Random-effects and conditional fixed-effects overdispersion models

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1 Title xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models Sytax Radom-effects ad coditioal fixed-effects overdispersio models xtbreg depvar varlist weight if exp i rage, re j fe i(varame) irr ocostat oskip exposure(varame) offset(varame) level(#) maximize optios Populatio-averaged model xtbreg depvar varlist weight if exp i rage, pa i(varame) irr robust ocostat exposure(varame) offset(varame) level(#) xtgee optios maximize optios iweights, aweights, ad pweights are allowed for the populatio-averaged model ad iweights are allowed i the radom-effects ad fixed-effects models; see [U] weight. Note that weights must be costat withi paels. xtbreg shares the features of all estimatio commads; see [U] 23 Estimatio ad post-estimatio commads. Sytax for predict Radom-effects ad coditioal fixed-effects overdispersio models predict type ewvarame if exp i rage, xb j stdp ooffset Populatio-averaged model predict type ewvarame if exp i rage, mu j rate j xb j stdp ooffset These statistics are available both i ad out of sample; type predict ::: if e(sample) ::: if wated oly for the estimatio sample. Descriptio xtbreg estimates radom-effects overdispersio models, coditioal fixed-effects overdispersio models, ad populatio-averaged egative biomial models. Here radom-effects ad fixed-effects apply to the distributio of the dispersio parameter, ad ot to the x term i the model. I the radom-effects ad fixed-effects overdispersio models, the dispersio is the same for all elemets i the same group (i.e., elemets with the same value of the i() variable). I the radom-effects model, the dispersio varies radomly from group to group such that the iverse of the dispersio has a Beta(r s) distributio. I the fixed-effects model, the dispersio parameter i a group ca take o ay value, sice a coditioal likelihood is used i which the dispersio parameter drops out of the estimatio. 383

2 384 xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models By default, the populatio-averaged model is a equal-correlatio model; xtbreg assumes corr(exchageable). See [R] xtgee for details o this optio to fit other populatio-averaged models. Optios re requests the radom-effects estimator. re is the default if oe of re, fe, adpa is specified. fe requests the coditioal fixed-effects estimator. pa requests the populatio-averaged estimator. i(varame) specifies the variable ame that cotais the uit to which the observatio belogs. You ca specify the i() optio the first time you estimate or use the iis commad to set i() beforehad. After that, Stata will remember the variable s idetity. See [R] xt. irr reports expoetiated coefficiets e b rather tha coefficiets b. For the egative biomial model, expoetiated coefficiets have the iterpretatio of icidece rate ratios. robust (pa oly) specifies the Huber/White/sadwich estimator of variace is to be used i place of the IRLS variace estimator; see [R] xtgee. This alterative produces valid stadard errors eve if the correlatios withi group are ot as hypothesized by the specified correlatio structure. It does, however, require that the model correctly specifies the mea. As such, the resultig stadard errors are labeled semi-robust istead of robust. Note that although there is o cluster() optio, results are as if there were a cluster() optio ad you specified clusterig o i(). ocostat suppresses the costat term (itercept) i the model. oskip specifies that a full maximum-likelihood model with oly a costat for the regressio equatio be estimated. This costat-oly model is used as the base model to compute a likelihood-ratio 2 statistic for the model test. By default, the model test uses a asymptotically equivalet Wald 2 statistic. For may models, this optio ca sigificatly icrease estimatio time. exposure(varame) ad offset(varame) are differet ways of specifyig the same thig. exposure() specifies a variable that reflects the amout of exposure over which the depvar evets were observed for each observatio; l(varame) with coefficiet costraied to be 1 is etered ito the regressio equatio. offset() specifies a variable that is to be etered directly ito the regressio equatio with coefficiet costraied to be 1; thus exposure is assumed to be e varame. level(#) specifies the cofidece level, i percet, for cofidece itervals. The default is level(95) or as set by set level; see[u] 23.5 Specifyig the width of cofidece itervals. xtgee optios specifies ay other optios allowed by xtgee for family(biom) lik(log); see [R] xtgee. maximize optios cotrol the maximizatio process; see [R] maximize. Usethetrace optio to view parameter covergece. Optios for predict xb calculates the liear predictio. This is the default for the radom-effects ad fixed-effects models. stdp calculates the stadard error of the liear predictio. mu ad rate both calculate the predicted probability of depvar. mu takes ito accout the offset(). rate igores those adjustmets. mu ad rate are equivalet if you did ot specify offset(). mu is the default for the populatio-averaged model.

