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Multilevel Regression Models

- sean f. reardon
- 17 june, 2004

Outline

- What/Why Multilevel Regression?
- The Basic Multilevel Regression Model
- Growth Models
- A Taste of Advanced Topics

Part I.A.

- What are multilevel data and multilevel analysis?

What are multilevel data?

- Multilevel data are data where observations are

clustered in units - Observations within the same unit may be more

similar than observations in separate units, on

average - What effect does this have on estimation and

statistical inference?

Examples of multilevel data with contextual

clustering

- Observations of students, clustered within

schools - Observations of siblings, clustered within

families - Observations of individuals, clustered within

countries, states, or neighborhoods

Examples of multilevel data with intra-person

clustering

- Repeated test scores, clustered within students
- Multiple measures of a latent construct,

clustered within persons

Other examples of multilevel data

- Patients, clustered within doctors
- Coefficient estimates, clustered within studies

(meta-analysis) - Widget sizes, clustered within factories
- And so on

What is multilevel regression analysis?

- Also called
- Hierarchical Linear Models
- Mixed Models
- Multilevel Models
- Growth Models
- Slopes-as-Outcomes Models

Multilevel Regression Models

- A form of regression models
- Used to answer questions about the relationship

of context to individual outcomes - Used to estimate both within-unit and

between-unit relationships (and cross-level

interactions) - e.g., within- vs. between-school relationships

between SES and achievement

Part I.B. Whats wrong with OLS?

The OLS Model

Assumptions of OLS

- Linearity
- Errors are normally distributed
- Errors are homoskedastic
- Errors are uncorrelated/independent
- Knowing the error term for one observation is not

informative of the error term of any other

observation

Some Example Data

- Data from Early Childhood Longitudinal

Study-Kindergarten Cohort (NCES, 1998-2004) - Longitudinal study of 21,000 kindergarten

students in K class of 1998-99 - Followed through fifth grade (2003-04)

ECLS-K data

- Subsample
- 399 kindergarten students
- sampled from 17 schools
- Math Score
- Fall kindergarten math test scores
- Administered 2-3 months into school year
- Age
- Age in months at time of math assessment
- Ranges from 60-79 months

What is the relationship between age and math

scores?

- Note this is NOT a growth model
- It is a cross-sectional model
- A growth model requires repeated measures, so we

can observe intra-individual growth

OLS RegressionMath on Age

. reg math age Source SS df

MS Number of obs

422 -------------------------------------------

F( 1, 420) 32.38 Model

1765.41947 1 1765.41947 Prob gt F

0.0000 Residual 22896.5737 420

54.5156517 R-squared

0.0716 ------------------------------------------

- Adj R-squared 0.0694 Total

24661.9932 421 58.5795563 Root

MSE 7.3835 -----------------------------

-------------------------------------------------

math Coef. Std. Err. t

Pgtt 95 Conf. Interval ------------------

--------------------------------------------------

--------- age .4956666 .0871016

5.69 0.000 .3244572 .666876

_cons -10.91381 6.049008 -1.80 0.072

-22.80391 .9762943 ----------------------------

--------------------------------------------------

OLS RegressionMath on Age

- Next look at the residuals from this model.
- Are they homoscedastic? Normally distributed?

Independent?

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- Residuals look correlated with each other within

schools - Formal test of this dependence ANOVA

Random Effects ANOVA

One-way Analysis of Variance for

resid Residuals

Number of obs 422

R-squared 0.1638 Source

SS df MS F Prob gt

F ------------------------------------------------

------------------------- Between s_id1

3750.169 17 220.59817 4.65

0.0000 Within s_id1 19146.405 404

47.392091 ----------------------------------------

--------------------------------- Total

22896.574 421 54.386161

Intraclass Asy.

correlation S.E. 95 Conf. Interval

------------------------------------------

------ 0.13487 0.05204

0.03287 0.23687 Estimated SD of

s_id1 effect 2.718128

Estimated SD within s_id1 6.884191

Est. reliability of a s_id1 mean

0.78517 (evaluated at n23.44)

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Consequences of Non-Independence of Residuals

- The computation of standard errors in OLS depends

on the assumption of independence of errors - If errors are not independent, then standard

errors will, in general, be too small (so the

probability of Type I errors is larger than it

should be)

Two extreme examples

- n individuals observed from each of K schools

(total of nK observations) - if Yik Yjk for all i and j in school k, then

knowing k completely determines Y, so there are

really only K unique observations - In this case, we can just treat each school as a

single observation (with outcome Y.k), and use

OLS on the sample of K schools

Two extreme examples

- n individuals observed from each of K schools

(total of nK observations) - if Yik?Yjk for all i and j and k, then knowing k

tells us nothing about Y, so there are really nK

unique observations - In this case, there is no dependence of the

errors, so we can use OLS on the sample of nK

students.

