1. Advanced quantitative methods for social scientists (2017–2018) LC & PVK
Session 2 Multilevel analysis in Stata (with a focus on random slope models for comparative research)
Louis Chauvel
University of Luxembourg, PEARL Institute for Research on Socio-Economic Inequality (IRSEI)

2. Outline Background Method with example: the PISA survey
Chauvel L, Leist AK. Socioeconomic hierarchy and health gradient in Europe: the role of income inequality and of social origins. International Journal for Equity in Health. 2015;14:132. doi: /s y.
Chauvel L, Hartung A, More Inequality, More Viscosity? Intergenerational Mobility in International Comparison, March 31 – April 2, 2016: PAA Annual Meeting, Washington DC,
Outline
Background
Standard multiple regressions versus random effects models
Fixed effects and random effects
Basics on notations in multilevel analysis
2-Level models / random effect / random slope
Generalization: Higher level models and cross-classified models
Method with example: the PISA survey
Fitting models random effects and random slopes
Post-estimation techniques: BLUPs, Multilevel tools (mlt)
Understanding and presenting results
Examples of publication
Further developments on panel analysis
xtmixed as a pervasive command

3. Main references R Stata
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks: Sage Publications.
Rabe-Hesketh, S., and A. Skrondal  Multilevel and longitudinal modeling using STATA. Stata Press.
Gelman, A., and J. Hill  Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press. 
Stata R

4. Multilevel (2L) data structure
Simple example: 2 level data
That is …
Level 2
Level 1
Country
Country
Country
Country
Country
Country 1
I I I I4
Country 2
I I I I4
Country 3
I I I I4
Country 4
I I I I4
NB: Minimum 20 level-2 groups

5. Typical example: PISA 2012
Educational performance at age circa 15 Many countries (68) Parental/family backgrounds Performance variation by country Influence of parental background (Social Reproduction) by country Explanation of Social Reproduction variation? Country GDP/capita, gini etc. The old solution: series of standard OLS
To open the dataset and prepare it …
* PROGRAM SEGMENT 0
To process the old solution …
* PROGRAM SEGMENT 1

6. FRANCE !
LUX
HKG
ALBANIA

7. matrix R=J(1,5,.)
levelsof cco
foreach i of numlist `r(levels)' {
di `i'
ta cnt if `i'==cco
capture {
quietly: reg PV1READ ST04Q01 f1 stdage if `i'==cco
matrix A=e(b)
noisily matrix li A
matrix C=`i',A
matrix R=R \ C }
mat li R
preserve
clear
svmat R
gen CountryScore=R5
gen SocReproduction=R3
gen cco=R1
two scatter SocR Cou, ml(cco)
reg SocR Cou
reg SocR Cou if R1!=1
restore

8. Multilevel Data: why?
Multilevel models respect the structure of data we have
1. Clustered data and correlated errors in each cluster
2. ML relaxes assumption of uncorrelated (independent) errors
3. Partitioning variance-covariance components
Question: At what level is most of the variance?
Conceptually: Different levels and their effects?
Statistically: Are your data clustered?
Empirically: are there variations both at L1 and L2?
… And we can “easily” refine the models

9. Fixed Effects Model (FEM) & Random Effects (REM)
J groups
For i cases within j groups
aj is a separate intercept for each group
at within-group, equivalent to:
“within group” model : all variables are centered around mean of each group.
In practice : FEM = J replications of standard OLS Models
With dummy variable approach => group differences as a fixed effect
* PROGRAM SEGMENT 2

10. Random Effects Alternatively, treat effects as random effect
No estimates for each case, but model them
A simple random intercept model
Notation from Rabe-Hesketh & Skrondal
Where b is the main intercept
Zeta (z) 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
NB: Minimum 20 level-2 groups

11. xtreg syntax xtreg PV1READ ST04Q01 f1 stdage, i(cco) fe
* PROGRAM SEGMENT 4
*Comparing FE and RE models
xtreg PV1READ ST04Q01 f1 stdage, i(cco) fe
Dependant variable
X-explanatory variables
level 2 group variable FE or RE model

12. Usual Solution => Hausman Specification Test
Best Model?
Fixed effects most consistent as N grows very large
But less efficient than random effects when low within-group variation (big between group variation) and small sample size (not PISA…)
Usual Solution => Hausman Specification Test
Hausman Specification Test: tool help evaluate fit of fixed vs. random effects
Logic: Both fixed & random effects models are consistent if models are properly specified
However, some model violations cause random effects models to be inconsistent
Ex: if X variables are correlated to random error
In short: Models should give the same results… If not, random effects may be biased
If results are similar, use the most efficient model: random effects
If results diverge, odds are that the random effects model is biased. In that case use fixed effects…

13. Hausman Specification Test
Strategy: Estimate both fixed & random effects models
Save the estimates each time
Finally invoke Hausman test
Ex (here with the “old” xtreg stata command):
xtreg PV1READ ST04Q01 f1 stdage, i(cco) fe
est store femod
xtreg PV1READ ST04Q01 f1 stdage, i(cco) re
est store remod
esttab femod remod
hausman femod remod
* PROGRAM SEGMENT 4
*Comparing FE and RE models

14. Linear Fixed Intercepts Model
. xtreg PV1READ ST04Q01 parentalbckgrnd stdage, i(cco) fe
Fixed-effects (within) regression Number of obs =
Group variable: ccode Number of groups =
R-sq: within = Obs per group: min =
between = avg =
overall = max =
F(3,413120) =
corr(u_i, Xb) = Prob > F =
PV1READ | Coef. Std. Err t P>|t| [95% Conf. Interval]
ST04Q01 |
parentalbckgrnd |
stdage |
_cons |
sigma_u |
sigma_e |
rho | (fraction of variance due to u_i)
F test that all u_i=0: F(66, ) = Prob > F =
SD of u (intercepts); SD of e; intra-class correlation

15. Linear Random Intercepts Model
. xtreg PV1READ ST04Q01 parentalbckgrnd stdage, i(cco) re
Random-effects GLS regression Number of obs =
Group variable: ccode Number of groups =
R-sq: within = Obs per group: min =
between = avg =
overall = max =
Wald chi2(3) =
corr(u_i, X) = 0 (assumed) Prob > chi =
PV1READ | Coef. Std. Err z P>|z| [95% Conf. Interval]
ST04Q01 |
parentalbckgrnd |
stdage |
_cons |
sigma_u |
sigma_e |
rho | (fraction of variance due to u_i)
Assumes normal uj, uncorrelated with X vars
SD of u (intercepts); SD of e; intra-class correlation

16. Hausman Specification Test
Example: Pisa read score fe vs re
. hausman femod remod
---- Coefficients ----
| (b) (B) (b-B) sqrt(diag(V_b-V_B))
| femod remod Difference S.E.
ST04Q01 |
parentalbc~d |
stdage |
b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from xtreg
Test: Ho: difference in coefficients not systematic
chi2(3) = (b-B)'[(V_b-V_B)^(-1)](b-B) =
Prob>chi2 =
(V_b-V_B is not positive definite)
Direct comparison of coefficients…
Non-significant p-value indicates that models yield similar results… OK

17. Within & Between Effects / Centering
Why do we do Multilevel models?  To understand the role of inequality Between and Within countries
So “Centering” variables both grand mean and group mean centering
Grand mean centering: computing variables as deviations from overall mean
Should be systematically done for X variables
Group mean centering: computing variables as deviation from group mean
Useful for decomposing within vs. between effects  relative role of inequality between and within countries
Often in conjunction with aggregate group mean vars.

18. Within & Between Effects
You can estimate BOTH within- and between-group effects in a single model
Strategy: Split a variable (e.g., household possession score) into two new variables…
1. Group mean household possession score
2. Within-group deviation from mean household possession score
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 betwparentalbckgrnd=mean(parentalbckgrnd), by(cco)
gen withinparentalbckgrnd=parentalbckgrnd-betwparentalbckgrnd
xtreg PV1READ ST04Q01 stdage betw withi, i(cco) re
* PROGRAM SEGMENT 5
*Assessing within and between effects

19. Linear Random Intercepts Model
. xtreg PV1READ ST04Q01 stdage betw withi, i(cco) re
Random-effects GLS regression Number of obs =
Group variable: ccode Number of groups =
R-sq: within = Obs per group: min =
between = avg =
overall = max =
Wald chi2(4) =
corr(u_i, X) = 0 (assumed) Prob > chi =
PV1READ | Coef. Std. Err z P>|z| [95% Conf. Interval]
ST04Q01 |
stdage |
betwparentalbckgrnd |
withinparentalbckgrnd |
_cons |
sigma_u |
sigma_e |
rho | (fraction of variance due to u_i)
Parental background has huge effect both within and between

20. Generalizing: Random Coefficients (=Random slopes)
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!
Random Coefficient Model
Which can be written as:
Where zeta-1 is a random intercept component = differences between countries
Zeta-2 is a random slope component = country specific inequality effect

21. Linear Random Coefficient Model
Rabe-Hesketh & Skrondal
Both intercepts and slopes vary randomly across j groups
PV1READ
Inequality between countries
vary randomly
Inequality within country
parentalbckgrnd

22. xtmixed syntax
* PROGRAM SEGMENT 6
* a first random slope model
xtmixed – allows random intercepts & slopes
“Mixed” models refer to models that have both fixed and random components
xtmixed [depvar] [fixed equation] || [random eq], options
xtmixed PV1READ ST04Q01 stdage || cco: parentalbckgrnd , iter(5) diff mle cov(unstr)
Dependant variable
fixed effect variables
RE Level 2 variable slope variable
estimation options
cov(unstructured)
cov(unstr) relaxes constraints regarding covariance among random effects (See Rabe-Hesketh & Skrondal) Stata default treats random terms (intercept, slope) as totally uncorrelated… not always reasonable

23. Example: PISA 2012
. xtmixed supportenv age male dmar demp educ incomerel ses || country: , mle
Mixed-effects ML regression Number of obs =
Group variable: ccode Number of groups =
Obs per group: min =
avg =
max =
Wald chi2(2) =
Log likelihood = Prob > chi =
PV1READ | Coef. Std. Err z P>|z| [95% Conf. Interval]
ST04Q01 |
stdage |
_cons |
.../...

24. Ex: PISA 2012 (cont’d)
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
ccode: Unstructured |
sd(parent~d) |
sd(_cons) |
corr(parent~d,_cons) |
sd(Residual) |
LR test vs. linear regression: chi2(3) = 1.5e+05 Prob > chi2 =
“cons” (constant) are intercepts for countries
“parent^d” for the slopes
Non-zero SDs indicates that both intercepts and slopes vary
If some of the estimates are not significant  you can simplify the model

25. What about the random slopes?
Slopes = within country parental background gradient of inequality

26. What about the random slopes?
* PROGRAM SEGMENT 8
* like 6 with BLUP predictors of intercepts and slopes
best linear unbiased predictions (BLUPs)
slopes
intercepts

27. Multilevel Model Notation
Random coeff (random slope) can be expressed in a single equation:
Random Coefficient Model
However, it is common to separate levels:
Level 1 equation
Gamma = constant
u = random effect
Here, we specify a random component for level-1 constant & slope
Intercept equation
Slope Equation

28. Cross-Level Interactions
Does context (i.e., level-2) influence the effect of level-1 variables?
Example: Effect of country inequality (gini) on lower achievements
Can you think of others?

29. Cross-level interactions
Idea: specify a level-2 variable that affects a level-1 slope
Level 1 equation
Intercept equation
Slope equation with interaction
Cross-level interaction:
Level-2 variable Z affects slope (B2) of a level-1 X variable Coefficient g3 reflects size of interaction (effect on B2 per unit change in Z)

30. Cross-level Interactions
Cross-level interaction in single-equation form:
Random 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.

31. 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
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)

32. Advice about building models
Raudenbush & Bryk 2002
Start building the level 1 model first
Then build level 2 model
Keeping a close eye on level 2 N.