3 xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models 385 ooffset is relevat oly if you specified offset(varame) for xtbreg. It modifies the calculatios made by predict so that they igore the offset variable; the liear predictio is treated as x it b rather tha x it b + oset it. Remarks xtbreg is a coveiece commad if you wat the populatio-averaged model. Typig. xtbreg :::, ::: pa exposure(time) is equivalet to typig. xtgee :::, ::: family(biom) lik(log) corr(exchageable) exposure(time) Thus, alsosee[r] xtgee for iformatio about xtbreg. By default, or whe re is specified, xtbreg estimates a maximum-likelihood radom-effects overdispersio model. Example You have (fictioal) data o ijury icidets icurred amog 20 airlies i each of 4 years. (Icidets rage from major ijuries to exceedigly mior oes.) The govermet agecy i charge of regulatig airlies has ru a experimetal safety traiig program ad, i each of the years, some airlies have participated ad some have ot. You ow wish to aalyze whether the icidet rate is affected by the program. You choose to estimate usig radom-effects egative biomial regressio because the dispersio might vary across the airlies because of uidetified airlie-specific reasos. Your measure of exposure is passeger miles for each airlie i each year.. xtbreg i_ct iprog, i(airlie) exposure(pmiles) irr olog Radom-effects egative biomial Number of obs = 80 Group variable (i) : airlie Number of groups = 20 Radom effects u_i ~ Beta Obs per group: mi = 4 avg = 4.0 max = 4 Wald chi2(1) = 2.04 Log likelihood = Prob > chi2 = i_ct IRR Std. Err. z P> z [95% Cof. Iterval] iprog pmiles (exposure) /l_r /l_s r s Likelihood ratio test versus pooled: chi2(1) = Prob > chi2 = I the output above, the /l r ad /l s lies refer to l(r) ad l(s), where the iverse of the dispersio is assumed to follow a Beta(r s) radom distributio. The output also icludes a likelihood-ratio test which compares the pael estimator with the pooled estimator (i.e., a egative biomial estimator with costat dispersio).

4 386 xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models You fid that the icidece rate for accidets is ot sigificatly differet for participatio i the program ad that the pael estimator is sigificatly differet from the pooled estimator. We may alteratively estimate a fixed-effects overdispersio model:. xtbreg i_ct iprog, i(airlie) exposure(pmiles) irr fe olog Coditioal fixed-effects egative biomial Number of obs = 80 Group variable (i) : airlie Number of groups = 20 Obs per group: mi = 4 avg = 4.0 max = 4 Wald chi2(1) = 2.11 Log likelihood = Prob > chi2 = i_ct IRR Std. Err. z P> z [95% Cof. Iterval] iprog pmiles (exposure) Example Reruig our previous example i order to fit a robust equal-correlatio populatio-averaged model:. xtbreg i_ct iprog, i(airlie) exposure(pmiles) eform robust pa Iteratio 1: tolerace = Iteratio 2: tolerace = Iteratio 3: tolerace = 2.914e-07 GEE populatio-averaged model Number of obs = 80 Group variable: airlie Number of groups = 20 Lik: log Obs per group: mi = 4 Family: egative biomial(k=1) avg = 4.0 Correlatio: exchageable max = 4 Wald chi2(1) = 1.28 Scale parameter: 1 Prob > chi2 = (stadard errors adjusted for clusterig o airlie) Semi-robust i_ct IRR Std. Err. z P> z [95% Cof. Iterval] iprog pmiles (exposure) We may compare this with a pooled estimator with cluster robust variace estimates:

5 xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models 387. breg i_ct iprog, exposure(pmiles) robust cluster(airlie) irr olog Negative biomial regressio Number of obs = 80 Wald chi2(1) = 0.60 Log likelihood = Prob > chi2 = (stadard errors adjusted for clusterig o airlie) Robust i_ct IRR Std. Err. z P> z [95% Cof. Iterval] iprog pmiles (exposure) /lalpha alpha Saved Results Scalars xtbreg, re saves i e(): e(n) umber of observatios e(ll c) log likelihood, compariso model e(k) umber of estimated parameters e(df m) model degrees of freedom e(k eq) umber of equatios e(chi2) model 2 e(k dv) umber of depedet variables e(p) model sigificace e(n g) umber of groups e(chi2 c) 2 for compariso test e(g mi) smallest group size e(r) value of r i Beta(r,s) e(g avg) average group size e(s) value of s i Beta(r,s) e(g max) largest group size e(ic) umber of iteratios e(ll) log likelihood e(rc) retur code e(ll 0) log likelihood, costat-oly model Macros e(cmd) xtbreg e(opt) type of optimizatio e(cmd2) xt re e(chi2type) Wald or LR; type of model 2 test e(depvar) ame of depedet variable e(chi2 ct) Wald or LR; type of model 2 test e(title) title i estimatio output correspodig to e(chi2 c) e(ivar) variable deotig groups e(offset) offset e(wtype) weight type e(distrib) Beta; the distributio of the e(wexp) weight expressio radom effect e(method) estimatio method e(predict) program used to implemet predict e(user) ame of likelihood-evaluatio program Matrices e(b) coefficiet vector e(v) variace covariace matrix of the estimators Fuctios e(sample) marks estimatio sample

