Dynamic Modeling of Behavior Change

Transcription

1 Dynamic Moeling of Behavior Change H. T. Banks, Keri L. Rehm, Karyn L. Sutton Center for Research in Scientific Computation Center for Quantitative Science in Biomeicine North Carolina State University Raleigh, NC an Christine Davis, Lisa Hail, Alexis Kuerbis, Jon Morgenstern Substance Abuse Services Department of Psychiatry Columbia University Meical Center 8 Fort Washington Ave. HP-4 NYC, NY 3 November, Abstract We consier a conceptual an quantitative moeling approach for investigating ynamic behavior change. While the approach is applicable to behavior change in eating isorers, smoking, substance abuse an other behavioral isorers, here we present our novel ynamical systems moeling approach to unerstan the processes governing an iniviual s behavior in the context of problem rinking. Recent avances in technology have resulte in large intensive longituinal ata sets which are particularly well suite for stuy within such frameworks. However, the lack of previous work in this area (specifically, on the inter- an intra-personal factors governing rinking behavior of iniviuals) reners this a aunting an unique challenge. As a result, issues which are typically routine in mathematical moeling require consierable effort such as the etermination of key quantities of interest, an the timescale on which to represent them. We iscuss the construction of an initial mathematical moel for two starkly istinct iniviuals an make a case for the potential for such efforts to help in unerstaning the unerlying mechanisms responsible for behavior change in problem rinkers. AMS Subject Classification: 9E, 34K9, 6G. Key Wors: Mathematical psychology, inverse problems, elay ifferential equations, behavior change, moel comparison.

2 Introuction In the stuy of rug an alcohol abuse, there has been extensive information collecte on substance use, participant s willingness to change behavior, an participant s success in a particular treatment. Fiel experts have formulate various ieas concerning factors that control a patient s motivations an behavior. However, the relative contributions of these riving factors, an specifically the mechanisms for behavior change, are unclear. These interacting factors, which are inherently complex an nonlinear, change over time. From a causal perspective, it is natural to explore these issues within the framework of a mathematical moel, which enables us to escribe these processes quantitatively as a ynamical system. Automate ata collection systems have enable the collection of large ata sets involving both substantial numbers of patients as well as time points (warranting the term intensive longituinal ata sets ). We seek to moel rinking behavior an associate behavior change as informe by such a ataset, Project MOTION, to gain insight into the unerlying mechanisms. Certainly, the application of mathematical moeling to unerstan learning processes an social behavior is not a new fiel an there have been fruitful efforts in the past. Briefly, but not exhaustively, some relevant works inclue those of Frey an Lau [6] in which a system of integral equations was use to escribe ways in which governments make ecisions. Van Geert for example in [4] an [4], employs technically iverse tools in the form of a logistical moel an a cellular automaton moel to unerstan cognitive evelopment of chilren. Notably, the work of Stephen Grossberg, summarize nicely in [9] an [], is a significant contribution to the theory of learning in the context of cognitive mechanisms as well as associate neural network ynamics. We note that there has been other recent work resulting from the collaborations of fiel experts in substance abuse an mathematicians. These efforts have been largely at the population-level, or specifically, the characterization of iniviuals accoring to their use ( heavy, light, abstaining, etc.) similar to epiemiological or ecological moels. The goal is then to investigate which scenarios result in overall less or more rinking/substance use in the population. Some contributions along these lines are that of Scribner et al., [37] an Ackleh et al., [], which focus on stuent populations an exploring campus alcohol policies with the goal of reucing wetness. This work was further extene in Rasul et al. [33], where the moel was calibrate (via parameter estimation) to a stuent population an specific alcohol policy scenarios were explore. Another series of efforts exploits ieas from theoretical epiemiology. In 4, Gorman et al., [7] outline the ways in which stuies on alcohol coul be improve via ynamic systems moeling an control theory. This note came out of a meeting on Ecological Moeling of Alcohol-Relate Behavior sponsore by the NIAAA an possibly began the collaboration that gave rise to a series of papers. Sanchez s initial paper [35], classifying iniviuals as susceptible (nonrinkers, S), rinkers (D), an recovere alcohol users (R) has the same structure as a typical SIR moel. The ifference is that the relapse term, (recruitment of R to D) is moele by an interaction term, giving rise to a backwar bifurcation. This implies that recovere alcohol users can easily relapse through even few social interactions with other rinkers. This work was extene in [3], to explore the role of nonlinear relapse among networks of rinking communities with varying connectivities. Mubayi (in [3, 3]) extens this work to examine the contagion of rinking (using the same basic SDR principles), among iniviuals in low- an high-risk communities, an explore the role of resience times in each community. Also, it is shown that social contacts or extent of mixing within communities rastically affects the outcomes. In [3], the eterministic moel was extene

