Variability and predictability in sleep patterns of chronic insomniacs
Annie Vallières, PhD, Neuropsychological Institute of Biological Research, Sackler Institute of Psychobiological Research, Southern General Hospital, 2nd floor, 1345 Govan Road, Glasgow G51 4TF, UK. Tel.: +44 (0) 141 232-7699; fax: +44 (0) 141 232-7697; e-mail: firstname.lastname@example.org, email@example.com
Sleep of chronic insomniacs is often characterized by extensive night-to-night variability. To date, no study has examined this variability with long series of daily sleep data. The present study examined night-to-night variability with a sample of 106 participants meeting DSM-IV diagnostic criteria for persistent primary insomnia. Participants completed daily sleep diaries for an average of 31 days (range: 18–56). Sleep efficiency, sleep onset latency and wake after sleep onset were derived from this measure. Despite evidence of extensive night variability, results showed that sleep patterns could be classified in three clusters. The first one was characterized by a high probability of having poor sleep, the second one by a low and decreasing probability, and the third one by a constant median probability of having a poor sleep, which is an unpredictable sleep pattern. In the first cluster, poor sleep was expected each night for patients with a predominance mixed insomnia including the three insomnia subtypes. In the second cluster, patients presented moderate insomnia, sleep-onset latency below the threshold level and a predominance of sleep-maintenance insomnia. In the third pattern, poor nights seemed unpredictable for patients with moderate to severe insomnia associated with the lowest proportion of sleep-maintenance insomnia. Overall, sleep was predictable for about two-thirds of individuals, whereas it was unpredictable for about one-third. These findings confirm the presence of extensive variability in the sleep of chronic insomniacs and that poor sleep may be predictable for some of them. Additional research is needed to characterize those sleep patterns in terms of clinical features and temporal course.
Variability and Predictability in Sleep Patterns of Chronic Insomniacs
Insomnia is a widespread complaint estimated to affect at least 10% of the adult population (Ohayon, 2002). It is characterized by difficulties initiating and/or maintaining sleep, or by a non-restorative sleep, and is associated with a marked distress or a significant impairment in daytime functioning for at least three nights per week (American Psychiatric Association, 1994). These diagnostic criteria suggest that insomnia does not necessarily occur each night. Clinical evidence also indicates that night-to-night variability and unpredictability of sleep are considered classic features of insomnia. Indeed, aetiological models of insomnia recognize these characteristics as central in the configuration of insomnia (e.g. Espie, 2002; Morin, 1993).
Over the years, several empirical studies have confirmed the patients’ clinical report of extensive night-to-night variability (Coates et al., 1981; Edinger et al., 1991, 1997; Frankel et al., 1976; Mullaney et al., 1980). Overall, these studies mostly observed sleep variability while comparing good sleepers and sleep profiles of insomniacs. They concluded that sleep variability was part of the insomnia symptomatology. Other studies addressing the accuracy of actigraphy in evaluating insomnia have also documented sleep variability (Hauri and Wisbey, 1992; Vallières and Morin, 2003). Their results showed that variability among nights was not present for all individuals and that some insomnia sufferers experienced similar consecutive poor nights of sleep. Yet, another group of studies evaluating effectiveness of cognitive-behavioural therapy (CBT) for chronic insomnia also reported night-to-night variability (Edinger et al., 1992; Espie et al., 1989; Vallières et al., in press). Interestingly, although the nightly variability was present at baseline, it decreased at post-treatment. CBT thus appeared efficacious at reducing the variability, hence making sleep more predictable.
In addition to these empirical findings, some theoretical elements also support the notion that sleep might follow specific patterns. First, according to the aetiological model of Espie (2002), insomnia takes place when the sleep regulation system, which includes homeostasis and circadian rhythm processes, is affected by several psychological, environmental or physiological factors. Once this system is altered, it does not necessary mean that it follows a random mode generating unpredictable poor nights. Indeed, a dysfunction might be predictable and generate specific sleep patterns despite some night-to-night variability. Nonetheless, it is possible that a fully damaged sleep regulation system generate unpredictable poor nights. Secondly, although studies suggested that sleep might be more predictable when treatment reduced variability (Espie et al., 1989; Edinger et al., 1992; Vallières et al., in press), it is conceivable that the prediction of poor nights is also possible before treatment. For instance, an individual may experience a very poor night followed by two moderately poor nights, and, then, by a good night and so on. The observation of such series would show an extensive night-to-night variability as well as a sleep pattern in which these four nights would occur repetitively over and over. In order to identify such sleep pattern, long series of data might be more suitable than the usual 2-week of sleep diary data used to establish a baseline level. Finally, taking into account that sleep may be predictable, would represent more adequately some patients’ clinical reports of a good recovering night after several poor nights even when insomnia is persistent (Morin and Espie, 2003).
