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A BRIEF INTRODUCTION TO MULTILEVEL MODELS
Leslie Rutkowski, PhD Assistant Professor of Inquiry Methodology Counseling & Educational Psychology School of Education Indiana University WIM Seminar February 2015
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Multilevel data • Often participants of studies are nested within specific
contexts • Patients treated in hospitals • Firms operate within countries • Families live in neighborhoods • Students learn in classes within schools
• Data stemming from such research designs have a
multilevel or hierarchical structure.
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Terminology • HLM • Multilevel modeling (MLM) • Random effects models • Variance components • Mixed effects modeling
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More terminology • Macro
• Macro-level units • Macro units • Primary units • Clusters • Level 2 units
• Micro • Micro-level units • Micro units • Secondary units • Elementary units • Level 1 units
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HLM: Simple model • One L1 predictor with a random intercept & a random
slope: • HLM form:
• Linear mixed model (LMM) form:
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Notation, notation, notation
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Notation, notation, notation, II
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Group Level Similarities • If we use traditional linear regression, we assume:
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Group Level Similarities • But more often we have:
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Group Level Similarities • Students within a school are somewhat similar • Students between schools are different • Why? (absolutely not comprehensive)
• Teacher factors • Pedagogical approaches • Training
• School factors • Public vs. Private • Safety
• Community factors • Parental involvement • Average SES
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Implications? • It is often (but not always) important to take into account
the group level dependencies in analyses. • Why?
• Most (traditional) assumptions are violated. • We might miss some very important group effects. • The level of group dependency is important on its own.
• When do we care about group-level dependencies?
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Single vs. multilevel regression
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How different are the relationships? MATH452 = 448.28 + 38.57*books. MATH517 = 518.99 + 4.78*books. MATH529 = 487.81 + 10.09*books. MATH548 = 461.95 + 16.55*books. MATH577 = 393.50 – 7.82*books. MATH604 = 493.21 + 17.81*books. MATH619 = 498.76 – 1.93*books. MATH622 = 465.68 + 5.35*books. MATH677 = 506.50 + 1.24*books. MATH1479= 480.86 + 29.54*books. MATH1673= 605.29 + 11.16*books.
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Ignoring data structure
• We can easily have such a situation:
y = 0.5x + 6
y = 0.5x + 1
y = 0.5x + 8
y = -1x + 21.619
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Within vs. Between
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Null/Empty/Baseline Model • Useful first step in model building / hypothesis testing
process. • With the empty model we learn if there are between group
differences • Yes? Multilevel approach is warranted. ( ) • No? A standard, plain vanilla regression is sufficient. ( )
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Simplest Multilevel Model: Null • Null model, empty model, fully unconditional model:
• Where
• There are no predictors • Linear mixed model: • Yij is random because U0j & Rij are random
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Deconstructed • Intercept is composed of two parts: • Overall (fixed) mean: • Random group effect:
• This is random difference for group j from . • Individual deviation / residual deviation:
• This is random difference for student i from • Variance components:
• Decomposed into two parts:
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In words • Groups (j) are regarded as a sample from a population of
groups. • We can say that the intercept coefficient depends on j. • This is an important first step b/c it provides a basic
partition of the variance. • If the intercept does not depend on j, then it is best to take
an OLS approach to analyzing the data. • Usually – it’s possible that there are no intercept differences but
there are slope differences. When could this happen?
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What are we modeling? • We know that we have a sample from a larger population of schools and students.
• We see that the means are quite different. • Do we have sufficient evidence to support a multilevel approach?
452 517 529 548 577 604 619 622 677 1479 1673
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Math_Score
SCHOOL ID
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How similar/different?: Intraclass correlation • Measure of similarity between two randomly chosen level one
units within a randomly chosen level two unit
• Proportion of variance in the outcome that is between groups
• Proportion of variance in outcome explained by group differences.
• Provides justification for a multilevel modeling approach
• is the population between group variance
• is the population within group variance
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Null Model ICC
• Estimated parameters:
• ICC from this example: 3170.88/(3170.88+ 4680.16) = 0.4039
• 40% of the variance in mathematics scores can be attributed to between school differences. The remainder lies within the school.
Parameter Value
3170.88
4680.16
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Intercepts as outcomes models • These are also referred to as random intercepts models. • A couple of possibilities:
• Some level one predictors, no level two predictors; • Some level one predictors, some level two predictors. • Some level two predictors, no level one predictors
• But what do these look like?
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Add a level-1 predictor • Add to our empty model a predictor for number of books
in the home:
• Number of books in the home is an historic proxy for SES (goes back to FIMS - http://www.iea.nl/fims.html)
• Additional variable may help to explain variance in achievement – this is at the heart of what we are trying to do.
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RI model with L1 Books • Adding a predictor to the model gives us:
• As a linear mixed model:
Specifically:
Abstractly:
HLM or MLM
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Based on the linear mixed model:
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Estimated parameters • We are estimating 4 parameters:
• Intercept • Books effect • Between groups intercept variance • Within-groups variance
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Results • Edited output from SAS:
• And the residual ICC:
• Notice this is a bit smaller than the empty model. Why?
Value SE Value SE
Fixed EffectsIntercept 514.03 17.62 515.62 14.59Books -- -- 17.99 3.70
Random EffectsIntercept 3170.88 2115.35Residual 4680.16 4336.00
Null Model Add Books
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Adding Level 2 Effects
• Only level 1 variable forces the between and within regressions for a particular effect to be equal.
• Does this seem reasonable? (1.0 = .25?)
