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Titlestata.comglm -Generalized linear modelsDescriptionQuic kstar tMen uSyntax

Options

Remar ksand e xamples

Stored results

Methods and f ormulas

Acknowledgments

Ref erences

Also see

Description

glmfits generalized linear models. It can fit models by using eitherIRLS(maximum quasilikelihood) or Newton-Raphson (maximum likelihood) optimization, which is the default. See[U] 27 Overview of Stata estimation commandsfor a description of all of Stata"s estimation commands, several of which fit models that can also be fit usingglm.

Quick start

Model ofyas a function ofxwhenyis a proportion

glm y x, family(binomial) Logit model ofyevents occurring in 15 trials as a function ofx glm y x, family(binomial 15) link(logit) Probit model ofyevents as a function ofxusing grouped data with group sizesn glm y x, family(binomial n) link(probit) Model of discreteywith user-defined familymyfamilyand linkmylink glm y x, family(myfamily) link(mylink) Bootstrap standard errors in a model ofyas a function ofxwith a gamma family and log link glm y x, family(gamma) link(log) vce(bootstrap) Menu Statistics>Generalized linear models>Generalized linear models (GLM) 1

2glm - Generaliz edlinear models

Syntax

glmdepvarindepvars if in weight ,options optionsDescriptionModel family(familyname)distribution ofdepvar; default isfamily(gaussian) link(linkname)link function; default is canonical link forfamily()specified

Model 2

noconstantsuppress constant term exposure(varname)include ln(varname) in model with coefficient constrained to 1 offset(varname)includevarnamein model with coefficient constrained to 1 constraints(constraints)apply specified linear constraints asisretain perfect predictor variables mu(varname)usevarnameas the initial estimate for the mean ofdepvar init(varname)synonym formu(varname)

SE/Robust

vce(vcetype)vcetypemay beoim,robust,clusterclustvar,eim,opg, bootstrap,jackknife,hackernel,jackknife1, orunbiased vfactor(#)multiply variance matrix by scalar# disp(#)quasilikelihood multiplier scale(x2jdevj#)set the scale parameter

Reporting

level(#)set confidence level; default islevel(95) eformreport exponentiated coefficients nocnsreportdo not display constraints displayoptionscontrol columns and column formats, row spacing, line width, display of omitted variables and base and empty cells, and factor-variable labeling

Maximization

mluse maximum likelihood optimization; the default irlsuse iterated, reweighted least-squares optimization of the deviance maximizeoptionscontrol the maximization process; seldom used fisher(#)use the Fisher scoring Hessian or expected information matrix (EIM) searchsearch for good starting values noheadersuppress header table from above coefficient table notablesuppress coefficient table nodisplaysuppress the output; iteration log is still displayed collinearkeep collinear variables coeflegenddisplay legend instead of statistics glm- Generaliz edlinear models 3 familynameDescriptiongaussianGaussian (normal) igaussianinverse Gaussian binomial varnameNj#NBernoulli/binomial poissonPoisson nbinomial #kjmlnegative binomial loglog logitlogit probitprobit cloglogcloglog power#power opower#odds power nbinomialnegative binomial logloglog-log logclog-complementindepvarsmay contain factor variables; see[U] 11.4.3 Factor variables. depvarandindepvarsmay contain time-series operators; see[U] 11.4.4 Time-series varlists. bayes,bootstrap,by,collect,fmm,fp,jackknife,mfp,mi estimate,nestreg,rolling,statsby, stepwise, andsvyare allowed; see[U] 11.1.10 Prefix commands. For more details, see[B AYES]bayes: glm and [FMM] fmm: glm. vce(bootstrap),vce(jackknife), andvce(jackknife1)are not allowed with themi estimateprefix; see [MI]mi estimate. Weights are not allowed with thebootstrapprefix; see[R] bootstrap. aweights are not allowed with thejackknifeprefix; see[R] jackknife. vce(),vfactor(),disp(),scale(),irls,fisher(),noheader,notable,nodisplay, and weights are not allowed with thesvyprefix; see[SVY] svy. fweights,aweights,iweights, andpweights are allowed; see[U] 11.1.6 weight. noheader,notable,nodisplay,collinear, andcoeflegenddo not appear in the dialog box. See[U] 20 Estimation and postestimation commandsfor more capabilities of estimation commands.

Options

Model family(familyname)specifies the distribution ofdepvar;family(gaussian)is the default. link(linkname)specifies the link function; the default is the canonical link for thefamily() specified (except forfamily(nbinomial)).

Model 2

noconstant,exposure(varname),offset(varname),constraints(constraints); see[ R]Esti- mation options.constraints(constraints)is not allowed withirls.