When do we need multilevel regression?

- In the intermediate case, where knowing the

school gives us some, but not complete

information about Y. - e.g., test scores vary both within and between

schools - e.g., individuals vary within and between

neighborhoods - e.g., mood varies both within individuals (over

time) and between individuals

Intermission I

Part II.A.

- Farewell OLS

What we know so far

- Two observations within the same unit may be more

similar than two observations chosen at random - If the regression model does not explain all of

the between-unit differences (and it is unlikely

that they will), we will have correlated errors

within units - This is a violation of the independence of

residuals assumption in OLS - At a minimum, this results in incorrect standard

errors (too small)

How do we allow dependence in the regression

model?

- We want a model that explicitly allows the level

of the outcome variable to vary across level-two

units - For example, we want to let the mean reading

score differ across schools - So lets write a model that allows this

Some notation

- i indexes level-one units (people within schools,

observations within persons) - j indexes level-two units (e.g., schools, if we

have students nested within schools) - We will use r to denote a level-one residual, and

u to denote a level-two residual

Farewell OLS Our first multilevel model

- Instead of
- Lets write

Farewell OLS Our first multilevel model

Farewell OLS Our first multilevel model

Outcome for observation i in unit j

Farewell OLS Our first multilevel model

Outcome for observation i in unit j

Intercept

Farewell OLS Our first multilevel model

Outcome for observation i in unit j

Value of X for observation i in unit j

Intercept

Coefficient

Farewell OLS Our first multilevel model

Outcome for observation i in unit j

Residual term specific to unit j

Value of X for observation i in unit j

Intercept

Coefficient

Farewell OLS Our first multilevel model

Residual term specific to observation i in unit j

Outcome for observation i in unit j

Residual term specific to unit j

Value of X for observation i in unit j

Intercept

Coefficient

Farewell OLS Our first multilevel model

Residual term specific to observation i in unit j

Outcome for observation i in unit j

Residual term specific to unit j

Value of X for observation i in unit j

Intercept

Coefficient

What is uj?

- A residual term
- Specific to unit j
- Common to all observations in unit j
- Subscript j, no subscript i
- Interpretation the difference between the

overall intercept and the intercept in unit j

What is rij?

- A residual term
- Specific to observation i in unit j
- Has a mean of 0, so any part of ?ij that is

common to all observations within j has been

removed - So the rijs may be independent
- Not guaranteed to be independent

Features of this model

- Note that ?ij uj rij
- We also have
- Var(?ij) Var(uj rij)
- Var(uj) Var(rij) 2Cov(uj,rij)
- Var(uj) Var(rij)
- We will come back to variance decomposition later

Features of this model

- The level of Yij after adjusting for Xij may

vary across the units - We have made no assumptions yet about the

distribution of the ujs or the rijs. - The relationship between X and Y does not depend

on j (?1 does not depend on j)

So how do we estimate this model?

- We want an estimate of ?1 , the relationship

between Xij and Yij. - Two approaches
- Fixed Effects estimator
- Random Effects estimator

Another way to write this model

where

The fixed effects estimator

- We have absorbed the level-two error terms (the

ujs) into the intercept - Now each aggregate unit has its own intercept so

between-unit variation is accounted for in the

intercepts - This solves the dependence problem with the rijs

(they may still not be independent, but not

because of unexplained variation

between-level-two units)

The fixed effects estimator

- Three methods of obtaining the fixed effects

estimator ?1 from this model - Dummy variables for each unit
- Change or difference scores
- Deviations from mean unit values
- All three are mathematically equivalent
- All can be estimated via OLS, with some

adjustment of the degrees of freedom

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The random effects estimator

- We treat the variance between units as consisting

of parameter variance (true variance between

units) and error variance (extra variance

produced because of sampling) - We treat units with larger samples as having more

reliable estimated unit means

The random effects estimator

- So our estimate of the unit mean for a particular

unit is a weighted average of the unit mean

estimated in a fixed effects model and the

overall meanour estimates are shrunken toward

the grand mean - Lets see a picture of this

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Part II.B.