3 to consier variability in social interactions of rinkers, an increasing/ecreasing levels of rinking. Then the istribution of rinking levels uner prevention, intervention, an a combination of both was presente an iscusse. Research on substance use over the past several ecaes has attempte to ientify preictors of successful achievement of abstinence an moeration. There are four sets of categories or constructs that have been broaly ientifie as being relate to sustaining the status quo or prompting changes in rinking behavior. These categories are: ) stable characteristics of the patient such as age, gener, an rinking history, ) moo an affect, 3) environmental factors, such as social networks an stressful events, an 4) internal process factors, such as motivation to change, commitment to changing, self-efficacy, etc. The MOTION ata set is comprise of information collecte via an interactive voice recoring (IVR) from rinkers who wante to reuce their rinking, although not necessarily cease rinking altogether. Ieally, each iniviual answere this 4-question survey once per ay for eight weeks, or 56 ays, an gave information about inter- an intrapersonal factors that coul potentially influence their rinking as well as their rinking behavior in the past 4 hrs an their commitment to avoi heavy rinking or rinking at all in the next ay. Within Project MOTION, both the IVR-base aily survey an the fixe assessments were attempts to collect ata across these broa categories in orer to inform the process of change for iniviuals aiming to moerate their rinking. These ata were collecte from participants both within the context of a brief treatment an through inepenent self-monitoring. Data yiele from each of these measures an iniviual items were utilize to help create an informe picture/guie to how iniviuals attempt to change their rinking an how the trajectories of change may iffer across groups. Generally, the use of mathematical moeling to learn about a physical or biological process is one as an iterative process in which an initial moel is formulate an then compare with observe or experimental ata. This comparison gives insight into any iscrepancies between the observe processes an moel preictions, suggesting moifications an/or refinements to the moel. This process is repeate until the moel provies sufficient information to capture key aspects of behavior of the observe system an to answer questions of interest. However, in most applications, previous moels have been evelope an/or there is previous knowlege of quantitative behavior of unerlying processes, so moeling is typically initiate with an iea of the appropriate basic framework an possible mechanisms. This is not the case for this particular problem, an the moeling here is with the goal of working towar such a framework. As such, moel terms representative of possible mechanisms are consiere on the basis of how well the moel solutions then reprouce the observe trens in ata. Only after several such efforts are successful are the use of more precise inverse problem methos, such as parameter estimation, moel sensitivity analyses, etc., appropriate as they rely on the mathematical moel being a relatively goo escription of the observe processes. Currently, there are also some challenges in the observation process, or measurements, that woul impee the application of such methos. While the aim of the current work was simply to attempt to evelop a mathematical moel that might ai in our unerstaning of any behavioral mechanisms relevant to rinking behavior, the potential is more far-reaching as we feel that similar approaches woul be valuable to unerstan behavior changes in ieting an eating isorers, smoking, substance abuse, etc. A cohesive theory of how an why iniviuals change as they moify their behavior long-term is lacking not only in problem rinking but in other areas of behavioral isorer. In such investigations it is not expecte 3

4 that all mechanisms will be at work in all iniviuals as there is consierable variability, but there are likely groups or cohorts of people who change similarly. Working towar mechanism-base escriptions of behavior change, our an similar efforts coul potentially help clinicians ientify patients as part of a characterize cohort an thereby potentially improve therapy. Seconary to this goal was the aim of improving upon current ata collection proceures an stuy esigns, which can greatly improve our ability to learn about these processes an make them more amenable to mathematical approaches. As we shall iscuss, the evelopment of the first two initial moels has been enlightening, an some immeiate changes to propose became clear. Further changes representing somewhat of a paraigm shift are iscusse in the closing remarks of this paper. There is no straightforwar way of organizing ata on such a large variety of aspects of aily life into a reasonable number of variables to stuy within a mathematical moeling framework. To begin formulating conceptual variables, we applie linear methos such as calculating coefficients of etermination, linear interpolation via least-squares approximation an principal component analysis. However, these methos o not provie useful information on relationships between the responses to questions. It is sensible to construct our moel variables on questions base on similar ieas use in the IVR. We shall examine how these categories relate to each other, an how to best use this ata in formulating a moel. Our initial interpretation of this ata is base on knowlege an preictions from the fiel of substance abuse an recovery, visually assessing qualitative trens between similar questions within topical categories, an grouping them into variables. Once variables were ecie upon, qualitative trens between them were examine to etermine the existence an nature of relationships between them within a single patient. We constructe a first moel base on prior knowlege of hypothesize relationships between the moel variables using knowlege of substance abuse therapy an relationships observe in the ata from Project MOTION. We iscuss the construction of this initial moel along with moifications to it which improve agreement between solutions an ata. One of these is supporte by the use of a moel comparison statistic, an example of one way that inverse problem methos can help to refine these moels once their solutions are reasonably close to ata. Builing upon lessons learne from the initial patient s moel, we evelope a moel for another patient, thereby emonstrating use of ynamic moels as a means of escribing an stuying these processes within iniviuals. Finally, we comment on issues with the current ata set an suggest improvements to future stuies so that these ata sets are better able to quantify the unerlying processes. Project MOTION: Rationale, Proceures, an Data Motivational interviewing (MI) has been emonstrate to be an effective stan-alone intervention for alcohol use isorers (AUD). The consistency, magnitue, an urability of its effects, especially given its brevity, suggest powerful mechanisms of behavior change (MOBC) are operating to reuce rinking. Thus, gaining a better unerstaning of the unerlying MOBC in MI is important. However, existing MOBC stuies of MI have yiele limite an contraictory finings. Project MOTION (escribe in greater etail in [9] an []) aime to rigorously examine MOBC in MI by improving on prior methos an using an enhance conceptual framework that consiere nonspecific therapy factors an self-change mechanisms. The treatment conitions were labele Full MI (FMI) an Spirit-Only MI (SOMI). General avertising online an in local meia was use to recruit 89 participants seeking to 4

