Metodološk zvezk, Vol. 1, No. 1, 2004, 213-223 Latent Varable Mxture Modellng of Treated Drug Msuse n Ireland Paul Cahll and Brendan Buntng 1 Abstract Ths study provdes analyses and profles of llegal drug usage n the Republc of Ireland. Two questons are addressed: a) can ndvduals be grouped nto homogeneous classes based upon ther type of drug consumpton, and b) how do these classes dffer n terms of other key background varables? The data reported n ths study s from the Natonal Drug Treatment Reportng System database n the Republc of Ireland. All analyses were carred out n collaboraton wth the Drug Msuse Research Dvson (the Irsh REITOX / EMCDDA focal pont). Ths database contans nformaton on all 6994 ndvduals who receved treatment for drug problems n the Republc of Ireland durng 2000. The analyss was conducted n four steps. Frst, a sngle class model was examned n order to establsh the respectve probablty assocated wth each drug type. Second, a seres of uncondtonal latent class models was examned. Ths was done to establsh the optmal number of latent classes requred to descrbe the data, and to establsh the relatve sze of each latent class. From ths analyss the condtonal probabltes for each ndvdual, wthn a gven class, were examned for typcal profles. Thrd, a seres of condtonal models was then examned n terms of key predctors (age and early school leavers). Ths analyss was conducted usng MPlus 2.13. In the fnal stage of the research, the parameter estmates obtaned from the multnomal logstc regresson model (that was prevously used to express the probablty of an ndvdual beng n a gven latent class, condtonal on a seres of covarates) were graphcally modelled wthn EXCEL and the respectve functons descrbed. The results from ths analyss wll be descrbed n terms of a) the proflng of typcal serous drug msuse n Ireland, b) the clusterng of drug types and, c) the respectve mportance of key background varables. The varous profles obtaned are dscussed n terms of health care strateges n Ireland. 1 School of Psychology, Unversty of Ulster at Magee Campus, Northern Ireland; p.cahll@ulster.ac.uk; bp.buntng@ulster.ac.uk Ths research has been awarded a Health Servces Research fellowshp and s beng carred out n collaboraton wth the Drug Msuse Research Dvson, Health Research Board, Dubln, Ireland.
214 Paul Cahll and Brendan Buntng 1 Introducton The constant battle between the competng needs of health-care and socal resources necessary for effectve drug treatment has been apparent for the last three decades. These resources are not only fnancal: people, facltes, expertse and even tme are all fnte, and the most effcent use of them needs to be found wthn the drug-treatment servces. The underlyng constructs of drug msuse and ts characterstcs can only be nvestgated ndrectly usng observable measures and ndcators. Latent varable mxture modellng allows for unobserved heterogenety n the sample (Muthén and Muthén, 2003) where dfferent ndvduals can be seen to belong to dfferent subpopulatons, and can uncover some of the unque characterstcs of ths hdden and vulnerable populaton. Over the last three decades large scale evaluaton studes Drug Abuse Reportng Program (DARP) (Smpson and Sells, 1990), and the Natonal Treatment Outcome Research Study (NTORS) (Gossop, Martn, and Stewart, 2003) have examned the effectveness of drug treatment, n both the UK and US. Despte these consderable efforts, addcton-treatment researchers are stll confronted wth many of the same core questons as those of thrty years ago. These questons nclude: can ndvduals be grouped nto homogeneous classes based upon ther type of drug consumpton, and how do these classes dffer n terms of other key background varables? 2 Method 2.1 Partcpants Partcpants were 6994 ndvduals who receved treatment for problem drug use n the Republc of Ireland durng the 2000 calendar year. Data returns for the 6994 clents attendng treatment servces durng 2000 were provded by 154 treatment servces (O Bren, Kelleher, and Cahll, 2002). 2.2 Measures The Natonal Drug Treatment Reportng System (NDTRS) s an epdemologcal database on treated drug msuse n the Republc of Ireland. The reportng system was orgnally developed n lne wth the Pompdou Group s Defntve Protocol