4glm - Generaliz edlinear models

asisforces retention of perfect predictor variables and their associated, perfectly predicted observations

and may produce instabilities in maximization; see [ R]probit. This option is allowed only with optionfamily(binomial)with a denominator of 1. mu(varname)specifiesvarnameas the initial estimate for the mean ofdepvar. This option can be useful with models that experience convergence difficulties, such asfamily(binomial)models with power or odds-power links.init(varname)is a synonym.

SE/Robust

vce(vcetype)specifies the type of standard error reported, which includes types that are derived from

asymptotic theory (oim,opg), that are robust to some kinds of misspecification (robust), that allow for intragroup correlation (clusterclustvar), and that use bootstrap or jackknife methods (bootstrap,jackknife); see[ R]vceoption. In addition to the standardvcetypes,glmallows the following alternatives: vce(eim)specifies that theEIMestimate of variance be used. vce(jackknife1)specifies that the one-step jackknife estimate of variance be used. vce(hackernel#)specifies that a heteroskedasticity- and autocorrelation-consistent (HAC) variance estimate be used.HACrefers to the general form for combining weighted matrices to form the variance estimate. There are three kernels built intoglm.kernelis a user-written program or one of nwestjgallantjanderson #specifies the number of lags. If#is not specified,N2 is assumed. If you wish to specify vce(hac:::), you musttssetyour data before callingglm. vce(unbiased)specifies that the unbiased sandwich estimate of variance be used. vfactor(#)specifies a scalar by which to multiply the resulting variance matrix. This option allows you to match output with other packages, which may apply degrees of freedom or other small-sample corrections to estimates of variance. disp(#)multiplies the variance ofdepvarby#and divides the deviance by#. The resulting distributions are members of the quasilikelihood family. This option is allowed only with option irls. scale(x2jdevj#)overrides the default scale parameter. This option is allowed only with Hessian (information matrix) variance estimates. By default,scale(1)is assumed for the discrete distributions (binomial, Poisson, and negative binomial), andscale(x2)is assumed for the continuous distributions (Gaussian, gamma, and inverse Gaussian). scale(x2)specifies that the scale parameter be set to the Pearson2(or generalized2) statistic divided by the residual degrees of freedom, which is recommended by

McCullagh and Nelder

1989
as a good general choice for continuous distributions. scale(dev)sets the scale parameter to the deviance divided by the residual degrees of freedom. This option provides an alternative toscale(x2)for continuous distributions and overdispersed or underdispersed discrete distributions. This option is allowed only with optionirls. scale(#)sets the scale parameter to#. For example, usingscale(1)infamily(gamma) models results in exponential-errors regression. Additional use oflink(log)rather than the defaultlink(power -1)forfamily(gamma)essentially reproduces Stata"sstreg, dist(exp) nohrcommand (see[ ST]streg) if all the observations are uncensored. glm- Generaliz edlinear models 5

Reporting

level(#); see[ R]Estimation options. eformdisplays the exponentiated coefficients and corresponding standard errors and confidence intervals. Forfamily(binomial) link(logit)(that is, logistic regression), exponentiation results are odds ratios; forfamily(nbinomial) link(log)(that is, negative binomial regression) and forfamily(poisson) link(log)(that is, Poisson regression), exponentiated coefficients are incidence-rate ratios. nocnsreport; see[ R]Estimation options. sformat(%fmt), andnolstretch; see[ R]Estimation options.

Maximization

mlrequests that optimization be carried out using Stata"smlcommands and is the default.

irlsrequests iterated, reweighted least-squares (IRLS) optimization of the deviance instead of Newton-

Raphson optimization of the log likelihood. If theirlsoption is not specified, the optimization is carried out using Stata"smlcommands, in which case all options ofml maximizeare also available. nrtolerance(#),nonrtolerance, andfrom(initspecs); see[ R]Maximize. These options are seldom used. Setting the optimization method totechnique(bhhh)resets the defaultvcetypetovce(opg). If optionirlsis specified, onlymaximizeoptionsiterate(),nolog,trace, andltolerance() are allowed. Withirlsspecified, the convergence criterion is satisfied when the absolute change in deviance from one iteration to the next is less than or equal toltolerance(), where ltolerance(1e-6)is the default. fisher(#)specifies the number of Newton-Raphson steps that should use the Fisher scoring Hessian orEIMbefore switching to the observed information matrix (OIM). This option is useful only for Newton-Raphson optimization (and not when usingirls). searchspecifies that the command search for good starting values. This option is useful only for Newton-Raphson optimization (and not when usingirls). The following options are available withglmbut are not shown in the dialog box: noheadersuppresses the header information from the output. The coefficient table is still displayed.

notablesuppresses the table of coefficients from the output. The header information is still displayed.

nodisplaysuppresses the output. The iteration log is still displayed. collinear,coeflegend; see[ R]Estimation options.collinearis not allowed withirls.

Remarks and examplesstata.com

Remarks are presented under the following headings:

General use

Variance estimators

User-defined functions

6glm - Generaliz edlinear models

General use

glmfits generalized linear models ofywith covariatesx: g

E(y)=x,yF

g()is called the link function, andFis the distributional family. Substituting various definitions forg()andFresults in a surprising array of models. For instance, ifyis distributed as Gaussian (normal) andg()is the identity function, we have

E(y) =x,yNormal

or linear regression. Ifg()is the logit function andyis distributed as Bernoulli, we have logit

E(y)=x,yBernoulli

or logistic regression. Ifg()is the natural log function andyis distributed as Poisson, we have ln

E(y)=x,yPoisson

or Poisson regression, also known as the log-linear model. Other combinations are possible. Althoughglmcan be used to perform linear regression (and, in fact, does so by default), this regression should be viewed as an instructional feature;regressproduces such estimates more quickly, and many postestimation commands are available to explore the adequacy of the fit; see [R]regressand[ R]regress postestimation. In any case, you specify the link function by using thelink()option and specify the distributional family by usingfamily(). The available link functions are

Link functionglmoptionidentitylink(identity)

loglink(log) logitlink(logit) probitlink(probit) complementary log-loglink(cloglog) odds powerlink(opower#) powerlink(power#) negative binomiallink(nbinomial) log-loglink(loglog) log-complementlink(logc) Define=E(y)and=g(), meaning thatg()mapsE(y)to=x+offset. glm- Generaliz edlinear models 7

Link functions are defined as follows:

identityis defined as=g() =. logis defined as=ln(). logitis defined as=ln=(1), the natural log of the odds. probitis defined as= 1(), where1()is the inverse Gaussian cumulative. cloglogis defined as=lnln(1). opoweris defined as==(1)n1=n, the power of the odds. The function is generalized so thatlink(opower 0)is equivalent tolink(logit), the natural log of the odds. poweris defined as=n. Specifyinglink(power 1)is equivalent to specifying link(identity). The power function is generalized so that0ln(). Thus,link(power

0)is equivalent tolink(log). Negative powers are, of course, allowed.

nbinomialis defined as=ln=(+k), wherek=1 iffamily(nbinomial)is specified, k=#kiffamily(nbinomial#k)is specified, andkis estimated via maximum likelihood if family(nbinomial ml)is specified. loglogis defined as=lnfln()g. logcis defined as=ln(1).

The available distributional families are

FamilyglmoptionGaussian (normal)family(gaussian)

inverse Gaussianfamily(igaussian)

Bernoulli/binomialfamily(binomial)

Poissonfamily(poisson)

negative binomialfamily(nbinomial) gammafamily(gamma) family(normal)is a synonym forfamily(gaussian). The binomial distribution can be specified as 1)family(binomial), 2)family(binomial#N), or 3)family(binomialvarnameN). In case 2,#Nis the value of the binomial denominatorN, the number of trials. Specifyingfamily(binomial 1)is the same as specifyingfamily(binomial). In case 3,varnameNis the variable containing the binomial denominator, allowing the number of trials to vary across observations. The negative binomial distribution can be specified as 1)family(nbinomial), 2)fam- ily(nbinomial#k), or 3)family(nbinomial ml). Omitting#kis equivalent to specifying family(nbinomial 1). In case 3, the value of#kis estimated via maximum likelihood. The value kenters the variance and deviance functions. Typical values range between 0.01 and 2; see the technical note belo w. You do not have to specify bothfamily()andlink(); the defaultlink()is the canonical link for the specifiedfamily()(except fornbinomial):

8glm - Generaliz edlinear models

Family Default link

family(gaussian) link(identity) family(igaussian) link(power -2) family(binomial) link(logit) family(poisson) link(log) family(nbinomial) link(log) family(gamma) link(power -1) If you specify bothfamily()andlink(), not all combinations make sense. You may choose from the following combinations: identity log logit probit cloglog power opower nbinomial loglog logcGaussian x x x inverse Gaussian x x x binomial x x x x x x x x x

Poisson x x x

negative binomial x x x x gamma x x xTechnical note Somefamily()andlink()combinations result in models already fit by Stata. These arequotesdbs_dbs14.pdfusesText_20

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