Comparison of methods for modelling a count outcome with excess zeros: an application to Activities of Daily Living (ADL-s)

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1 Comparson of methods for modellng a count outcome wth excess zeros: an applcaton to Actvtes of Daly Lvng (ADL-s) Paola Zannotto, Emanuela Falaschett To cte ths verson: Paola Zannotto, Emanuela Falaschett. Comparson of methods for modellng a count outcome wth excess zeros: an applcaton to Actvtes of Daly Lvng (ADL-s). Journal of Epdemology and Communty Health, BMJ Publshng Group, 2010, 65 (3), pp.205. < /jech >. <hal > HAL Id: hal Submtted on 6 Feb 2011 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Ttle: Comparson of methods for modellng a count outcome wth excess zeros: an applcaton to Actvtes of Daly Lvng (ADL-s) Paola Zannotto 1 and Emanuela Falaschett 1 1 Department of Epdemology & Publc Health, UCL, Correspondng author: Paola Zannotto Department of Epdemology & Publc Health, UCL 1-19 Torrngton Place WC1E 7HB London, UK tel +44(0) fax +44(0) e-mal: p.zannotto@ucl.ac.uk Word count: 2,890 1

3 Abstract Background: count outcomes are commonly encountered n many epdemology applcatons, and are often characterzed by a large proporton of zeros. Whle lnear or logstc regresson models have often been used to analyse count outcomes, the resultng estmates are lkely to be neffcent, nconsstent or based. Methods: data are from wave1 of the Englsh Longtudnal Study of Ageng (ELSA). The man outcome measure s number of dffcultes (rangng from 0 to 6) wth Actvtes of Daly Lvng (ADL-s) such as dressng, walk across a room, bathng, eatng, gettng n and out of bed and usng the tolet. We ftted four regressons models specfcally developed for count outcomes: Posson, NB, ZIP and ZINB. We then compared these models usng the Lkelhood Rato test of overdsperson, the Vuong test, and graphcal methods. Results: The plots of predctons showed that overall, the ZINB model ft best. Although the ZINB and the ZIP models showed smlar ft the LR test provded strong evdence that the ZINB mproves the ft over the ZIP. Increasng number of dffcultes wth ADL-s was assocated wth far/poor self-reported health, lmtng longstandng llness and physcal nactvty. The probablty of not havng any dffculty wth ADL-s decreases wth a lmtng longstandng llness, ncreasng age, no educaton, far/poor self-reported health and wth not lvng wth the partner. Concluson: models specfcally developed for count outcomes wth excess zeros such as ZINB can provde better nsghts nto the nvestgaton of the factors assocated wth the number of dffcultes wth Actvtes of Daly Lvng. Key word: count outcome, actvtes of daly lvng, old age. 2

4 INTRODUCTION Many studes have addressed the ssue of functonal dependence and ts rsk factors among older adults [1-8]. A measure of functonal dependence wdely used n epdemology studes of older adults s the dffcultes to perform Actvtes of Daly Lvng (ADL-s) such as bathng, dressng, walk across a room, eatng, gettng n or out of bed and usng a tolet [9]. From these tems a varable s derved to assess the number of dffcultes n dong ADL-s, whch ranges from 0 (to ndcate no dffcultes) to 6 (to ndcate dffcultes wth all 6 actvtes of daly lvng). Ths outcome has been used as a contnuous varable, or as a dchotomous varable for comparson between those reportng zero dffcultes wth ADL-s and those reportng 1 to 6 dffcultes wth ADL-s. [5-8] However, usng ths varable as a contnuous outcome n a lnear regresson would not be approprate and the categorsaton of the varable to be used n logstc regresson may result n loss of nformaton. Count outcomes are commonly encountered n many epdemology applcatons, and are often characterzed by a large proporton of zeros. Whle lnear or logstc regresson models have often been used to analyse count outcomes, the resultng estmates are lkely to be neffcent, nconsstent or based. [10, 11] Several models belongng to the famly of Generalsed Lnear Models are avalable for performng regressons wth count outcomes [12]. Such models are the Posson and negatve bnomal (NB) as well as zero-nflated Posson (ZIP) and zero-nflated negatve bnomal (ZINB) specfcally developed for count outcomes wth excess zeros and dsperson. [10-21] In recent years there has been ncreasng nterest n the use of these models n research studes, and few examples are avalable n epdemology and publc health lterature. [22-27] The am of ths paper s to examne and compare the utlty of the Posson, negatve bnomal (NB), ZIP and ZINB regressons for modellng a count outcome, wth partcular focus on applcaton n epdemology and publc health research. For ths purpose we use data on the number of dffculty wth ADL-s from a natonal sample of older adults, partcpants of the frst wave of the Englsh Longtudnal Study of Ageng (ELSA). 3