Latent Varable Mxture Modellng... 215 and subsequently refned n accordance wth the Treatment Demand Indcator Protocol (European Montorng Centre for Drugs and Drug Addcton & Pompdou Group, 2000). Drug treatment data are vewed as an ndrect ndcator of drug msuse. These data are used at natonal and European levels to provde nformaton on the characterstcs of clents enterng treatment, and on patterns of drug msuse, such as types of drugs used and consumpton behavours. They are valuable from a publc health perspectve to assess needs and to plan and evaluate servces (EMCDDA, 2002: 23). The montorng role of the NDTRS s recognsed by the Irsh Government, and data collecton for the Natonal Drug Treatment Reportng System s one of the actons dentfed by the government for mplementaton by all health authortes: All treatment provders should co-operate n returnng nformaton on problem drug use to the NDTRS (Department of Toursm, Sport and Recreaton 2001: 118). 2.3 Analyses The basc premse of the latent class analyses, relevant to ths study s that the covaraton found among the observed varables s due to each observed varable s relatonshp to the latent varable (Hagenaars and McCutcheon, 2002). Fgure 1: Latent classes wth covarates. In ths study, the observed drug consumpton behavours (u) are dchotomous n nature, representng the presence or absence of the partcular drug n each
216 Paul Cahll and Brendan Buntng ndvdual s drug usng career. C denotes the latent classes to be determned. Once the optmal number of latent classes s calculated, the key background varables of age (contnuous) and early school leavers (dchotomous) are ntroduced. The Bayesan Informaton Crteron (BIC) together wth the Akake Informaton Crteron (AIC) were used to determne the optmal number of uncondtonal latent classes necessary to adequately ft the data. The Bayesan nformaton crteron, as proposed by Schwarz (1978), calculates the crteron for one or more ftted model obects for whch a log-lkelhood value can be obtaned. -2*log-lkelhood + npar*log(nobs) (1.1) where npar represents the number of parameters and nobs the number of observatons n the ftted model. BIC s a "parsmony crtera" used to perform model comparson and dscrmnaton. A seres of condtonal models was examned n terms of key predctors. Multnomal logstc regresson s an extenson of bnary logstc regresson that allows for the comparson of more than one contrast, smultaneously estmatng the log odds of a number of contrasts. Logstc regresson apples maxmum lkelhood estmaton after transformng the dependent nto a logt varable (the natural log of the odds of the dependent occurrng or not). In ths way, logstc regresson estmates the probablty of βa certan event occurrng. In the multnomal logt model t s assumed that the log-odds of each response follow a lnear model. The logt model usng the baselne-category logts wth predctors x has the form π ( x ) ' log = α + x, (2.1) π ( x ) J where β α s a constant and ' s a vector of regresson coeffcents of x, for J = 6, = 1, 2,, J-1, (Agrest, 2002). Ths model s analogous to a logstc regresson model, except that the probablty dstrbuton of the response s multnomal nstead of bnomal and we have J-1 equatons nstead of one. The multnomal logt model may also be wrtten n terms of the orgnal probabltes π rather than the log-odds. Startng from Equaton 2.1, then π ( x ) = J 1 αβ'β ( α ) + x ' ( x ) exp 1 + exp + k = 1 k k (2.2) where x are the explanatory varables age (years) and early school leavers, and = 1, 2, 5 for each of the sx latent classes respectvely. The denomnator
Latent Varable Mxture Modellng... 217 remans the same for each probablty, and the numerators for all sum to the denomnator, π =. The parameters for the baselne category n the logt ( x ) 1 expressons equal to zero, therefore the model wth category J as the baselne for the demonnator (equaton 2.1), has α β= 0, (Hosmer and Lemeshow, 2000). J = J 2.4 Instruments / Psychometrc software Database preparaton and data cleanng were completed usng SPSS (11.0). There were no mssng data. MPlus 2.13 was used, and all analyses were carred out n a sngle-step. MSExcel 2000 was used graphcally to llustrate the multnomal logstc regresson. 3 Results 0,8 0,7 0,6 Reponse Probabltes 0,5 0,4 0,3 0,2 0,1 0 heron cannabs benzo's ecstasy methadone cocane Drug of Msuse alcohol amphetamnes other opates hallucnogens hypnotcs volatle nhalants other drugs Fgure 2: Postve response probabltes for the thrteen drug types. Fgure 2 llustrates the respectve probabltes assocated wth each of the ten drug types among ndvduals recevng drug treatment durng 2000. The maorty of those seekng treatment (75%) reported heron as one of ther drugs of msuse. Cannabs was the next most probable drug of msuse among those seekng treatment (42%), and only one percent of those recevng treatment n 2000 reported use of volatle nhalants.
