View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

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?

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

- 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)

(No Transcript)

(No Transcript)

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

(No Transcript)

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

(No Transcript)

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

(No Transcript)

(No Transcript)

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

(No Transcript)

(No Transcript)

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

(No Transcript)

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.

(No Transcript)

(No Transcript)

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-)

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Multilevel Regression Models" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!