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A cautionary note regarding count models of alcohol consumption in randomized controlled trials

A cautionary note regarding count models of alcohol consumption in randomized controlled trials Background: Alcohol consumption is commonly used as a primary outcome in randomized alcohol treatment studies. The distribution of alcohol consumption is highly skewed, particularly in subjects with alcohol dependence. Methods: In this paper, we will consider the use of count models for outcomes in a randomized clinical trial setting. These include the Poisson, over-dispersed Poisson, negative binomial, zero- inflated Poisson and zero-inflated negative binomial. We compare the Type-I error rate of these methods in a series of simulation studies of a randomized clinical trial, and apply the methods to the ASAP (Addressing the Spectrum of Alcohol Problems) trial. Results: Standard Poisson models provide a poor fit for alcohol consumption data from our motivating example, and did not preserve Type-I error rates for the randomized group comparison when the true distribution was over-dispersed Poisson. For the ASAP trial, where the distribution of alcohol consumption featured extensive over-dispersion, there was little indication of significant randomization group differences, except when the standard Poisson model was fit. Conclusion: As with any analysis, it is important to choose appropriate statistical models. In simulation studies and in the motivating example, the standard Poisson was not robust when fit to over-dispersed count data, and did not maintain the appropriate Type-I error rate. To appropriately model alcohol consumption, more flexible count models should be routinely employed. Background In this setting, estimating differences between treatment Count outcomes are common in randomized studies of group and control group is of primary interest. alcohol treatment. Subjects may be queried about their daily consumption of alcohol, measured as a number of A challenge in modeling consumption outcomes is to drinks over a recent period [1] (typically 30 days), and appropriately account for the distribution of drinking. these values are used to estimate average drinking per day. These distributions are characterized by a large number of zeros (abstinent subjects) along with a long right tail Page 1 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 (heavy drinking subjects). An extensive literature uniquely specifies this distribution, and is equal to the describes models for counts [2-8], and they have been expected value (mean) and variance (i.e. E[Y ] = Var(Y ) = ij ij commonly applied in economic analyses, traffic acci- λ for all i and j). The maximum likelihood estimate ij dents, and health services utilization. Many routines are now available in general purpose statistical software (e.g. (MLE) of λ is given by Y . In this setting, the test of ran- i i Stata) [8]. A natural model for counts is the single-param- domized group effects for the Poisson model is a test of eter Poisson distribution. One disadvantage of the Pois- the null hypothesis that λ = λ . 1 2 son is that it makes strong assumptions regarding the distribution of the underlying data (in particular, that the One limitation of this model is that it may be overly sim- mean equals the variance). While these assumptions are plistic and may not provide an adequate fit to consump- tenable in some settings, they are less appropriate for alco- tion data of the type that we consider. The constraint that hol consumption. Extensions of the Poisson, such as the the variance is equal to the mean may lead to incorrect test over-dispersed Poisson, negative binomial and two stage results. (hurdle) or zero inflated models have been proposed [2- 5]. Consider as an example the data from the ASAP study con- trol group at 3 months. For this dataset, non-integer count Our methods are motivated by the analysis of the ASAP values are possible. These arise when subjects consume a (Addressing the Spectrum of Alcohol Problems) study, a number of drinks not divisible by 30 (in the case of 30- randomized clinical trial comparing a brief motivational day assessments). One approach in this situation would interview to usual care for a sample of inpatients with be to model the number of drinks consumed in a 30 day unhealthy alcohol use at an urban hospital [9]. These sub- period, or utilize the non-integer values. Sometimes even jects were followed to see if there were differences in the 30 day value is non-integer because people report a drinking outcomes that could be attributed to rand- drink size that is then translated into standard drinks. The omized group assignment. maximum likelihood estimates of the probability distri- butions remains the same for non-integer values, though In this paper, we will demonstrate the limitations of the it is necessary to move each non-integer observed value to standard Poisson model in the presence of over-disper- the next integer (using a ceiling function) to be plotted. sion. We begin by describing several count models for For the models that we discuss, we can plug non-integer alcohol outcomes, compare their performance in a series values into the software and still get sensible results. of simulated randomized trials, apply them to the ASAP study, and conclude with some general recommenda- tions. Figure 1 displays the observed distribution and superim- posed Poisson with λ = Y = 4.98 using the prcounts 1 1 Methods routine in Stata [8]. The axis for the number of drinks per Statistical methods for the analysis of count outcomes day after 3 months was limited to 25 drinks to improve We begin by introducing notation to be used throughout. readability (the maximum observed count was 48.6). Let Y denote the number of events for the jth subject (j = ij There is a pronounced lack of fit to this model, particu- 1,..., n ) in the ith group (i = 1, 2), where n is the number i i of subjects in the ith group. Typically in a randomized trial larly for values of less than 10 drinks per day. For the ASAP n and n are approximately equal. 1 2 data, the assumption that the mean is equal to the vari- ance is not tenable. In fact, the observed variance (71.7) is The Poisson distribution is one of the simplest models for more than an order of magnitude larger than the mean. count data. Let λ indicate the average number of events ij Also, note that there is some evidence for digit preference (in this case drinks consumed) in a given time interval for (even numbers are more common than odd numbers). subject j in group i, where f(Y = k|λ ) is the probability of ij ij observing k events. The Poisson distribution [8,10] is One approach to loosen the restrictive variance assump- denoted: tion involves use of an empirical (or robust or sandwich) variance estimator [11-13] to account for the over-disper- exp(−λλ ) sion. This more flexible extension of the Poisson allows ij ij PY(|== k λ) ij ij the variance to be unconstrained. The over-dispersed Pois- k! son option is available in a number of general purpose for k = 0, 1, 2, ..., i = 1, 2, and j = 1,..., n where λ > 0 and i ij statistics packages (e.g. the robust option in Stata). we assume that λ = λ for all j (i.e. all subjects in a given ij i Another approach is to fit a negative binomial (two group have the same rate of drinking). The λ parameter parameter) count model (NB) [5-8,10]. One common Page 2 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 observed Poisson predicted 0 5 10 15 20 25 # of drinks per day