218 Paul Cahll and Brendan Buntng Tests of Model Ft (Informaton Crtera) 61000 60000 59000 58000 57000 56000 55000 54000 53000 52000 51000 50000 Numbers of classes Akake (AIC) Bayesan (BIC) 1 Class 2 Classes 3 Classes 4 Classes 5 Classes 6 Classes 7 Classes Akake (AIC) 60177,287 54040,653 53784,39 53337,598 53138,994 53038,709 53093,581 Bayesan (BIC) 60286,373 54225,679 54065,355 53714,502 53611,838 53607,492 53758,303 Fgure 3: Inf. Crteron (BIC&AIC) testng model ft for seven model-solutons. Table 1: Sx latent classes, the average class probabltes by class and the assocated probabltes of problem drug use. Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class n = 950 577 1013 999 183 3272 sze % = 13.6 8.3 14.5 14.3 2.6 46.8 Average Class Probabltes by Class Class 1 0.759 0.213 0.026 0.001 0.001 0.000 Class 2 0.099 0.847 0.026 0.000 0.023 0.005 Class 3 0.028 0.023 0.784 0.000 0.029 0.136 Class 4 0.001 0.000 0.010 0.989 0.000 0.000 Class 5 0.004 0.053 0.069 0.000 0.842 0.032 Class 6 0.001 0.005 0.141 0.000 0.035 0.818 Msuse of: Probabltes of Problem Use of Drug Heron 0.065 0.079 0.973 0.856 0.517 1.000 Methadone 0.010 0.000 0.008 1.000 0.000 0.000 Other Opates 0.000 0.015 0.053 0.038 0.397 0.033 Cocane 0.215 0.028 0.292 0.118 0.068 0.097 Amphetamnes 0.429 0.021 0.060 0.009 0.000 0.000 Ecstasy 0.842 0.363 0.319 0.047 0.012 0.000 Hypnotcs 0.006 0.110 0.017 0.015 0.026 0.001 Benzo s 0.018 0.033 0.370 0.356 0.665 0.296 Hallucnogens 0.168 0.018 0.049 0.006 0.000 0.000 Volatle 0.007 0.101 0.008 0.003 0.000 0.000 nhalants Cannabs 0.883 0.930 0.654 0.250 0.216 0.122 Alcohol 0.391 0.365 0.014 0.022 0.248 0.018 Other drugs 0.010 0.008 0.006 0.011 0.082 0.007 Clubbers Novces Polydrug users Methadon e msusers Benzo msusers Heron msusers
Latent Varable Mxture Modellng... 219 3.1 Goodness of ft The AIC and BIC ft ndces mprove as addtonal latent classes are added to the model, (for example, a three-class soluton BIC value 54065.36, entropy of 0.849, whereas the 6-class soluton yelds BIC value of 53607.492, wth entropy of 0.767). As the BIC ft-ndex plateaus there s no addtonal beneft n ncreasng the number of latent classes n the model. BIC and AIC logc dctate that the optmal model s one wth the lowest value, as s evdent from Fgure 3, the sxclass soluton was selected as the optmal model. 3.2 Latent classes Table 1 llustrates the unque characterstcs for each of the sx homogeneous latent classes, together wth the average class probabltes and anecdotal labels used to descrbe them. Class 5 s relatvely small n sze (n= 183), however ths class proves very valuable n terms of the emergng trends n drug consumpton patterns and renforces anecdotal evdence (EMCDDA, 2002). Class 1, ttled Clubbers conssts of 950 ndvduals (13.6%), charactersed by a hgh probablty of cannabs and ecstasy use (0.88 and 0.84 respectvely) wth alcohol and amphetamnes (speed) the next most probable drugs of msuse (0.39 and 0.43). The Clubbers are the class most lkely to msuse hallucnogenc drugs such as LSD or magc mushrooms (16.8) and are unlkely to use the harder drugs, such as heron and methadone (0.07 and 0.01). Class 2, enttled Novces represents 8.3% of the total sample (n=577). Indvduals n ths class have a very hgh probablty of cannabs use (0.93) and are also most lkely to msuse volatle nhalants such as glue and/or solvents (0.10). There s a low probablty of msuse of harder drugs such as amphetamnes, cocane or heron (0.02, 0.03 and 0.08). Class 3, Polydrug users (n = 1013, 14.5%) are those recevng treatment for multple drugs of msuse, n ths case three or more drugs. Ths class has a very hgh probablty of msuse of heron (0.97), wth the next most probable drug of msuse beng cannabs (0.65). Indvduals classed as Polydrug users are the most lkely to abuse cocane (0.29). The 999 ndvduals (14.3%) n class 4 have a probablty of 1.0 of msusng methadone. These ndvduals have a hgh probablty of heron use (0.86) and benzodazepnes (0.36) n combnaton wth methadone. The harder drugs domnate ths class and there s a low probablty of abuse of softer drugs (ecstasy 0.05 and amphetamnes 0.01) as well as alcohol (0.02). Class 5 s the smallest of all the classes representng 183 ndvduals (2.6%). Indvduals n ths Benzodazepne msusers class msuse