5 METHODS The sample The data are from the frst wave of the Englsh Longtudnal Study of Ageng (ELSA), a panel study where the same ndvduals are followed and re-ntervewed every two years. The techncal detals of ths study and the results of prmary analyses have been publshed elsewhere [28, 29] and are also avalable at the web ste of the Insttute of Fscal Studes ( Brefly the ELSA sample was drawn from people who had taken part n the Health Survey for England (HSE) n 1998, 2000 or 2001 and were born before March The HSE samples are selected to be representatve of people lvng n prvate households n England. A total of 11,392 elgble sample members took part n wave1 ( ), gvng a response rate of 67%. Partcpants gave ther nformed consent to take part n the study. Ethcal approval for ELSA was gven by the London Mult-centre Research Ethcs Commttee (MREC/04/2/006). Measures Informaton on dffculty wth Actvtes of Daly Lvng (ADL-s) was collected durng the ntervew usng the health-assessment questonnare. [6] The orgnal scale of ADL-s was developed by Katz et al,. [9] Respondents were shown a card and the followng text was read to them: Here are a few everyday actvtes. Please tell me f you have any dffcultes wth these because of a physcal, mental, emotonal or memory problem. Exclude any dffcultes they expect to last less than three months. Because of a health or memory problem, do you have any dffculty n dong any of the actvtes on ths card?. The followng sx ADL-s tems were lsted n the card: 1. Dressng, ncludng puttng on shoes and socks 2. Walk across a room 3. Bathng or showerng 4. Eatng, such as cuttng up food 5. Gettng out of bed 6. Usng tolet, ncludng gettng up or down From 11,220 vald answers to ths queston a varable s derved to count the number of dffculty wth ADL-s (range 0 to 6). 4

6 Varables ncluded n the models as potental predctors for ADL-s were age, sex, martal status, educaton level, smokng status, weekly alcohol consumpton, physcal actvty, lmtng longstandng llness and self-reported general health. Apart from age, all the varables were entered nto the models as dummy. Analyss Posson, negatve bnomal, zero-nflated Posson (ZIP) and zero-nflated negatve bnomal (ZINB) regressons models for count outcomes were ftted to the data n order to examne and compare whch model gves the best ft for the data. The Posson dstrbuton s a probablty dstrbuton for non-negatve ntegers. Let y be a random varable whch ndcates the number of tmes an event occurred and µ the expected count. The Posson dstrbuton specfes the relatonshp between the expected count µ and the probablty of observng any observed count y: μ y e μ Pr(Y = y μ) = for y = 0, 1, 2, (1) y! where µ s the mean and the varance of the dstrbuton, whch s known as equdsperson. [17] The mean (µ) of the dstrbuton strongly controls the shape of the dstrbuton. When the mean s small the most common count s zero. For example, f µ =0.5, then the probablty of obtanng a count of zero s 0.61, a count of one s 0.33, a count of two s 0.12, and as the mean ncreases, the probablty of a zero count becomes less than zero. The dstrbuton s therefore hghly skewed towards the lowest value, zero. The Posson regresson model can be represented as: Pr(Y μ y μ e = y X ) = (2) y! where Y X P(µ ) followng a Posson dstrbuton; and where µ = E(Y X ) = exp{ β 0 + β 1 X 1 + β 2 X β p X p } (3) Y s the count for the th subject, X ={ 1+ X 1 + X X p } s a vector of covarates plus 1 and β 0...β p the regresson coeffcents to be estmated. 5

