PSC 8185: Multilevel Modeling Fitting Random Coefficient Binary Response Models in Stata

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PSC 8185: Multilevel Modeling Fitting Random Coefficient Binary Response Models in Stata Consider the following two-level model random coefficient logit model. This is a Supreme Court decision making model, where level-1 units are justices votes and level-2 units are court cases (justices votes (3544) nested within 399 cases). For binary dependent variables, a Bernoulli sampling model is used, and I use a logit link here. For the logit link, first define Pr(Y ij =1) = p ij, which is the probability of a liberal vote for choice i in case j. Then define ij as the log-odds of p ij : ij = log[p ij / (1 p ij )]. This allows one to specify the log-odds as a linear function of the level-1 independent variables. [Level-1 eq.] [Level-2 eqs.] ij = 0j + j PREF ij 0j = 00 + 01 SAL j + 02 COMP j + 03 ISSUEFAM j + 04 INFO j + 05 USAMICUS j + 06 USPARTY j + 07 STAT j + u 0j j = 10 + 11 SAL j + 12 COMP j + 13 ISSUEFAM j + 14 INFO j + 15 USAMICUS j + 16 USPARTY j + 17 STAT j + u 1j Reduced-form Representation of Model 2: ij = Effects on case outcome Avg. effect of preferences Cross-level interactions Error components 00 + 01 SAL j + 02 COMP j + 03 ISSUEFAM j + 04 INFO j + 05 USAMICUS j + 06 USPARTY j + 07 STAT j + 10 *PREF ij + 11 SAL j *PREF ij + 12 COMP j *PREF ij + 13 ISSUEFAM j*pref ij + 14 INFO j *PREF ij + 15 USAMICUS j *PREF ij + 16 USPARTY j *PREF ij + 17 STAT j *PREF ij + u 0j + u 1j *PREF ij 1

Fitting the Two-Level Model using Full Maximum Likelihood (Adaptive Quadrature) in GLLAMM set more off *ADAPTIVE QUADRATURE MODELS *Generate a constant; gllamm needs this gen cons=1 *Specify random intercept and random coefficient eq cons: cons eq pref: lagpctlib_use *Specify the level-2 variables that are a function of the random intercept eq f1: salience_nyt_cen complex_3pt_cen lnissuecount_cen infoenv_01_cen usamicus_cen usparty_cen statutory_cen *Specify the level-2 variables that are a function of the random coefficient eq f2: salience_nyt_cen complex_3pt_cen lnissuecount_cen infoenv_01_cen usamicus_cen usparty_cen statutory_cen *Model command; start with 4 quadrature points, then 8, then 12, etc.; use *start values from the previous model (that s the from(a) command) geqs(f1 f2) adapt nip(4) geqs(f1 f2) adapt nip(8) from(a) geqs(f1 f2) adapt nip(12) from(a) est store lag_adapt12 geqs(f1 f2) adapt nip(16) from(a) est store lag_adapt16 geqs(f1 f2) adapt nip(20) from(a) est store lag_adapt20 *Present results in a table form est table lag_adapt12 lag_adapt16 lag_adapt20, se p varwidth(34) stats(n ll aic) 2

. *ADAPTIVE QUADRATURE MODELS GLLAMM OUTPUT [Suppressing output from models using 4 and 8 quadrature points.]. gllamm dir lagpctlib_use if lagpctlib_use~=., i(caseid) l(logit) f(binom) nrf(2) eqs(cons pref) geqs(f1 f2) adapt nip (12) from(a) Running adaptive quadrature Iteration 0: log likelihood = -1433.9683 Iteration 1: log likelihood = -1433.9498 Iteration 2: log likelihood = -1433.9498 Adaptive quadrature has converged, running Newton-Raphson Iteration 0: log likelihood = -1433.9498 Iteration 1: log likelihood = -1433.9498 (backed up) Iteration 2: log likelihood = -1433.9496 Iteration 3: log likelihood = -1433.9496 number of level 1 units = 3544 number of level 2 units = 399 Condition Number = 15.170465 gllamm model log likelihood = -1433.9496 - dir Coef. Std. Err. z P> z [95% Conf. Interval] -------------+---------------------------------------------------------------- lagpctlib_~e 4.469579.3126479 14.30 0.000 3.8568 5.082357 _cons -.3927239.2204633-1.78 0.075 -.824824.0393763 - Variances and covariances of random effects ***level 2 (caseid) var(1): 15.246868 (2.1409271) cov(2,1):.47004543 (1.589734) cor(2,1):.03597937 var(2): 11.194171 (2.0465321) Regressions of latent variables on covariates random effect 1 has 7 covariates: salience_nyt_cen: 1.3087985 (.53243653) complex_3pt_cen: -.23641807 (.91949432) lnissuecount_cen:.0830284 (.21363707) infoenv_01_cen: -3.2768549 (2.4414709) usamicus_cen: -1.2216683 (.55349651) usparty_cen: -1.6855654 (.58330776) statutory_cen:.20317709 (.45585559) 3

