Multilevel Models 3

Sociology 229A, Class 10

Copyright © 2008 by Evan SchoferDo not copy or distribute without permission

Announcements

• Final class!

• Papers due today

• Topics:• Presentations• Multilevel models• EHA: Shared Frailty• EHA: Heterogeneous Diffusion Models.

Multilevel Data

• Simple example: 2-level dataClass Class Class Class Class Class

• Which can be shown as:

Class 1

S1 S2 S3

Class 2

S1 S2 S3

Class 3

S1 S2 S3

Level 2

Level 1

Review: Multilevel Strategies

• Problems of multilevel models• Non-independence; correlated error• Standard errors = underestimated

• Solutions:– Each has benefits, disadvantages…

• 1. OLS regression• 2. Aggregation (between effects model)• 3. Robust Standard Errors• 4. Robust Cluster Standard Errors• 5. Dummy variables (Fixed Effects Model)• 6. Random effects models

– Intercept only; slopes; cross-level interactions

Review: Fixed Effects Model (FEM)

• Fixed effects model:

ijijjij XY • For i cases within j groups

• Therefore j is a separate intercept for each group

• It is equivalent to solely at within-group variation:

jijjijjij XXYY )(• X-bar-sub-j is mean of X for group j, etc• Model is “within group” because all variables are

centered around mean of each group.

Review: Random Effects

• Issue: The dummy variable approach (ANOVA, FEM) treats group differences as a fixed effect

• Alternatively, we can treat it as a random effect• Don’t estimate values for each case, but model it

– Like “e” in a regression equation

• This requires making assumptions– e.g., that group differences are normally distributed with a

standard deviation that can be estimated from data

• BUT, ignoring slope variability is also an assumption…

Review: Random Effects

• A simple random intercept model– Notation from Rabe-Hesketh & Skrondal 2005, p. 4-5

ijjijY 0

Random Intercept Model

• Where is the main intercept• Zeta () is a random effect for each group

– Allowing each of j groups to have its own intercept– Assumed to be independent & normally distributed

• Error (e) is the error term for each case– Also assumed to be independent & normally distributed

• Note: Other texts refer to random intercepts as uj or j.

Linear Random Intercepts Model. xtreg supportenv age male dmar demp educ incomerel ses, i(country) re

Random-effects GLS regression Number of obs = 27807Group variable (i): country Number of groups = 26

R-sq: within = 0.0220 Obs per group: min = 511 between = 0.0371 avg = 1069.5 overall = 0.0240 max = 2154

Random effects u_i ~ Gaussian Wald chi2(7) = 625.50corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

------------------------------------------------------------------------------ supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | -.0038709 .0008152 -4.75 0.000 -.0054688 -.0022731 male | .0978732 .0229632 4.26 0.000 .0528661 .1428802 dmar | .0030441 .0252075 0.12 0.904 -.0463618 .05245 demp | -.0737466 .0252831 -2.92 0.004 -.1233007 -.0241926 educ | .0857407 .0061501 13.94 0.000 .0736867 .0977947 incomerel | .0090308 .0059314 1.52 0.128 -.0025945 .0206561 ses | .131528 .0134248 9.80 0.000 .1052158 .1578402 _cons | 5.924611 .1287468 46.02 0.000 5.672272 6.17695-------------+---------------------------------------------------------------- sigma_u | .59876138 sigma_e | 1.8701896 rho | .09297293 (fraction of variance due to u_i)------------------------------------------------------------------------------

Assumes normal uj, uncorrelated with X vars

SD of u (intercepts); SD of e; intra-class correlation

Review: Choosing Models• Which model is best?

• Fixed effects are most consistent under a wide range of circumstances

– But, can be a problem if your interest is between-group variation

• Random Effects = more efficient– But, runs into problems if specification is poor– Esp. X variables correlated with random error

• Hausman Specification Test: A tool to help evaluate fit of fixed vs. random effects

• Logic: Both fixed & random effects models are consistent if models are properly specified

• In short: Models should give the same results… If not, random effects may be biased.

Within & Between Effects

• Issue: What is the relationship between within-group effects and between-group effects?

• FEM models within-group variation• BEM models between group variation (aggregate)

– Usually they are similar• Ex: Student skills & test performance• Within any classroom, skilled students do best on tests• Between classrooms, classes with more skilled

students have higher mean test scores– BUT…

Within & Between Effects

• But: Between and within effects can differ!• Ex: Effects of wealth on attitudes toward welfare• At the country level (between groups):

– Wealthier countries (high aggregate mean) tend to have pro-welfare attitudes (ex: Scandinavia)

• At the individual level (within group)– Wealthier people are conservative, don’t support welfare

• Result: Wealth has opposite between vs within effects!– Watch out for ecological fallacy!!!