In summary, it is well acknowledged in the literature that night-to-night variability is inherent to primary insomnia. Also, sleep in insomnia is considered unpredictable despite some empirical and clinical indices suggesting that sleep might be predictable at least for some patients. In addition, none of the previous studies specifically addressed this question within series of daily sleep data over more than 2 weeks. Therefore, the present study examined the night-to-night variability in long series of data within a sample of chronic insomniacs. It is hypothesized that consecutive poor nights of sleep are predictable for some patients and unpredictable for some others.
The sample included participants from three studies conducted by our research group since 1997. Results of two studies are presented elsewhere (Morin et al., 2003; Vallières et al., in press), whereas the third one is still recruiting. Study 1 examined the influence of stress, coping skills, and arousal on sleep patterns of insomniacs and good sleepers whereas studies 2 and 3 evaluated combined treatment in insomniac population. For each study, participants were recruited through newspaper advertisements and physician referrals.
General inclusion criteria were (i) presenting insomnia defined as a sleep-onset latency (SOL), wake after sleep onset (WASO), or early morning awakening longer than 30 min per night, for a minimum of three nights per week; (ii) insomnia duration of at least 6 months; (iii) report of significant distress or daytime impairment; (iv) cessation, at least 1 month prior to experimentation, of any sleep or other psychotropic medication that could have an impact on sleep; and (v) being 18 years old or older.
General exclusion criteria were (i) presence of another sleep disorder such as sleep apnoea or circadian rhythm disorder; (ii) evidence that insomnia was related to a medical condition; (iii) presence of major depression, anxiety disorder, alcohol or substance abuse, or another psychopathology; (iv) currently in psychotherapy; and (v) regular use of medication interfering with sleep. In addition to those criteria, the present study included only participants who had completed at least 21 consecutive days of sleep diaries with no more than three missing data.
The final sample included a total of 106 participants (57.6% women), 37 participants were from study 1, 22 from study 2 and 47 from study 3. Mean age was 45.5 years (SD = 9.7; range, 23–71), mean education level was 14.8 years (SD = 3.1; range, 7–24) and mean insomnia duration was 12.8 years (SD = 11.6; range, 0.5–52). In studies 2 and 3, participants completed daily sleep diaries for an average of 31 days (SD = 12, range 18–56) before receiving insomnia treatment. Treatments delivered in these two studies were either a combinations of drug and cognitive behaviour therapy (CBT) for insomnia or CBT alone. In the case of study 2, participants were included in a multiple baseline across subjects design whereas they were included in a randomized control trial in study 3. This latter study is still recruiting at the present time. Overall, 7.5% of the total sample of participants presented initial insomnia, 16.0% sleep-maintenance insomnia, 3.0% terminal insomnia and 73.5% mixed insomnia.
First, a 20-min telephone interview was conducted to determine participants’ eligibility. Subsequently, a multistep screening evaluation composed of a semi-structured sleep history interview to diagnose insomnia, the Structured Clinical Interview for DSM-IV (SCID-IV; First et al., 1997) to evaluate the presence of psychological disorders and a physical examination was performed. The Insomnia Interview Schedule (IIS; Morin, 1993) was used in the three studies to assess sleep history and insomnia complaints. In addition, positive answers to specific questions regarding other sleep disorders in the IIS were used to evaluate presence of other sleep disorders. In studies 2 and 3, the SCID-IV (First et al., 1997) was used to diagnose depression or anxiety disorders whereas in study 1 (Morin et al., 2003) no SCID was performed, but participants were excluded based on their scores of both Beck's inventories (Beck Anxiety Inventory (BAI), Beck et al., 1988a; Beck Depression Inventory (BDI), Beck et al., 1988b). Exclusion resulted if scores were higher than 30 for BAI and higher than 23 for BDI.
Participants completed daily sleep diaries throughout the different studies. For the purpose of this report, only baseline diaries prior to treatment were used. The sleep diary was completed upon arising each morning. The diaries monitor several sleep parameters including bedtime, arising time, SOL, number of awakenings and subjective sleep quality. Outcome variables derived from this measure and used in this study are SOL, WASO and sleep efficiency (SE; ratio of total sleep time to time in bed × 100).
The Insomnia Severity Index (ISI; Morin, 1993) is a seven-item scale evaluating perceived insomnia severity. Each item is ratings on a five-point scale (0–4). The total score ranges from 0 to 28 and higher scores indicate more severe insomnia. The ISI has adequate psychometric properties (Bastien et al., 2001; Smith and Trinder, 2001). The BDI was also administered in each study to assess depression symptoms. The BAI was administered in studies 1 and 3 only, to assess anxiety symptoms.
Participants completed self-report measures and daily sleep diaries. Measures were mailed to participants in each study for various durations across protocols. For study 1, the instruction was to complete daily sleep diaries for 3 weeks and to mail them back each week after completion. For studies 2 and 3, participants were asked to complete baseline sleep diaries and to fax them every week after completion. For more details on studies procedure, see Morin et al. (2003) for study 1 and Vallières et al. (in press) for study 2. For the purpose of the present study, no objective sleep data will be presented and only subjective data from sleep diaries will be retained. Each participant thus presents a series of sleep diary nights in which each night is dichotomized into either ‘poor’ or ‘good’. Therefore, data are transformed into 106 series (one per participant) of nights labelled as either poor or good night. As the purpose of this study is to evaluate the prediction of poor night, nights labelled as poor should not raise doubt about the fact that they are effectively poor. Given that using the standard criteria of 30 min could increased the risk of arbitrary assignation of a poor or good night, the criteria used to dichotomize nights were thus strengthened. A poor night was then defined as SOL and/or WASO equal to or longer than 60 min associated with a SE equal to 80% or less, as derived from daily sleep diaries. Nights that did not match both criteria were considered as good nights.
Data analysis plan
All data were carefully inspected to identify missing data and outliers and to assess normality (Tabachnick and Fidell, 2001). Computations of missing data percentage yielded an averaged of 2.7 missing nights (7.6%) per participant (n = 106). In addition, there was no relationship between number of nights and percentage of missing data (r(105) = −0.05, P = 0.64). Following statistical guidelines in these particular set of data cases (Roth, 1994), no missing data imputation was performed and only complete sequences of two, three, or four nights were included in the computation of conditional probabilities. Descriptive and inferential statistics were completed using SAS 8.2 statistical software (SAS Institute, 2001). Alpha level was fixed at 5% (two-tailed) for all inferential tests.
Consecutive daily sleep data were conceptualized as time-series data. Each night was dichotomized as either a good or a poor night according to criteria described in the procedure section. Conditional probabilities to have a poor night after 1, 2 or 3 consecutive poor nights were computed for each participant. This analysis is used to predict the probability of an event, which in this study context is a poor night, rather than the magnitude of this event, which in sleep would be the severity. However, the classic conditional probability formula was not appropriate as some time-series had non-consecutive (missing) data. Thus, the formula was slightly modified to take into account only consecutive sleep data (see Appendix 1). Conditional probabilities to have a poor night after 1, 2 or 3 consecutive poor nights were submitted to an exploratory k-means (least squares) cluster analysis in order to identify subgroups of participants showing similar levels of conditional probabilities. Solutions ranging from 2 to 5 clusters were investigated. The final solution was selected based on three criteria: (i) the parsimony of the solution, (ii) the sample size of each cluster, and (iii) the clinical interpretability of each cluster. One-way anovas, mixed model analysis and chi-square tests were computed to compare clusters on demographics, clinical, sleep and psychological measures. Mixed model analysis was performed comparing the variability of sleep across clusters (3) and the severity of night (2) (fixed effects) while controlling for patient covariance (random effects).
Sleep patterns and level of predictability
When computed on the total sample, conditional probabilities suggested that sleep was unpredictable. Indeed, conditional probabilities to have a poor night after 1, 2 or 3 poor nights were respectively [p(p|p) = 0.50, p(p|pp) = 0.48, and p(p|ppp) = 0.48]. These results indicated that there was always about 50% chances to experience a poor night following a poor night.
