Statistical Definition of Relapse: Case of Family Drug Court

For a published version of this paper please see Alemi F, Haack M, Nemes S. Statistical definition of relapse: case of family drug court.  Addict Behav. 2004 Jun;29(4):685-98.


After treatment, a patient’s return to drug use can be seen either as a temporary event or as a return to persistent use. There is no formal standard for distinguishing persistent drug use from an occasional relapse. This lack of standardization persists even though the consequences of either interpretation are life altering. In a drug court or regulatory situation, for example, misinterpreting relapse as return to drug use could lead to incarceration, loss of child custody or loss of employment. A clinician who mistakes a client’s relapse for persistent drug may fail to adjust treatment intensity to client’s needs. An empirical and standardized method for distinguishing relapse from persistent drug use is needed. This paper provides a tool for clinicians and judges to distinguish relapse from persistent use based on statistical analyses of patterns of client’s drug use. A control chart is created for time-in-between relapses. The paper shows how a statistical limit can be calculated by examining either the client’s history or other clients in the same program. If client’s time-in-between relapse exceeds the statistical limit, then the client has returned to persistent use. Otherwise, the drug use is a temporary. To illustrate the method, it is applied to data from three family drug courts. The approach allows the estimation of control limits based on the client’s as well as the court’s historical patterns. The approach also allows comparison of courts based on recovery rates.


Relapse prevention interventions are effective in reducing drug use (Irvin, Bowers, Dunn, & Wang, 1999). Different investigators have offered numerous psychological and biological models to explain how relapse occurs (Connors, Maisto & Donovan, 1996). It is generally accepted that “how a problem is understood dictates how it is responded to (Saunders & Houghton, 1996).” When a patient relapses, his or her perception that this is a temporary slip will make a significant difference in how optimistic the patient remains and how soon the patient returns to avoiding drug use (Marlatt & Gordon, 1985).

In addition, the clinician’s attribution of client’s drug use to relapse or persistent use may influence the course of treatment: When a clinician identifies relapse, the immediate step is to analyze the situational predictors of relapse and implement the appropriate relapse prevention program. But when there is a pattern of relapse suggesting a return to persistent drug use, the clinician may change the treatment modality to more intensive interventions such as the addition of pharmacotherapy (Annis, 1991). Furthermore, the factors that affect minor relapse or persistent drug use are often different and clinicians need to adjust their treatment to reflect the causes of drug use. Minor relapse is often connected to social pressures while more persistent drug use is often connected to negative feelings (Hodgins, El-Guebaly, Armstrong, 1995).

Similarly, when a positive urine screen is used as evidence in the court, the judge may order further counseling or treatment. But when there has been a pattern that suggests return to pretreatment persistent drug use, the judge may take more drastic actions. In a US court, this can mean jail time in a criminal proceeding or loss of child custody in a civil family proceeding. Because the consequences can be severe in this and other settings, clinicians, judges and patients need a method for distinguishing temporary relapse from return to persistent drug use. This paper provides a statistical procedure for doing so.

The definition of relapse remains elusive. The Webster’s New Collegiate Dictionary (1983) defines relapse as “a recurrence of symptoms after a period of improvement.” Marlatt and Gordon (1985) define relapse as a “transitional process, a series of events that may or may not be followed by the return to baseline levels of the target behavior.” This definition does not distinguish a temporary relapse that involves a few occasions of use from a return to persistent drug use. At what point of reoccurrence of symptoms does one call it a return to baseline levels of drug use as opposed to temporary relapse? Many clinicians and investigators have set arbitrary rules regarding what is considered persistent drug use. Some clinicians may argue that any use is return to drug use, which in essence says there is no temporary relapse but every drug use is persistent use. Others (e.g. Hser, Yamaguchi, Chen and Anglin, 1995) arbitrarily assume that daily use is return to drug use. Miller (1996) and others have argued that a less dichotomous definition of relapse is needed. Donovan (1996) identified conceptual and operational problems with the definition of relapse including: 1) whether relapse was assessed after the fact or before; 2) whether there are biases in recalling relapse events; and, 3) whether a single event or a pattern of drug use is considered relapse. Longabaugh and colleagues (1996) found that despite considerable cross training, three independent groups of clinicians and researchers failed to consistently categorize patients’ relapses. They called for a better definition of relapse.