– Issue: Such dynamics often result from omitted level-1 variables (omitted variable bias)

• Ex: If we control for individual “political conservatism”, effects may be consistent at both levels…

Within & Between Effects / Centering

• Multilevel models & “centering” variables

• Grand mean centering: computing variables as deviations from overall mean

• Often done to X variables• Has effect that baseline constant in model reflects

mean of all cases– Useful for interpretation

• Group mean centering: computing variables as deviation from group mean

• Useful for decomposing within vs. between effects• Often in conjunction with aggregate group mean vars.

Within & Between Effects• You can estimate BOTH within- and between-

group effects in a single model• Strategy: Split a variable (e.g., SES) into two new

variables…– 1. Group mean SES– 2. Within-group deviation from mean SES

» Often called “group mean centering”

• Then, put both variables into a random effects model• Model will estimate separate coefficients for between

vs. within effects

– Ex:• egen meanvar1 = mean(var1), by(groupid)• egen withinvar1 = var1 – meanvar1• Include mean (aggregate) & within variable in model.

Within & Between Effects. xtreg supportenv meanage withinage male dmar demp educ incomerel ses, i(country) mle

Random-effects ML regression Number of obs = 27807Group variable (i): country Number of groups = 26

Random effects u_i ~ Gaussian Obs per group: min = 511 avg = 1069.5 max = 2154

LR chi2(8) = 620.41Log likelihood = -56918.299 Prob > chi2 = 0.0000

------------------------------------------------------------------------------ supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- meanage | .0268506 .0239453 1.12 0.262 -.0200812 .0737825 withinage | -.003903 .0008156 -4.79 0.000 -.0055016 -.0023044 male | .0981351 .0229623 4.27 0.000 .0531299 .1431403 dmar | .003459 .0252057 0.14 0.891 -.0459432 .0528612 demp | -.0740394 .02528 -2.93 0.003 -.1235873 -.0244914 educ | .0856712 .0061483 13.93 0.000 .0736207 .0977216 incomerel | .008957 .0059298 1.51 0.131 -.0026651 .0205792 ses | .131454 .0134228 9.79 0.000 .1051458 .1577622 _cons | 4.687526 .9703564 4.83 0.000 2.785662 6.58939

Between & within effects are opposite. Older countries are MORE environmental, but older people are LESS. Omitted variables? Wealthy European countries with strong green parties have older populations!

• Example: Pro-environmental attitudes

Generalizing: Random Coefficients

• Linear random intercept model allows random variation in intercept (mean) for groups

• But, the same idea can be applied to other coefficients• That is, slope coefficients can ALSO be random!

ijijjijjij XXY 2211

Random Coefficient Model

ijijjjij XY 2211

Which can be written as:

• Where zeta-1 is a random intercept component• Zeta-2 is a random slope component.

Linear Random Coefficient Model

Rabe-Hesketh & Skrondal 2004, p. 63

Both intercepts and slopes vary randomly across j groups

Random Coefficients Summary

• Some things to remember:• Dummy variables allow fixed estimates of intercepts

across groups• Interactions allow fixed estimates of slopes across

groups

– Random coefficients allow intercepts and/or slopes to have random variability

• The model does not directly estimate those effects– Just as we don’t estimate coefficients of “e” for each case…

• BUT, random components can be predicted after you run a model

– Just as you can compute residuals – random error– This allows you to examine some assumptions (normality).

STATA Notes: xtreg, xtmixed

• xtreg – allows estimation of between, within (fixed), and random intercept models

• xtreg y x1 x2 x3, i(groupid) fe - fixed (within) model• xtreg y x1 x2 x3, i(groupid) be - between model• xtreg y x1 x2 x3, i(groupid) re - random intercept (GLS)• xtreg y x1 x2 x3, i(groupid) mle - random intercept (MLE)

• xtmixed – allows random slopes & coefs• “Mixed” models refer to models that have both fixed and

random components• xtmixed [depvar] [fixed equation] || [random eq], options• Ex: xtmixed y x1 x2 x3 || groupid: x2

– Random intercept is assumed. Random coef for X2 specified.

