Log Negative Binomial Regression Using the GENMOO" Procedure SAS/STAT" Software

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1 Abstract Log Negative Binomial Regression Using the GENMOO" Procedure SAS/STAT" Software Joseph M. Hilbe Oepartment of Sociology, Arizona State University, Tempe, AZ The negative binomial model is a member of the GLM..type exponential family of distributions. The Iog-linked form of the distribution allows useful modeling of many types of overdispersed count data. The mathematical properties of the negative binomial are derived and a GLM-Iype Log Negative Binomial (thb) macro is developed using the SASlSTA" GENMOOo Procedure. Applications In terms of.-irnulation data and a clinical study are presented and modeled. KEY WORDS: SAS/sTAT. GENMOD Procedure. Generalized linear model. Poisson. Overdispersion, Negative binomial. Log negative binomial 1. Introduction The SASISTAr-GENMOO Procedure, inaugurated with SASe Version allows full generalized linear modeling (GLM) capability. Gaussian. binomial, Poisson, gamma, and inverse GaUssian models are built into the Procedure. But more important. allowance is also made for the creation of user-defined models. This feature enables one to extend standard GLMs to incorporate models which may be more appropriate for certain types of data situations. This is particularily the case when dealing with overdispersed discrete count data. Discrete count data is typically modeled using Poisson regression. However, there are various assumptions upon which the Poisson model is based. one being the equivalence of the mean and variance. When the value of the variance does exceed that of the mean, the data is said to be overdispersed. A measure of overdispersion is provided by the "dispersion" statistic. given in SAS output as VALUEIDF. In fact. SAS displays two dispersion values, one based upon the Chi2 statistic and the other upon the fmal deviance. A well specified Poisson model will have both dispersion values approximating 1.0. However. simulation studies have demonstrated that the Chil-baSed dispersion is a more optimal guage of overdispersion than i. the deviance-ba.ed... ti.tic (Hilbe. 1994a). I shall not discuss the various causes of overdispersion; there is much in the literature that addresses this difficulty. However, Ule essential problem with overdispersed models is that the standard errors of the predictors are deflated. When this occurs. predictors may appear to signirlcantly contribute to a model when in fact they do not Except for scaling standard errors by the value of the dispersion. hence creating a quasi... ikelihood model, the Poisson model cannot appropriately deal with overdispersed data. Unfortunately. overdispersion is a common occurrence in real data sets and statisticians have had few ifany alternative modeling tools to accommodate this problem. Log..fmked negative binomial regression (LNB) can be used to model many types of overdispersed count data situations. I shall first briefly describe the mathematical properties of the standard negative binomial and its noncanonieal Iog-Unked fonn and shall then present a LNB SAS macro. The macro will be examined using both synthetic data and an actual clinical study application. 2. Mathematical properties of the negative binomial The negative binomial probability distribution can be thought of as the probability that there will be y failures before the rth success in a series of Bernoulli trials. Taken in this manner, the negative binomial pdf appears as fy(y;i,p) i In the exponential family form of this is formulated as: exp{«yo-b(o))/lx(q») + c(y,q>)) 1199

2 The mean, II. and variance, V, are detennined respectively by differentiating b(9), r1n(p), and b'(9): Il = b' (6) V = b" (6) = r(l-p)/p = r(1-p)/p2 = Il + 1l 2 /r Obs~rving that 8 i& 1n(1~), the linear precflctor, t) or 8, is translated in tenns of II and r as: Hence the canonical inverse link function is a = -In «Il+r) Ill) Reparameterizing r so that it is directly proportional to p2 in the second term of the variance function. we derme k I: 1lf. Parameterized in terms of t. results of the above appear as: 6 or 11 Il V = -In(l+llkll) = l/k(e-"-l) Il + kll 2 The negative binomial pdf. parameterized in terms of II and k, can be formulated as: r(y+l/k) (klljy In exponential form the above appears as: f(y;il,k), = exp( yln(l/(l+l/kll» - l/k In(l+kll) +... } From this the Iog-likelihood and deviance functions can be derived: LL(Il;y,k), = yln(kll)-(y+l/k)ln(l+kll)+lnr(y+l/k)-lnr(y+l)-lnr(l/k) D/2 (Y;Il,k), yln(y/il) - (l+ky)/k In((l+ky)l(l+kll)) where lor Is the Iogilamma function. Knowledge of the link 9, inverse link p. variance. and deviance function are required when constructing a GLM algorithm. A comprehensive mathematical examination of the above relationships can be found in Hilbe (19948). 3. The GLM log negative binomial algorithm The methodology of generalized linear models (GLMs) requires that estimation be in terms of II; Le. in the case of the negative binomial k is placed into the variance and deviance function as a known constant as... varies. Placed within the common GLM estimation scheme the canonical negative binomial GLM paradigm can be structured as follows: Il = (y+mean(y»/2 1* Nelder initialization *1 11 = -In(llk)l+l) 1* link (k parameterization) *1 } WHILE (abs(adev»tolerance) { w Il+kll' /* variance function */ z 11 + (Y-Il) /w - offset 1* derivative * I ~ = (X'wX)-'X'wz 1* regression *1 11 = X'~ + offset 1* linear predictor *1 Il l/k(e- 8-1) 1* inverse link *1 olddev = dey dey = 2 ~ (yln(ylll) - (l+ky)/k In((l+ky)l(l+kll))} adev = dey - olddev 1200