Providers of clinical services in the treatment of addictions have viewed substance use disorders within the framework of a medical model. Physicians, therapists, patients and their families often agree that addiction is characterized by chronic use and a long-term trajectory with periods of controlled abstinence punctuated by periods of symptom exacerbations, a process seen in other chronic diseases such as diabetes or asthma (McLellan, Lewis, O'Brien & Kleber, 2000). The advent of evidence-based treatment no longer permits any health care service provider, including those in addiction treatment, to make decisions regarding a patient’s care based on intuition, educated guesses, or good intentions. Approaching addiction as a chronic disease necessitates a methodology for distinguishing temporary relapse from persistent use.

Current Study

Quantification, or a statistical definition, can differentiate a relapse from return to persistent use. The use of statistical methods in understanding relapse data is not new (Hedeker and Mermelstein, 1996). In 1995, Hser, Yamaguchi, Chen and Anglin constructed time series hazard functions to understand relapse rates. Statistical approaches address the common mistake of equating a single episode of use with full-blown return to the pretreatment baseline use. Relapse is not viewed dichotomously as success versus failure, or progress versus dead end but in probabilistic terms of return to baseline use.

Assessing relapse using the statistical tool presented here can delineate a profile of the patient. This profile can be used to inform the court, the clinician and the patient about the severity of the relapse and to identify the appropriate relapse intervention programs. Different programs could be tailored to the needs of patients who have a one-time episode of use and those who return to a baseline pattern of persistent use.

Within the context of a chronic disease trajectory, relapse is not unexpected in the course of treatment. Periods of abstinence, and within a certain margin of predictability, the chance for relapse can be anticipated with planned intervention. If, however, the frequency of relapse extends beyond an upper limit, the patient has a pattern of use beyond what could be expected by chance. The upper limit of what could be expected by mere chance is referred to as Upper Control Limit and is one of the concepts that will be defined and explained in this paper.

Statistical Definition of Relapse

Abstinence can be thought of as a probabilistic repeating event. One can then test if any new pattern of abstinence is different from a historical pattern by calculating 95% confidence intervals for historical probabilities of relapse. A patient’s drug use can be classified as either temporary relapse or return to persistent use by examining if the pattern is statistically different from historical standards. For example, drug use for one day may be thought of as a chance deviation from a pattern of abstinence. However, as the days of drug use increase, the probability of observing these events by mere chance deviation from a pattern of abstinence decreases. At some point, drug use is so persistent that we no longer can assume that it is a deviation from a pattern of abstinence. At this point, the client has returned to persistent drug use.

If persistent drug use is defined as a statistically significant deviation from a pattern of abstinence, then statistical process control charts can be used to detect temporary relapses and persistent drug use. In particular, g-charts, or “relapse charts,” can be employed to display the pattern of use by the patient. These charts have been discussed in detail elsewhere (Benneyan, 2001). Here we show their application to analysis of drug use patterns.

Step one: Test of Assumption of Relapse Chart

Before charting the data for a patient, it is important to test that the data follow certain assumptions. A relapse chart assumes that the length of relapse (calculated as the time between two consecutive abstinences) has a geometrically decaying shape, sort of the inverse of an exponentially growing curve. To show this, assume that a patient who has kept daily records of his abstinence for 20 weeks. He has used drugs on the 5th, 10th and 15th through 18th week. Table 1 shows the data on this patient.

Table 1: Case History for A Hypothetical Patient

A histogram of length of relapses can allow us to visually test the assumption of a geometrically decaying shape. Figure 1 shows this histogram. It shows 11 times of no relapse, 2 times of relapses of one week and 1 time of relapse of three weeks. The shape of the histogram shows a geometrically decaying shape, meaning that longer stretches of relapse are increasingly rarer.

Figure 1:  Histogram of Length of Relapse

Step two: Calculation of Statistical Upper Control Limit

The upper control limit, UCL, is the point at which the length of relapse is so large that it cannot be expected by mere chance deviation from the underlying pattern of abstinence. The limit is set so that 99% of the data points for a patient who has abstained from drug use would typically fall below the limit. In 1% of cases, a patient who has a pattern of abstinence may be erroneously classified as persistent drug user. Lower or higher confidence intervals can be set based on how stringent the judge or the clinician wants to be in avoiding erroneous conclusions.

Let “ALR” signify the average length of relapse, and calculate it as the number of weeks of relapse divided by the number of weeks of success.

For the case in Table 1, the average length of relapse is calculated by dividing the total number of weeks of relapse, 5, by the number of weeks abstinent, 15.

The upper control limit, UCL, is the limit below which 99% of the data should fall. It is calculated as:

UCL = ALR + 3 [ALR*(ALR+1)]0.5

The upper control limit for the case in Table 1 is calculated as:

UCL = .33 + 3 (.33*1.33)0.5

UCL = 2.32

If the client is followed for 100 weeks, only once should the relapse rate be higher than 2.32 weeks. In the above case, the client was followed for 20 weeks. The likelihood of observing relapses longer than 2.32 weeks should be relatively small and therefore observing such long relapses should raise concern that the relapse signifies a new pattern of using by the client.

Step three: Displaying and interpreting the data

In a relapse chart, the length of relapse is plotted against weeks since admission. To assist in the interpretation of the chart, the upper control limit is superimposed on the plot. Figure 2 shows the relapse chart for the case presented in Table 1. Points below the upper control limit are chance variation from the general pattern of abstinence. In this case, despite occasional drug use, the underlying abstinence is repeating. Points above the control limit have less than 1% probability of occurring due to chance alone. They represent changes in the underlying repetition of abstinence. Therefore, for these points we can assume that the client’s pattern of drug use has changed. In Figure 2, the first two occasions of drug use (relapse prior to weeks 6 and 9 on the x-axis) are below the control limit. They represent no change in the underlying repetition of abstinence. Thus, these two episodes of drug use suggest temporary relapses. However the last drug use (relapse after week 15 on the x-axis) is above the control limit. The length of this drug use is too long to be expected from random chance variation from the history of this client. It represents a change in the repetition of abstinence. Therefore, this point no longer suggests a temporary relapse but a return to persistent drug use.

Figure 2:  Relapse Chart

Abstinence Rates

A relapse chart shows whether clients have maintained their patterns of abstinence. This is not to say that the pattern of abstinence displayed in the chart is acceptable or desirable. Some judges and clinicians insist on zero tolerance policy and do not accept any relapses. Others argue that an occasional relapse is part of the recovery. Both groups can benefit from more precise estimates of probabilities of abstinence. The relapse chart proposed in this paper allows us to provide a quantitative estimate of probability of abstinence. If, as before, the average length of relapse is given by ALR, then the probability of abstinence, p, is given by:

p = 1 / (1+ALR)

In the case example, being followed in this paper, the probability of abstinence is calculated as:

p = 1 / (1+.33) = 0.75

To understand whether 75% chance is a reasonable probability of abstinence, keep in mind that the probability of abstinence for patients who never use drugs is equal to one. For those who are currently using drugs, the probability of abstinence is equal to zero. As patients move towards recovery and have more abstinent weeks, the probability increases. A 75% rate of abstinence is better than weekly drug use but does not represent recovery. A much higher rate is needed. At the same time, a 100% rate of abstinence may be rare. Since drug addiction is a chronic disease, the probability of abstinence may get very close to one but there is always a remote chance of a resumption of drug use. For example, for a patient who relapses once in three months, the probability of abstinence is 99%. Abstinence rates are markers for recovery and can be used by clinicians and judges to understand the extent of recovery.

