The Journals of Gerontology Series A: Biological Sciences and Medical Sciences 62:415-419 (2007)
© 2007 The Gerontological Society of America
Utility of the Mean Cumulative Function in the Analysis of Fall Events
Meghan G. Donaldson,
Boris Sobolev,
Lisa Kuramoto,
Wendy L. Cook,
Karim M. Khan and
Patti A. Janssen
1 Department of Health Care and Epidemiology, 2 Centre for Clinical Epidemiology and Evaluation, Vancouver Coastal Health Research Institute, 3 Division of Geriatric Medicine, Faculty of Medicine, and 4 Department of Family Practice, Faculty of Medicine, University of British Columbia, Vancouver, British Columbia.
Address correspondence to Meghan G. Donaldson, MSc, PhD (C), University of British Columbia, Health Care and Epidemiology, 713-828 West 10th Avenue, Vancouver, BC V5Z 1L8, Canada. E-mail: meghangd{at}interchange.ubc.ca
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Abstract
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Background. Falls are the most common cause of injury among elderly people; half of those people fall recurrently. The objective of these simulation studies was to describe the Mean Cumulative Function (MCF) and to evaluate the utility of the MCF in detecting differences between groups experiencing different patterns of event intensities.
Methods. We specified 250 participants per group with a maximum follow-up time of 365 days. A participant could experience 0, 1, 2, 3, or 4 falls. In the baseline experiment, Groups A and B had an average intensity of 60 and 90 days to the first fall event. These event intensities remained constant for events 2–4. Group C represents a short term "strong" initial impact of the intervention modeled for falls 1 and 2, with an average intensity of one fall per 117 days; however, the intervention wanes to "moderate" for falls 3 and 4 with an average intensity of one fall per 90 days. Group D represents a long-term "strong" impact of the intervention modeled by an average intensity of one fall per 117 days for all subsequent events.
Results. The MCF was able to detect differences between groups that had varying intensities of subsequent falls. In Group A, all participants experienced at least one fall, whereas Groups B, C, and D had 4, 9, and 15 participants, respectively, who did not experience any falls. The proportion of participants who had 4 falls declined from 84% to 40% in Groups A and D, respectively. When Group A was compared to Group D, the MCF difference detected the prevention of, on average, one fall per person within 175 days.
Discussion. A novel instrument for this field of clinical research—the MCF—allows investigators to compare the average number of falls per participant when the intervention reduces the intensity of subsequent falls.
FALLS are the most common cause of injury among elderly persons. It is estimated that one in three persons older than 65 years falls at least once each year, a proportion that increases to one in two people older than 80 years (1,2). Almost half of those persons who fall experience the event recurrently (3,4).
The epidemiological literature has identified a number of risk factors for falls including age and number of falls in the previous year. Certain populations that seem to be particularly vulnerable to falls include residents in long-term care facilities and community-dwelling persons who present to the emergency department with a fall. Although the literature suggests that falls can be prevented if intervention is directed to subgroups within the 65-year-old-and-older population, for example, persons presenting to the emergency department, there is also strong evidence that falls can be prevented in a population of community-dwelling older persons who have been identified to be at risk for falls due to age alone (5–8). In this population, there is particularly strong evidence that persons who undertake specific strength and balance retraining programs have approximately 30% fewer falls than do controls (5–8).
Interventions designed to prevent falls may also decrease the risk of subsequent fall events in an older population (5,9). However, it remains unclear whether such interventions prevent future falls from ever occurring, whether they delay subsequent falls, or both. Robertson and colleagues (10) have argued that evaluating the efficacy of falls interventions should include analysis of all falls for each person rather than simply the number of people who fall. Without these data, it is difficult to evaluate how many falls the intervention would prevent or how long it would take for an intervention to begin to take effect.
Methodologically, the possibility that an event may occur more than once in an individual over a given period of observation makes falls analysis challenging. There are several alternatives to survival analysis of the first event, such as the Andersen–Gill regression model, the marginal model (Wei Lin and Weissfeld), and negative binomial regression (10,11). Somewhat surprisingly, the Mean Cumulative Function (MCF), useful in describing the average number of events occurring in one individual within a certain time and efficient in comparing intensity of events between groups (12), has not previously been used to analyze multiple events in falls research (12).
It remains unknown whether the MCF can detect differences between groups experiencing different patterns of event intensities. To learn how the MCF performs when an intervention affects the intensity of consecutive events with different patterns, we conducted simulation studies comparing MCFs among samples with different intensities of consecutive falls. We aimed to evaluate the performance of the MCF through the random simulation of multiple fall events per person.