- The basic multilevel model

The basic multilevel model

- Random effects ANOVA is the simplest random

effects model - The random effects model is a very simple kind of

multilevel model - So we are building up here to the multilevel model

The multilevel model as a random effects model

- We write the random effects model as
- Yij ?0 uj rij
- We can also write it as
- Yij ?0j rij
- .
- .

The multilevel model as a random effects model

- We write the random effects model as
- Yij ?0 uj rij
- We can also write it as
- Yij ?0j rij
- .
- .

Level-1 model

The multilevel model as a random effects model

- We write the random effects model as
- Yij ?0 uj rij
- We can also write it as
- Yij ?0j rij
- ?0j ?00 uj
- .

Level-1 model

The multilevel model as a random effects model

- We write the random effects model as
- Yij ?0 uj rij
- We can also write it as
- Yij ?0j rij
- ?0j ?00 uj
- (here were using the ?00 notation where before

we used ?0 this is the notation of HLM)

Level-1 model

Level-2 model

HLM Notation (Null Model)

- Level-1 model
- Yij ?0j rij
- Level-2 model
- ?0j ?00 uj
- Mixed model
- Yij ?00 uj rij

HLM Notation

- Level-1 model
- Yij ?0j ?1jXij rij
- Level-2 model
- ?0j ?00 uj
- ?1j ?10
- Mixed model
- Yij ?00 ?10Xij uj rij

HLM Notation

- Mixed model
- Yij ?00 ?10Xij uj rij

Structural part of the model

HLM Notation

- Mixed model
- Yij ?00 ?10Xij uj rij

Structural part of the model

Stochastic (random) part of the model

HLM Notation

- Mixed model
- Yij ?00 ?10Xij uj rij

Fixed Effects

HLM Notation

- Mixed model
- Yij ?00 ?10Xij uj rij

Random Effect

Fixed Effects

Part II.C.

- Variance Decomposition

Reminder The unconditional (null) random effects

model

- The one-way random effects ANOVA model
- Yij ?00 uj rij
- We can also write it as
- Yij ?0j rij
- ?0j ?00 uj
- Useful as a baseline model
- Allows us to decompose the variance

Mixed (composite) model

Level-1 model

Level-2 model

Variance decomposition

- Var(Yij) Var(uj) Var(rij)
- ?00 ?2
- Intraclass Correlation (?) the proportion of the

total variance in Yij that is between level-2

units - ? ?00 /(?00 ?2)

Multilevel Analyses

- Analytic Problems
- Explain variation in means across units
- Estimate within- and between-unit relationships
- Distinguish contextual from compositional

variation in means across units - Explain how and why within-unit relationships

differ across units

Explaining variation in means across units

- Why do some schools have higher mean achievement

levels? - Why do some hospitals have lower mortality rates?
- Why do some countries have higher infant

mortality rates?

Explaining variation in means across units

- Means-as-outcomes regression (MLM)
- Yij ?0j rij
- ?0j ?00 ?01Wj uj
- where Wj is a variable indicating some

characteristic of unit j (no i subscript) - Wj may be inherent to level-2
- School curriculum, doctor/patient ratio, regime

type - Wj may be a compositional property of unit j
- School racial composition, patient diagnosis

composition, average maternal education level

Note why dont we just compute the means of Y in

each unit and use OLS at level 2?