Family Drug Court Cases

Child abuse and neglect cases are commonly processed in civil family court. In accordance with the US Adoption and Safe Families Act (ASFA), a law passed in 1997, family courts manage cases by making a temporary placement order for the child for one year and requiring the parent to enter drug treatment. However, the obstacles to success are enormous. At the end of the year, most frequently the parent has made little progress in obtaining and participating in treatment and the court has little information to guide its decision regarding permanent placement of the child. As a result, the fate of many children can rest on a parent’s urine screens, the only objective evidence that the court has.

Recently a new model of family court, family drug treatment court, has evolved across the country in part because of the success of the criminal drug courts but also in response to the tight timelines mandated by ASFA. Family drug treatment courts intervene in the lives of parents and their children, providing treatment and supervision that might not otherwise be available in the general court system. Treatment involves the entire family, not just the parents (Drug Strategies, 1999). Despite improved access to substance abuse treatment and wrap around services however, family drug treatment courts must still decide the fate of children in families with parents who suffer from a chronic disease that is fraught with the potential for relapse. Therefore in a family drug treatment court, a parent’s urine screens may still be a deciding factor. Positive urine screens indicate failure to achieve abstinence or relapse.


Date were collected data from three Family Drug Courts in three different States: Florida, Kansas and New York. Courts differed significantly in the type of clients they supervised, the relationship between the district attorney and the defense lawyers, the sanctions they used, the length of drug testing they required and many other procedures. All three courts required drug testing but modified the frequency of the testing based on client’s recovery and cooperation. C Court records and court staff’s private notes were reviewed to identify when clients were tested and what the results were. Court staff accessed the data for us and completed a court record review form that was developed by the evaluation team in conjunction with court staff and a lawyer. Cases were selected consecutively for court record data collection. Table 2 shows the number of cases examined at each court and the number of drug tests conducted on each client.

Table 2: Number of Cases and Tests

Court A had 24 cases, Court B 21 and Court C had 20 cases in which drug test results were available at the time of the study. Table 2 shows that the courts differed regarding length of drug testing. Court B had the shortest length of drug testing per patient. Courts A and C were similar. The large standard deviation values also show that there was significant variation among the cases in the same court regarding length of drug testing. Some patients received few and others received many more tests. This fits with court procedures of conducting more drug tests on non-compliant clients.

Distribution of Abstinence Rates in Three Courts

Courts also differed regarding length of relapses. Table 3 shows the average length of relapse at each of the three courts. The Upper Control Limit shows the statistical limit (99% of cases would have relapses shorter than this limit). Court A had higher average length of relapse than the other courts. Court C had the lowest length of relapse. An average patient was abstinent in 31% of the tested weeks in Court A and 46% of tested weeks in Court B. One way to understand if these rates are better than expected is to compare them to experience of other treatment programs. Nemes, Wish and Messina (1999) found high rates of completion in their study of therapeutic communities (ranging from 33%-38%) with primarily cocaine users. This rate of completion exceeds those found by previous studies. McCusker (1995) reported that studies have repeatedly found that longer programs have lower completion rates, ranging from 7% to 15%. Condelli and DeLeon (1993) have found that nearly half of all program participants abandon long term TCs within the first three months after admission. Previous studies have indicated that success depends largely on the length of stay in a therapeutic community; however, it has also been found that success is more closely related to a client’s completion of the program (Heit, 1991). Nemes et al. (1999) also found that most improved outcomes have been seen in clients who completed all phases of treatment.

Table 3: Average Drug Test Results for Clients in Each Court

Figure 3 shows the distribution of abstinence rates in the three family drug courts. Average length of relapse, ALR, for each of the 65 cases in the three courts were calculated. Using the estimated ALR, we obtained the probability of abstinence for each case in the courts was obtained. Figure 3 shows a histogram of our findings.

Figure 3:  Distribution of Abstinence in Three Courts

Figure 3 shows that in these courts more than 20% of cases were abstinent in 55% of weeks tested. Less than 10% relapsed nearly every time tested. More than 10% of cases were abstinent nearly every time tested (group marked as “More” in the graph). It also shows considerable variability among the cases. In the following, we show how a judge can use this information to classify minor relapse and significant return to persistent drug use for a specific case in his or her court.

Application to Relapse of Specific Case

The relapse chart in Figure 4 shows the data from one patient, coded as MITC05, in one of the courts. For weeks 1 through 8 the patient did not relapse. But following these weeks the patient relapsed once for 8 weeks, then for 2 weeks and then again for 5 weeks. These data show a patient who has relapsed three times for a length of 15 weeks and has had 16 weeks of drug free tests. The average length of relapse was .93 weeks. The probability of an abstinent week for this patient was 0.52, significantly below 1. In 48% of weeks the patient relapsed. The Upper Control Limit calculated from this patient’s own pattern was 4.98 weeks. Relapses starting in weeks 10 and 25 exceed the Upper Control Limit. The relapse starting in week 21 does not. The chances of observing one point above the Upper Control Limit is less than 1% and therefore the judge may decide that the relapses that exceed the control limit represent a new pattern of drug use for this patient. These two relapse were so extensive that they could not be due to mere chance variation from previous patterns of this patient’s abstinence.

Figure 4:  Case MITC-05

How would this case compare to other cases in the same court? So far control limits have been set based on the client’s historical patterns. In short, the method allows for comparison the client to herself to see if she has changed. The limits can also be set on other basis. The limits could be set based on patterns of others in the same court. The upper control limits calculated from cases in the three courts are presented in Table 3. When the case MITC-05 had a relapse of 9 weeks, it exceeded the pattern in all three courts. When the case had a relapse of 5 weeks, it exceeded the pattern in court A and B. And when it had a relapse of 2 weeks, it did not exceed the pattern in any of the three courts. Judges can use the control chart to compare the case at hand to success rates of others under their supervision.


A methodology has been presented for examining clients’ drug use over time. The procedure can be used to examine relapse during a particular treatment program or across several treatment programs to create a visual picture of what is known as the client’s “drug treatment career” (Hser et al., 1997). The procedure provides a number of statistical summary parameters that can be used in comparing effectiveness of treatment programs. It also provides a method of distinguishing a relapse from return to drug use.

Perhaps because relapse associated with addiction has such a strong behavioral component that is often assigned a moral value, decisions about the appropriate course of treatment when symptoms occur have been subjective. Thus, when a patient experiences a return of symptoms, or relapse; regardless of the severity or circumstances, he or she is usually considered a “treatment failure” and at best is readmitted to the treatment facility to repeat the initial treatment process at the same level of care provided at baseline entry to the system. At worst, the client may be discouraged from re-entering the healthcare system at any level due the interpretation of relapse as treatment failure.

A large body of literature describes what needs to be done to prevent relapse (Friedmann, Saitz and Samet 1998). In theory much progress has been made. But in practice, few US treatment programs or courts actively focus on relapse prevention. Patients who relapse are simply readmitted to the system from the point at which they started, regardless of their individual pattern, frequency, or intensity of symptoms. This practice may be detrimental in several respects. First the patient, returning to the initial treatment program after a relapse, usually reports experiencing further guilt, shame, and loss of self-esteem (Marlatt & Gordon, 1985). Secondly, returning to baseline treatment services prevents the patient from receiving a treatment approach consistent with emerging clinical needs.

Specifically in the family drug court setting, we recommend that judges and clinicians establish different intensities of treatment and sanctions depending on the nature of relapse. For temporary relapse, patients may need a review of situational determinants of relapse and the decision to alter those conditions. For persistent drug use, patients may need a change in modality of treatment, such as the addition of pharmacotherapy for co-existing psychiatric disorders. Cohen (1999) suggests an analysis of the client’s background in terms of risk for HIV, and history of sexual, physical, or emotional abuse.

Another consideration for assessing the need for referral to strategic relapse prevention programs as opposed to automatically returning the patient to initial baseline treatment is cost effective utilization of resources. Mandating a patient to a long-term setting or an expensive inpatient drug treatment facility may actually deter her from receiving the relapse prevention skills and intervention she needs.


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