at 3 months Observed value of drinks per day for the control group of the ASAP study at 3 months, plus the estimated Poisson fit to these data ( = 4.98) Figure 1 Observed value of drinks per day for the control group of the ASAP study at 3 months, plus the estimated Poisson fit to these data ( λ = 4.98). parametrization of the negative binomial distribution is The negative binomial model is attractive because it given by: allows the relaxation of strong assumptions regarding the relationship between the mean and the variance. This flex- ibility comes at some cost, since a two-parameter model is PY(|== k λθ, ) ij i i inherently more complicated to interpret. − −1 θ k −1 −1 ⎛ ⎞ ⎛ ⎞ Γ() θ + k θ λ i i i Other models have been proposed that allow for an extra ⎜ ⎟ ⎜ ⎟ −1 −1 −1 ⎜ ⎟ ⎜ ⎟ ΓΓ () θ (k++ 1) θλ θλ + abundance of subjects with no consumption. In alcohol i ⎝ ii ⎠ ⎝ ii ⎠ consumption outcomes, there may be subjects who are "non-susceptible" (e.g. abstinent). These "zero-inflation" where Γ(·) denotes the Gamma function, λ > 0 and θ > i i (or "hurdle") models account for subjects who are struc- 0. We note that E[Y ] = λ and Var(Y ) = λ+ λ * θ = λ * tural zeros (e.g., abstinent subjects thought of as "non- ij i ij i i i i susceptible") [2,3]. Conditional on being susceptible (1 + λ * θ ) for all i and j and that Var(Y ) > E[Y ]. It can i i ij ij (with some probability), the distribution is assumed to be be shown that the negative binomial can be derived in Poisson or negative binomial. terms of a Poisson random variable where the parameter λ varies according to a gamma distribution. Page 3 of 9 (page number not for citation purposes) probability 0.00 0.05 0.10 0.15 0.20 BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Zero-inflated Poisson (ZIP) models [3] separately esti- process was repeated 2500 times for each set of parame- mate a parameter p that governs the proportion of non- ters, where E[Y ] = λ = 5 (taken from the ASAP control i i α level of 0.05 was used. For the simulation susceptible subjects in the ith group: group) and an of Poisson data the variance was equal to the mean. Neg- ative binomial distributions were simulated using three fY(|== k λ,p) ij i i arbitrary variances (13.3, 40 and 70), with the latter value exp(−λλ ) comparable to the observed variance from the ASAP con- ii Ik()=+01 p (− p) , ii trol group. The zero-inflated model had a probability of k! 0.2 of being a structural zero, and Poisson with λ = 5 oth- erwise. The true distributions for the simulations are dis- for 0 <p < 1 and λ > 0 where I(k = 0) is equal to 1 when k i i played in Figure 2. Models were fit using the Poisson = 0, and equal to 0 otherwise. By distinguishing Always-0 distribution, over-dispersed Poisson using an empirical (with probability p ) and Not Always-0 group (with proba- variance estimator, negative binomial and zero inflated bility (1 - p ) * exp(-λ )) for abstainers and drinkers who i i Poisson. We estimated the probability that each model didn't drink during the reporting period, respectively, it rejected the null hypothesis and constructed a 99% confi- can incorporate an overabundance of zeros [8]. Condi- dence interval around this estimate. The code for the sim- tional on being a Not Always-0, counts are given by the ulations is available upon request from the first author. Poisson distribution. This approach has been generalized to a regression framework, and implemented in general ASAP study purpose statistical software (e.g. zip in Stata). The ASAP study was a randomized clinical trial of the effectiveness of a brief motivational intervention [14] on In many settings, the assumption that after accounting for alcohol consumption among a group of hospitalized the zeros the remaining counts are Poisson may not be patients at Boston Medical Center. Details of the recruit- tenable. The zero-inflated negative binomial (ZINB) ment procedures, inclusion criteria, description of sample allows for over-dispersion in this manner, though at the and results of the RCT have been published [15]. The cost of more parameters. Institutional Review Board of Boston University Medical Center approved this study, and the Institutional Review Another approach to the modeling of count data involves Board of Smith College approved the secondary analyses. use of a linear model (assuming that the observations are After consenting to enroll, all subjects received an inter- approximately Gaussian). While this is an extremely flex- viewer-administered baseline assessment prior to rand- ible model that is typically robust to misspecification omization into the control or intervention group. Subjects (since the mean and variance are not linked), the linear were randomly assigned to control or intervention group model is less attractive because it may predict negative val- using a blocked randomization procedure. Intervention ues of drinking given the skewness of the distribution. Use subjects participated in a brief motivational interview of a linear model is also inefficient if the variance is a func- with a counselor (less than half an hour). Control subjects tion of the mean. received usual care. Simulation study Follow-up was planned at 3-month and 12-month time- To better understand the behavior of these methods in a points. Because the subjects came from a transient and known situation, we conducted a series of simulation hard-to-reach population, the researchers employed studies with parameters derived from the motivating exhaustive techniques to track subjects over the follow-up example. These simulation studies were designed to period. The two primary alcohol-related outcomes were address the question of whether or not the models were measures of alcohol consumption and linkage to appro- robust to misspecification of the underlying count distri- priate alcohol treatment; for these secondary analyses we bution. More formally, we wanted to assess whether these focus solely on treatment differences in alcohol consump- models preserved the appropriate Type-I error rate (the tion. The outcome of interest was the average number of probability of rejecting the null hypothesis when it is true) standard drinks consumed per day in the past thirty days when there are no true differences between groups (i.e. do as reported using the Timeline Followback method [1] at they reject the null at the appropriate α level). the 3 and 12-month interviews. For the purpose of this secondary analysis we consider the 3 month time point; For each set of parameters within a simulation, 100 obser- similar results were seen utilizing 12 month data (not vations were generated in each of two groups, to mimic a reported here). randomized clinical trial setting. The amount of alcohol consumption, in drinks per day was the outcome. For Eight models were fit comparing treatment to control for each simulated dataset a series of models (Poisson, nega- the ASAP study: tive binomial and zero-inflated Poisson) were fit. This Page 4 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 lambda=5 NB (Var=70) NB (Var=40) NB (Var=13.3) ZIP (Var=8) Poisson (Var=5) 0 5 10 15 20 25 # of drinks per day Graphical display of the five di m p Figure 2 = ial [NB13, Var 0.2, Var = 8]= 13 ) ], negative st binom ributioins, al [NB40, Var all with rate p = 4a 0ram ], ne ega ter 5, used in the simulations tive binomial [NB 70, Var = 70] (Poisson [Var = and zero-inflated 5], negative bi Poisson [ZI noP - , Graphical display of the five distributions, all with rate parameter 5, used in the simulations (Poisson [Var = 5], negative bino- mial [NB13, Var = 13], negative binomial [NB40, Var = 40], negative binomial [NB 70, Var = 70] and zero-inflated Poisson [ZIP, p = 0.2, Var = 8]). Poisson standard Poisson model, WILCOXON Wilcoxon-Mann-Whitney, a non-parametric two-sample comparison procedure suitable for ordinal Over-dispersed Poisson Poisson model with empirical data, and ("robust") variance estimator, PERMUTE two-sample permutation test. NB negative binomial, Results ZIP zero-inflated Poisson, shared inflation parameter esti- Simulation studies mated for both randomized groups (p = p ), In the simulation studies we assessed the behavior of 1 2 models when the null hypothesis was true (there were no ZINB zero-inflated negative binomial, shared inflation differences between alcohol consumption for groups 1 parameter estimated for both randomized groups (p = and 2). We note that the ZIP model failed to converge for p ), more than a quarter of the simulations from the standard Poisson distribution. This is likely due to the fact that TTEST two-sample unequal variance t-test, many datasets had no zeros whatsoever (for the Poisson distribution with λ = 5, the probability that a dataset has no zeros whatsoever is equal to (1 - exp(-5)) = 0.51). Page 5 of 9 (page number not for citation purposes) probability 0.00 0.05 0.10 0.15 0.20 0.25 BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Table 1: Estimated probability (and 99% CI) of rejecting the null hypothesis when there is no true difference between groups for a variety of statistical models and underlying distributions (results that do not include the alpha level of 0.05 are bolded) Analysis model fit True Distribution: Poisson ODP NB ZIP Poisson (Var = 5) .053 (.041,.064) .054 (.042,.066) .047 (.036,.058) .055* (.043,.067) NB (Var = 13) .225 (.204,.247) .049 (.038,.060) .049 (.038,.060) .050 (.039,.061) NB (Var = 40) .467 (.441,.493) .047 (.036,.058) .044 (.033,.055) .046 (.036,.057) NB (Var = 70) .584 (.558,.609) .052 (.041,.063) .048 (.037,.059) .062 (.049,.074) ZIP (Var = 8) .179 (.159,.199) .058 (.046,.070) .031 (.022,.040) .051 (.040,.063) all distributions except ZIP have E[Y ] = λ = 5, for ZIP E[Y ] = 0.8 * 5 = 4. i i ODP (over-dispersed Poisson); NB (negative binomial); ZIP (zero-inflated Poisson) * For the true distribution under the Poisson, the ZIP model failed to converge for n = 672 of the simulations. Table 1 displays the estimated Type I error rate (when an overall response rate of 79%. Table 2 displays the dis- there is no difference between the groups) when α was set tribution of drinks per day at baseline and 3-month fol- to 0.05. The negative binomial model was conservative low-up separately for each group. As noted earlier, when the underlying data were zero-inflated. When the drinking outcomes are highly skewed to the right, with underlying distributions were not Poisson, the Poisson some extremely large values. These extreme values are model did not maintain the appropriate Type I error rate. plausible given the large number of dependent drinkers in When the count models were over-dispersed by a factor of the sample, many of whom have developed tolerance (the more than 2 (i.e. Var(Y ) > 2 * E[Y ]), the Poisson model need to consume large amounts of alcohol to induce i i rejected more than 22% of the time. When the over-dis- effects). We also note that reported drinking quantities persion was more extreme (factor of 8 and 14), the Type I decreased for both groups between baseline and 3-month error rate was 47% and 58%, respectively. The severe lack outcome. of robustness of the Poisson model in this setting is a seri- ous concern. Table 3 displays the results from the ASAP study using a variety of count models. Use of the Poisson model yielded ASAP study a statistically significant p-value, in contrast to the other Of 341 subjects enrolled in the clinical trial, 169 subjects methods (all other p-values > 0.45). were randomized to the control group and the other 172 into the intervention group. The mean age of the subjects Figure 3 displays the observed and predicted counts for was 44.3 (SD = 10.7). Twenty-nine percent were women, the Poisson, negative binomial, and ZIP models, while 45% were Black, 39% White, 9% Hispanic, and 7% Other. Figure 4 displays the plot of (observed minus expected) Sixty-three percent were unemployed during the past for the Poisson, negative binomial and ZIP models for the three months and 25% of the subjects were homeless at control group. The standard Poisson model provides an one point during the past three months. Four percent of unsatisfactory fit, and is not appropriate for the analysis of the subjects met criteria for current (past year) alcohol this dataset. The fit of the zero-inflated Poisson is abuse and 77% were alcohol dependent. improved, particularly for modeling the probability of no drinking, but remains unsatisfactory over most of the We analyze the 3-month follow-up data for which 271 remaining values. The negative binomial provides an subjects were observed (141 control, 130 treatment), for excellent fit for these data, and that there is no indication Table 2: Distribution of drinking outcome by timepoint and randomization group Base line 3 Months C (n = 169) T (n = 72) C (n = 141) T (n = 130) MIN 0.17 0 0 0 25th percentile 1.14 1.32 0.17 0.13 MEDIAN 3.47 3.85 1.8 1.6 75th percentile 8.23 9.12 6.1 5.7 MAX 61.77 60 48.6 38.43 mean (SD) 6.95 (9.58) 6.68 (8.44) 4.98 (8.47) 4.36 (6.47) Page 6 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Table 3: p-values for the ASAP randomization group effect at 3 months for a variety of count models MODEL p-value Poisson .018 over-dispersed Poisson .489 Negative binomial .458 zero-inflated Poisson .542 zero-inflated negative binomial .489 t-test .495 Wilcoxon .805 Permutation .746 that any further zero-inflation is needed, since the model In this setting, there was little indication from the already overpredicts zeros (hence the predicted values for observed plots that there were significant group differ- the NB and ZINB would be identical). ences. As seen in the simulation studies, the Poisson may observed (C) Poisson pred. (C) observed (T) Poisson pred. (T) Wilcoxon p−value=0.80 Poisson p−value=0.02 0 5 10 15 20 25 0 5 10 15 20 25 # of drinks per day at 3 months # of drinks per day at 3 months Negative binomial pred. (C) zero−inflated Poisson pred. (C) Negative binomial pred. (T) zero−inflated Poisson pred. (T) Negative binomial p−value=0.46 ZIP p−value=0.54 0 5 10 15 20 25 0 5 10 15 20 25 # of drinks per day at 3 months # of drinks per day at 3 months Wilco Figure 3 Observed an xon, Po d p issr oedicted n, negative bi values fr nomial om th and ze e ASAP study ro-inflate at 3 months d Poisson for control and treatment groups for each of four models: Observed and predicted values from the ASAP study at 3 months for control and treatment groups for each of four models: Wilcoxon, Poisson, negative binomial and zero-inflated Poisson. Page 7 of 9 (page number not for citation purposes) probability probability 0.00 0.10 0.20 0.00 0.10 0.20 probability probability 0.00 0.10 0.20 0.00 0.10 0.20 BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Poisson ]HURïLQIODWHG3RLVVRQ QHJDWLYHELQRPLDO REVHUYHGïH[SHFWHG 0 5 10 15 20 25 # of drinks per day at 3 months Obser and z Figure 4 eved minus ex ro-inflated Poisson pected values from the ASAP study at 3 months as a function of count for the Poisson, negative binomial Observed minus expected values from the ASAP study at 3 months as a function of count for the Poisson, negative binomial and zero-inflated