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METHODS
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MCF
Nelson (13) introduced the MCF as a method to summarize the average number of events occurring in one individual within a certain time period in a population exposed to censoring events such as losses to follow-up and termination of the study, assuming that the time to events is independent from the time of censoring. If an individual can only experience one event, the MCF is equivalent to the proportion of individuals ever experiencing an event. To define the MCF, consider n individuals who are observed over a period of time and may experience fall events at times t1, t2, t3...tN. At distinct moment tj when an event or censoring occurs, the time-dependent MCF(tj) is calculated as:
where ek is the number of events occurring at the time tk and nk–1 is the number of persons at risk just beyond time tk–1. The number of persons at risk at time tk–1 is equal to the total number initially at risk minus those who were censored prior to time tk–1. In this setting, the risk set only decreases when an individual is removed from follow-up.
Simulation Studies
We performed a series of Monte Carlo simulation studies to investigate the utility of the MCF in detecting differences between four groups with different patterns of event intensity. For each group we generated a random sample of "times of falls" (days from enrolment) among 250 participants per group over the period of 365 days. For each participant, using an exponential distribution, we generated the number of days until the 1st fall using the 1st fall intensity, the number of days between the 1st and 2nd fall using the 2nd fall intensity, the number of days between the 2nd and 3rd fall using the 3rd fall intensity, and the number of days between the 3rd and 4th fall using the 4th fall intensity. This calculation resulted in the distribution of participants experiencing 0, 1, 2, 3, or 4 falls during the period of simulation.
The patterns of event intensity for two groups were specified based on published data. We chose a constant average intensity of one fall per 60 and 90 days for all subsequent events in Group A (control, Otago trial) and B (intervention, Otago trial), respectively (5). To our knowledge, no falls prevention studies have specifically examined the effect of the intervention on changes in the intensity of subsequent fall events. Therefore, for Groups C and D we arbitrarily applied a 30% rate reduction in subsequent falls, a decrease considered clinically significant and also cost effective (9). The intervention groups thus were defined as follows: Group B represents the intervention group, in which a long-term "moderate" effect of the intervention is modeled by an average intensity of one fall per 90 days for all subsequent events (5). Group C represents the plausible situation in an intervention, where patients comply at the outset but adhere less well with the intervention over time. To reflect this situation, the intensity of subsequent falls in Group C represents an intervention where a short-term "strong" initial impact of the intervention is modeled for falls 1 and 2, each with an average intensity of one fall per 117 days. However, the effect of intervention wanes to "moderate" for falls 3 and 4 with an average intensity of one fall per 90 days. Finally, Group D represents an intervention where a long-term "strong" impact of the intervention is modeled by an average intensity of one fall per 117 days for all subsequent events (Table 1).
Analysis
Data were analyzed using the SAS RELIABILITY Procedure (SAS Institute, Inc., Cary, NC).The MCF graph was generated using the MCF function in SAS. We compared Groups B, C, and D to Group A (control) using the MCF difference and Nelson's 95% confidence interval (CI) (13). The MCF and the MCF difference were plotted for each comparison group. We established the approximate day when the MCF curves began to separate by determining the day at which the 95% CI of the difference no longer encompassed zero. We established the approximate day when the MCF difference curve was at its maximum.
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RESULTS
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The results of the simulations indicate that the MCF was able to detect differences between groups that had varying intensities of subsequent falls. During the 365-day period of simulation, the control group (Group A) had a total of 937 fall events; Groups B, C, and D had 805, 725, and 699 fall events, respectively (Table 2). In Group A, all participants experienced at least one fall, whereas Groups B, C, and D had 4, 9, and 15 participants, respectively, who did not experience any falls during the observation period. The proportion of participants who had four falls declined from 84% to 40% in Groups A and D, respectively.
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Table 2. Distribution of Participants by Number of Falls (Percent), Average Number of Falls per Participant, and Number of Prevented Falls by Intervention Group.
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The MCF calculated for the control and the intervention groups showed that the average number of falls experienced in 365 days reduced from 3.75 falls per participant in Group A to 2.75 falls per participant in Group D. The MCF difference calculated between Groups A and D showed that the particular intervention attributed to Group D would prevent one fall per person within 175 days on average. In addition, the MCF difference calculated between the control and the intervention groups showed that the average number of prevented falls per participant increased from Group B to Group D, with Group D nearly achieving an additional 0.5 saved falls compared to Group B. Finally, Group B (moderate long-term effect) achieved its maximum benefit at day 200, whereas Groups C (strong short-term effect, followed by moderate effect) and D (strong long-term effect) achieved their maximum benefit 25 days earlier and with greater impact (Table 2).