7 One problem wth Posson regresson s over-dsperson, whch means that the assumpton of equalty between the mean and the varance s not often adapted to observed data, as the varance often exceeds the mean. [17] Ths leads to underestmaton of the standard errors of the regresson estmates, confdence ntervals that are too narrow, and p-values that are too small. The negatve bnomal (NB) regresson model s an alternatve to the Posson model when the count data are over-dspersed n relaton to the mean. [30, 31] The NB regresson adds an error ε to equaton (3) as follow: µ = E(Y X ) = exp{ β 0 + β 1 X 1 + β 2 X β p X p + ε } (4) where ε s assumed to be uncorrelated wth X and exp{ ε } has a gamma dstrbuton wth mean 1 and varance α 2. It follows that: Pr(Y μ exp{ ε } e μ = y μ, ε ) = (5) y! y If α=0 then the NB (5) reduces to the Posson (2). [15,17] Therefore the napproprateness of the Posson regresson relatve to the NB regresson s determned by the statstcal sgnfcance of the α parameter. In addton to dsperson, data often dsplay a greater number of zero observatons than expected from the Posson model, ths problem s known as excess zeros. [17] Zero-nflated count models provde a way of modellng dsperson and the excess zeros by changng the mean structure to allow zeros to be generated by two dstnct processes. [13, 21] Zero-nflated count models assume that there are two latent groups: the frst group s the group of zeros and the second s the group of non-zeros. The frst group generates only zeros whle the second s ether a Posson or a negatve bnomal dstrbuton. In the frst stuaton we wll have a ZIP regresson represented as: Pr(Y p + (1 p )e = y X ) = μ y (1 p )e μ y! μ y y = 0 1 (6) 6

8 where p s the probablty of beng an extra zero, whch s determned, n our applcaton, by a logstc model. Substtutng the negatve bnomal for the Posson yelds the zeronflated negatve bnomal (ZINB). All analyses were run usng Stata verson 10. We frst looked at how the probablty dstrbutons, whch underpn the four models, ft the observed data; to do so, we have run the Posson, NB, ZIP and ZINB regressons wthout any ndependent varables and compared graphcally the predcted proportons (from models wth ntercept only) wth the observed proportons from the ADL-s score. The Posson, NB, ZIP and ZINB regressons wth covarates were then ftted to the data. Because the NB reduces to the Posson when α=0 these models can be compared usng the Lkelhood Rato test for overdsperson avalable n Stata, to test the H 0 :α=0. Smlarly because the ZINB and ZIP are nested, [15] t s possble to use the Lkelhood Rato test for overdsperson, whch s easy to compute n Stata. [15, p:285] Comparsons of the Posson wth the ZIP and comparsons of the NB wth the ZINB nvolve non-nested comparsons whch can be assessed usng the Vuong test for non-nested models, [32] avalable n Stata. In addton, to obtan graphcal llustraton of ft, we plotted the observed mnus predcted probabltes across models, adjusted for covarates. The same set of covarates was used to ft both the logstc part and the ntensty part of the ZIP and ZINB models. The results from the best model were then compared to results from a logstc regresson (wth same predctors), n whch ADL-s was analysed as a dchotomous varable for comparson between those reportng 1 to 6 dffcultes and those reportng no dffcultes wth ADL-s. Statement of Ethcs: We certfy that all applcable nsttutonal and governmental regulatons concernng the ethcal use of human partcpants were followed durng ths research. The survey was approved by the approprate Research Ethcs Commttee. RESULTS Table 1 presents the dstrbuton of the outcome varable and predctor varables. Of the 11,220 respondents, 79% report no dffcultes wth ADL-s. 7

9 Fgure 1 shows how the probablty dstrbutons, whch underpn the four models, ft the observed proportons from the ADL-s score. The Posson model was a poor ft, whle the predcted values of the NB model were very close to the observed. The ZIP and ZINB models gave a good predcton of zeros. In fact, 85% of the observed zeros were predcted by both models. The ZINB model produced the best ft for the entre range of the ADL-s values. The lkelhood test for overdsperson comparng the NB to the Posson, whch tests H 0 :α=0, yelds a statstc of The estmate of α s 1.1 (s.e. 0.06) whch s sgnfcantly dfferent from 0, therefore the NB model s favored over the Posson. The ZIP regresson yelded a log-lkelhood of ; whle the ZINB yelded a loglkelhood of wth an estmate of α equals to 0.25 (s.e. 0.05). The ZIP and ZINB models are nested so they can be compared by usng the lkelhood test for overdsperson to test H 0 :α=0, whch yelds a statstc of 51.1 whch provde evdence for preferrng the ZINB over the ZIP. The Posson and ZIP and also the NB and ZINB are not nested; therefore the models cannot be compared usng the Log-lkelhood test. We used the Vuong test for non-nested models. The Vuong test for the ZIP vs the Posson yelded a z equals to 14.85, whch supported the ZIP model over the Posson (p<0.0001); smlarly, the Vuong test supported the ZINB model over NB (p<0.0001). 8