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And With Our Data
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Adding Level 2 Effects • If we add group mean for books, the between and within groups can differ.
Hierarchical linear model (HLM):
Linear mixed model (LMM):
If , then the between and within relationships are not different
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Estimated parameters • Now we are estimating 5 parameters:
• Intercept • Books effect • School average effect of books • Intercept variance • Within variance component
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Models compared
• What changed?
• And the ICC:
Value SE Value SE Value SEFixed Effects
Intercept 514.03 17.62 515.62 14.59 365.39 26.67books -- -- 17.99 3.70 16.30 3.75Mbooks 76.13 12.89
Random EffectsIntercept 3170.88 1460.23 2115.35 1021.63 574.51 363.05Residual 4680.16 432.74 4336.00 402.16 4345.33 403.97
Null Model Add Books Add Mbooks
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Random slopes
Forcing slopes to be equal across groups
Allowing slopes to be unequal across groups
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How does the model look?
• With one micro-variable:
• Where
• And
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What are all these parts? First, what’s new?
1. A random component for the slope:
2. Variance components – Before: – Now, add slope variance: – And intercept/slope covariance:
How many more parameters are we estimating?
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Putting it all together… • Linear Mixed Model:
• And:
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And how do we interpret ?
• If this were a representative situation:
• What do you think?
2 Groups
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Intra-class correlation • In a random intercept model, recall the ICC:
• Now, variance and covariance of Yij
• So, the ICC is also dependent on the value of x
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Ex 1: Random slopes, 1 predictor • Going back to our earlier example
• Yij = mathij & xij = booksij
• Level 1:
• Level 2:
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Results
Value SE Value SE Value SEFixed Effects
Intercept 514.03 17.62 515.62 14.59 505.46 3.10books -- -- 17.99 3.70 12.71 0.73
Random EffectsIntercept 3170.88 2115.35 2157.87Int/Slope 36.33Slope 11.29Residual 4680.16 4336.00 3426.69
Null Model Random Intercept Random Slope
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Plot of Groups
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And the ICC? • Depends on books, so let’s choose a couple of
reasonable values: • Most students fall between (-2.36, 2.36)
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L2 Predictors & Cross Level Interactions • As in random intercept models, we can include level-2
predictors • Means of level one variables
• School SES • School attitude toward math
• Natural level two predictors • School resources (books, facilities) • Principal reports of student behavior problems
• These effects can predict the intercept, the slope or a combination
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Simple CLI Model • Adding a bit to what we had before:
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Linear Mixed Model • Combined:
• Notice the term. • This is a “cross-level” interaction • We are trying to explain the slope
• Does the slope for group j depend on some level 2 predictor? • Example: does the effect of student language use depend on the
average SES of the school?
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Variance components • Notice that our variance components do not change
With all of the usual assumptions
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Many other possibilities for L2 • We could also have:
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Cross-Level Example Level 1: Level 2: Combined:
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Results • Interpretation?
Value SE Value SE Value SEFixed Effects
Intercept 514.03 17.62 505.46 3.10 412.91 3.10books -- -- 12.71 0.73 14.08 2.53Mbooks 47.68 3.36books*Mbooks -0.71 1.24
Random EffectsIntercept 3170.88 2157.87 836.09Int/Slope 36.33 73.56Slope 11.29 39.51Residual 4680.16 3426.69 3426.69
Null Model Random Slope CLI Model
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Multilevel models • Just a VERY brief overview • A major methodological advance
• Allows for the possibility of randomly varying intercepts • Randomly varying slopes • We can decompose the variance within and between higher level
units. • We can fit cross level interactions
• We can test rich and complex theories
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What do we gain? • Statistically efficient estimates of regression coefficients. • Correct standard errors, confidence intervals, and
significance tests. • Can use covariates measured at any of the levels of the
hierarchy. • We can test hypotheses about homogeneity or
heterogeneity of groups.
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Lots of other topics • Centering – what and why • Testing variance components • Longitudinal analysis • Cross-classified models
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Now, on to SAS!
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PROC MIXED • Basic Syntax:
1. PROC MIXED options 2. CLASS statement 3. MODEL statement 4. RANDOM statement. 5. TITLE statement.
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PROC MIXED PROC MIXED <options go here> ; <various commands on the following lines> Options: DATA=<name of sas data set>: what SAS data set to use. NOCLPRINT: don’t print classification levels/information. COVTEST: hypothesis tests for variances (& covariances). METHOD: estimation method to use.
ML = maximum likelihood REML = restricted maximum likelihood (REML is the default)
EMPIRICAL: empirical / Hubert-White / sandwich estimators.
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CLASS, MODEL, BY statements • CLASS statement indicates variables that are class /
categorical / nominal / discrete • MODEL is of the form
• MODEL response = fixed predictors / options • SOLUTION requests a solution for the fixed effects
• BY statement produces a separate analysis for all of the “by” variables (e.g. country)
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RANDOM & TITLE statement • RANDOM specifies the random effects, RANDOM random predictors / options • Option SOLUTION produces solution for random effects • Option type=UN gives variance and covariance components • Option SUBJECT= specifies the grouping variable
• TITLE adds a title to the procedure TITLE ‘title here’
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ODS OUTPUT • Writes a data file based on your requests. • Requires certain statements in the PROC MIXED syntax.
• See TABLE 56.22 in SAS Help file for details. • In general: ods output <table name> = • Example: ods output solutionf=f_full covparms=c_full;
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SAS Examples: • Null model:
• RI with L1 predictor: