use "simulated_multilevel_longitudinal_data.dta", clear3 Cross-Classified Models
3.1 Introduction
A two level multilevel model imagines that Level 1 units are nested in Level 2 units. A three level multilevel model imagines that Level 1 units are nested in Level 2 units, which are in turn nested in Level 3.
A cross-classified model imagines that the nesting is not hierarchical, but rather that there are two sets of clusters or nestings in which individuals may be nested.
3.2 Get Data
3.3 Cross Classified Model
We can treat these random effects as being cross classified.
This might be useful if we had data where individuals lived in different countries at different times.
However, because id is in fact nested inside country, in this case, estimating the random effects as cross classified will be more time consuming, but will give us equivalent results to a three level model.
3.3.1 Standard (Less Computationally Efficient) Syntax
The below syntax will take a very long time to run with the full sample, and thus we have commented it out.
* mixed outcome t warmth physical_punishment || _all: R.country || _all: R.id
* est store crossed1The documentation notes that we can use a much more computationally efficient version of the above command, which is what we do in these notes. The user can verify that both versions of the command will produce equivalent results.
In fact, at the end of handout we verify the similarity of both sets of syntax using a random sample.
3.3.2 Cross Classified With Computationally Efficient Syntax
mixed outcome t warmth physical_punishment || _all: R.country || id:
est store crossed2 // store crossed effects resultPerforming EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -28516.314
Iteration 1: Log likelihood = -28516.277
Iteration 2: Log likelihood = -28516.277
Computing standard errors ...
Mixed-effects ML regression Number of obs = 9,000
Grouping information
-------------------------------------------------------------
| No. of Observations per group
Group variable | groups Minimum Average Maximum
----------------+--------------------------------------------
_all | 1 9,000 9,000.0 9,000
id | 3,000 3 3.0 3
-------------------------------------------------------------
Wald chi2(3) = 1168.69
Log likelihood = -28516.277 Prob > chi2 = 0.0000
-------------------------------------------------------------------------------------
outcome | Coefficient Std. err. z P>|z| [95% conf. interval]
--------------------+----------------------------------------------------------------
t | .9434605 .065866 14.32 0.000 .8143654 1.072556
warmth | .9053924 .0380439 23.80 0.000 .8308277 .9799572
physical_punishment | -1.014385 .0499354 -20.31 0.000 -1.112257 -.916514
_cons | 50.8301 .4123007 123.28 0.000 50.022 51.63819
-------------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects parameters | Estimate Std. err. [95% conf. interval]
-----------------------------+------------------------------------------------
_all: Identity |
var(R.country) | 3.429974 .930313 2.015668 5.836634
-----------------------------+------------------------------------------------
id: Identity |
var(_cons) | 8.608872 .4757699 7.725107 9.59374
-----------------------------+------------------------------------------------
var(Residual) | 26.02862 .4752444 25.11363 26.97695
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 1260.84 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.3.4 Three Level Model
mixed outcome t warmth physical_punishment || country: || id: // 3 level w/ random intercepts only
est store threelevel // store random intercept modelPerforming EM optimization ...
Performing gradient-based optimization:
Iteration 0: Log likelihood = -28516.314
Iteration 1: Log likelihood = -28516.277
Iteration 2: Log likelihood = -28516.277
Computing standard errors ...
Mixed-effects ML regression Number of obs = 9,000
Grouping information
-------------------------------------------------------------
| No. of Observations per group
Group variable | groups Minimum Average Maximum
----------------+--------------------------------------------
country | 30 300 300.0 300
id | 3,000 3 3.0 3
-------------------------------------------------------------
Wald chi2(3) = 1168.69
Log likelihood = -28516.277 Prob > chi2 = 0.0000
-------------------------------------------------------------------------------------
outcome | Coefficient Std. err. z P>|z| [95% conf. interval]
--------------------+----------------------------------------------------------------
t | .9434605 .065866 14.32 0.000 .8143654 1.072556
warmth | .9053924 .0380439 23.80 0.000 .8308277 .9799572
physical_punishment | -1.014385 .0499354 -20.31 0.000 -1.112257 -.916514
_cons | 50.8301 .4123007 123.28 0.000 50.022 51.63819
-------------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects parameters | Estimate Std. err. [95% conf. interval]
-----------------------------+------------------------------------------------
country: Identity |
var(_cons) | 3.429974 .930313 2.015668 5.836634
-----------------------------+------------------------------------------------
id: Identity |
var(_cons) | 8.608872 .4757699 7.725107 9.59374
-----------------------------+------------------------------------------------
var(Residual) | 26.02862 .4752444 25.11363 26.97695
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 1260.84 Prob > chi2 = 0.0000
Note: LR test is conservative and provided only for reference.3.5 Nice Table of Results of Three Level and Cross Classified Model
etable, estimates(threelevel crossed2), ///
showstars showstarsnote /// show stars and note
column(estimate) // column is modelnameinvalid 'showstars'
r(198);
r(198);3.6 Verification of Syntax Equivalence for Cross Classified Model
keep if family <= 5 // random sample of families
quietly mixed outcome t warmth physical_punishment || _all: R.country || _all: R.id
est store crossed1A // less efficient syntax
quietly mixed outcome t warmth physical_punishment || _all: R.country || id:
est store crossed2A // more efficient syntax
etable, estimates(crossed1A crossed2A) ///
showstars showstarsnote /// show stars and note
column(estimate) // column is modelname(8,550 observations deleted)
------------------------------------------------------
crossed1A crossed2A
------------------------------------------------------
time 0.745 ** 0.745 **
(0.281) (0.281)
parental warmth in past week 0.871 ** 0.871 **
(0.160) (0.160)
physical punishment in past week -1.262 ** -1.262 **
(0.206) (0.206)
Intercept 51.755 ** 51.755 **
(1.009) (1.009)
var(R_country) 2.245 2.245
(1.319) (1.319)
var(R_id) 5.425
(1.843)
var(e) 23.638 23.638
(1.933) (1.933)
var(_cons) 5.425
(1.843)
Number of observations 450 450
------------------------------------------------------
** p<.01, * p<.05