STATA Notes: xtreg, xtmixed• Random intercepts

• xtreg y x1 x2 x3, i(groupid) mle– Is equivalent to

• xtmixed y x1 x2 x3 || groupid: , mle• xtmixed assumes random intercept – even if no other

random effects are specified after “groupid”

– But, we can add random coefficients for all Xs:• xtmixed y x1 x2 x3 || groupid: x1 x2 x3 , mle cov(unstr)

– Useful to add: “cov(unstructured)”• Stata default treats random terms (intercept, slope) as

totally uncorrelated… not always reasonable• “cov(unstr) relaxes constraints regarding covariance

among random effects (See Rabe-Hesketh & Skrondal).

STATA Notes: GLLAMM

• Note: xtmixed can do a lot… but GLLAMM can do even more!

• “General linear & latent mixed models”• Must be downloaded into stata. Type “search gllamm”

and follow instructions to install…

– GLLAMM can do a wide range of mixed & latent-variable models

• Multilevel models; Some kinds of latent class models; Confirmatory factor analysis; Some kinds of Structural Equation Models with latent variables… and others…

• Documentation available via Stata help– And, in the Rabe-Hesketh & Skrondal text.

Random intercepts: xtmixed. xtmixed supportenv age male dmar demp educ incomerel ses || country: , mle

Mixed-effects ML regression Number of obs = 27807Group variable: country Number of groups = 26

Obs per group: min = 511 avg = 1069.5 max = 2154Wald chi2(7) = 625.75Log likelihood = -56919.098 Prob > chi2 = 0.0000

------------------------------------------------------------------------------ supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | -.0038662 .0008151 -4.74 0.000 -.0054638 -.0022687 male | .0978558 .0229613 4.26 0.000 .0528524 .1428592 dmar | .0031799 .0252041 0.13 0.900 -.0462193 .0525791 demp | -.0738261 .0252797 -2.92 0.003 -.1233734 -.0242788 educ | .0857707 .0061482 13.95 0.000 .0737204 .097821 incomerel | .0090639 .0059295 1.53 0.126 -.0025578 .0206856 ses | .1314591 .0134228 9.79 0.000 .1051509 .1577674 _cons | 5.924237 .118294 50.08 0.000 5.692385 6.156089------------------------------------------------------------------------------[remainder of output cut off] Note: xtmixed yields identical results to xtreg , mle

• Example: Pro-environmental attitudes

Random intercepts: xtmixed supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | -.0038662 .0008151 -4.74 0.000 -.0054638 -.0022687 male | .0978558 .0229613 4.26 0.000 .0528524 .1428592 dmar | .0031799 .0252041 0.13 0.900 -.0462193 .0525791 demp | -.0738261 .0252797 -2.92 0.003 -.1233734 -.0242788 educ | .0857707 .0061482 13.95 0.000 .0737204 .097821 incomerel | .0090639 .0059295 1.53 0.126 -.0025578 .0206856 ses | .1314591 .0134228 9.79 0.000 .1051509 .1577674 _cons | 5.924237 .118294 50.08 0.000 5.692385 6.156089------------------------------------------------------------------------------------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]-----------------------------+------------------------------------------------country: Identity | sd(_cons) | .5397758 .0758083 .4098899 .7108199-----------------------------+------------------------------------------------ sd(Residual) | 1.869954 .0079331 1.85447 1.885568------------------------------------------------------------------------------LR test vs. linear regression: chibar2(01) = 2128.07 Prob >= chibar2 = 0.0000

xtmixed output puts all random effects below main coefficients. Here, they are “cons” (constant) for groups defined by “country”, plus residual (e)

• Ex: Pro-environmental attitudes (cont’d)

Non-zero SD indicates that intercepts vary

Random Coefficients: xtmixed. xtmixed supportenv age male dmar demp educ incomerel ses || country: educ, mle[output omitted] supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | -.0035122 .0008185 -4.29 0.000 -.0051164 -.001908 male | .1003692 .0229663 4.37 0.000 .0553561 .1453824 dmar | .0001061 .0252275 0.00 0.997 -.0493388 .049551 demp | -.0722059 .0253888 -2.84 0.004 -.121967 -.0224447 educ | .081586 .0115479 7.07 0.000 .0589526 .1042194 incomerel | .008965 .0060119 1.49 0.136 -.0028181 .0207481 ses | .1311944 .0134708 9.74 0.000 .1047922 .1575966 _cons | 5.931294 .132838 44.65 0.000 5.670936 6.191652------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]-----------------------------+------------------------------------------------country: Independent | sd(educ) | .0484399 .0087254 .0340312 .0689492 sd(_cons) | .6179026 .0898918 .4646097 .821773-----------------------------+------------------------------------------------ sd(Residual) | 1.86651 .0079227 1.851046 1.882102------------------------------------------------------------------------------LR test vs. linear regression: chi2(2) = 2187.33 Prob > chi2 = 0.0000

• Ex: Pro-environmental attitudes (cont’d)

Here, we have allowed the slope of educ to vary randomly across countries

Educ (slope) varies, too!

