Random Effects Logit Models
The Stata manual has data on union membership from the NLS for 4434 women who were 14-24 in 1968 and were observed between 1 and 12 times. A copy of the data is in the WWS509 website.
Note that unionXt is an interaction term computed
as union*t0 where t0=year-70.
. use http://data.princeton.edu/wws509/datasets/union, clear (NLS Women 14-24 in 1968)
Logit Estimates
We first compute logit estimates for later comparison, fitting the same model as in [R] xtlogit with clustered standard errors
. logit union age grade not_smsa south southXt
Iteration 0: log likelihood = -13864.23
Iteration 1: log likelihood = -13550.511
Iteration 2: log likelihood = -13545.74
Iteration 3: log likelihood = -13545.736
Logistic regression Number of obs = 26200
LR chi2(5) = 636.99
Prob > chi2 = 0.0000
Log likelihood = -13545.736 Pseudo R2 = 0.0230
------------------------------------------------------------------------------
union | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
age | .0099931 .0026737 3.74 0.000 .0047527 .0152335
grade | .0483487 .0064259 7.52 0.000 .0357541 .0609432
not_smsa | -.2214908 .0355831 -6.22 0.000 -.2912324 -.1517493
south | -.7144461 .0612145 -11.67 0.000 -.8344244 -.5944678
southXt | .0068356 .0052258 1.31 0.191 -.0034067 .0170779
_cons | -1.888256 .113141 -16.69 0.000 -2.110009 -1.666504
------------------------------------------------------------------------------
. estimates store logit
. logit union age grade not_smsa south southXt, cluster(id)
Iteration 0: log pseudolikelihood = -13864.23
Iteration 1: log pseudolikelihood = -13550.511
Iteration 2: log pseudolikelihood = -13545.74
Iteration 3: log pseudolikelihood = -13545.736
Logistic regression Number of obs = 26200
Wald chi2(5) = 161.37
Prob > chi2 = 0.0000
Log pseudolikelihood = -13545.736 Pseudo R2 = 0.0230
(Std. Err. adjusted for 4434 clusters in idcode)
------------------------------------------------------------------------------
| Robust
union | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
age | .0099931 .0039563 2.53 0.012 .0022389 .0177473
grade | .0483487 .013936 3.47 0.001 .0210346 .0756628
not_smsa | -.2214908 .0713568 -3.10 0.002 -.3613475 -.0816341
south | -.7144461 .0919865 -7.77 0.000 -.8947362 -.5341559
southXt | .0068356 .0070241 0.97 0.330 -.0069314 .0206026
_cons | -1.888256 .207871 -9.08 0.000 -2.295676 -1.480837
------------------------------------------------------------------------------
. estimates store cluster
Random Intercepts
The next step is to fit a random-intercepts model and compare results
. xtlogit union age grade not_smsa south southXt, i(id)
Fitting comparison model:
Iteration 0: log likelihood = -13864.23
Iteration 1: log likelihood = -13550.511
Iteration 2: log likelihood = -13545.74
Iteration 3: log likelihood = -13545.736
Fitting full model:
tau = 0.0 log likelihood = -13545.736
tau = 0.1 log likelihood = -12926.225
tau = 0.2 log likelihood = -12419.526
tau = 0.3 log likelihood = -12003.162
tau = 0.4 log likelihood = -11656.844
tau = 0.5 log likelihood = -11367.53
tau = 0.6 log likelihood = -11129.716
tau = 0.7 log likelihood = -10947.266
tau = 0.8 log likelihood = -10845.532
Iteration 0: log likelihood = -10947.312
Iteration 1: log likelihood = -10557.296
Iteration 2: log likelihood = -10540.582
Iteration 3: log likelihood = -10540.367
Iteration 4: log likelihood = -10540.367
Iteration 5: log likelihood = -10540.366
Random-effects logistic regression Number of obs = 26200
Group variable: idcode Number of groups = 4434
Random effects u_i ~ Gaussian Obs per group: min = 1
avg = 5.9
max = 12
Wald chi2(5) = 227.30
Log likelihood = -10540.366 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
union | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
age | .0093936 .004454 2.11 0.035 .000664 .0181232
grade | .0867878 .0176345 4.92 0.000 .0522247 .1213508
not_smsa | -.2519379 .082334 -3.06 0.002 -.4133095 -.0905663
south | -1.163769 .1114164 -10.45 0.000 -1.382141 -.945397
southXt | .023245 .0078497 2.96 0.003 .0078599 .0386302