The overall pattern of the MCF difference curves, used to compare control to intervention groups, was similar. All MCF difference curves initially rose steeply to reach a peak and then gradually declined until the end of the period of observation (Figure 1). All groups experienced a gradual decline in the MCF difference after the maximum was achieved; however, the MCF difference and the 95% CI values remained above zero indicating that the benefits of the intervention persisted but to a lesser degree. Regardless of the intervention, after achieving their maximum MCF difference, all curves gradually declined until the end of the observation period. The MCF difference slope from time zero to time at maximum MCF difference was calculated to demonstrate the number of prevented falls per week. The number of falls prevented per week from baseline until the MCF difference reached its maximum was 0.03, 0.04, and 0.05 for Groups B, C, and D, respectively. The drop of the MCF difference from the time when the MCF difference was at a maximum to the end of the period of observation was calculated to demonstrate the number of additional falls permitted per week in each group. The drop from maximum MCF difference by 365 days was 0.013, 0.01, and 0.009 additional falls per week in Groups B, C, and D, respectively.
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Figure 1. The Mean Cumulative Function (MCF) (left) and the MCF difference (right). I: Group A compared to Group B; II: Group A compared to Group C; III: Group A compared to Group D
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DISCUSSION
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Whether falls prevention interventions prevent falls, delay falls, or both is not known. One of the challenges in assessment of falls prevention programs is that falls may occur more than once in the same individual. We have demonstrated the use of a novel instrument for this field of clinical research—the MCF—which allows investigators to compare the average number of falls per person between intervention groups when the intervention reduces the intensity of subsequent falls.
In this study we generated four different samples of fall times by simulating different patterns of event intensity. As expected, the most noticeable difference in the MCF was between the control and the intervention group in which the intensity of falls decreased by 30% for falls 1–4 (Group D). The MCF difference between the two groups showed that the particular intervention attributed to Group D would prevent, on average, one fall per person within 175 days. These simulation results suggest that these interventions prevented falls, as several participants in Groups B–D did not experience any falls during 365 days, whereas all participants in Group A experienced at least one fall.
Regardless of whether the intervention was sustained across all falls (long-term effect), as in Groups B and D, or if the intervention was initially strong and then weakened over time, as in Group C, after achieving its maximum MCF difference, all curves declined gradually until the end of the observation period. That is, after the maximum MCF difference was achieved, the benefit was not maintained, regardless of the simulated intervention. This information may help to guide interventions by providing information as to when exercise programs aimed at preventing falls should be re-evaluated in the study population. This concern may be addressed by adjusting the frequency, intensity, and time and/or type of exercise being delivered (14). This idea warrants further exploration and could be achieved by applying the MCF method to data from intervention studies.
Although rarely used in health applications, the MCF has several benefits for analyzing multiple events in fall prevention literature. As the MCF is a nonparametric method, it does not require assumptions about the underlying time to event distribution (12) or independence of multiple events (15). The MCF is the average number of falls occurring in one individual over a given period; it is concerned with units (individuals) experiencing events. The Andersen–Gill, marginal Cox regression, and negative binomial models, conversely, are concerned with the average rate of events over a given period. Although the MCF does not allow risk adjustment to be handled within the RELIABILITY Procedure in SAS, methods for multivariate regression analyses of the MCF are available (16).
The MCF method assumes noninformative censoring—that is, individual histories of sample individuals are statistically independent of their censoring times. Although Nelson (12) has argued that censoring in recurrent event studies is a feature of data collection and not of the study population, the noninformative censoring assumption may not be applicable to falls intervention studies.
It is noteworthy that moving to a long-term care facility and death are major reasons for participants withdrawing from falls studies (17). Therefore, if continuing to live independently is a hypothetical positive effect of balance and strength retraining, the intervention may increase the number of very frail patients who remain through the study period. Because these patients are also more likely to have a fall, the intervention group may have more observed falls compared to the study in which losses to follow-up occur independently. Therefore, informative censoring may result in an underestimation of the intervention effect.
Also, several baseline factors are associated with withdrawals: having more than one fall in the previous year, having a fall that occurred indoors (compared to falls occurring outdoors), and having impaired cognition (17). These factors should be taken into consideration through appropriate randomization procedures.
Conclusion
The MCF provides at least four benefits. First, the MCF allows researchers to interpret how many falls an intervention would prevent, on average, in one individual over a given time period, compared to a usual care group. Second, the MCF naturally accommodates different follow-up times among study participants—the usual case in randomized trials. Third, using the MCF can provide some evidence as to how long it takes for an intervention to begin to take effect. Finally, given the emphasis on economic evaluation, the MCF is also applicable to costs accrued over time (12).
We recommend using the MCF to compare samples when the intensity of subsequent events varies. This method helps to answer the central research question posed in falls prevention studies: Can an intervention reduce the average number of falls sustained by an individual over a period of observation and, if so, by what magnitude?
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Acknowledgments
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This study was supported by the Canadian Institutes for Health Research, the Michael Smith Foundation for Health Research, the Canada Research Chair Program, and the Western Regional Training Centre for Health Services Research.
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Footnotes
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Decision Editor: Luigi Ferrucci, MD, PhD
Received February 16, 2006
Accepted June 22, 2006
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References
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Abstract
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