6 388 xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models xtbreg, fe saves i e(): Scalars e(n) umber of observatios e(ll) log likelihood e(k) umber of estimated parameters e(ll 0) log likelihood, costat-oly model e(k eq) umber of equatios e(df m) model degrees of freedom e(k dv) umber of depedet variables e(chi2) model 2 e(n g) umber of groups e(p) model sigificace e(g mi) smallest group size e(ic) umber of iteratios e(g avg) average group size e(rc) retur code e(g max) largest group size Macros e(cmd) xtbreg e(method) requested estimatio method e(cmd2) xt fe e(user) ame of likelihood-evaluator program e(depvar) ame of depedet variable e(opt) type of optimizatio e(title) title i estimatio output e(chi2type) Wald or LR; type of model 2 test e(ivar) variable deotig groups e(offset) offset e(wtype) weight type e(predict) program used to implemet predict e(wexp) weight expressio Matrices e(b) coefficiet vector e(v) variace covariace matrix of the estimators Fuctios e(sample) marks estimatio sample xtbreg, pa saves i e(): Scalars Macros e(n) umber of observatios e(deviace) deviace e(n g) umber of groups e(chi2 dev) 2 test of deviace e(g mi) smallest group size e(dispers) deviace dispersio e(g avg) average group size e(chi2 dis) 2 test of deviace dispersio e(g max) largest group size e(tol) target tolerace e(df m) model degrees of freedom e(dif) achieved tolerace e(chi2) model 2 e(phi) scale parameter e(df pear) degrees of freedom for Pearso 2 e(cmd) xtgee e(ivar) variable deotig groups e(cmd2) xtbreg e(vcetype) covariace estimatio method e(depvar) ame of depedet variable e(chi2type) Wald; type of model 2 test e(family) egative biomial(k=1) e(offset) offset e(lik) log; lik fuctio e(balpha) e(corr) correlatio structure e(predict) program used to implemet predict e(scale) x2, dev, phi, or#; scale parameter Matrices e(b) coefficiet vector e(v) variace covariace matrix of the e(r) estimated workig correlatio matrix estimators Fuctios e(sample) marks estimatio sample

7 xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models 389 Methods ad Formulas xtbreg is implemeted as a ado-file. xtbreg reports the populatio-averaged results obtaied by usig xtgee, family(breg) lik(log) to obtai estimates. See [R] xtgee for details o the methods ad formulas. For the radom-effects ad fixed-effects overdispersio models, we let y it be the cout for the t th observatio i the ith group. We begi with the model y it j it Poisso( it ), where it j i Gamma( it 1= i ) with it = exp(x it + offset it ) ad i is the dispersio parameter. This yields the model Pr(Y it = y it j i ) = ;( it + y it ) ;( it );(y it + 1) i it yit i 1 + i Lookig at withi-group effects oly, this specificatio yields a egative biomial model for the ith group with dispersio (variace divided by the mea) equal to 1 + i ; i.e., costat dispersio withi group. Note that this parameterizatio of the egative biomial model differs from the default parameterizatio of breg, which has dispersio equal to 1 + exp(x + offset); see[r] breg. For a radom-effects overdispersio model, we allow i to vary radomly across groups; amely, we assume that 1=(1 + i ) Beta(r s). The joit probability of the couts for the ith group is Pr(Y i1 = y i1 ::: Y ii = y ii ) = The resultig log likelihood is l L = X w i l ;(r + s) + l ; i=1 ; l ; r + s + Z i Y P = ;(r + s);(r + i P i ;(r);(s);(r + s + k=1 i ik + r + X k=1 k=1 y ik + Pr(Y it = y it j i ) f ( i ) d i Y i P i it);(s + P y it) i it + y it) ik + l ; h s + k=1 ;( it + y it ) ;( it );(y it + 1) y ik ; l ;(r) ; l ;(s) l ;( it + y it ) ; l ;( it ) ; l ;(y it + 1) where it = exp(x it + offset it ) ad w i is the weight for the ith group. For the fixed-effects overdispersio model, we coditio the joit P probability of the couts for each group o the sum of the couts for the group (i.e., the observed i y it). This yields Pr(Y i1 = y i1 ::: Y ii = y ii P i Y it = P i y it) The coditioal log likelihood is l L = X w i (l ; i=1 + h = ;(P i P i it);( P y it + 1) i it + y it) ;( P i it! + l ; y it + 1! Y i l ;( it + y it ) ; l ;( it ) ; l ;(y it + 1) ; l ; ;( it + y it ) ;( it );(y it + 1) i ) i! X it + y it i

8 390 xtbreg Fixed-effects, radom-effects, & populatio-averaged egative biomial models See Hausma et al. (1984) for a more thorough developmet of the radom-effects ad fixed-effects models. Note that Hausma et al. (1984) use a that is the iverse of the we have used here. Refereces Hausma, J., B. H. Hall, ad Z. Griliches Ecoometric models for cout data with a applicatio to the patets R & D relatioship. Ecoometrica 52: Liag, K.-Y. ad S. L. Zeger Logitudial data aalysis usig geeralized liear models. Biometrika 73: Also See Complemetary: Related: Backgroud: [R] licom, [R] predict, [R] test, [R] testl, [R] vce, [R] xtdata, [R] xtdes, [R] xtsum, [R] xttab [R] breg, [R] xtgee, [R] xtpois [U] 16.5 Accessig coefficiets ad stadard errors, [U] 23 Estimatio ad post-estimatio commads, [U] Obtaiig robust variace estimates, [R] xt