Poisson. not have preserved the appropriate Type I error rate due to other models, which had highly non-significant results). the extremely large values of drinking for some subjects. The unrealistic assumption that the expected rate of drink- The Appendix includes the Stata commands to fit these ing is the same for all subjects may partially account for models and the output, along with the code to generate the poor fit of the Poisson distribution. We caution observed and predicted plots using the prcounts routine. against use of the Poisson for this analysis. The negative binomial fit particularly well, and we saw no evidence for zero-inflation. Discussion and conclusion A number of models have been proposed for the analysis of count data, and these models are now available in gen- In settings where there are excess zeros, zero-inflation eral purpose statistical packages. We have described these models are attractive. One advantage of these models is methods in the context of modeling reports of alcohol that they can estimate the probability of being a zero as a consumption, where a large proportion of respondents function of covariates, as well as allowing the rate param- report no drinking, and a small number of respondents eter to be a function of covariates. In an alcohol study, the typically account for an extreme amount of drinking. intervention may be hypothesized to affect the abstinence proportion as well as the rate parameter for drinkers. Ad- For the analysis of the ASAP study, we found that the hoc methods in this setting might involve estimating the standard Poisson had an extremely poor fit, and yielded a proportion of drinkers at follow-up, and in a separate statistically significant p-value (in contrast to all of the model, estimating the amount of drinking amongst the Page 8 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 subset of subjects who reported any drinking. A more References 1. Sobell LC, Sobell MB: Timeline follow-back: a technique for principled approach involves the simultaneous estima- assessing self-reported alcohol consumption. In Measuring Alco- = p ) and the tion of the zero-inflation factor (testing p 1 2 hol Consumption: Psychosocial and Biochemical Methods Edited by: Litten rate parameter (testing λ = λ ). Slymen and colleagues [2] RZ, Allen JP. Totowa: Humana Pr, Inc.; 1992:41-69. 1 2 2. Slymen DJ, Ayala GX, Arredondo EM, Elder JP: A demonstration of adopted this approach by simultaneously fitting separate modeling count data with an application to physical activity. models for what they describe as the "logistic" component Epidemiologic Perspectives & Innovations 2006, 3(3):1-9. 3. Lambert D: Zero-inflated Poisson regression, with an applica- and the "Poisson" component, and this approach is also tion to defects in manufacturing. Technometrics 1992, 34:1-14. detailed in books by Winkelmann [7] as well as Cameron 4. Cameron AC, Trivedi PK: Regression analysis of count data Cambridge, and Trivedi [4]. UK: Cambridge University Press; 1998. 5. Gardner W, Mulvey EP, Shaw EC: Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative The results of the simulation studies and the secondary binomial models. Quantitative Methods in Psychology 1995, analyses of the ASAP study demonstrated the importance 118(3):392-404. 6. Hilbe JM: Negative binomial regression: modeling overdispersed count data of appropriately modeling count outcomes. We caution Cambridge: Cambridge University Press in press. against the use of the standard Poisson model when the 7. Winkelmann R: Econometric analysis of count data fourth edition. Ber- lin: Springer-Verlag; 2003. mean and variance are not equal. Extensions of the Pois- 8. Long JS, Freese J: Regression models for categorical dependent variables son (incorporating an over-dispersion parameter or use of using Stata Texas: Stata Press Publication; 2003. the negative binomial distribution and/or zero-inflated 9. Saitz R, Freedner N, Palfai TP, Horton NJ, Samet JH: The severity of unhealthy alcohol use in hospitalized medical patients: the models) are now available in general purpose statistical spectrum is narrow. Journal of General Internal Medicine 2006, software, and address many of the shortcomings of the 21(4):381-5. overly simplistic Poisson model. 10. McCullagh P, Nelder JA: Generalized linear models Chapman & Hall; 11. Zeger SL, Liang KY: Longitudinal data analysis for discrete and As always, analysts are obliged to look at their data and continuous outcomes. Biometrics 1986, 42:121-130. 12. Liang KY, Zeger SL: Longitudinal data analysis using general- utilize models that provide an appropriate fit in their sit- ized linear models. Biometrika 1986, 73:13-22. uation. In particular, for models of alcohol consumption, 13. Huber PJ: The behavior of maximum likelihood estimates attention should be paid to the functional form of the out- under non-standard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1967, 1:221-233. come to ensure that underlying assumptions of the meth- 14. Miller WR, Rollnick S: Motivational interviewing: Preparing people to ods utilized are met. change addictive behavior second edition. New York: Guilford Press; 15. Saitz R, Palfai TP, Cheng DM, Horton NJ, Freedner N, Dukes K, Krae- Authors' contributions mer KL, Roberts MS, Guerriero RT, Samet JH: Brief intervention NH conceived of the project and provided overall guid- for medical inpatients with unhealthy alcohol use: A rand- omized-controlled trial. Annals of Internal Medicine 2007, ance, in addition to reviewing and interpreting analyses, 146(3):167-176. and drafting the manuscript. EK participated in the draft- ing of the manuscript, and carried out analyses and simu- Pre-publication history lations. RS led the ASAP study and participated in the The pre-publication history for this paper can be accessed drafting of the manuscript. All authors read and approved here: the final version of the manuscript. http://www.biomedcentral.com/1471-2288/7/9/prepub Additional material Additional File 1 Appendix. Stata code and results for count models. Click here for file [http://www.biomedcentral.com/content/supplementary/1471- 2288-7-9-S1.pdf] Publish with Bio Med Central and every scientist can read your work free of charge "BioMed Central will be the most significant development for disseminating the results of biomedical researc h in our lifetime." Acknowledgements Sir Paul Nurse, Cancer Research UK This research was supported in part by the National Institute on Alcohol Your research papers will be: Abuse and Alcoholism R01-AA12617, the Smith College Summer Research Program and the Howard Hughes Medical Institute. Thanks to Jessica Rich- available free of charge to the entire biomedical community ardson for editorial assistance, Emily Shapiro and Min Zheng for assistance peer reviewed and published immediately upon acceptance with simulations and Joseph Hilbe and Jeffrey Samet for helpful comments cited in PubMed and archived on PubMed Central on an earlier draft. yours — you keep the copyright BioMedcentral Submit your manuscript here: http://www.biomedcentral.com/info/publishing_adv.asp Page 9 of 9 (page number not for citation purposes) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png BMC Medical Research Methodology Springer Journals

A cautionary note regarding count models of alcohol consumption in randomized controlled trials

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Springer Journals
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Copyright © 2007 by Horton et al; licensee BioMed Central Ltd.
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Medicine & Public Health; Theory of Medicine/Bioethics; Statistical Theory and Methods; Statistics for Life Sciences, Medicine, Health Sciences
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Abstract

Background: Alcohol consumption is commonly used as a primary outcome in randomized alcohol treatment studies. The distribution of alcohol consumption is highly skewed, particularly in subjects with alcohol dependence. Methods: In this paper, we will consider the use of count models for outcomes in a randomized clinical trial setting. These include the Poisson, over-dispersed Poisson, negative binomial, zero- inflated Poisson and zero-inflated negative binomial. We compare the Type-I error rate of these methods in a series of simulation studies of a randomized clinical trial, and apply the methods to the ASAP (Addressing the Spectrum of Alcohol Problems) trial. Results: Standard Poisson models provide a poor fit for alcohol consumption data from our motivating example, and did not preserve Type-I error rates for the randomized group comparison when the true distribution was over-dispersed Poisson. For the ASAP trial, where the distribution of alcohol consumption featured extensive over-dispersion, there was little indication of significant randomization group differences, except when the standard Poisson model was fit. Conclusion: As with any analysis, it is important to choose appropriate statistical models. In simulation studies and in the motivating example, the standard Poisson was not robust when fit to over-dispersed count data, and did not maintain the appropriate Type-I error rate. To appropriately model alcohol consumption, more flexible count models should be routinely employed. Background In this setting, estimating differences between treatment Count outcomes are common in randomized studies of group and control group is of primary interest. alcohol treatment. Subjects may be queried about their daily consumption of alcohol, measured as a number of A challenge in modeling consumption outcomes is to drinks over a recent period [1] (typically 30 days), and appropriately account for the distribution of drinking. these values are used to estimate average drinking per day. These distributions are characterized by a large number of zeros (abstinent subjects) along with a long right tail Page 1 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 (heavy drinking subjects). An extensive literature uniquely specifies this distribution, and is equal to the describes models for counts [2-8], and they have been expected value (mean) and variance (i.e. E[Y ] = Var(Y ) = ij ij commonly applied in economic analyses, traffic acci- λ for all i and j). The maximum likelihood estimate ij dents, and health services utilization. Many routines are now available in general purpose statistical software (e.g. (MLE) of λ is given by Y . In this setting, the test of ran- i i Stata) [8]. A natural model for counts is the single-param- domized group effects for the Poisson model is a test of eter Poisson distribution. One disadvantage of the Pois- the null hypothesis that λ = λ . 1 2 son is that it makes strong assumptions regarding the distribution of the underlying data (in particular, that the One limitation of this model is that it may be overly sim- mean equals the variance). While these assumptions are plistic and may not provide an adequate fit to consump- tenable in some settings, they are less appropriate for alco- tion data of the type that we consider. The constraint that hol consumption. Extensions of the Poisson, such as the the variance is equal to the mean may lead to incorrect test over-dispersed Poisson, negative binomial and two stage results. (hurdle) or zero inflated models have been proposed [2- 5]. Consider as an example the data from the ASAP study con- trol group at 3 months. For this dataset, non-integer count Our methods are motivated by the analysis of the ASAP values are possible. These arise when subjects consume a (Addressing the Spectrum of Alcohol Problems) study, a number of drinks not divisible by 30 (in the case of 30- randomized clinical trial comparing a brief motivational day assessments). One approach in this situation would interview to usual care for a sample of inpatients with be to model the number of drinks consumed in a 30 day unhealthy alcohol use at an urban hospital [9]. These sub- period, or utilize the non-integer values. Sometimes even jects were followed to see if there were differences in the 30 day value is non-integer because people report a drinking outcomes that could be attributed to rand- drink size that is then translated into standard drinks. The omized group assignment. maximum likelihood estimates of the probability distri- butions remains the same for non-integer values, though In this paper, we will demonstrate the limitations of the it is necessary to move each non-integer observed value to standard Poisson model in the presence of over-disper- the next integer (using a ceiling function) to be plotted. sion. We begin by describing several count models for For the models that we discuss, we can plug non-integer alcohol outcomes, compare their performance in a series values into the software and still get sensible results. of simulated randomized trials, apply them to the ASAP study, and conclude with some general recommenda- tions. Figure 1 displays the observed distribution and superim- posed Poisson with λ = Y = 4.98 using the prcounts 1 1 Methods routine in Stata [8]. The axis for the number of drinks per Statistical methods for the analysis of count outcomes day after 3 months was limited to 25 drinks to improve We begin by introducing notation to be used throughout. readability (the maximum observed count was 48.6). Let Y denote the number of events for the jth subject (j = ij There is a pronounced lack of fit to this model, particu- 1,..., n ) in the ith group (i = 1, 2), where n is the number i i of subjects in the ith group. Typically in a randomized trial larly for values of less than 10 drinks per day. For the ASAP n and n are approximately equal. 1 2 data, the assumption that the mean is equal to the vari- ance is not tenable. In fact, the observed variance (71.7) is The Poisson distribution is one of the simplest models for more than an order of magnitude larger than the mean. count data. Let λ indicate the average number of events ij Also, note that there is some evidence for digit preference (in this case drinks consumed) in a given time interval for (even numbers are more common than odd numbers). subject j in group i, where f(Y = k|λ ) is the probability of ij ij observing k events. The Poisson distribution [8,10] is One approach to loosen the restrictive variance assump- denoted: tion involves use of an empirical (or robust or sandwich) variance estimator [11-13] to account for the over-disper- exp(−λλ ) sion. This more flexible extension of the Poisson allows ij ij PY(|== k λ) ij ij the variance to be unconstrained. The over-dispersed Pois- k! son option is available in a number of general purpose for k = 0, 1, 2, ..., i = 1, 2, and j = 1,..., n where λ > 0 and i ij statistics packages (e.g. the robust option in Stata). we assume that λ = λ for all j (i.e. all subjects in a given ij i Another approach is to fit a negative binomial (two group have the same rate of drinking). The λ parameter parameter) count model (NB) [5-8,10]. One common Page 2 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 observed Poisson predicted 0 5 10 15 20 25 # of drinks per day at 3 months Observed value of drinks per day for the control group of the ASAP study at 3 months, plus the estimated Poisson fit to these data ( = 4.98) Figure 1 Observed value of drinks per day for the control group of the ASAP study at 3 months, plus the estimated Poisson fit to these data ( λ = 4.98). parametrization of the negative binomial distribution is The negative binomial model is attractive because it given by: allows the relaxation of strong assumptions regarding the relationship between the mean and the variance. This flex- ibility comes at some cost, since a two-parameter model is PY(|== k λθ, ) ij i i inherently more complicated to interpret. − −1 θ k −1 −1 ⎛ ⎞ ⎛ ⎞ Γ() θ + k θ λ i i i Other models have been proposed that allow for an extra ⎜ ⎟ ⎜ ⎟ −1 −1 −1 ⎜ ⎟ ⎜ ⎟ ΓΓ () θ (k++ 1) θλ θλ + abundance of subjects with no consumption. In alcohol i ⎝ ii ⎠ ⎝ ii ⎠ consumption outcomes, there may be subjects who are "non-susceptible" (e.g. abstinent). These "zero-inflation" where Γ(·) denotes the Gamma function, λ > 0 and θ > i i (or "hurdle") models account for subjects who are struc- 0. We note that E[Y ] = λ and Var(Y ) = λ+ λ * θ = λ * tural zeros (e.g., abstinent subjects thought of as "non- ij i ij i i i i susceptible") [2,3]. Conditional on being susceptible (1 + λ * θ ) for all i and j and that Var(Y ) > E[Y ]. It can i i ij ij (with some probability), the distribution is assumed to be be shown that the negative binomial can be derived in Poisson or negative binomial. terms of a Poisson random variable where the parameter λ varies according to a gamma distribution. Page 3 of 9 (page number not for citation purposes) probability 0.00 0.05 0.10 0.15 0.20 BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Zero-inflated Poisson (ZIP) models [3] separately esti- process was repeated 2500 times for each set of parame- mate a parameter p that governs the proportion of non- ters, where E[Y ] = λ = 5 (taken from the ASAP control i i α level of 0.05 was used. For the simulation susceptible subjects in the ith group: group) and an of Poisson data the variance was equal to the mean. Neg- ative binomial distributions were simulated using three fY(|== k λ,p) ij i i arbitrary variances (13.3, 40 and 70), with the latter value exp(−λλ ) comparable to the observed variance from the ASAP con- ii Ik()=+01 p (− p) , ii trol group. The zero-inflated model had a probability of k! 0.2 of being a structural zero, and Poisson with λ = 5 oth- erwise. The true distributions for the simulations are dis- for 0 <p < 1 and λ > 0 where I(k = 0) is equal to 1 when k i i played in Figure 2. Models were fit using the Poisson = 0, and equal to 0 otherwise. By distinguishing Always-0 distribution, over-dispersed Poisson using an empirical (with probability p ) and Not Always-0 group (with proba- variance estimator, negative binomial and zero inflated bility (1 - p ) * exp(-λ )) for abstainers and drinkers who i i Poisson. We estimated the probability that each model didn't drink during the reporting period, respectively, it rejected the null hypothesis and constructed a 99% confi- can incorporate an overabundance of zeros [8]. Condi- dence interval around this estimate. The code for the sim- tional on being a Not Always-0, counts are given by the ulations is available upon request from the first author. Poisson distribution. This approach has been generalized to a regression framework, and implemented in general ASAP study purpose statistical software (e.g. zip in Stata). The ASAP study was a randomized clinical trial of the effectiveness of a brief motivational intervention [14] on In many settings, the assumption that after accounting for alcohol consumption among a group of hospitalized the zeros the remaining counts are Poisson may not be patients at Boston Medical Center. Details of the recruit- tenable. The zero-inflated negative binomial (ZINB) ment procedures, inclusion criteria, description of sample allows for over-dispersion in this manner, though at the and results of the RCT have been published [15]. The cost of more parameters. Institutional Review Board of Boston University Medical Center approved this study, and the Institutional Review Another approach to the modeling of count data involves Board of Smith College approved the secondary analyses. use of a linear model (assuming that the observations are After consenting to enroll, all subjects received an inter- approximately Gaussian). While this is an extremely flex- viewer-administered baseline assessment prior to rand- ible model that is typically robust to misspecification omization into the control or intervention group. Subjects (since the mean and variance are not linked), the linear were randomly assigned to control or intervention group model is less attractive because it may predict negative val- using a blocked randomization procedure. Intervention ues of drinking given the skewness of the distribution. Use subjects participated in a brief motivational interview of a linear model is also inefficient if the variance is a func- with a counselor (less than half an hour). Control subjects tion of the mean. received usual care. Simulation study Follow-up was planned at 3-month and 12-month time- To better understand the behavior of these methods in a points. Because the subjects came from a transient and known situation, we conducted a series of simulation hard-to-reach population, the researchers employed studies with parameters derived from the motivating exhaustive techniques to track subjects over the follow-up example. These simulation studies were designed to period. The two primary alcohol-related outcomes were address the question of whether or not the models were measures of alcohol consumption and linkage to appro- robust to misspecification of the underlying count distri- priate alcohol treatment; for these secondary analyses we bution. More formally, we wanted to assess whether these focus solely on treatment differences in alcohol consump- models preserved the appropriate Type-I error rate (the tion. The outcome of interest was the average number of probability of rejecting the null hypothesis when it is true) standard drinks consumed per day in the past thirty days when there are no true differences between groups (i.e. do as reported using the Timeline Followback method [1] at they reject the null at the appropriate α level). the 3 and 12-month interviews. For the purpose of this secondary analysis we consider the 3 month time point; For each set of parameters within a simulation, 100 obser- similar results were seen utilizing 12 month data (not vations were generated in each of two groups, to mimic a reported here). randomized clinical trial setting. The amount of alcohol consumption, in drinks per day was the outcome. For Eight models were fit comparing treatment to control for each simulated dataset a series of models (Poisson, nega- the ASAP study: tive binomial and zero-inflated Poisson) were fit. This Page 4 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 lambda=5 NB (Var=70) NB (Var=40) NB (Var=13.3) ZIP (Var=8) Poisson (Var=5) 0 5 10 15 20 25 # of drinks per day Graphical display of the five di m p Figure 2 = ial [NB13, Var 0.2, Var = 8]= 13 ) ], negative st binom ributioins, al [NB40, Var all with rate p = 4a 0ram ], ne ega ter 5, used in the simulations tive binomial [NB 70, Var = 70] (Poisson [Var = and zero-inflated 5], negative bi Poisson [ZI noP - , Graphical display of the five distributions, all with rate parameter 5, used in the simulations (Poisson [Var = 5], negative bino- mial [NB13, Var = 13], negative binomial [NB40, Var = 40], negative binomial [NB 70, Var = 70] and zero-inflated Poisson [ZIP, p = 0.2, Var = 8]). Poisson standard Poisson model, WILCOXON Wilcoxon-Mann-Whitney, a non-parametric two-sample comparison procedure suitable for ordinal Over-dispersed Poisson Poisson model with empirical data, and ("robust") variance estimator, PERMUTE two-sample permutation test. NB negative binomial, Results ZIP zero-inflated Poisson, shared inflation parameter esti- Simulation studies mated for both randomized groups (p = p ), In the simulation studies we assessed the behavior of 1 2 models when the null hypothesis was true (there were no ZINB zero-inflated negative binomial, shared inflation differences between alcohol consumption for groups 1 parameter estimated for both randomized groups (p = and 2). We note that the ZIP model failed to converge for p ), more than a quarter of the simulations from the standard Poisson distribution. This is likely due to the fact that TTEST two-sample unequal variance t-test, many datasets had no zeros whatsoever (for the Poisson distribution with λ = 5, the probability that a dataset has no zeros whatsoever is equal to (1 - exp(-5)) = 0.51). Page 5 of 9 (page number not for citation purposes) probability 0.00 0.05 0.10 0.15 0.20 0.25 BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Table 1: Estimated probability (and 99% CI) of rejecting the null hypothesis when there is no true difference between groups for a variety of statistical models and underlying distributions (results that do not include the alpha level of 0.05 are bolded) Analysis model fit True Distribution: Poisson ODP NB ZIP Poisson (Var = 5) .053 (.041,.064) .054 (.042,.066) .047 (.036,.058) .055* (.043,.067) NB (Var = 13) .225 (.204,.247) .049 (.038,.060) .049 (.038,.060) .050 (.039,.061) NB (Var = 40) .467 (.441,.493) .047 (.036,.058) .044 (.033,.055) .046 (.036,.057) NB (Var = 70) .584 (.558,.609) .052 (.041,.063) .048 (.037,.059) .062 (.049,.074) ZIP (Var = 8) .179 (.159,.199) .058 (.046,.070) .031 (.022,.040) .051 (.040,.063) all distributions except ZIP have E[Y ] = λ = 5, for ZIP E[Y ] = 0.8 * 5 = 4. i i ODP (over-dispersed Poisson); NB (negative binomial); ZIP (zero-inflated Poisson) * For the true distribution under the Poisson, the ZIP model failed to converge for n = 672 of the simulations. Table 1 displays the estimated Type I error rate (when an overall response rate of 79%. Table 2 displays the dis- there is no difference between the groups) when α was set tribution of drinks per day at baseline and 3-month fol- to 0.05. The negative binomial model was conservative low-up separately for each group. As noted earlier, when the underlying data were zero-inflated. When the drinking outcomes are highly skewed to the right, with underlying distributions were not Poisson, the Poisson some extremely large values. These extreme values are model did not maintain the appropriate Type I error rate. plausible given the large number of dependent drinkers in When the count models were over-dispersed by a factor of the sample, many of whom have developed tolerance (the more than 2 (i.e. Var(Y ) > 2 * E[Y ]), the Poisson model need to consume large amounts of alcohol to induce i i rejected more than 22% of the time. When the over-dis- effects). We also note that reported drinking quantities persion was more extreme (factor of 8 and 14), the Type I decreased for both groups between baseline and 3-month error rate was 47% and 58%, respectively. The severe lack outcome. of robustness of the Poisson model in this setting is a seri- ous concern. Table 3 displays the results from the ASAP study using a variety of count models. Use of the Poisson model yielded ASAP study a statistically significant p-value, in contrast to the other Of 341 subjects enrolled in the clinical trial, 169 subjects methods (all other p-values > 0.45). were randomized to the control group and the other 172 into the intervention group. The mean age of the subjects Figure 3 displays the observed and predicted counts for was 44.3 (SD = 10.7). Twenty-nine percent were women, the Poisson, negative binomial, and ZIP models, while 45% were Black, 39% White, 9% Hispanic, and 7% Other. Figure 4 displays the plot of (observed minus expected) Sixty-three percent were unemployed during the past for the Poisson, negative binomial and ZIP models for the three months and 25% of the subjects were homeless at control group. The standard Poisson model provides an one point during the past three months. Four percent of unsatisfactory fit, and is not appropriate for the analysis of the subjects met criteria for current (past year) alcohol this dataset. The fit of the zero-inflated Poisson is abuse and 77% were alcohol dependent. improved, particularly for modeling the probability of no drinking, but remains unsatisfactory over most of the We analyze the 3-month follow-up data for which 271 remaining values. The negative binomial provides an subjects were observed (141 control, 130 treatment), for excellent fit for these data, and that there is no indication Table 2: Distribution of drinking outcome by timepoint and randomization group Base line 3 Months C (n = 169) T (n = 72) C (n = 141) T (n = 130) MIN 0.17 0 0 0 25th percentile 1.14 1.32 0.17 0.13 MEDIAN 3.47 3.85 1.8 1.6 75th percentile 8.23 9.12 6.1 5.7 MAX 61.77 60 48.6 38.43 mean (SD) 6.95 (9.58) 6.68 (8.44) 4.98 (8.47) 4.36 (6.47) Page 6 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Table 3: p-values for the ASAP randomization group effect at 3 months for a variety of count models MODEL p-value Poisson .018 over-dispersed Poisson .489 Negative binomial .458 zero-inflated Poisson .542 zero-inflated negative binomial .489 t-test .495 Wilcoxon .805 Permutation .746 that any further zero-inflation is needed, since the model In this setting, there was little indication from the already overpredicts zeros (hence the predicted values for observed plots that there were significant group differ- the NB and ZINB would be identical). ences. As seen in the simulation studies, the Poisson may observed (C) Poisson pred. (C) observed (T) Poisson pred. (T) Wilcoxon p−value=0.80 Poisson p−value=0.02 0 5 10 15 20 25 0 5 10 15 20 25 # of drinks per day at 3 months # of drinks per day at 3 months Negative binomial pred. (C) zero−inflated Poisson pred. (C) Negative binomial pred. (T) zero−inflated Poisson pred. (T) Negative binomial p−value=0.46 ZIP p−value=0.54 0 5 10 15 20 25 0 5 10 15 20 25 # of drinks per day at 3 months # of drinks per day at 3 months Wilco Figure 3 Observed an xon, Po d p issr oedicted n, negative bi values fr nomial om th and ze e ASAP study ro-inflate at 3 months d Poisson for control and treatment groups for each of four models: Observed and predicted values from the ASAP study at 3 months for control and treatment groups for each of four models: Wilcoxon, Poisson, negative binomial and zero-inflated Poisson. Page 7 of 9 (page number not for citation purposes) probability probability 0.00 0.10 0.20 0.00 0.10 0.20 probability probability 0.00 0.10 0.20 0.00 0.10 0.20 BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 Poisson ]HURïLQIODWHG3RLVVRQ QHJDWLYHELQRPLDO REVHUYHGïH[SHFWHG 0 5 10 15 20 25 # of drinks per day at 3 months Obser and z Figure 4 eved minus ex ro-inflated Poisson pected values from the ASAP study at 3 months as a function of count for the Poisson, negative binomial Observed minus expected values from the ASAP study at 3 months as a function of count for the Poisson, negative binomial and zero-inflated Poisson. not have preserved the appropriate Type I error rate due to other models, which had highly non-significant results). the extremely large values of drinking for some subjects. The unrealistic assumption that the expected rate of drink- The Appendix includes the Stata commands to fit these ing is the same for all subjects may partially account for models and the output, along with the code to generate the poor fit of the Poisson distribution. We caution observed and predicted plots using the prcounts routine. against use of the Poisson for this analysis. The negative binomial fit particularly well, and we saw no evidence for zero-inflation. Discussion and conclusion A number of models have been proposed for the analysis of count data, and these models are now available in gen- In settings where there are excess zeros, zero-inflation eral purpose statistical packages. We have described these models are attractive. One advantage of these models is methods in the context of modeling reports of alcohol that they can estimate the probability of being a zero as a consumption, where a large proportion of respondents function of covariates, as well as allowing the rate param- report no drinking, and a small number of respondents eter to be a function of covariates. In an alcohol study, the typically account for an extreme amount of drinking. intervention may be hypothesized to affect the abstinence proportion as well as the rate parameter for drinkers. Ad- For the analysis of the ASAP study, we found that the hoc methods in this setting might involve estimating the standard Poisson had an extremely poor fit, and yielded a proportion of drinkers at follow-up, and in a separate statistically significant p-value (in contrast to all of the model, estimating the amount of drinking amongst the Page 8 of 9 (page number not for citation purposes) BMC Medical Research Methodology 2007, 7:9 http://www.biomedcentral.com/1471-2288/7/9 subset of subjects who reported any drinking. A more References 1. Sobell LC, Sobell MB: Timeline follow-back: a technique for principled approach involves the simultaneous estima- assessing self-reported alcohol consumption. In Measuring Alco- = p ) and the tion of the zero-inflation factor (testing p 1 2 hol Consumption: Psychosocial and Biochemical Methods Edited by: Litten rate parameter (testing λ = λ ). Slymen and colleagues [2] RZ, Allen JP. Totowa: Humana Pr, Inc.; 1992:41-69. 1 2 2. Slymen DJ, Ayala GX, Arredondo EM, Elder JP: A demonstration of adopted this approach by simultaneously fitting separate modeling count data with an application to physical activity. models for what they describe as the "logistic" component Epidemiologic Perspectives & Innovations 2006, 3(3):1-9. 3. Lambert D: Zero-inflated Poisson regression, with an applica- and the "Poisson" component, and this approach is also tion to defects in manufacturing. Technometrics 1992, 34:1-14. detailed in books by Winkelmann [7] as well as Cameron 4. Cameron AC, Trivedi PK: Regression analysis of count data Cambridge, and Trivedi [4]. UK: Cambridge University Press; 1998. 5. Gardner W, Mulvey EP, Shaw EC: Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative The results of the simulation studies and the secondary binomial models. Quantitative Methods in Psychology 1995, analyses of the ASAP study demonstrated the importance 118(3):392-404. 6. Hilbe JM: Negative binomial regression: modeling overdispersed count data of appropriately modeling count outcomes. We caution Cambridge: Cambridge University Press in press. against the use of the standard Poisson model when the 7. Winkelmann R: Econometric analysis of count data fourth edition. Ber- lin: Springer-Verlag; 2003. mean and variance are not equal. Extensions of the Pois- 8. Long JS, Freese J: Regression models for categorical dependent variables son (incorporating an over-dispersion parameter or use of using Stata Texas: Stata Press Publication; 2003. the negative binomial distribution and/or zero-inflated 9. Saitz R, Freedner N, Palfai TP, Horton NJ, Samet JH: The severity of unhealthy alcohol use in hospitalized medical patients: the models) are now available in general purpose statistical spectrum is narrow. Journal of General Internal Medicine 2006, software, and address many of the shortcomings of the 21(4):381-5. overly simplistic Poisson model. 10. McCullagh P, Nelder JA: Generalized linear models Chapman & Hall; 11. Zeger SL, Liang KY: Longitudinal data analysis for discrete and As always, analysts are obliged to look at their data and continuous outcomes. Biometrics 1986, 42:121-130. 12. Liang KY, Zeger SL: Longitudinal data analysis using general- utilize models that provide an appropriate fit in their sit- ized linear models. Biometrika 1986, 73:13-22. uation. In particular, for models of alcohol consumption, 13. Huber PJ: The behavior of maximum likelihood estimates attention should be paid to the functional form of the out- under non-standard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1967, 1:221-233. come to ensure that underlying assumptions of the meth- 14. Miller WR, Rollnick S: Motivational interviewing: Preparing people to ods utilized are met. change addictive behavior second edition. New York: Guilford Press; 15. Saitz R, Palfai TP, Cheng DM, Horton NJ, Freedner N, Dukes K, Krae- Authors' contributions mer KL, Roberts MS, Guerriero RT, Samet JH: Brief intervention NH conceived of the project and provided overall guid- for medical inpatients with unhealthy alcohol use: A rand- omized-controlled trial. Annals of Internal Medicine 2007, ance, in addition to reviewing and interpreting analyses, 146(3):167-176. and drafting the manuscript. EK participated in the draft- ing of the manuscript, and carried out analyses and simu- Pre-publication history lations. RS led the ASAP study and participated in the The pre-publication history for this paper can be accessed drafting of the manuscript. All authors read and approved here: the final version of the manuscript. http://www.biomedcentral.com/1471-2288/7/9/prepub Additional material Additional File 1 Appendix. Stata code and results for count models. Click here for file [http://www.biomedcentral.com/content/supplementary/1471- 2288-7-9-S1.pdf] Publish with Bio Med Central and every scientist can read your work free of charge "BioMed Central will be the most significant development for disseminating the results of biomedical researc h in our lifetime." Acknowledgements Sir Paul Nurse, Cancer Research UK This research was supported in part by the National Institute on Alcohol Your research papers will be: Abuse and Alcoholism R01-AA12617, the Smith College Summer Research Program and the Howard Hughes Medical Institute. Thanks to Jessica Rich- available free of charge to the entire biomedical community ardson for editorial assistance, Emily Shapiro and Min Zheng for assistance peer reviewed and published immediately upon acceptance with simulations and Joseph Hilbe and Jeffrey Samet for helpful comments cited in PubMed and archived on PubMed Central on an earlier draft. yours — you keep the copyright BioMedcentral Submit your manuscript here: http://www.biomedcentral.com/info/publishing_adv.asp Page 9 of 9 (page number not for citation purposes)

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