_cons | -3.360131 .2586306 -12.99 0.000 -3.867038 -2.853225
-------------+----------------------------------------------------------------
/lnsig2u | 1.749534 .0469964 1.657423 1.841645
-------------+----------------------------------------------------------------
sigma_u | 2.398317 .0563561 2.290366 2.511356
rho | .6361486 .0108779 .6145729 .6571902
------------------------------------------------------------------------------
Likelihood-ratio test of rho=0: chibar2(01) = 6010.74 Prob >= chibar2 = 0.000
. estimates store xtlogit
. estimates table logit xtlogit, eq(1:1)
----------------------------------------
Variable | logit xtlogit
-------------+--------------------------
#1 |
age | .00999311 .00939361
grade | .04834865 .08678776
not_smsa | -.22149081 -.25193788
south | -.71444608 -1.1637691
southXt | .0068356 .02324502
_cons | -1.8882564 -3.3601312
-------------+--------------------------
lnsig2u |
_cons | 1.7495341
----------------------------------------
Except for age, we see that the subject-specific estimates are larger in magnitude than in the marginal logit model.
The odds of being in a union in 1970 are 69% lower for a woman who lives in the south than for one who doesn't, everything else being equal. The effect declined over time, and by 1988 the odds of belonging to a union were 53% lower in the south than elsewhere.
In contrast the logit model estimates the effect as 51% lower odds in 1970 and 45% lower in 1988. These estimates can be interpreted as population average effects.
Given how different the coefficients are, it doesn't make much sense to compare standar errors. Generally speaking, though, they increase as one moves from the logit model to robust standard errors to the estimates based on the random intercept model.
Intra-class Correlation
Stata reports the intraclass correlation as 0.636. This coefficient pertains to a latent variable reflecting propensity to belong to a union, rather than manifest union membership. The correlation between this propensity in any two years for the same individual is 0.64. We can also say that 64% of the variance in the propensity to belong to a union can be attributed to individuals.
Using the xtrho command we can compute the
correlation in actual union membership in any two years
for a woman with a median linear predictor:
. xtrho Measures of intra-class manifest association in random-effects logit Evaluated at median linear predictor Measure | Estimate [95% Conf.Interval] -----------------+------------------------------------ Marginal prob. | .225847 .218973 .232833 Joint prob. | .125072 .117026 .133375 Odds ratio | 8.29305 7.64627 9.00295 Pearson's r | .423617 .4039 .443193 Yule's Q | .784785 .768686 .800059
We estimate a probability of 23% of belonging to a union in any given year and 12% of belonging in two years, much more than one would expect under independence. The correlation is reflected in an odds ratio of 7.7, so for women at the median the odds of belonging to a union at t_2 are 7.7 times as high for those who belonged to a union at t_1 than for those who didn't. Pearson's r is 0.41 and Yule's Q is 0.77.
These measures can be computed for women whose observed characteristics
make then more or less likely to belong to a union by using the
detail option:
. xtrho, detail Measures of intra-class manifest association in random-effects logit Evaluated with linear predictor set at selected percentiles Measure | p1 p25 p50 p75 p99 -----------------+------------------------------------------------------------ Marginal prob. | .10807 .161144 .225847 .251047 .309243 Joint prob. | .046724 .079682 .125072 .14408 .190535 Odds ratio | 10.3122 9.09452 8.29305 8.08413 7.73468 Pearson's r | .36357 .39737 .423617 .431096 .444279 Yule's Q | .8232 .801873 .784785 .779836 .771028
The correlation as measured by the odds ratio or Yule's Q is higher when women are less likely to belong to a union, but the opposite is true if one uses Pearson's r.
For a more detailed discussion of this post-estimation command see muy paper with Elo in the Stata Journal 3(1):32--46 (2003).