Explaining variation in means across units

- Called means-as-outcomes because the Wjs can

only explain mean differences in Yij across units

(Wj only predicts the intercept, not the slope) - uj is now a level-2 residual
- We can compute an R2 at both levels of the model
- ?00 from the null model is the total level-2

variance that can be explained - R2between ?00 (null) - ?00 (model)/?00

(null) - R2within ?2 (null) - ?2(model)/?2 (null)

Example (null model)

- The outcome variable is MATH1
- Final estimation of fixed effects
- -------------------------------------------------

--------------------------- - Standard

Approx. - Fixed Effect Coefficient Error

T-ratio d.f. P-value - -------------------------------------------------

--------------------------- - For INTRCPT1, B0
- INTRCPT2, G00 20.369682 0.129474

157.327 867 0.000 - -------------------------------------------------

--------------------------- - Final estimation of variance components
- -------------------------------------------------

---------------------------- - Random Effect Standard Variance

df Chi-square P-value - Deviation Component
- -------------------------------------------------

---------------------------- - INTRCPT1, U0 3.25337 10.58439

867 3733.93573 0.000 - level-1, R 6.46230 41.76128
- -------------------------------------------------

----------------------------

Example Means-as-outcomes model

- The outcome variable is MATH1
- Final estimation of fixed effects
- -------------------------------------------------

--------------------------- - Standard

Approx. - Fixed Effect Coefficient Error

T-ratio d.f. P-value - -------------------------------------------------

--------------------------- - For INTRCPT1, B0
- INTRCPT2, G00 19.585257 0.133633

146.560 866 0.000 - S_PRIVAT, G01 3.691165 0.286361

12.890 866 0.000 - -------------------------------------------------

--------------------------- - Final estimation of variance components
- -------------------------------------------------

---------------------------- - Random Effect Standard Variance

df Chi-square P-value - Deviation Component
- -------------------------------------------------

---------------------------- - INTRCPT1, U0 2.87151 8.24560

866 3070.97445 0.000

Part II.D.

- Variable Centering

Variable Centering

- An important topic with major implications for

fitting and interpreting multilevel models

Variable Centering

- An important topic with major implications for

fitting and interpreting multilevel models - which we will not have time to cover today.

Part II.E.

- Random Coefficients and
- the Full 2-Level Model

Individual-Level Model

- Yij ?0j ?1jX1ij ?2jX2ij ?KjXKij rij

Slope on X1 for unit j

Slope on XK for unit j

Slope on X2 for unit j

Intercept for unit j

Outcome for person i in unit j

Contextual Model

- Yij ?0j ?1jX1ij ?2jX2ij ?KjXKij rij
- ?0j ?00
- ?1j ?10
- ?2j ?20
- ?Kj ?K0

In OLS, the intercept and slopes are fixed they

are the same in all units

Contextual Questions

- Does the intercept vary across units?
- Can we predict the intercepts using level-2

covariates (Zs)? - Do the slopes vary across units?
- Can we predict the slopes using level-2

covariates (Zs)?

Does the intercept vary across units?

- Yij ?0j ?1jX1ij ?2jX2ij ?KjXKij

rij - ?0j ?00 u0j
- ?1j ?10
- ?2j ?20
- ?Kj ?K0

In the random effects model, the intercept varies

around some grand mean intercept (?00), and the

slopes are fixed they are the same in all units

Test H0 Var(u0j) 0

Can we predict the intercepts?

- Yij ?0j ?1jX1ij ?2jX2ij ?KjXKij

rij - ?0j ?00 ?01Z1 ?02Z2 ?0MZM u0j
- ?1j ?10
- ?2j ?20
- ?Kj ?K0

Here, the Zms predict the intercept.

Test H0 ?0m 0

Do the slopes vary across units?

- Yij ?0j ?1jX1ij ?2jX2ij ?KjXKij

rij - ?0j ?00 u0j
- ?1j ?10 u1j
- ?2j ?20 u2j
- ?Kj ?K0 uKj

The intercept and each of the slopes varies

around thei grand means (the ?k0s)

Test H0 Var(ukj) 0

Can we predict the slopes?

- Yij ?0j ?1jX1ij ?2jX2ij ?KjXKij

rij - ?0j ?00 ?01Z1 ?02Z2 ?0MZM u0j
- ?1j ?10 ?11Z1 ?12Z2 ?1MZM u1j
- ?2j ?20 ?21Z1 ?22Z2 ?2MZM u2j
- ?Kj ?K0 ?K1Z1 ?K2Z2 ?KMZM uKj

Here, the Zms predict the slopes.

Test H0 ?km 0

Example

- ECLS-K Fall Kindergarten data
- 8,799 white and black students in 807 schools

(618 public, 189 private schools) - SES measured by standarized SES variable
- Outcome is Fall K math score

Research Questions

- What is the within-school relationship between

race and SES and math scores? - Do average math scores vary across schools?
- Are math scores higher in private schools?
- Does the relationship between SES and math scores

vary across schools? - Is the relationship between SES and math scores

lower in private schools?