220 Paul Cahll and Brendan Buntng benzodazepnes, such as valum, dalmane and rohypnol (1.0). Included n ths class are ndvduals who also msuse other opates, such as codene and dstalgescs (0.40) or alcohol (0.25). Anecdotal evdence suggests the msuse of benzodazepnes as an emergng trend (EMCDDA, 2002), and thereby ths evdence from 2000 warrants close nvestgaton. Class 6 s the largest of all the latent classes wth 3272 ndvduals, representng 46.8% of the sample. Ths class, enttled Heron msusers s domnated by the abuse of heron (1.0). For members of ths class, f more than one drug was beng msused, the next most common drugs together wth heron were benzodazepnes (0.30) and cannabs (0.12). The average class probabltes ndcate the level of accurate classfcaton wthn each of the latent classes (dagonal, Table 1). The class probabltes on the off-dagonal represent msclassfcatons between each of the classes, for example, the msclassfcaton between Methadone addcts (Class 4) and Clubbers (Class1) was 0.001. Respondents were assgned to classes based on ther probablty of ncluson n that class; t was ths condtonal probablty score that was correlated wth covarates for the multnomal logstc regresson. In ths way, the probablty of beng ncluded n a specfc class was not constraned to be certan (1.0), as ths would have ntroduced assgnment error. 3.3 Multnomal logstc regresson Table 2: Multnomal logstc regresson equatons for the latent classes regressed on age (x) and early school leavers (y). Age α + β x, Age and Early School Leavers α + β x + β y Class 1 equaton = 1.853-0.129*(age) 0.094-0.121*(age) + 0.840*(school) Class 2 equaton = 2.808-0.18* (age) 1.133-0.130*(age) + 0.207*(school) Class 3 equaton = -0.339-0.02*(age) -0.651-0.020*(age) +0.193*(school) Class 4 equaton = -1.124 +0.0*(age) -0.663-0.006*(age) -0.204*(school) Class 5 equaton = -4.703 +0.084*(age) -5.594 +0.091*(age) +0.348*(school) The estmated probabltes of class membershp for a gven age (Equaton 2.2) equal ( 0.129x ) exp 1.853 π = 1 1+ exp( 1.853 0.129x ) + exp(2.808 0.18x ) + exp( 0.339 0.02x ) + exp( 1.124 + 0.0x ) + exp( 4.703 + 0.084x )
Latent Varable Mxture Modellng... 221 exp(2.808 0.18x ) π = 2 1+ exp(1.853 0.129x ) + exp(2.808 0.18x ) + exp( 0.339 0.02x ) + exp( 1.124 + 0.0x ) + exp( 4.703 + 0.084x ) through to 1 π = 6 1+ exp( 1.853 0.129x ) + exp(2.808 0.18x ) + exp( 0.339 01.02x ) + exp( 1.124 + 0.0x ) + exp( 4.703 + 0.084x ) The 1 term n each of the denomnators and n the numerator of the π 6 represents exp( α + β ) usng α β = 0, where J s class 6. The sx probabltes J J J = J add to 1 as the numerators sum to the common denomnator. The Novces class domnates the younger ages and almost dsappears by the early thrtes. The probablty of membershp n the Clubbers class peaks at 20 years of age and declnes slowly through the thrtes. The Methadone and Polydrug classes reman very robust over the spectrum of ages. From the earlytwentes through to the late-fortes, ndvduals recevng treatment for drug msuse are most lkely to be members of the heron addcts class. For older drug users, (40 years +), there s a sharp rse n the probablty of membershp of the Benzodazepne msuser class. Fgure 4 llustrates the multnomal logstc regresson paths for each of the classes regressed upon age (contnuous varable) graphcally modelled wthn MSExcel. 0,7 0,6 Probablty of class ncluson 0,5 0,4 0,3 0,2 0,1 'Clubbers' 'Novces' 'Polydrug users' 'Methadone msusers' 'Benzo msusers' 'Heron msusers' 0 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 Age Fgure 4: Sx Latent Classes regressed on Drug Users Age. Fgure 5 llustrates the regresson paths for each of the sx latent classes regressed on drug user s age, and also regressed on those who left school before aged 15, and those who stayed n school after the age of 15 years. All the sold lnes represent those who left school early. Indvduals who stayed n school longer were more lkely to be members of the Clubbers class (recreatonal drugs use), whle the probablty of early school leavers beng n the Novces class declnes sharply n the late teens. It s also apparent that early school leavers are