5 reuce but not stop rinking. Participants were consiere eligible for the stuy if they were: () between the ages of 8 an 65; () ha an estimate average weekly consumption of greater than 5 or 4 stanar rinks per week for women an men, respectively; an (3) ha a primary alcohol use isorer (AUD). Participants were exclue from the stuy if they: () presente with significant substance use or a current substance use isorer (for any substance other than alcohol, marijuana, nicotine or caffeine), which was efine for our assessment purposes as greater than once weekly use in the past month; () presente with a serious psychiatric illness or substantial suicie or violence risk; (3) emonstrate clinically severe alcoholism, as evience by physical withrawal symptoms or a history of serious withrawal symptoms; (4) were legally manate to complete a substance abuse treatment program; (5) reporte social instability (e.g., homeless); (6) expresse a esire to achieve abstinence at baseline; or (7) expresse a esire or intent to obtain aitional substance abuse treatment while in the stuy. Treatment was elivere in 4 sessions that laste between 45 minutes to an hour long at weeks,, 4, an 8. Participants were blin to conition assignment, meaning they were not tol which therapy they woul be receiving. All participants were followe for a total of 9 weeks an participate in 9 minute assessments at weeks,, 4 an 8. Participants in MI an SOMI were calle an aitional 4 weeks post-treatment (week ) to collect rinking ata via the Timeline Follow Back (TLFB, escribe further below). Follow up rates for assessments at weeks, 4, 8 an were %, 96%, 9.%, an 68% respectively. All participants were aske to complete a aily telephone survey at the en of each ay for the uration of the eight weeks (one week of baseline information an seven weeks of treatment) of the stuy, for a total of 56 possible ays. At the en of their initial screening visit (a week prior to ranomization), a research assistant (RA) provie each participant a 5 minute training session on how to use the Interactive Voice Recoring (IVR) system, a system evelope using TELESAGE SmartQ 5., a software package specifically esigne for the aministration of automate surveys [39]. Each ay, participants respone to a series of questions about potential meiators of rinking behavior such as moo, commitment to not rinking heavily or not at all, an confience to o so, in aition to the number an type of rinks they consume in the last 4 hours. Once familiar with the system, the aily IVR session require approximately 5-minutes to complete. Each participant was provie a toll-free phone number an an anonymous participant ientification number to ensure confientiality. The IVR system coul be accesse between 4: p.m. an : p.m. This time perio was juge to be when participants most likely woul be able to reflect on their alcohol use that occurre past the prior ays assessment an before most iniviuals woul be likely to consume large amounts of alcohol. This time winow ha the avantage of proviing consistent report timing an facilitate compliance by creating routines for participants. If participants faile to call into the system by 8: p.m., an automate reminer call was mae. Participants ata were coe as missing if they were not able to complete a call. This aily questionnaire was consiere a form of self monitoring one s rinking. 3 Interpretation of Data to Inform Moeling While the wealth of information in this rich ata set presents a unique opportunity to examine aspects of rinking behavior change longituinally, it also presents challenges. In contrast to other areas, such as the physical sciences, in which the state variables representing key players in the processes are more obvious, no such information is known. In aition, the relevant timescales on 5

6 which the variables are changing an therefore shoul be observe are unclear. It is possible that some of the measures, if not eeme to be important on the timescale of the current ata set, may be riving other aspects of rinking behavior either on a shorter or longer timescale than that on which we choose to focus. As we initially approache the task of eveloping ynamical systems moels of these processes, we neee to assess the information content in the Project MOTION ata. Iniviuals rinking behavior, an their response to treatment, is highly variable, an it is likely that one moel escribing all iniviuals is not possible. At any rate, to be so ambitious with first attempts at iscerning key mechanisms is unreasonable an focusing on groups of iniviuals who respon in similar ways is a more feasible approach. With that in min, we proceee to organize the ata in ways that woul best inform moeling efforts, escribe in much greater etail in [9]. Given the longituinal nature an frequency of the IVR responses along with the esire to unerstan change, it is natural to focus on those ata for the evelopment of a ynamic moel. We also consiere which of the fixe assessment battery provies novel or equivalent information to that in the IVR. While these ata are not sufficiently informative on a longituinal basis to be consiere as main components of a moel at this stage, it is possible that these ata may have supplemente possible missing responses from the IVR, or the novel ata coul be information that shoul be inclue in future IVR-type questionnaires. The etails of this exploration are given in [9], but ultimately, we i not use any of this information in the moels presente here. The timeline followback responses prove to be useful in that we were able to etermine criteria upon which to ivie the subjects into responers, non-responers, or neither. We efine responers as those iniviuals who emonstrate at least a 4% ecrease in their rinking by their last treatment session or the week 8 assessment. Non-responers were ientifie as emonstrating no change in their rinking or increasing their rinking by the week 8 assessment. Thus, there are several iniviuals who woul be classifie neither as a responer nor as a non-responer. We thought it pruent to focus on iniviuals with the esire behavior-changing mechanisms in place, or at least those who exhibite the esire behavior changes, for initial moeling efforts. Therefore, we chose to focus on the eight subjects classifie as responers. With these iniviuals ata, we etermine an initial grouping of IVR ata questions (as it is not reasonable to consier each of the 4 questions as inepenent information, or variables) base on intuition, categories the investigators ha in min when constructing the IVR questionnaire, an previous hypotheses of interactions of key factors. 4 Initial moel formulation In the hopes of ientifying meaningful groups of similar IVR items to be use as the basis for moel variables, we turne to common statistical techniques that are frequently use in the analysis of such ata sets among psychologists. The techniques employe were least-squares regression, principal component analysis, an factor analysis. We note that these test for static relationships, an therefore we woul not necessarily expect them to be very informative concerning processes that may change nonlinearly in time. These results an an explanation of the methos are containe in [9] an more succinctly in []. The static methos use to search for affine relationships between variables faile to prouce strong evience for any linear patterns. While regression moels were informative in that some existing weak relationships between variables, or observe ata categories, were reveale, the nature 6