Random Coefficients: xtmixed

• What if the random intercept or slope coefficients aren’t significantly different from zero?

• Answer: that means there isn’t much random variability in the slope/intercept

• Conclusion: You don’t need to specify that random parameter

– Also: Models include a LRtest to compare with a simple OLS model (no random effects)

• If models don’t differ (Chi-square is not significant) stick with a simpler model.

Random Coefficients: xtmixed• What are random coefficients doing?

• Let’s look at results from a simplified model– Only random slope & intercept for education

34

56

78

Fitt

ed

valu

es:

xb

+ Z

u

0 2 4 6 8highest educational level attained

Model fits a different slope & intercept for each group!

Random Coefficients

• Why bother with random coefficients?– 1. A solution for clustering (non-independence)

– Usually people just use random intercepts, but slopes may be an issue also

– 2. You can create a better-fitting model– If slopes & intercepts vary, a random coefficient model may fit

better– Assuming distributional assumptions are met– Model fit compared to OLS can be tested….

– 3. Better predictions– Attention to group-specific random effects can yield better

predictions (e.g., slopes) for each group» Rather than just looking at “average” slope for all groups.

Random Coefficients

• 4. Multilevel models explicitly put attention on levels of causality

• Higher level / “contextual” effects versus individual / unit-level effects

• A technology for separating out between/within• NOTE: this can be done w/out random effects

– But it goes hand-in-hand with clustered data…

• Note: Be sure you have enough level-2 units!

– Ex: Models of individual environmental attitudes• Adding level-2 effects: Democracy, GDP, etc.

– Ex: Classrooms• Is it student SES, or “contextual” class/school SES?

Multilevel Model Notation

• So far, we have expressed random effects in a single equation:

ijijjijjij XXY 2211

Random Coefficient Model

• However, it is common to separate levels:

Gamma = constant

u = random effect

Here, we specify a random component for level-1 constant & slope

ju111 Intercept equation

ju222 Slope Equation

ijijij XY 21

Level 1 equation

Multilevel Model Notation• The “separate equation” formulation is no

different from what we did before…• But it is a vivid & clear way to present your models• All random components are obvious because they are

stated in separate equations• NOTE: Some software (e.g., HLM) requires this

– Rules:• 1. Specify an OLS model, just like normal• 2. Consider which OLS coefficients should have a

random component– These could be the intercept or any X (slope) coefficient

• 3. Specify an additional formula for each random coefficient… adding random components when desired

Cross-Level Interactions

• Does context (i.e., level-2) influence the effect of level-1 variables?– Example: Effect of poverty on homelessness

• Does it interact with welfare state variables?

– Ex: Effect of gender on math test scores• Is it different in coed vs. single-sex schools?

– Can you think of others?

Cross-level interactions

• Idea: specify a level-2 variable that affects a level-1 slope

ju111 Intercept equation

ijijij XY 21

Level 1 equation

jj uZ 2322 Slope equation with interaction

Cross-level interaction:

Level-2 variable Z affects slope (B2) of a level-1 X variable

Coefficient 3 reflects size of

interaction (effect on B2 per unit change in Z)

Cross-level Interactions

• Cross-level interaction in single-equation form:

ijijjijjij XXY jij32211 ZXRandom Coefficient Model with cross-level interaction

– Stata strategy: manually compute cross-level interaction variables

• Ex: Poverty*WelfareState, Gender*SingleSexSchool• Then, put interaction variable in the “fixed” model

– Interpretation: B3 coefficient indicates the impact of each unit change in Z on slope B2

• If B3 is positive, increase in Z results in larger B2 slope.

Cross-level Interactions. xtmixed supportenv age male dmar demp educ income_dev inc_meanXeduc ses || country: income_mean , mle cov(unstr)

Mixed-effects ML regression Number of obs = 27807Group variable: country Number of groups = 26

supportenv | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- age | -.0038786 .0008148 -4.76 0.000 -.0054756 -.0022817 male | .1006206 .0229617 4.38 0.000 .0556165 .1456246 dmar | .0041417 .025195 0.16 0.869 -.0452395 .0535229 demp | -.0733013 .0252727 -2.90 0.004 -.1228348 -.0237678 educ | -.035022 .0297683 -1.18 0.239 -.0933668 .0233227 income_dev | .0081591 .005936 1.37 0.169 -.0034753 .0197934inc_meanXeduc| .0265714 .0064013 4.15 0.000 .0140251 .0391177 ses | .1307931 .0134189 9.75 0.000 .1044926 .1570936 _cons | 5.892334 .107474 54.83 0.000 5.681689 6.102979------------------------------------------------------------------------------