Example (cont.)

- See HLM command files lecture8a-g.hlm and

corresponding HLM output files lecture8a-g.txt - We will meet in the lab 2/11/04.

Intermission II

Part III.A.

- Growth Modeling

Growth Models

- Allow us to model development over time and to

investigate correlates of between-person

variation in growth trajectories over time. - The fitted model describes an expected growth

trajectory for each person, rather than a single

expected value on an outcome measure

Examples

- Modeling inter-individual changes in some outcome
- School achievement, income, attitudes
- Modeling growth in national characteristics
- GNP, population, etc
- Modeling change in organizational outcomes
- Business profits, hospital mortality rates

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The Growth Model

- Made up of a within-unit model of change and a

between-unit model of inter-individual variation

in change - Requires repeated measures of outcome within each

unit - Requires multilevel error structure since errors

are likely not independent

Within-unit model of change

- Outcome varies as a function of time
- Yit fi(time) rit
- Simple case f is linear
- Yit ?0i ?1i(timeit) rit

Within-unit model of change

- Outcome varies as a function of time
- Yit fi(time) rit
- Simple case f is linear
- Yit ?0i ?1i(timeit) rit

Intercept for unit i

Within-unit model of change

- Outcome varies as a function of time
- Yit fi(time) rit
- Simple case f is linear
- Yit ?0i ?1i(timeit) rit

Growth slope for unit i

Intercept for unit i

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Between-unit model

- Model between-unit differences in growth

trajectories - Yit ?0i ?1i(timeit) eit
- ?0i ?00 ?01(Xi) r0i
- ?1i ?10 ?11(Xi) r1i

Between-unit model of intercept

Between-unit model of slope

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Within-unit model for change

- Simple case f is linear
- Yit ?0i ?1i(timeit) rit
- Need to specify zero-point for time
- Pick a point that is interpretable and

substantively meaningful for your study - e.g., age time in school, time since institution

opened, calendar time, etc - Affects estimation and interpretation of the

intercept

Defining time

- TOLERANCEit ?0i ?1i(AGEit) eit
- ?0i ?00

r0i - ?1i ?10

r1i - Final estimation of fixed effects
- -------------------------------------------------

--------------------------- - Standard

Approx. - Fixed Effect Coefficient Error

T-ratio d.f. P-value - -------------------------------------------------

--------------------------- - For INTRCPT1, B0
- INTRCPT2, G00 -0.081187 0.511521

-0.159 15 0.876 - For TIME slope, B1
- INTRCPT2, G10 0.130812 0.043074

3.037 15 0.009 - -------------------------------------------------

---------------------------

Defining time

- TOLERANCEit ?0i ?1i(AGEit-11) eit
- ?0i ?00

r0i - ?1i ?10

r1i - Final estimation of fixed effects
- -------------------------------------------------

--------------------------- - Standard

Approx. - Fixed Effect Coefficient Error

T-ratio d.f. P-value - -------------------------------------------------

--------------------------- - For INTRCPT1, B0
- INTRCPT2, G00 1.357750 0.074445

18.238 15 0.000 - For TIME slope, B1
- INTRCPT2, G10 0.130812 0.043074

3.037 15 0.009 - -------------------------------------------------

---------------------------

Modeling Inter-personal variation in growth

trajectories

- TOLERANCEit ?0i ?1i(AGEit-11) eit
- ?0i ?00

?01(MALEi) r0i - ?1i ?10

?11(MALEi) r1i - Final estimation of fixed effects
- -------------------------------------------------