10 Table 1 Sample characterstcs Number of dffcultes wth ADL-s % % 2 4.9% 3 2.6% 4 1.7% 5 0.9% 6 0.4% Mean Age of partcpants 65.2 (s.e.) (0.10) Females 54.5% No educatonal qualfcaton 42.6% Not lvng wth the partner 31.3% Far or poor health 27.1% Lmtng longstandng llness 35.0% Current smoker 17.8% Current drnker 28.1% Physcally nactve 65.7% 9

11 Fnally, Fgure 2 shows the observed mnus the predcted probabltes at each count, for each model. Ponts above 0 on the y-axs ndcate more observed count than predcted; whle those below 0 ndcate more predcted counts than observed. From the graph t s clear that the Posson regresson does not predct well the average number of zeros. The NB model s a substantal mprovement over the Posson; however, t underestmates the proportons at 1. Both, the ZIP and ZINB models ft the data qute well; however, the ZIP predcts more 1s and less 2s and 3s than the ZINB. Based on formal tests and graphcal methods, we prefer the ZINB modellng approach. Results from the ZINB model are shown n table 2. The frst three columns report respectvely the raw coeffcents, p-values and coeffcents for the factor change n the odds (for unt ncrease n the dependent varable) of beng n the group wthout any dffculty wth ADL-s, modelled wth a logt model. The last three columns report the raw coeffcents, p-values and the factor change n the expected count for those who have one and more dffculty wth ADL-s, modelled wth a Negatve Bnomal. Increasng age, not havng educatonal qualfcaton, not lvng wth the partner, havng a lmtng longstandng llness, reportng far or poor health were all assocated wth decrease odds of reportng zero dffculty wth ADL-s. Among those wth one and more dffculty wth ADL-s, havng a lmtng longstandng llness, reportng far or poor health and beng physcally nactve ncrease the expected rate of reportng dffcultes wth ADL-s, whle drnkng alcohol decrease the expected rate of reportng dffculty wth ADL-s. Table 2 ZINB regresson coeffcents for the number of dffcultes wth ADL-s 10

12 b (s.e.) Age (0.002) Females (0.049) No educatonal qualfcaton (0.049) Not lvng wth the partner (0.051) Far or poor health (0.058) Lmtng longstandng llness (0.082) Current smoker (0.059) Current drnker (0.060) Physcally nactve (0.109) Logt porton a Negatve Bnomal porton p-value exp(b) b (s.e.) p-value exp(b) (0.006) (0.104) (0.103) (0.112) (0.124) (0.132) (0.124) (0.124) (0.175) value (95% CI) s.e. α (0.167, 0.375) 0.05 a Models the probablty of no dffculty wth ADL-s Results from a logstc regresson (Table 3) comparng those wth 1 to 6 dffcultes wth ADL-s those wth 0 dffcultes show that ncreasng age, not havng educatonal 11

13 qualfcaton, not lvng wth the partner, havng a lmtng longstandng llness, reportng far or poor health and beng physcally nactve were all assocated wth ncreased odds of reportng one and more ADL-s. Table 3 Results from a logstc regresson for the rsk of havng one to sx dffcultes wth ADL-s b (s.e.) Age (0.003) Females (0.058) No educatonal qualfcaton (0.059) Not lvng wth the partner (0.061) Far or poor health (0.061) Lmtng longstandng llness (0.063) Current smoker (0.073) Current drnker (0.067) Physcally nactve (0.075) p-value exp(b) DISCUSSION 12