7 of these relationships were not mae clear. Results from principal component analysis i not yiel any unique trens in patient ata. First, PCA in each iniviual i not ientify possible subsets of responses to group together. Moreover, the groups create i not explain the variance in the ata better than the original variables. Finally the groupings of variables that PCA prouce i not follow any patterns when comparing iniviuals. We also performe factor analysis, a metho similar to PCA an iscusse by R. J. Rummel in [34], on the ata. While factor analysis may be implemente using any one of a variety of methos, we employe the maximum likelihoo metho. The factor analysis resulte in a so-calle Heywoo case, an anomaly in communality estimates, suggesting that factor analysis is not an appropriate metho to use on this ata set. In factor analysis the covariance of the assume error in the measurements is estimate in an iterative process when using the maximum likelihoo metho, so if a value on the iagonal of this covariance matrix becomes less than or equal to, the maximum likelihoo metho is converging to an inappropriate result. It is likely that the non-normal istribution of ata an the small number of possible values for most observations negatively affecte the estimations of covariance components an factor loaings. Our static linear analysis i not show any clear patterns between IVR questions. Some of the questions inclue in the Project MOTION IVR ask about reunant or closely relate information. Thus, we began with a reasonable grouping of IVR questions to base our moel variables on the categorical groupings of IVR questions. It is reasonable to represent composite scores of the categorical groupings (an hence the variables) as fluctuating on a continuum as oppose to a iscrete set of numbers, as a iscrete set may not appropriately reflect how an iniviual is feeling. For example, a person may not be simply happy or sa, an there are many egrees or intensities at which happiness or saness may be felt that o not correspon to five igits. Discrete scales are more amenable to questionnaires, particularly when aministere via the telephone; however when eveloping a ynamic moel, the use of a continuum of values is preferre, an likely more reasonable, as it can be argue. Similar arguments can be mae concerning the true nature of the ynamics of the processes in time, so we will consier the moel as a function of continuous time as oppose to iscrete time. We thus assume that only the ata collection times an observation values are iscrete, while the unerlying processes of interest change in a continuous manner. We etermine that a triweekly timescale, seen in Figure, woul be most suitable to observe the processes. Daily ata appeare to have too many fluctuations, so that any trens (an hence, mechanisms or ynamics) were not evient, an ata average by week appeare to not manifest sufficient ynamics, or that the information was essentially average out. A triweekly timescale appeare to be a reasonable level of refinement for observing the ata. It consists of 3 time points per week: weeken (Friay night through Sunay prior to IVR call), Sunay an Monay, an Tues through Thurs. 4. Patient 69 moel We initially chose to focus on patient 69 since, base on the TLFB assessment seen in Figure, it appeare that his rinking significantly change uring the observe time perio in a seemingly systematic way. That is, there appeare to be some heavy rinking initially that the iniviual graually reuce so that there was consistently an noticeably less rinking at the en of the observation perio. This patient respone consistently to the IVR system an appeare to generally 7

8 4 Daily rink counts Weekay Weeken 9 8 Triweekly rink averages 7 Drinks Drinks Day Weekly rink averages, Day Drinks Day Figure : Daily (top), triweekly (mile) an weekly (bottom) rink counts for patient 69. Re triangles inicate weeken ays, blue ots inicate weekays. exhibit characteristics that the clinicians woul ientify as inicators of successful behavior change. Causal relationships between potential variables were etermine base on visual analysis of the ata, an categorical moels base upon these relationships were constructe for a select four responing patients, incluing patient 69. The categorical moel is a schematic representation of moel variables an their relationships. In general, whether categories were inclue epene entirely on whether the category appeare to be irectly relate to rinking (represente as aily averages over the or 3 ay time perio). Relationships were consiere among the variables that were initially selecte as being pertinent to the preliminary categorical moel. We i not iscuss possible relationships among variables that i not appear to be rinking relate, an therefore were not consiere in these categorical moels. Consequently, it coul be that some of the variables are relate to each other but that relationship is not represente in the categorical moel unless each of them is relate to rinking. Soli lines in the schematics for categorical moels enote strong relationships while weaker relationships are represente with otte lines. Arrows inicate causality or irection. The representation of the categorical moel for patient 69 can be seen in Figure 3. The graphs of the ata from which we ecie there are noticeable relationships are shown in Figure 4. All relationships for this iniviual seeme to be relatively strong when present. 8

9 7 6 Drinks for 69 TLFB IVR 5 Sum Drinks time (wks) Figure : Weekly totals of rinks as reporte by patient 69 in the Timeline Followback (TLFB) assessment an the IVR system. The initial moel, given by ( ) t A(t) = a χ (G>) G(t + s)κ (s)s + a 3χ (D>)D(t) τ a 4 (Q(t) + χ (Q>) Q (t)) a 5 L(t) () ( ) t G(t) = a A(t + s)κ (s)s ( + c χ W (t) )A G () τ [ ( ) ] t D(t) = a 34Q(t) a 3,5 L(t) a 3 exp G G(t + s)κ 3 (s)s G D (3) D τ 3 t Q(t) = a ( ) 43 + χw (t) χ (D>) D(t) [ ( a 4 exp A Q min(, (A(t τ 4 ) A Q)) ) ( χ W (t) )A ] Q (4) [ ( ) ] t L(t) = a 5G(t) a 53 D(t) a 5 exp A min(a L, (A(t τ 5 ) A L)) A L. (5) L was erive from patterns in the IVR ata an confirme by plausible explanations of rinking behavior processes. It inclues all possible mechanisms consiere in these investigations an we i not think it likely that all terms woul remain as we procee in an iterative moeling effort [, Chapter ]. However, there are some key features that will likely be necessary to inclue in even the simplest mathematical representation of iniviual-level behavior change in general an specifically concerning one s alcohol intake. Namely, these are nonlinearities, elaye an/or cumulative effects, an time-epenent threshol behavior. To illustrate such moel components an their use, we consier the equation () escribing the 9

10 Limit L(t) Drinks A(t) Guilt G(t) Comm to Quit Q(t) Desire D(t) Figure 3: Categorical moel for patient 69 base on hypothesize relationships in the IVR ata. Strong relationships are represente by soli lines, an weak relationships are represente by ashe lines. All relationships are taken to be biirectional. rate of change of guilt: ( ) t G(t) = a A(t + s)κ (s)s ( + c χ W (t) )A G. τ This equation is the only one that was not moifie as the moel solutions were compare to ata (escribe in Section 4.), an exhibits most of the features liste above. It can be interprete as the iniviual s guilt, or more specifically his feeling that his rinking over the previous ay was excessive, increases with the number of rinks consume since τ ays ago, if they excee the threshol ( + χ W (t))a G. We note that since A(t) is the rinking rate, the number of rinks consume from τ ays ago to the present time is represente by τ A(t + s)s. The function κ (s) is use to effectively weight times at which rinking may have a more or less significant effect on his guilt. For example, if κ (s) increases with s going from τ to, then this woul inicate that his rinking rate at the present time more strongly influences his guilt than that in the past. This particular iniviual s (an likely most iniviuals ) iea of excessive rinking changes from the weekay to the weeken, an thus the function χ W (t), which is equal to one uring the weeken an zero uring the week, changes the iniviual s level of acceptable rinking or his threshol from A G uring the week to ( + c )A G uring the weeken.