random effect 2 has 7 covariates: salience_nyt_cen: 1.4321357 (.63688317) complex_3pt_cen: -.37590127 (1.0727686) lnissuecount_cen:.52559336 (.25446341) infoenv_01_cen: -.23535991 (2.9681077) usamicus_cen: -.7769956 (.6769883) usparty_cen: -1.3340209 (.70159345) statutory_cen: -.6900348 (.54179413). est store lag_adapt12 [Suppressing output from models using 16 and 20 quadrature points.].. est table lag_adapt12 lag_adapt16 lag_adapt20, se p varwidth(34) stats(n ll aic) --------------------------------------------------------------------------- Variable lag_ada~12 lag_ada~16 lag_ada~20 dir lagpctlib_use 4.4695786 4.4671294 4.4646248.3126479.31198263.31178696 _cons -.39272387 -.38596767 -.38979855.22046332.22660269.22829931 0.0749 0.0885 0.0877 cas1_1 cons 3.9047238 3.8868616 3.8851496.27414577.26850628.26867348 cas1_2 lagpctlib_use 3.3436029 3.3329991 3.3345159.30571974.30443987.30496215 cas1_2_1 _cons.12037866.12164403.119327.40697491.4042363.40362999 0.7674 0.7635 0.7675 f1 salience_nyt_cen 1.3087985 1.3005355 1.3042134.53243653.53845044.54107965 0.0140 0.0157 0.0159 complex_3pt_cen -.23641807 -.23030577 -.23172214.91949432.93965606.94469984 0.7971 0.8064 0.8062 lnissuecount_cen.0830284.08238374.08341868.21363707.22026835.22284323 0.6975 0.7084 0.7082 infoenv_01_cen -3.2768549-3.2372372-3.2487924 2.4414709 2.4672407 2.4796085 0.1795 0.1895 0.1901 usamicus_cen -1.2216683-1.2033215-1.2095659 4

.55349651.56258063.56695643 0.0273 0.0324 0.0329 usparty_cen -1.6855654-1.6638382-1.6713425.58330776.59268672.59686642 0.0039 0.0050 0.0051 statutory_cen.20317709.20579869.20441877.45585559.46632308.47066661 0.6558 0.6590 0.6641 f2 salience_nyt_cen 1.4321357 1.4270938 1.4280513.63688317.63558729.63648104 0.0245 0.0247 0.0249 complex_3pt_cen -.37590127 -.37517886 -.3744169 1.0727686 1.0706272 1.0717017 0.7260 0.7260 0.7268 lnissuecount_cen.52559336.52337804.52371888.25446341.25367009.25423543 0.0389 0.0391 0.0394 infoenv_01_cen -.23535991 -.24728057 -.24405967 2.9681077 2.9606287 2.9653711 0.9368 0.9334 0.9344 usamicus_cen -.7769956 -.77136087 -.77295672.6769883.67531714.67646903 0.2511 0.2534 0.2532 usparty_cen -1.3340209-1.3306189-1.3306759.70159345.7000968.70095192 0.0572 0.0574 0.0576 statutory_cen -.6900348 -.68659908 -.68781565.54179413.54078971.54172083 0.2028 0.2042 0.2042 Statistics N 3544 3544 3544 ll -1433.9496-1434.0637-1434.0918 aic 2905.8993 2906.1274 2906.1837 --------------------------------------------------------------------------- legend: b/se/p 5

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