--------------------------- - Standard

Approx. - Fixed Effect Coefficient Error

T-ratio d.f. P-value - -------------------------------------------------

--------------------------- - For INTRCPT1, B0
- INTRCPT2, G00 1.355556 0.102740

13.194 14 0.000 - MALE, G01 0.005016 0.155328

0.032 14 0.975 - For TIME slope, B1
- INTRCPT2, G10 0.102333 0.058323

1.755 14 0.101 - MALE, G11 0.065095 0.088177

0.738 14 0.473 - -------------------------------------------------

---------------------------

Parameters of the growth model

- Yit ?0i ?1i(timeit) eit
- ?0i ?00 ?01(Xi) r0i
- ?1i ?10 ?11(Xi) r1i

Structural parameters of the growth model

- ?0i true intercept for individual i
- ?1i true slope for individual i
- ?00 population average intercept (for

individuals with X0) - ?01 population average difference in level-one

intercept for individuals with one unit

difference in X - ?10 population average slope (for individuals

with X0) - ?01 population average difference in level-one

slope for individuals with one unit difference in

X

Stochastic (random) parameters of the growth model

- Var(eit) ?e2 level1 residual variance
- Var(r0i) ?02 level2 residual variance in true

intercept (?0i) - Var(r1i) ?12 level2 residual variance in true

slope (?1i) - Cov(r0i, r1i) ?01 level2 residual covariance

in true intercept (?0i) and true slope (?1i)

Part III.B.

- Additional Issues in
- Growth Modeling

Growth Modeling Issues

- Timing of observations
- Centering the time variable
- Variable numbers of observations
- Missing observations
- Time-varying covariates
- Slope-intercept covariances
- Non-linear growth curves

Timing of observations

- If each person (unit) has the same number of

observations, and if the timing of observations

is the same for all units,the data (and design)

are said to be balanced. - In this case, the growth model is equivalent to a

repeated measures ANOVA - But if not, the growth model is mode flexible

than the repeated measures ANOVA model

Centering time in growth models

- Group-mean centering time results in unbiased

estimate of average within-person growth rate - Any other centering of time results in biased

estimate of average within-person growth rate if

individuals mean times are correlated with their

mean outcomes - In balanced design, mean time is the same for all

persons, so centering does not affect slope

estimate - In unbalanced design, it may be necessary to

center time - Centering time affects interpretation of the

intercept

Variable numbers of observations

- Number and timing of observations may differ by

design or because of missingness - Types of missingness (see SW p. 157-159)
- MCAR missing completely at random
- CDD covariate dependent dropout
- MAR missing at random
- The growth model estimates are unbiased under any

of these types of missingness

Time-varying covariates

- So far, we have considered only the relationship

between stable person-level covariates and the

growth trajectory - What about time-varying covariates?
- A time-varying covariate is a covariate whose

value changes over time - Examples from SW chapter 5

Examining the covariance matrix in growth models

- Yit ?0i ?1i(timeit) eit
- ?0i ?00 ?01(Xi) r0i
- ?1i ?10 ?11(Xi) r1i
- Var(r0i) ?02 level2 residual variance in true

intercept (?0i) - Var(r1i) ?12 level2 residual variance in true

slope (?1i) - Cov(r0i, r1i) ?01 level2 residual covariance

in true intercept (?0i) and true slope (?1i)

The slope-intercept covariance

- Do individuals with high initial values of Y have

faster growth rates of Y? - Is r0i correlated with r1i?
- It depends on how we center the time variable
- It is possible to observe any correlation (-1 to

1) between r0i and r1i depending where we define

the intercept.

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Non-linear growth curves

- So far we have assumed that each individuals

growth trajectory is linear (constant growth rate

over time) - Now we consider cases where growth may be

non-linear - Polynomial curve
- Piecewise linear
- Discontinuous

Polynomial growth curves

- Quadratic growth trajectory
- Yit ?0i ?1i(timeit) ?2i(time2it) eit
- ?0i ?00 r0i
- ?1i ?10 r1i
- ?2i ?20 r2i

Piecewise linear growth curves

- Piecewise growth trajectory
- Yit ?0i ?1i(time1it) ?2i(time2it)

eit - ?0i ?00 r0i
- ?1i ?10 r1i
- ?2i ?20 r2i

Discontinuous growth curves

- Discontinuous growth trajectory
- Yit ?0i ?1i(timeit) ?2i(eventit) eit
- ?0i ?00 r0i
- ?1i ?10 r1i
- ?2i ?20 r2i

Discontinuous growth curves

- Discontinuous growth trajectory (with

time-varying growth rate) - Yit ?0i ?1i(timeit) ?2i(eventit)

?3i(timeiteventit) eit - ?0i ?00 r0i
- ?1i ?10 r1i
- ?2i ?20 r2i
- ?3i ?30 r3i

Intermission III

Part IV.A.