11 Wken rinks Wkay rinks 4 3 Wken guilt Wkay guilt 4 3 Wken esire Wkay esire Wken limit Wkay limit 6 5 Wken quit Wkay quit Figure 4: IVR ata for variables eeme important for the evelopment of a mathematical moel moel moification Attempting to examine how well solutions agree with ata is nontrivial. Numerous parameters are not reaily psychologically or emotionally interprete, thus complicating the estimation of feasible ranges for their values. While we may be able to anticipate the moel solution of each iniviual term, there is no way to anticipate the nature of solutions of the moel when all terms are inclue simultaneously. Further, we o not expect that all terms will be inclue in a final moel which we consier the best escription for the system as justifie by the current ata set. Therefore, we took a reuctionist approach. We began with the simplest system of rinking behavior that we anticipate capable of proucing solutions with reasonably close behavior to that seen in the ata. Once we reprouce the ata as well as possible with the simple moel, we ae aitional components if necessary to capture features in the ata not explaine by the current moel. This approach correspons to a general moeling philosophy of incluing as much complexity in the moel as neee, but no more, to capture the key features manifeste in the ata. The overall goal of the project is to unerstan rinking behavior, so the simplest moel necessarily inclues the equation for the rinking rate ta(t), an mechanisms riving an suppressing alcohol consumption. To have one without the other woul lea to strictly increasing or ecreasing alcohol consumption, an from the IVR ata, we know this is not the case. We focuse on the esire D(t) an guilt G(t) (which may be more accurately interprete as a norm violation measure, but still referre to here in shorthan as guilt) variables. These were ientifie as two variables that appeare to have the strongest relationships with rinks in the IVR ata. Therefore, the simple moel initially consiere is

12 t A(t) = a χ G>G (G(t) G ) + a 3 D(t), (6) [ ] t G(t) = a A(t + s) ( + c χ W (t) )A, (7) t D(t) = a 3χ G>G (G(t) G ) (8) Immeiately upon inspection of the equations (6) - (8) there is an issue in that the only riving force in the alcohol equation is the esire to rink. But the equation governing esire, equation (8), results in a non-increasing time course for esire D(t). Thus, as written, esire will not provie a riving force for rinking as it will not increase beyon its initial value. Also in the original moel, the only way for the iniviual s esire to increase is through a negation of essentially controlling mechanisms. Therefore, the moel was missing a mechanism to rive the patient s rinking, an upon further consultation with the ata, a notable weeken/weekay pattern in both the esire an rinking ata was iscerne. That is, the iniviual s esire to rink an his rinking increase going into the weeken an remaine elevate uring the weeken, an ecrease as the weeken ens. Thus, we inclue the term c h(ˆt) in the t D(t) equation, where ˆt = t mo 7 an h(ˆt) is given by (ˆt.5).5 ˆt < (Fri a.m. thru Fri p.m.) (ˆt.5) ˆt <.5(Fri p.m. thru Sat a.m.) h(ˆt) = (ˆt 3.5) 3.5 ˆt < 4(Sun a.m. thru Sun p.m.). (ˆt 4.5) 4 ˆt < 4.5(Sun p.m. thru Mon a.m.) else A possible interpretation of this mechanism is iscusse briefly in Section 4.4. Setting a 3 = so that the esire equation is just t D(t) = c h(ˆt) allows us to see the effect of this weeken-epenent mechanism h(t) on esire, shown in Figure 5. This form of the esire equation appeare to agree well with the esire ata as seen in the bottom panel of Figure 6. Therefore, we have goo reason to think that this subject s esire is escribe well by our representation of the weeken/weekay mechanism alone. We note that not all iscussions of ata fitting in this ocument are actually referring to a least-squares type of fitting, but rather in some cases the choosing of parameter values an manual ajustment to see the effects on the solution as compare with ata, i.e., a type of simulation-base sensitivity analysis for moel response with respect to parameter values. The parameter values must be somewhat close, an/or with feasible ranges etermine to hope for any optimization routine to be able to minimize the ifference between the moel an ata, thereby resulting in reasonable parameter estimates. Therefore, the manual fitting iscusse here is a necessary first step prior to estimating parameters via inverse problem methos. Once the esire ata was explaine well, we examine the epenence of the iniviual s rinking on esire, which we ha hypothesize as the riving mechanism. It became apparent that to have that term in the ta(t) equation epen on D(t) was flawe since any positive value of D(t) will result in an increase in rinking rate (i.e., rinking is accelerate). What is more accurate is that positive but possibly waning esire, for example on a Sunay, woul mean that the iniviual is consuming alcohol albeit at a slower rate, an thus, their rate of rinking is positive A(t) >,

13 .5 D/t = h(t) D(t) Figure 5: Solution of t D(t) = c h(ˆt) an c (ˆt) with c = 4.5 an D() =.5. but ecreasing (say, as they go into the work week) ta(t) <. This is irectly proportional to the rate of esire, not the value of the esire variable. Intuitively, it oes make sense that the rate of change of alcohol consumption shoul be more irectly relate to the rate of change of esire, an thus, the number of rinks shoul be correlate with the iniviual s esire level. This is a subtle point, but it oes require attention to maintain the fielity of the processes represente in the mathematical moel. The simple moel is then more reasonably given by t A(t) = a χ G>G (G(t) G D ) + a 3 t, (9) [ ] t G(t) = a A(t + s)s ( + c χ W (t) )A, () t D(t) = a 3χ G>G (G(t) G ) + c h(t). () Then the system may be reuce by the substitution of the equation td(t) into the secon term of equation ta(t), resulting in t A(t) = a χ G>G (G(t) G ) + a 3 ( a 3 χ G>G (G(t) G ) + c h(t)) [ ] t G(t) = a A(t + s)s ( + c χ W (t) )A. 3