222 Paul Cahll and Brendan Buntng more lkely to be members of the Heron class over all of the ages and less lkely to be members of the Benzodazepne msuser class n later years. It should be noted that the actual number of drug users n treatment decreases wth age and although the probablty of membershp of the Benzodazepne msusers class ncreases wth age, ths may be as a functon of a declnng proporton n the other latent classes. 0,6 Probablty of class ncluson 0,5 0,4 0,3 0,2 0,1 'Clubbers_1' 'Clubbers_2' 'Novces_1' 'Novces_2' 'Polydrug users_1' 'Polydrug users_2' 'Methadone msusers_1' 'Methadone msusers_2' 'Benzo msusers_1' 'Benzo msusers_2' 'Heron msusers_1' 'Heron msusers_2' 0 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 Age Fgure 5: Sx Latent Classes regressed on Drug User s Age, and those who left school before age 15 sold lne_1) and those who stayed n school after age 15 years (broken lne_2). 4 Conclusons The current study utlsed latent varable mxture models to examne profles of drug msusers and ncorporated these methodologes for the analyss of treated drug msuse n the Republc of Ireland Usng both the Bayesan and Akake nformaton crtera (BIC and AIC) a sxclass soluton was selected as the optmal number of latent classes best-fttng the data. From the subsequent uncondtonal latent class analyses, the characterstcs of each of the classes were uncovered and each of the typcal profles was developed. The dverse drug msusng behavours of the 6994 ndvduals who receved treatment for drug msuse were structured nto sx dstnct and homogeneous classes (Clubbers n=950, Novces n=577, Polydrug users n=1013, Methadone msusers n=999, Benzodazepne msusers n=183 and Heron msusers n=3272). Gven the complex nature of problems assocated wth drug msuse, ths research hghlghted the need to avod a sngle treatment modalty for problem drug use, and showed that a range of treatment optons may need to be consdered for the unque characterstcs of each class. A seres of condtonal models was then examned n terms of the key covarates 1) age and 2) age and early school leavers. The parameter estmates
Latent Varable Mxture Modellng... 223 obtaned from the multnomal logstc regresson were graphcally modelled wth MSExcel and the respectve functons descrbed. Stemmng from a better understandng of the probablty of an ndvdual beng n a gven latent class dependent on age (contnuous) and early school leavers (dchotomous), nterventon strateges both reactve (health care polces) and proactve (socal and educatonal ntatves) may be more fully evaluated. References [1] Agrest, A. (2002): Categorcal Data Analyss (2 nd ed). NY: Wley and Sons. [2] Department of Toursm, Sport and Recreaton. (2001): Buldng on Experence. Natonal Drugs Strategy 2001-2008. Dubln: The Statonery Offce. [3] EMCDDA (2002): European montorng centre on drugs and drug addcton. Annual Report on the State of the Drugs Problem n the European Unon. Luxembourg: Offce for Offcal Publcatons of the European Communtes. [4] EMCDDA and Pompdou Group (2000): Treatment Demand Indcator: Standard Protocol 2.0. Lsbon: European Montorng Centre for Drugs and Drug Addcton. [5] Gossop, M., Marsden, J., and Stewart, D. (2001): NTORS. After Fve Years. The Natonal Treatment Outcome Research Study. London: Natonal Addcton Centre. [6] Hagenaars, J.A. and McCutcheon, A.L. (Eds.) (2002): Appled Latent Class Analyss Models. UK: Cambrdge Unversty Press. [7] Hosmer, D.W. and Lemeshow, S. (2000): Appled Logstc Regresson. New York: John Wley and Sons. [8] Muthén, L. and Muthén B. (2003): MPlus short courses. Specal Topcs n Latent Varable Modellng Usng Mplus. Fulda, Germany. [9] O Bren, M., Kelleher, T., and Cahll, P. (2002): Trends n Treated Drug Msuse n Health Boards 1996-2000. Health Research Board, Dubln. [10] Schwarz, G. (1978): Estmatng the dmensons of a model. Annals of Statstcs, 6, 461-464. [11] Smpson, D.D. and Sells, S.B. (Eds.). (1990): Opod Addcton and Treatment: A 12-Year Follow-up. Malabar, Florda: Kreger.