- A Taste of Advanced Topics
- 3 Level Models

Examples of 3-level data

- Students gt classrooms gt schools
- Students gt schools gt districts
- Patients gt Doctors gt Hospitals
- Children gt Families gt Neighborhoods
- Repeated observations gt individuals gt contexts
- For example
- Repeated obs gt students gt classrooms gt schools gt

districts gt countries gt planets

Data may have more than 3 levelsbut the more

levels, the more data needed to model

relationships.

The 3-level null model

Level 1 model

Level 2 model

Level 3 model

3-level variance decomposition

- ?02 Var(eijk) true within level-1

variance (variance within j, between i) - ?002 Var(r0jk) true level-2 variance

(variance within k, between j) - ?0002 Var(u00k) true between k level-3

(variance within k, between j) - Var(Yijk) ?02 ?002 ?0002

3-level variance decomposition

- Remember the ICC from the 2-level model

Proportion of true variance in Y that lies

between clusters - ICC ?002 /(?02 ?002)
- We apply the same logic to the 3-level model
- Proportion of total variance that lies between

level-3 units - ?0002 /(?02 ?002 ?0002 )
- Proportion of level-1 level 2 variance that

lies between level-2 units - ?002 /(?02 ?002 )

The 3-level growth model

Part IV.B.

- A Taste of Advanced Topics
- Meta-Analysis

Meta-Analysis

- The problem we often have a lot of studies, each

trying to estimate the same parameter - effect of small classes on learning rates
- effect of welfare receipt on income, maternal

depression, child welfare, etc.

Multiple studies

- Suppose we conducted a number of similar studies

to estimate the effect of treatment T on outcome

Y. - Each study gives us an estimate of d, the

standardized effect of T on Y. - The estimates of d may vary across studies Why?
- We would like to estimate the true average effect

of T in the population

Possible reasons for varying estimates across

studies

- The ds may vary because of sampling variance

(each study is conducted with a different sample) - The ds may vary because of differences in the

populations of each study sample - The ds may vary because of differences in the

study design (e.g., different instruments) - The ds may vary because of differences in the

treatment studied (differences in implementation,

duration, etc.)

Approach 1

- In each study, we fit a regression model to

estimate the treatment effect ?1 - But the treatment effect may vary across studies
- Here ?10 is the true mean effect of T across

studies, and u1j is the deviation of the effect

in study j from this mean.

Approach 1

- If we had access to all the data from each study,

we could fit a multilevel model - But what if we dont have access to all the

original data?

Approach 2

- Recall, that in each study, we have a regression

model like this - Typically, each study will report the estimate of

?1 and some measure of its sampling variance (its

standard error) - Remember (lecture 4) that we estimate the grand

mean by weighting individual estimates by their

precision (the inverse of the variance of the

estimate) - We use the standard errors of the study-specific

effect estimates to construct these weights, so

we dont need the original data.

Key Points

- We need from each study an estimate of the

treatment effect and its sampling variance

(standard error) - The treatment effects must be measured in the

same metric across all studies

Part IV.C.

- A Taste of Advanced Topics
- Cross-Classified Data and Models

Cross-Classified Data

- Observations nested in multiple, non-hierarchical

units - e.g. persons nested in schools and neighborhoods
- patients nested in multiple doctors/clinics
- students nested in multiple classrooms
- repeated observations on relationship dynamics

nested in partners (where partner changes are

common)

Cross-Classified Data

- Yijk is outcome Y for person i in neighborhood j

and school k

Cross-Classified Data

- In 3-level hierarchical data, each observation

can be decomposed into - a grand mean (common to all observations)
- a row-specific (e.g. school) deviation from the

grand mean - a column-specific (e.g. nbhd) deviation from the

grand mean - a rowcolumn-specific deviation
- and individual deviation from the rowcolumn mean

Conclusion

Resources

- Textbooks
- Raudenbush Bryk (2002) Hierarchical Linear

Models. Sage. - Singer Willett (2003) Applied Longitudinal Data

Analysis - Multilevel Listserv
- http//www.nursing.teaching.man.ac.uk/staff/mcampb

ell/multilevel.html - http//www.jiscmail.ac.uk/lists/multilevel.html

Resources

- Software
- HLM
- MLWin
- SAS (PROC MIXED)
- SPSS
- Stata (-gllamm-)

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