3 A log negative binomial is constructed by substituting the Poisson variance, link, and inverse links in place of the canonical HB fonnulae. A variance adjustment of P/(1+kJJ) is placed into the algorithm direcuy following the derivative. The goal of the LHB algorithm is to adjust the value of k so that the chi2~sed or deviance based dispersion approximates 1, which is the optimal value for a Poisson model k represents Ule amount of positive heterogeneity in the otherwise Poisson count data. AS previously indicated, simulation studies (Hilbe, 1994a) have given support to the notion that the optimal LNB model results when k is adjusted in such a manner that the Chi2~sed dispersion. der-ned as (I(y-ll)2N(p)}/(df), approximates 1.0. I have designed a LNB SAS macro called hilbenbo which iteratively converges to a dispersion value of 1.0. k as well as parameter estimates are re-estimatec:l with each iteration. The initial value of k is taken as the inverse of the Poisson Chi2 dispersion. The algorithm is structured as:. Poisson Y <predictors> chi2 = l: (Y-Jl) '/Jl disp = chi2/df ex=l/disp j=l while (abs(adisp»tolerance) oldisp = disp LNB Y <predictors>, k=a chi2 = l: (Y-Jl)'1 (Jl+kJl') disp = chi2/df a = disp*a Adisp = disp - oldisp j = j+l 4. Log negative binomial regression: Synthetic data The LNB model is based upon the premise that events enter a period of observation with a gamma distribution. Noting that the Poisson and gamma variances are p and 1J 2 /r respectively, Ule negative binomial is considered as a Poisson-gamma mixture distribution with a variance ofp... 2/r.1 have previously reparameterized this as p+kp2 to allow a direct linear relationship in the second tenn. The parameter k can be regarded as a heterogeneity factor and is entered into the function as a known constant In fact. the standard negative binomial formulation requires that k be fixed and independent of p.lf it is parameterized in any other manner, then the distribution is not a member of a GLM type exponential family (Neider 1993). In order to evaluate the viability of the macro,l generated a observation array distributed as standard HB with a mean of 10 and a k value of 0.1. The data was created outside of SAS and saved as an external ASCII file called nbbig.raw. The mean and standard deviation of the HB distributed variable, called xnb. is: N Mean Std Dev Minimum Maximum o Recalling that V c P + 1qJ2, a value of k may be calculated as: k = «StD*StD)-Mean)/(Mean*Mean) In th.. c e, k. « " ) ( * ) =.1012 xnb "" be modeled.s a constant-only negative binomial regression using hilbenb with the following c;ode: Relevant output.. d"played: data nbbig; infile 'c:\msas_win\nbbig.raw'; input xnb 8-9; [hilbenb macro] %hilbenb(dsin=nbbig, yvar=xnb); 1201

4 Log Negative Binomial Regression Criterion DF Value Value/DF Deviance Scaled Deviance Pearson Chi-Square Scaled Pearson X2 Log Likelihood Log Negative Binomial Regression Parameter DF Estimate Std Err ChiSquare Pr>Chi INTERCEPT Number of iterations: 11 Estimated alpha The estimated value of alpha (k) is nearly identical to the value of k calculated based on the summary information. Exponentiating the estimated constant should result in the variable mean: The result is thoroughly consistent with what we expected. exp(2.3029) = Synthetic data sets may also be created which incorporate predictors. These more clearly simulate real data situations with the advantage that they may be manipulated to test a variety of data types. In order to evaluate the macro using complex data J created a observation data set having two positive random variates, xi and x2, as predictor. having Je$pective paramelervalues 0(0.5 and 3. t also specjf'.ed a constant with a value of 2. The linear predictor is calculated as: The inverse link yields the value of... Ip = *x1 + 3*x2 \l = exp (lp) SubmiUing II to a NB random number generator with a k value of.1 produces a LNB variate called xnb. Using the macro call: results in the following relevant output %hilbenb(dsin=nbbigx, yvar=xnb, xvars=x1 x2); Log Negative Binomial Regression Criterion DF Value Value/DF Deviance Scaled Deviance Pearson Chi-Square Scaled Pearson X Log Likelihood