14 Table : Parameter values ã c 8. A() ã 3 6 A G() a.8 c 4.5 D() -.5 If we take ã = a + a 3 an ã 3 = a 3 + c, the moel is further simplifie to t A(t) = ã χ G>G (G(t) G ) + ã 3 h(t), () [ ] t G(t) = a A(t + s)s ( + c χ W (t) )A. (3) The rinking an guilt ata are shown along with solutions to the moel ()-(3) with ã = in Figure 6, along with the esire ata an the solution of t D(t) = c h(t). The parameters an initial conitions are given in Table. The agreement between the solution an ata is goo in the sense that rinking episoes (with the exception of two) are preicte, an suggests that the iniviual s elevate rinking episoes may be primarily attribute to the weeken/weekay pattern. Further, it appears that his esire an rinking are closely linke, being that the same function h(t) can be use to reprouce both of the trajectories shown in the ata. While we suspect that the effect of guilt on rinking is relatively insignificant compare to the more ominant weeken-epenent effect seen in the esire ata, as reflecte in the parameter values in Table, we make use of an inverse problem technique in the next section to etermine if it is reasonable to exclue this term in the moel. 4

15 5 moel rinks ata wkay rnks ata wken rnks moel G(t i ) ata wkay G(t i ) ata wken G(t i ) moel D(t i ) ata wkay D(t i ) ata wken D(t i ) Figure 6: Agreement between the mathematical moel an patient 69 s triweekly IVR ata. 5

16 4.3 Moel Comparison Statistic The moel comparison statistic use here (given in [, Chapter 3] more precise an thorough iscussions of this test statistic can be foun in [4, 6, 7, 8]) can be use to etermine if a more complicate version of a simpler moel gives a statistically significant improvement in the agreement between moel solution an ata. That is, it is appropriate when comparing neste moels, or when one moel can be written as a special case of the other. As such, it is well suite to help etermine whether specific terms in moels can reasonably be simplifie, or even neglecte altogether. We employ it here to etermine whether the moel t A(t) = a χ G>G (G(t) G ) + a 3 h(t), [ ] t G(t) = a A(t + s)s ( + c χ W (t) )A. (where the tile s above parameters a an a 3 are neglecte here an throughout the rest of the paper) with a provies a statistically significant improvement, if any, to the moel with a = when comparing moel solutions with ata. The statistic involves the resiual sum of squares (RSS), which is a measure of the istance between the moel solution an ata an is given by RSS = n f () (t i ; θ) y () + f () (t i ; θ) y (), i= where moel parameters are represente by θ, {y () } n i= is the rinking ata, {f () (t i ; θ)} n { A(t i + s; θ)s} n i= compute with parameters θ, {y() } n i= is the guilt ata, {f () (t i ; θ)} n i i i= = i= = {G(t i ; θ)} n i= an n = n = 3 for the ata on the triweekly timescale. The corresponing moel values are also converte to the triweekly timescale in the equivalent way (averaging over or 3 ays as escribe immeiately before the beginning of Section 4.). We estimate moel parameters θ by minimizing the objective functional n J(θ) = i= A(t i + s)s y () i n + G(t i ) y (), (4) over a feasible parameter space Θ. In comparison, estimating parameters with the constraint a = is one by minimizing the same functional over a restricte parameter space Θ H = {θ Θ a = } which is a subspace of the original space (Θ H Θ). This is a reuction in the egrees of freeom in parameters by one, an in general, the resulting resiual sum of squares when allowing a to vary (minimizing over Θ) will be at least as small or smaller than when minimizing over Θ H, or fixing a =. The test statistic can be use to etermine whether this improvement, if any, is statistically significant an is efine by U = n ( J(θ H ) J(θ) ) where n = n + n. We performe this test in two ways: comparing resiuals from estimating moel parameters θ = (a, a ) T with estimating θ H = (, a ) T (where a = an a is allowe to vary), an 6 J(θ) i= i (5)

17 Table : Some values for the χ () istribution χ () confience.3 75%.7 9% % % % θ = (a, a 3 ) T compare with θ H = (, a 3 ) T. In the first case, the feasible parameter space is Θ = {[, ] [, ]}, an in the secon case Θ = {[, ] [8, ]}. The statistic when comparing the resiuals for θ = (a, a ) T an θ H = (, a ) T is U = 46(4.8 3.) This statistic may then be compare to a χ istribution with one egree of freeom (select values of which can be seen in Table ) to etermine whether we reject the null hypothesis. For this example, the null hypothesis is that the restricte parameter space (with a = ) is the appropriate one. We note this proceure oes not allow us to accept the null hypothesis, but only to either reject it or not. Therefore, we conclue that allowing a > oes not result in a statistically significant improvement in moel fit to ata. When beginning the minimization to estimate parameters θ = (a, a 3 ) T we initialize the proceure with a =.. However, the parameter value for a that minimize the objective functional was a =, an thus the resiuals (RSS) were the same. Therefore, there was no nee to compute the statistic an we can irectly interpret that there is no improvement in the fit. Similarly, when estimating all three parameters θ = (a, a 3, a ) T, as compare with estimating = (, a 3, a ) T, the ifference in resiuals is approximately 6, which results in a moel comparison statistic value (also approximately 6 ) that inicates a very insignificant improvement to the fit. Therefore, we cannot reject the null hypothesis with any reasonable level of confience. θ H Thus, it is reasonable to take a =, which amounts to excluing the term a χ G>G (G(t) G ) in the ta(t) equation for the final moel of patient Final 69 moel The moel that best escribes the ynamics in the patient 69 s ata is t A(t) = a 3h(t), (6) [ ] t G(t) = a A(t + s)s ( + c χ W (t) )A. (7) While there is always room for improvement in the evelopment of a moel, at this stage there is likely not much to be gaine from the inclusion of aitional mechanisms. We note that the moele processes are not as inherently preictable as those in biology, physics, or engineering. 7

18 Therefore, our goals an stanars for agreement between moel solutions an ata shoul be ajuste accoringly. We eem the fit seen in Figure 6 to be relatively goo since the episoes of rinking are, with the exception of two, accurately preicte by the moel. We surmise that it is not necessarily a reasonable goal to preict the number of rinks uring the episoes, but rather, whether or not they occur. In future moeling efforts, it may be that the episoes not preicte by the moel are especially instructive, as they may be inicative of mechanisms missing from the moel. This iniviual reuce his rinking substantially at the beginning of the stuy an not uring the IVR observation perio. Therefore, it is ifficult to speculate on the associate mechanisms accompanying his behavior change. However, the characterization of his rinking after reuction is instructive. The iniviual appeare to not rink much, an also exhibite a corresponing low esire to rink uring the week, but his esire an rinking increase substantially on the weekens. Incorporating the h(t) function accounte for this an allowe for consierable agreement between moel solutions an ata. One possible explanation an therefore justification of incluing this term (weekay/weeken) relates to the perceive availability, craving, an consumption literature. Fiel an Cox ([5]) in their review of this literature note that intensity of craving following cue exposure has been shown to vary epening on the participants belief that they woul be able to consume the substance. For example, numerous experiments have shown that smokers who are tol the opportunity to smoke is imminent exhibit a higher subjective craving than when they are tol they will not be able to smoke for several hours. Perceive availability has also been use to explain ifferences in responses to cues among substance users continuing to use substances, an those seeking to remain abstinent [4]. We know from the clinical recors that 69 set a goal of reuce rinking on the weekay, but woul allow himself to rink more heavily on the weeken. It may be that by clearly efining this limit, 69 change the perceive availability of heavy rinking an that his esire reflecte when he felt he coul rink. We can speculate from his clinical recor an ability to reuce rinking that 69 possesse strong internal control an perhaps this strong internal control was manifest in the ability to set limits such that there was a corresponing reuction in esire. By contrast, someone with weaker internal control might set a limit, but still experience strong esire. One puzzling question is why commitment an self-efficacy, variables hypothesize to be critical to control, o not appear in the equation. It may be that because 69 expecte to rink more on the weeken, he i not rate his commitment or self-efficacy lower uring these perios. Alternatively, it may be that these variables are important eterminants of change in the ecision to reuce rinking, but that once a plan has been set in motion, they recee as operative variables, thus their fluctuation is not eterminant to rink, unless the plan fails. We return to a fuller iscussion of these issues in the iscussion section. It is striking that this iniviual s rinking is escribe well by the weeken-epenent riving mechanism, an also that his esire mirrors the same behavior. This close link may be an inication that his internal control is very goo. That the other measure control variables (such as the confience an commitment to limit rinking, or the commitment to abstain completely from rinking) o not appear to capture this information may suggest that these questions are not goo inicators of internal control for this patient. Or it may be that there are other issues with the ata collection process that o not allow for those relationships to surface in the ata. At any rate, these efforts provie motivation to look at the measures of control further. The first categorical moel (Figure 3), which was provie as a schematic of hypothesize 8

19 Weeken-epenent Desire h(t) Rate of Drinking A(t) Guilt G(t) Figure 7: Upate categorical moel for patient 69 possible relationships between moel variables, can now be upate to reflect the moel (6) - (7), an is epicte in Figure 7. While we i not use inverse problem methos to go from the more complex moel to the resulting simple one, we i iterate between the moel, comparisons between solutions an ata, interpretation of ata, an using current knowlege an plausible hypotheses to inform the moel. Inee, each arrow in the categorical moel, inicative of terms in the mathematical moel, represents a hypothesis about the unerlying process. The ecision to keep, iscar, or refine the term is in a sense hypothesis testing, although it is not always one in the usual formal way escribe at the en of the last section. Typical statistical methos aressing hypothesis testing woul be limite in their ability to reveal the ynamic relationships we focus on here as this type of information is lost as ata is aggregate across iniviuals or responses (means, stanar eviations, etc.). Mathematical moeling provies a way for us to precisely formulate hypotheses concerning ynamic variables. If we are able to make use of inverse problem methos an estimate moel parameters, then we can use statistical methos in conjunction to test whether moel moifications provie a statistically significant ifference in the fit to ata. As illustrate above, these methos involve the calculation of a statistic base upon the resiuals (the sum of the ifferences between the moel solution an ata point) from two moels that have been fit to the same ata set. Other examples of use of such statistically base moel comparison techniques are given in [, Chapter 3] an [8, Chapter V]. It is encouraging to be able to arrive at such a relatively simple explanation of this patient s behavior, but this iniviual is atypical in a crucial way. Frequently among responing subjects in Project Motion, the iniviual experiences some pivotal or transformative perio of change after which their alcohol consumption pattern is significantly change. While patient 69 s rinking pattern is still not the picture of perfection (he continues to violate his norms on weekens), it oes not change substantially uring the time perio of the IVR. It woul likely be more instructive to stuy an iniviual who experiences this pivotal change uring the IVR perio so that we woul have observations before, uring, an after it to better characterize this process of change. While we shoul not expect a simple moel to escribe a typical problem rinker, (inee the close connection between their esire an actual rinking behavior is likely not present), our ability to methoically o so in this case emonstrates mathematical moeling as a useful tool in unerstaning these ata. Notably, it appears to provie a significant avantage over the use of static methos in which the ientification of important factors (without attempting to etermine precise relationships between them) relevant to the patient s rinking behavior prove ifficult. 9