14 Ths study nvestgated the utlty of the Posson, negatve bnomal (NB), ZIP and ZINB regressons for modellng the number of dffculty wth Actvtes of Daly Lvng (ADL-s) n a natonal sample of older adults, partcpants of the frst wave of the Englsh Longtudnal Study of Ageng (ELSA). There s a growng lterature on models specfcally developed for count outcomes, and several examples are avalable n epdemology and publc health research. [22-27] However, ths s the frst study to brng the utlty of these approaches to model a count measure of physcal functonng, ADL-s, wdely used n epdemology. We found that 79% of respondents n our study reported no dffcultes wth ADL-s. Posson regresson s often used for count data; however, we have shown that when the observed counts exhbt more varablty than what s predcted by the Posson (known as overdsperson), lke n our outcome varable, s better to use a dfferent model. To allow for overdsperson we used the NB regresson model. The NB provded a substantal mprovement over the Posson. To account for both dsperson and the excess zeros the ZIP and ZINB were ftted to the data. The ZIP and ZINB provded a smlar ft; however, we found that the ZINB modellng approach yelded the best ft. The choce of the ZINB however, cannot be made strctly on the bass of model fttng. [15, 17-18] In our applcaton t seems reasonable to assume that there s a separate populaton of zeros formed of those who dd not experence any dffculty n performng ADL-s, as results of beng less lmted n ther physcal functonng or smply healther people. We have also performed logstc regresson usng the dchotomous varable of ADL-s (comparng 0 dffculty wth ADL-s versus 1 1 and more dffculty wth ADL-s). Results from logstc regresson were consstent wth those found from the ZINB regresson, the excepton was only for physcal actvty. In the logstc regresson, beng physcally nactve was assocated wth ncreased odds of havng 1 to 6 dffcultes wth ADL-s. In the ZINB model, beng physcally nactve was not sgnfcantly assocated wth the odds of remanng wthout dffcultes for those wth zero dffcultes (Always zero group) but t was postvely assocated wth the odds of havng ncreasng number of dffcultes for those wth dffcultes (Not always zero group). Ths nformaton s not avalable wth logstc regresson. Also by usng a logstc regresson s not possble to consder the range of dffcultes wth ADL-s experenced; whle the ZINB model offers the advantage of modellng dffcultes wth ADL-s scores on a contnuum nstead of the dchotomzed outcome used n logstc regresson. Therefore the ZINB model allows assessng the 13

15 assocaton of the exposure varables wth dsease severty and not just the presence or absence of dsease.[13] The fndng that beng a current drnker s assocated wth a reducton n the number of dffculty wth ADL-s may be part explaned by the cross-sectonal nature of the data and n part by the nsuffcent nformaton avalable about drnkng frequency. Respondents were asked to report whether they drnk twce a day or more, whether they drnk less frequently than twce a day or whether they abstan completely from drnkng. Detaled nformaton about the number of daly alcohol consumpton would have provded a better understandng of the relatonshp between ths varable and ADL-s. Future research could nvestgate the utlty of the ZIP and ZINB regressons for modellng the number of dffculty wth ADL-s usng longtudnal data. In regresson analyss of count data, ndependent varables are often modelled by ther lnear effects under the assumpton of log-lnearty. However, the valdty of such an assumpton s dffcult to test due to the lack of readly accessble testng procedures. To conclude we have shown that models specfcally developed for count outcomes wth excess zeros such as ZIP and ZINB can provde better nsghts nto the nvestgaton of the factors assocated wth the number of dffcultes wth Actvtes of Daly Lvng (ADL-s). When dealng wth the number of dffcultes wth ADL-s we recommend the use of the ZIP and ZINB models rather dchotomzng the outcome whch yelds to loss of nformaton wth respect to the assocaton of exposure wth dsease severty. 14

16 What s already known on ths subject? Countng outcomes have often been used as contnuous varables, transformed to nduce normalty, or categorzed whch can yeld to based estmates or loss of nformaton. Alternatves models such as Posson, negatve bnomal (NB), zero nflated Posson (ZIP) and zero nflated negatve bnomal (ZINB) have been proposed whch better suted to countng processes. What does ths study add? Ths paper provdes a useful methodology to analyse a count outcome measures lke Actvtes of Daly Lvng (ADL-s) n epdemologcal studes. We recommend the use of the ZIP or ZINB models whch can provde better nsghts nto the nvestgaton of the factors assocated wth the number of dffcultes wth ADL-s. Acknowledgements: The data were made avalable through the UK Data Archve. Elsa was developed by a team of researchers based at Unversty College of London, the Natonal Centre for Socal Research and the Insttute for Fscal Studes. The data were collected by the Natonal Centre for Socal Research. Competng nterests: None. Fundng: The fundng s provded by the Natonal Insttute of Agng n the Unted States, and a consortum of UK Government departments coordnated by the Offce for Natonal Statstcs. The developers and funders of ELSA and the Archve do not bear any responsblty for the analyses or nterpretatons presented here. 15

17 Lcence Statement The correspondng Author has the rght to grant on behalf of all authors and does grant on behalf of all authors, an exclusve lcence (or non exclusve for government employees) on a worldwde bass to the BMJ Publshng Group Ltd and ts Lcensees to permt ths artcle (f accepted) to be publshed n JECH edtons and any other BMJPGL products to explot all subsdary rghts, as set n our lcence ( Fgures Legend Fgure 1 Predcted proportons from ntercept-only Posson, NB, ZIP and ZINB models compared wth the observed proportons from ADL-s score Fgure 2 Observed mnus predcted probabltes for four models 16

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