• Pro-environmental attitudes

Interaction: inc_meanXeduc has a positive effect… The education slope is bigger in wealthy countries

Note: main effects change. “educ” indicates slope when inc_mean = 0

Interaction between country mean income and individual-level education

Cross-level Interactions. xtmixed supportenv age male dmar demp educ income_dev inc_meanXeduc ses || country: income_mean , mle cov(unstr)

------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]-----------------------------+------------------------------------------------country: Unstructured | sd(income~n) | .5419256 .2095339 .253995 1.156256 sd(_cons) | 2.326379 .8679172 1.11974 4.8333 corr(income~n,_cons) | -.9915202 .0143006 -.999692 -.7893791-----------------------------+------------------------------------------------ sd(Residual) | 1.869388 .0079307 1.853909 1.884997------------------------------------------------------------------------------LR test vs. linear regression: chi2(3) = 2124.20 Prob > chi2 = 0.0000

• Random part of output (cont’d from last slide)

Random components:

Income_mean slope allowed to have random variation

Interceps (“cons”) allowed to have random variation

“cov(unstr)” allows for the possibility of correlation between random slopes & intercepts… generally a good idea.

Beyond 2-level models

• Sometimes data has 3 levels or more• Ex: School, classroom, individual• Ex: Family, individual, time (repeated measures)• Can be dealt with in xtmixed, GLLAMM, HLM• Note: stata manual doesn’t count lowest level

– What we call 3-level is described as “2-level” in stata manuals

– xtmixed syntax: specify “fixed” equation and then random effects starting with “top” level

• xtmixed var1 var2 var3 || schoolid: var2 || classid:var3– Again, specify unstructured covariance: cov(unstr)

Beyond Linear Models

• Stata can specify multilevel models for dichotomous & count variables– Random intercept models

• xtlogit – logistic regression – dichotomous• xtpois – poisson regression – counts • xtnbreg – negative binomial – counts • xtgee – any family, link… w/random intercept

– Random intercept & coefficient models– Plus, allows more than 2 levels…

• xtmelogit – mixed logit model• xtmepoisson – mixed poisson model

Panel Data

• Panel data is a multilevel structure• Cases measured repeatedly over time• Measurements are ‘nested’ within cases

Person 1

T2T1 T4T3 T5

Person 2

T2T1 T4T3 T5

Person 3

T2T1 T4T3 T5

Person 4

T2T1 T4T3 T5

– Obviously, error is clustered within cases… but…– Error may also be clustered by time

• Historical time events or life-course events may mean that cases aren’t independent

– Ex: All T1s and all T5s

• Ex: Models of economic growth… certain periods (e.g., Oil shocks of 1970s) affect all countries.

Panel Data

• Issue: panel data may involve clustering across cases & time

• Good news: Stata’s “xt” commands were made for this

• Allow specification of both ID and TIME clusters…• Ex: xtreg var1 var2 var3, mle i(countryid) t(year)

– Note: You can also “mix and match” fixed and random effects

• Ex: You can use dummies (manually) to deal with time-cultuering with a random effect for case ids

Panel Data: serial correlation

• Panel data may have another problem:• Sequential cases may have correlated error

– Ex: Adjacent years (1950 & 1951 or 2007 & 2008) may be very similar. Correlation denoted by “rho” ()

• Called “autocorrelation” or “serial correlation”

• “Time-series” models are needed• xtregar – xtreg, for cases in which the error-term is

“first-order autoregressive”• First order means the prior time influences the current

– Only adjacent time-points… assumes no effect of those prior

• Can be used to estimate FEM, BEM, or GLS model• Use option “lbi” to test for autocorrelation (rho = 0?).

Panel Data: Choosing a Model

• If clustering is mainly a nuisance:• Adjust SEs: vce(cluster caseid)• Or simple fixed or random effects

– Choice between fixed & random• Fixed is “safer” – reviewers are less likely to complain

– If hausman test works, random = OK, too

• But, if cross-sectional variation is of interest, fixed can be a problem…

– In that case, use random effects… and hope the reviewers don’t give you grief.

Panel Data: Choosing a Model

• If you have substantive interests in cross-level dynamics, mixed models are probably the way to go…

• Plus, you can create a better-fitting model– Allows you to relax the assumption that slopes are the same

across groups.