20 5 Patient 69 moel Patient 69 was selecte as the next moel subject because his rinking ata isplaye a clear an consistent ownwar tren. Therefore, we hope that information on any pivotal experience he may have ha preceing an uring his rinking reuction may be observe in the ata. Aitionally, this iniviual was homeless an unemploye when he enrolle in the stuy an was able to secure stable housing uring the IVR. Therefore, we suspecte that the supporting mechanisms of his behavior change woul be ifferent than those seen in patient 69. Namely, patient 69 appeare to exhibit strong internal control, whereas it is possible that patient 69 was able to reuce his rinking as a result of external factors. We consiere that there may be other mechanisms not consiere in the initial categorical moel or in the IVR ata categories, such as the weeken-epenent esire mechanism that subsequently was key to unerstaning patient 69 s esire an rinking ata. Of the ata categories inclue in the categorical moel for patient 69 (shown in Figure 8), we looke to single question responses for possible relationships with rinks, as the variables in the resulting moel for patient 69 are from single questions, iscusse in more etail in Section 7. In the en, the simple moel of patient 69 agree with the ominating characteristics ocumente in the supporting notes of the clinical team. As a result, we begin by using the clinical summary of the patient, along with those selecte to be in the initial categorical moel (Figure 8) to select for items in the IVR that may be most relevant to his rinking. We then looke for whether any ynamic relationships appeare to exist between these variables an his rinking ata, an thereby selecte for inclusion in the moel. We note that this moification of our approach - use of single items an clinical summary ata - is characteristic of the iterative nature of moeling. That is, it is common that each moeling exercise reveals new information which is then use to inform the next moel. 6 Initial moel for patient 69 We etermine that guilt G(t), loneliness L(t), confience C (t) to resist rinking heavily, commitment C (t) to resist rinking heavily, an commitment C 3 (t) to completely resist rinking are the best caniates for state variables for a moel that woul escribe patient 69 s rinking behavior. For the reaer s convenience, ata from these variables, overlaye with rinking ata, is isplaye in Figure 9. Not surprisingly as seen in Figure, patient 69 i not exhibit a clear weekay versus weeken pattern as suggeste in his case summary where it was note that he ha a lot of unstructure time (that he fille by rinking) ue to being unemploye. The timescale was kept consistent with the triweekly timescale of patient 69, as it is still the esire resolution. With the exception of guilt G(t), these variables were selecte base on the effects that they appear to have on rinking. That is, there were notable features in the evolution of these variables in time that seeme to correlate with a feature in the rinks consume over time, along with a plausible explanation for such an effect. For a complete ynamical system, we woul like to also be able to erive equations for all state variables (A(t), G(t), L(t), C (t), C (t), C 3 (t)), an therefore we woul nee to be able to attribute changes in these variables to changes in other state variables, or to other time-epenent functions as we i with the h(t) function for patient 69. The state variables (L(t), C (t), C (t), C 3 (t)) may epen on other variables not in the current moel (an therefore, not irectly relate to rinking), but may also be influence by external events an/or internal processes not capture in

21 Pleasant Events Drinks Moo Guilt Commitment Desire Figure 8: Categorical moel for patient 69 from initial efforts, iscusse further in [9]. 5 5 Drinks A(t+s)s Guilt G(t) Drinks A(t+s)s Conf to limit Drinks A(t+s)s Lonely Drinks A(t+s)s Comm to limit Drinks A(t+s)s Comm to abstain Figure 9: Data for initial 69 moel variables (left axis) overlaye with rinks (right axis).

22 4 wkay A(t+s)s wken A(t+s)s Figure : Weekay an weeken rinks plotte on the triweekly timescale. the IVR ata. For example, we know from 69 s clinical recor that he receive housing somewhere between ay an. This event likely contribute to the reuction in his rinking uring that perio, but was not capture in the IVR script. We return to the issue of missing ata in a later section of this paper. Regarless, it oes not appear that the fluctuations in these variables are relate in any clear way to the other variables in the moel as can be seen when comparing the ata for the supporting variables irectly. Therefore, we eeme it not likely that we woul be able to construct moel equations for the variables (L(t), C (t), C (t), C 3 (t)) epening on other moel variables. We constructe functions for these variables that capture key aspects of their ynamics, similar to the function h(t) that reflecte the observe weeken-weekay pattern for patient 69. That is, we constructe best fit lines through these ata (piecewise linear functions) an use these functions as inputs into the equations for rinking rate A(t) an guilt G(t). The use of the observe ynamics of the supporting variables in the moel equations for A(t) an G(t) that give a reasonable approximation to the observe ata thereby suggest that these ata are observations of ynamic processes. Further, the use of these ynamic ata to reprouce ynamics in rinking an guilt ata provie encouraging support for the use of mathematical moels to provie meaningful explanations for these observations. A revise categorical moel, or schematic of the hypothesize relationships between variables is given in Figure. Supporting variables, or variables that are use as inputs only are represente by boxes with otte borers, an causal relationships are again represente by arrows. The appearance of the lack of relationships between support variables in this schematic is not to say that they o not exist, but that we are unable to etermine them with our knowlege at this time an from the ata set on han. We constructe an initial moel by examining the iniviual s IVR response ata, consiering the previous experience moeling patient 69, an constructing quantitative relationships that represent mechanisms consistent with our knowlege of processes unerlying rinking behavior. We initially represente the mechanisms governing the state variables A(t) an G(t) by

23 Commitment to Abstain C 3 (t) Loneliness L(t) Rate of Drinking A(t) Guilt G(t) Confience to Limit C (t) Commitment to Limit C (t) Figure : Upate categorical moel, or schematic of hypothesize relationships amongst variables reflecte in the initial moel. 6. Drinking A t t A(t) = c + a 3 (L (t) L(t)) a 7 χ C3 >C3 C 3(t) a 5 χ {C >C } χ {C >C } (C (t) C) (C (t) C), (8) t G(t) = a equation ( ) A(t + s)s A (t). (9) τ The equation for A t, which is the change in the iniviual s rate of rinking with respect to time, is our primary focus since unerstaning the iniviual s rinking behavior is our goal. The first constant term, c represents the iniviuals relatively constant esire to rink, so is an ever-present source riving his rinking up. This is an unknown quantity, but we can etermine a range of values that this parameter coul possibly take on (a necessary first step in oing computations an also for parameter estimation, if we were to pursue it). The only other variable that causes an increase in rinking is the feeling of loneliness L(t) (perhaps relating to epression). Drinking appears to increase proportionally to loneliness, until it reaches a certain threshol, which may be interprete as perhaps an unbearable level. The first of these switches occurs aroun t =, an twice more at t = 3 an t = 49. It appears that the threshol level of loneliness oes not change until t = 3, at 3