Based on the three criteria mentioned previously, the k-means cluster analysis supported a three-cluster solution (R2 = 75.6%). Each cluster is mutually exclusive, which means that each participant is part of only one cluster. Moreover, each participant within a given cluster, experiences the same sleep pattern over time. The first cluster labelled ‘high probability pattern’ (HPP) included 22 participants (21% of the total sample) who displayed a predictable sleep pattern. Their mean probabilities (SD) to have a poor night after 1, 2 or 3 consecutive poor nights were high and constant [p(p|p) = 0.82, p(p|pp) = 0.84 and p(p|ppp) = 0.84] (SD = 0.12, 0.11 and 0.12 respectively). The second cluster, labelled ‘low probability pattern’ (LPP), comprised 45 participants (42.9% of the total sample) showing a low and decreasing probability to have a poor night following previous poor night(s) [p(p|p) = 0.30, p(p|pp) = 0.18 and p(p|ppp) = 0.00] (SD = 0.19, 0.20 and 0.00 respectively). This second cluster showed a predictable pattern of sleep given that after three poor nights of sleep they had 0% chance to experience a fourth consecutive poor night. The third cluster, labelled ‘unpredictable pattern’ (UP), contained 38 participants (36.1% of the total sample) showing a constant median probability to have either a poor night or a good night following poor nights [p(p|p) = 0.56, p(p|pp) = 0.56, and p(p|ppp) = 0.51] (SD = 0.11, 0.12 and 0.18 respectively). For the latter cluster, a poor night appears being unpredictable and unrelated to the number of previous poor nights.
Threshold and clusters appropriateness
Two analyses were performed to ensure that the 60-min criterion used does not create arbitrary assignation of good or poor night. Means and standard errors of night-to-night variability (i.e. SD computed for each participant) of sleep variables for each cluster are presented in Table 1. First, linear mixed models showed that clusters did not differ significantly regarding night-to-night variability in SOL, WASO and SE assessed (F1,97 = 1.54, 0.00 and 2.28, NS respectively). Also, these results showed that for each cluster poor nights are significantly different on each sleep variables compared to good nights (F1,97 = 95.79, 178.35 and 20.23, P < 0.0001 respectively). Second, percentage of night on the edge of the 60-min threshold was computed for each participant following these criteria: WASO or SOL of 60 ± 15 min and SE of 80% of ±5%. Results showed that 9.3% of nights of the overall sample, which included a total of 3563 nights, meet these near the threshold criteria. Also, only eight participants of 106 presented the highest percentage of nights near the threshold, which was from 20% to 30%. Four of these eight participants are in the LLP while the other four are in the UP. Therefore, the variability among nights in each cluster seems equivalent, poor and good nights are different, and none of the participant presents a constant pattern of near the threshold nights that could lead to arbitrary assignation.
Table 1. Means and standard errors of night-to-night variability (individual standard deviations) of sleep variables for poor nights and good nights among clusters
| Total||22.85 (3.20)||24.50 (2.13)||28.95 (2.31)|
| Poor nights||35.06a (4.29)||38.53a (3.03)||46.60a (3.26)|
| Good nights||10.65b (4.74)||10.47b (3.00)||11.31b (3.26)|
| Total||30.02 (2.26)||29.96 (1.51)||29.90 (1.63)|
| Poor night||46.12a (3.04)||42.14a (2.15)||44.01a (2.31)|
| Good night||13.91b (3.36)||17.77b (2.12)||15.80b (2.31)|
|Sleep efficiency (%)|
| Total||12.80 (0.97)||10.32 (0.64)||11.11 (0.70)|
| Poor night||14.77a (1.30)||11.54a (0.92)||14.02a (0.99)|
| Good night||10.83b (1.43)|| 9.09b (0.91)|| 8.21b (0.99)|
Differences among clusters
Means and standard deviations for sleep variables and clinical insomnia variables according to clusters are presented in Table 2. One way anovas revealed significant differences among clusters concerning SOL (F2,102 = 8.47, P = 0.0004), WASO (F2,102 = 18.40, P < 0.0001), and SE (F2,102 = 23.60, P < 0.0001). In addition, results showed that the ISI significantly differed among clusters (F2,64 = 4.45, P = 0.02). Multiple comparisons test revealed that HPP presented a longer WASO and a lowest SE (P < 0.05) than the two other clusters and that insomnia severity is higher for HPP and UP (P < 0.05). Also, LPP presented a shorter SOL compared with HPP and UP clusters (P < 0.05) and the highest SE (P < 0.05). One-way anovas revealed no significant difference among clusters regarding age (F2,102 = 1.40, P = 0.25), and education level (F2,102 = 0.41, P = 0.67). No association was found between the three clusters and gender ( = 1.83, P = 0.40). Although insomnia duration seemed to be higher for the cluster HPP, this was not significant either (F2,100 = 1.75, P = 0.18). Results showed that depression (F2,102 = 0.50, P = 0.61), and anxiety (F2,86 = 0.60, P = 0.55), symptoms did not differ among clusters.