5 Log Negative Binomial Regression Parameter DF Estimate Std Err Chi Square INTERCEPT Xl X Pr>Chi NUmber of iterations: 8 Estimated alpha This accords well with the parameters and value of k supplied to the original data. Employing the same model using Poisson regression results in estimates that are somewhat similar to those produced by the LNB model. but the.tandard error are overty optimiatic.. Standard goodnes.~f4it statistics also fail: the ValueIDF.tatistics. or dispersion. far exceed the ideal Poisson value of 1.0. criterion DF Value Value/DF Deviance 9997 Scaled Deviance 9997 Pearson Chi-Square Scaled Pearson X Log Likelihood Parameter DF Estimate Std Err ChiSquare Pr>Chi INTERCEPT Xl X The LNB macro appear. to consistently and appropriately model LHB data. It is also clear that the foremost feature of LNB regresskm is that it properly adjusts the standard errors for otherwise overdispersed Poisson count data. 5. Log negative binomial regression: An application Convulsive episodes reportedly occur in approximately 1% of all hospitalized adult patients. However. published studies on seizure rates and associated risk factors have primarily been characterized by s.mall s.ample sizes. The following study performed by Health Sciences Institute in Pennsylvania under a grant from Merck Human Health Division. examined computerized hospital patient discharge abstracts from three consecutive years of the National Hospital Discharge SUNey (NHDS. 1989"'1). Data was grouped by known risk factors. This type of data is typically submitted to a logistic or Poisson regress.ion. However, we found that the data was greatly overdisperaed and that all published goodnes.s~f..fit tests failed. The data was. -subsequently shown to be correctly specified as a LNB model with the log of the number of cases having a unique covariate pattern entered as an offset The final model includes the following significant risk factors of non-epiieptic seizures: cvd = cerebrovascular dis.ease adr = adverse drug reaction IIlJ II head injuries uti = uniary tract infections psych = psych. conditions including depression & schizophrenia The following command yields: tum ak:h = brain tumor = ak:helism %hilbenb(osin=seiz,yvar=nums,xvars=cvd adr tum inj uti alch psych,offvar=lnumr); 1203

6 Log Negative Binomial Regression Criterion DF Value Value/DF Deviance Scaled Deviance Pearson Chi-Square Scaled Pearson X2 Log Likelihood Parameter DF Estimate Std Err ChiSquare Pr>Chi INTERCEPT CVD ADR TUM INJ UTI ALCH PSYCH Number of iterations: 9 Estimated alpha Renal failure. widely thought to be a aignif'cant predictor of non-epileptic seizures, failed to enter into the model as previously expected. Incidence rate ralto. may be calculated by exponentiating the respective coerrlcien'ts. We {"mel that having a brain tumor increase. the risk of having a seizure nearty sixfold. 6. Summary Log negative binomial regression is a valuable member of the family of generarlzed tinear modela. For discrete data, when there Is independence between observation. and heterogeniety is not due to longitudinal effects, the LNB model appears to be U1e optimal GLM-type modeling tool Acknowledgements should like to acknowledge Ule kind assistance of Walter Linde.zwirble. Health outcomes Technologies, Doylestown. PA. His work on the random number generators as well as discussions on the nature of the negative binomial were most helpful Gordon Johnston, SAS Institute assisted in writing the SAS Ul8 macro and Richard Newbold, Health Outcomes Technologies, provided clinical insights regarding the seizure data. I also wish to acknowledge Merek HUman Health Division, Michael Steck., Director, for funding the seizure study. References Hilbe, J. (1993) GLM: A unif"jed power~ink based program including the negative binomial Stata Teehnlc.I BuYetln 14: HUbe, J. (19948) Log negative binomial regression as a generalized linear model, Technical Report 21, Committee on Statistics, Graduate College, Arizona State University. Hilbe, J. (1994b) Generalized linear models. The American Statistician. Forthcoming. Hilbe, J. (1994c) Modeling overdispersed count data using log negative binomial regression. Proceedings oftha Am~n stljtistical Society. Forthcoming. McCullagh, P. and Nekler. J. A. (1989) Generalized Unear Models. 2nd ed. New You: Chapman & HaiL Neider, J. (1993) Generalized Unear Models with Negative Binomial or 8eta-binomial Errors, unpubushed manuscript. 1204