicians, schools, census tracts, leaves
Level 2 unit 3
Level 2 unit 2
Level 2 unit 1
Level 1 (individual observation)
Patients, students, residents, leaf samples
5
Cluster analysis is a different topic finds
clusters in data
x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x
x x x x x x x x
6
Repeated Measures are also a Type of Clustered or
Multilevel Data
Individual subjects
Level 2 (cluster)
Person 3
Person 2
Person 1
4,3
2,1
3,1
1,2
2,2
1,3
2,3
3,3
1,1
Time 1
Level 1 (individual observation)
Observations at different times
7
Multilevel Data is Common in Outcomes Research

• Secondary data sets are often multilevel
• Patients clustered within physicians clustered
  within hospitals or clinics (hospital discharges)
• National health surveys (NHIS, NHANES) are
  stratified probability surveys
• Health interventions often randomize institutions
  or geographic areas
• Health policy changes are applied at geographic
  or institutional level

8
Characteristics of Clustered Data

• Measurements within clusters are correlated (eg,
  measures on same person are more alike than
  measurements across persons)
• Variables can be measured at each level
• The variance of the outcome can be attributed to
  each level
• Standard statistical models and tests are
  incorrect

9
Effects of Clustered Data

• The assumptions of independence and equal
  variance of standard statistics do not hold
• Standard errors for statistical testing will be
  incorrect
• Regression models cannot be fit using methods
  that assume independence of observations
• For example, ordinary least squares calculation
  of the regression line is incorrect

10
Example of Multilevel Data with a Linear Outcome
Variable

• PORT study of type II diabetes patients
  satisfaction with medical care
• Outcome score from 14 questionnaire items
• Sample of 70 physicians (level 2 sample)
• Sample of 1492 patients (level 1 sample)
• Mean 21.3 patients per physician
• Range from 5 to 45 patients per physician
• Two levels of covariates considered
• Physician years in practice, specialty (level 2)
• Patient age, gender (level 1)

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Clustered/Multi-level Data VarianceOutcome
Patient Satisfaction Score
Level 2 Physicians (N70)
MD3 mean74
MD2 mean58
MD1 mean81
79
55
61
68
74
75
85
77
81
Level 1 Patients (N1492)
Variance in the patient score divides into two
parts (1) the variance between physicans
?2B (2) the variance within the physicians
?2W So the total variance ?2B ?2W
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Intraclass Correlation Coefficient
The intraclass correlation coefficient (ICC) is a
measure of the correlation among the individual
observations within the clusters
It is calculated by the ratio of the between
cluster variance to the total variance
?2B / (?2B ?2W )
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Intraclass Correlation Coefficient (ICC)
MD3 mean74
MD2 mean58
MD1 mean81
74
58
58
74
74
74
81
81
81
Take extreme case where each MDs patients
have the same score no variance within the
physicians. So, ICC ?2B / ?2B ?2W ?2B /
?2B 0 1 perfect correlation within the
clusters.
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Methods of Analyzing Multilevel Data

• Use a single measure per cluster (e.g., mean
  satisfactions score) as the outcome variable
• Fit a model with indicator variables for each
  cluster (minus one)
• Fit a regression model with generalized
  estimating equations (GEE model)
• Fit a fixed effects conditional regression model
• Fit a random effects regression model

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Choice of Analysis Model Two Main Considerations

• What is the research question
• How many observations are there at level 2 and
  how many level 1 observations are there per level
  2 observation

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Choice of Analysis Model The Research Question

• What is the relationship of patient age to the MD
  satisfaction score? (level 1 predictor)
• What is the relationship between MD years in
  practice and the score? (level 2 predictor)
• How much variation is there in the mean
  satisfaction score between MDs adjusted for level
  1 and level 2 predictors? (level 2 variance)

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Method (1) Use mean satisfaction score for each
physician as outcome

• Single measure for each cluster
• simple, easy to understand
• loses information, power (N70, not 1492)
• ignores different variance of single outcome if
  clusters are different sizes
• no individual level variables except as mean
  values (eg, mean patient age)
• Only answers question 2 (MD years in practice)
  although can use mean patient age

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Method (2) Use dummy variable for each MD

• Dummy variable represents each MD effect
• treats each MD effect as equally well estimated
  but some of the clusters small (N5,7,8, etc.)
• If we had 70 MDs and only 200 patients, 69 dummy
  variables would use up too many degrees of
  freedom
• If we had only 10 MDs, it is a good choice
• Can only answer question 1 (relationship of
  patient age to satisfaction score)

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Method (3)Regression with Generalized
Estimating Equations (GEE)

• Estimates regression coefficients and variance
  separately to account for clustering
• Gives population average effect of age on
  satisfaction (marginal model)
• Analyst indicates correlation structure within
  the clusters
• Answers questions 1 and 2 but not 3
• Variation in patient satisfaction between MDs is
  not modeled separately

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Specifying Correlation within Clusters for GEE
Model

• Most common assumption is one correlation
  coefficient for all pairs of observations within
  the clusters called compound symmetry or
  exchangeable correlation structure
• Other assumptions about the correlation are
  possible (eg, correlation weakens with
  time/distance)
• The GEE regression will give good estimate of
  predictor coefficients even if the correlation
  specified is incorrect if you use the robust ses

21
Method (4) Use Conditional Regression Model with
Fixed Effects

• Looks within each MD to model the association
  between patient age and the score
• No coefficient for MD (conditioned out)
• Good choice if number of MDs large relative to
  number of patients (70 MDs, 200 patients)
• Matched pairs are analyzed with conditional
  regression
• Answers question 1, but not 2 and 3

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Method (5) Use a Random Effects Regression Model

• Predictor variables for both individual and
  cluster level variables
• Models variance associated with MD separately
  from variance within the clusters in patient
  satisfaction
• Improves estimate of MD effect by treating MD
  mean scores as random sample of scores
• Only model that answers all 3 questions

23
Fixed versus Random Effects

• Effects are random when the levels are a sample
  of a larger population
• have variation because sampled another sample
  would give different data
• Effects are fixed if they represent all possible
  levels/members of a population
• eg, male/female treatment groups all the
  regions of the U.S.

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Fixed versus Random Effects

• Effects can often be considered fixed or random
  depending on the research question
• If you want to generalize from the sample of
  doctors to other doctors, you would consider the
  doctors as a random effect
• If the doctors in your sample are the only ones
  you care about, you could consider doctors as a
  fixed effect

25
Random Effects Illustrated from the PORT Diabetes
Study

• In the MD satisfaction score example, begin by
  ignoring predictors such as the patients age and
  the physicians number of years in practice
• The overall mean patient satisfaction score for
  all 1492 patients was 67.7 (SD23.5)
• Separate means calculated for each physicians
  patients ranged from 53.4 to 87.1

26
Random Effects MD Score

• Consider the satisfaction score as composed of
  two parts the overall mean (?) plus or minus the
  difference from that overall mean of the mean
  score for each physician (?j)
• Each MDs difference, ?j, is a random effect
  because the 70 MDs represent a sample of
  possible MDs.
• If we sampled another 70, the ?js would be
  different

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A Simple Random Effects Model

• If we add a term for error associated with each
  individual patient, the model is
• yij ? ?j eij,
• where ? overall mean, ?j difference for MD,
  and eij individual error
• Model says there is random variation from the
  mean score at the level of MDs (level 2) plus
  variation at the level of patients (level 1)

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What does the random effects model do?

• Actual MD means vary from 53.4 to 87.1 and
  patient N for each MD varies from 5 to 45. Thus,
  actual MD means not very stable.
• Random effects model assumes MD mean scores are
  from an underlying normal distribution
• It uses the information from all the MDs and the
  characteristics of a normal distribution to
  estimate the true ?js

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Estimating the Random Effects

• In our example from the PORT study, raw means
  range 53.4 to 87.1
• Ordinary least squares estimates range 54.0 and
  87.9 (term for each MD, ANCOVA)
• The random effects estimates of the mean patient
  scores by MD ranged from 60.4 to 78.6 their SD
  was 4.94.
• so random effects are closer to the overall mean

30
Adding MD and Patient Predictors to the Simple
Model

• We want to examine the effect of patients age
  (level 1 variable) and MD years in practice
  (level 2) on the satisfaction score
• Specify a regression model with 2 predictor
  variables and a random effect for the MD
• Score for each MD is modeled both by adjusting
  for patients age and MD years in practice and by
  modeling the distribution of MD mean scores

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Final Random (or Mixed) Effects Regression Model

• Positive association with patient age (?0.15,
  p0.003, satisfaction score goes up with age)
• No association with MD years in practice (p0.69)
• Significant variance (24.4) in satisfaction score
  by MD (random effect)

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Summary

• Clustered data should not be analyzed with
  standard statistical methods and tests
• Reduction of outcome and predictors to one value
  per cluster is an option but loses information
• Choice of remaining methods (dummy variables,
  conditional regression, GEE, or random effects)
  depends on the research question and on the
  number of observations at each level

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Summary

• Research questions affect choice of method
• if only care about predictors, GEE models are a
  good alternative
• if question is about variation between clusters
  (level 2 variable), a model that produces random
  effects estimates is needed
• Number of clusters has to be large enough to
  estimate a random effect (N30)
• Small number of clusters can be handled with
  dummy variables

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Data Set for Homework

• CA hospitals CABG registry
• Patients (N28,555) clustered within hospitals
  (N80)
• Binary outcome alive/dead after 30 days
• Patient level characteristics and hospital
  characteristics
• Use STATA to answer questions syntax for the
  models supplied