Table 2. Means and standard deviations of sleep variables, insomnia severity and clinical variables among clusters
|HPP||54.2a (40.0)||83.4a (30.5)||56.4a (10.1)||20.7a (3.7)||16.1 (13.8)||9.3 (4.7)||8.9 (6.5)|
|LPP||26.9b (17.4)||44.8b (20.4)||75.2b (10.0)||16.9b (4.6)||13.5 (12.4)||9.7 (5.5)||7.2 (5.6)|
|UP||43.4a (27.6)||54.3b (25.5)||69.2c (11.4)||18.6a,b (3.6)||10.4 (8.9)||10.7 (6.4)||8.4 (6.6)|
Subtypes of insomnia for each cluster are presented in Table 3. A one-way anova revealed a significant difference among clusters concerning the number of subtypes of insomnia (F2,102 = 5.43, P = 0.006). Hence, mixed insomnia in the HPP cluster is more often composed of the three insomnia subtypes whereas for the two others clusters mixed insomnia included mainly two insomnia subtypes only. No difference was found in proportions of initial and terminal insomnia between clusters ( = 3.30, P = 0.19 and = 5.04, P = 0.08 respectively). However, proportions of individuals with maintenance insomnia differed among clusters ( = 8.09, P = 0.02). Indeed, there were significantly fewer individuals with maintenance insomnia in the UP cluster compared with HPP and LPP clusters.
Table 3. Insomnia subtypes across clusters
|HPP (n = 22)||21||2.32a (0.57)||54.6||100.0a||77.3|
|LPP (n = 45)||30||1.80b (0.66)||35.6||91.1a||51.1|
|UP (n = 38)||27||1.84b (0.64)||52.6||76.3b||50.0|
These findings confirm that sleep in primary insomnia might follow specific patterns despite the presence of an extensive night-to-night variability. Poor sleep was predictable for about two-thirds (64%) of individuals with chronic insomnia whereas it was unpredictable for about one-third (34%). The night-to-night variability among nights appears similar in each cluster and none of the participant presents a constant pattern of near the threshold nights, ensuring that nights labelled as poor were not arbitrary assigned. The first cluster identified (HPP), was characterized by more severe insomnia with sleep efficiency under 56% (±10%), and a predominance of mixed insomnia including the three insomnia subtypes. The second cluster (LPP) was characterized by a moderate insomnia, defined as sleep-onset latency below the threshold level, and a predominance of sleep-maintenance insomnia. In the third cluster (UP), poor nights seemed unpredictable. This pattern presented moderate to severe insomnia associated with the lowest proportion of participants with sleep-maintenance insomnia. Finally, people in each cluster were quite similar concerning insomnia duration, age, gender and anxiety or depression symptoms.
The present findings extend previous ones on the presence of an extensive night-to-night variability in insomnia (Coates et al., 1981; Edinger et al., 1991, 1997; Frankel et al., 1976; Hauri and Wisbey, 1992). While other studies demonstrated that it is more likely to predict sleep when sleep variability is reduced (Edinger et al., 1992; Espie et al., 1989; Vallières et al., in press), the present results support the fact that indices about sleep predictability can be identified despite the presence of night-to-night variability. Principal distinctions found between clusters concern a SOL under threshold level when the probability of having a poor night is low and decreasing, presence of sleep-maintenance insomnia when sleep is predictable and presence of mixed insomnia when having a poor night sleep is highly probable.
In trying to understand the sleep patterns found, it is important to consider that they were identified based on subjective report of sleep, which means on participants’ sleep perception. Therefore, difference in sleep perception might also account for identified patterns. Hence, paradoxical insomniacs or participants with a more negative thinking who tend to exaggerate their insomnia might be included into the HPP. It is likely that misperception leads to a constant severe evaluation of sleep. Furthermore, participants presenting a more accurate sleep perception or a more positive thinking could provide a more positive prediction of sleep and be in the LPP. While sleep perception might be well suited to explain HPP and LPP, it does not explain UP so readily. Participants in this cluster could be endorsing false beliefs pertaining to the unpredictability of sleep. Unfortunately, beliefs on unpredictability of sleep or sleep expectancy towards the following night were not assessed in the present study. The relationship between beliefs, sleep perception and sleep patterns thus deserves greater attention in further studies.
In addition to identifying clues distinguishing each sleep pattern, understanding why sleep may be predictable might help clarifying sleep patterns. The presence of two predictable sleep patterns suggests that the sleep regulation system including sleep homeostatis and circadian rhythm is not necessarily affected at the same level or at random among insomnia sufferers. One study showed that homeostasis pressure is weaker but operating for sleep-maintenance insomnia patients compared with good sleepers (Besset et al., 1998). Thus, the LPP identified might represent participants of Besset et al.'s study. This would suggest that the homeostatic pressure to sleep, despite being weaker, ended up by achieving a sufficient level to bypass all other factors contributing to a poor night. In the same way, for the high probability pattern, it can be hypothesized that, when the homeostasis process is below the threshold level to operate, it would not add up to enough pressure to generate a better night although there were several previous poor nights. As a consequence, insomnia is highly expected every night and, thus, becomes predictable. In both cases, it seems that level of sleep homeostatis dysregulation might be hypothesized to explain sleep predictability. This hypothesis deserves greater attention in other studies.
The current study presents some methodological limitations. First, as mentioned previously all data are based on subjective and self-reported measures. Despite the importance of daily sleep assessment (Coates et al., 1982), compliance to daily retrospective measure may be questionable. Daily objective measures of sleep would have been of great value to corroborate these findings. Further studies could use actigraphy, for example, as an alternative to polysomnography. Although not perfect, this assessment device could provide a more objective daily measure of sleep than the sleep diary. Secondly, reactivity to a self-report repeated measure could affect the external validity. Reactivity refers to the influence of the participant's awareness of their participation in the study (Kazdin, 1998). In some cases, participants may have enhanced severity in their self-reported sleep difficulties in order to be accepted in the treatment study. In other cases, daily self-report of sleep difficulties might have generated a natural improvement of sleep. Thirdly, the fact that the thresholds used to dichotomize sleep were more stringent than the usual 30 min, limits the interpretation towards ‘good’ sleep. For example, in the LPP cluster, it can be suggested that the fourth night is one of recovery but not necessarily a ‘good’ one. It would be of interest in subsequent study to predict good sleep using strengthening criteria to define a good night sleep. Finally, the present sample represents a fairly homogenous primary insomniac population who was in majority seeking treatment for their difficulties. As a result, this precludes generalization to other insomnia population.
In conclusion, despite the night-to-night variability, poor sleep in insomnia seems to be predictable in two-thirds of the sample. Therefore, unpredictability of sleep does not seem to be a general characteristic of all insomnia sufferers. Poor nights seem to be constant and predictable for patients with severe insomnia associated with mixed insomnia including the three insomnia subtypes. Predictability seems to be related to a high proportion of sleep-maintenance insomnia, which could indicate a greater influence of the homeostasis process. Identifying clear factors distinguishing sleep patterns could eventually be helpful to modify sleep perception and belief about sleep unpredictability during treatment. Nevertheless, additional research is needed to replicate these findings using concurrent subjective and objective daily sleep data, to identify other factors related to predictability, and to characterize those sleep patterns in insomnia in terms of diagnosis, clinical features and temporal course.
This research was supported by NIMH grant no. 60413 awarded to C. M. Morin, by a CIHR grant no. 49500 to C. H. Bastien, and by an FRSQ and an FFER scholarship awarded to A. Vallières.
Appendix 1. Conditional Probability Formula
The classic conditional probability formula states that the probability to observe a good night after a poor night is:
However, missing data within the series affected the number of available two-nights sequences for the numerator of the formula. Thus, the solution was to correct the denominator of the formula, and replace it by the correct number of available two-nights sequences for the participant. The modified conditional probability formula is: