Abstract
Objectives. We decomposed the total effect of coexisting diseases on a timed occurrence of an adverse outcome into additive effects from individual diseases.
Methods. In a cohort of older adults enrolled in the Precipitating Events Project in New Haven County, Connecticut, we assessed a longitudinal extension of the average attributable fraction method (LE-AAF) to estimate the additive and order-free contributions of multiple diseases to the timed occurrence of a health outcome, with right censoring, which may be useful when relationships among diseases are complex. We partitioned the contribution to death into additive LE-AAFs for multiple diseases.
Results. The onset of heart failure and acute episodes of pneumonia during follow-up contributed the most to death, with the overall LE-AAFs equal to 13.0% and 12.1%, respectively. The contribution of preexisting diseases decreased over the years, with a trend of increasing contribution from new onset of diseases.
Conclusions. LE-AAF can be useful for determining the additive and order-free contribution of individual time-varying diseases to a time-to-event outcome.
Attributable fraction (AF), also called attributable risk, of a risk factor has been extensively used to measure how much of the burden of an adverse health outcome might be reduced if a risk factor were eliminated.1–3 For example, calculation of the AF of hypertension on death is based on the death rate difference between the absence and the presence of hypertension. The AF is unadjusted for coexisting risk factors because it does not take into account other risk factors. When several risk factors exist simultaneously, the sum of the individual AFs usually exceeds the AF corresponding to eliminating all the involved risk factors. This is because the nonindependence or nonbalance of the coexisting risk factors leads to overlapping contributions to the occurrence of an adverse health outcome.
Adjusted AF is a calculation of the AF of a risk factor that takes into account coexisting risk factors and assumes that they are removed before the risk factor of interest. Adjusted AFs of individual risk factors also do not sum to their combined AF.4–7 Individual AFs derived by a sequential approach may sum to the total AF,8 but the sequential AF of a risk factor varies greatly across different removal orders. Assuming a particular removal order is often not plausible because the relationships among the risk factors may be nondirectional and complex (e.g., reciprocal, mediational, synergetic, counteracting), which limits the utility of methods based on sequential partitioning.
Broadly speaking, risk factors are the presence of a disease or nondisease exposure, and the outcome can be any adverse health event, including a disease that is different from any of the risk factors. We developed an approach to partitioning the combined effects of time-varying risk factors into additive, individual contributions to the occurrence of a timed or time-to-event outcome. In such longitudinal situations, a measure that quantifies a risk factor’s contribution to the outcome must account for possible overlapping effects among coexisting risk factors. An effective measure should be additive (the total contribution from the multiple risk factors combined equals the sum of the measures of contribution of the separate risk factors), should display symmetry (the measures of contribution should be independent of the order the coexisting risk factors were removed or added when the measures are calculated), and should account for the timing and duration of the risk factor exposures and the outcome. This third property is necessary because risk factors may be chronic, with long durations, or acute, with short durations prior to the occurrence of the adverse outcome over time.
The methods of AF and adjusted AF lack all 3 properties.5,9,10 The sequential methods lack the property of symmetry. By contrast, in a cross-sectional setting, the average attributable fraction (AAF) possesses the properties of additivity and symmetry.11–14 We extended use of AAF to situations with time-varying risk factors and timed or time-to-event outcomes while preserving additivity and symmetry, creating the longitudinal extension of AAF (LE-AAF). The key to achieving additivity and symmetry is averaging the contributions of a disease in all the possible removal orders of the coexisting diseases (i.e., it is the average of the sequentially adjusted AFs). By contrast, an adjusted AF or an AF derived from the sequential method only accounts for one particular order. Samuelsen and Eide developed attributable hazard fractions for time-to-event outcomes, but this measure does not possess the properties of additivity or symmetry.15
The LE-AAF of a risk factor reveals its fraction of contribution to the timed occurrence of the adverse outcome in the presence of multiple coexisting risk factors that may have a complex relationship. The LE-AAFs correctly allocate the overlapping effects among coexisting risk factors to individual risk factors. The LE-AAF method makes an important contribution by accurately assessing time-varying risk factors and timed occurrence of an outcome, including a time-to-event outcome.
Five commonly occurring and prevalent diseases and their association with time to death in a cohort of older adults illustrate the calculation of LE-AAFs. We considered the diseases as risk factors and death as the adverse outcome. Most deaths in older adults occur in persons with multiple coexisting diseases.16 These coexisting diseases have both separate and overlapping effects on death.17–19 The contribution of these coexisting diseases is not captured in current methods that assign cause of death according to medical decision rules applied to death certificates.20 The cause of death recorded on a death certificate thus reflects a medical decision on the overlapping effects of coexisting diseases rather than a quantitative allocation of overlapping effects.
METHODS
The Precipitating Events Project (PEP) cohort comprises 754 community-dwelling persons aged 70 years or older living in New Haven County, Connecticut, who at inception required no personal assistance in activities of daily living, were cognitively intact or had an available proxy, and had a life expectancy of at least 1 year. Because of this latter exclusion and to allow time to accumulate new diseases, we limited our analyses to the 713 participants who survived the first year. Recruitment for PEP took place between March 1998 and October 1999. The study is ongoing, and its design and cohort are described in more detail elsewhere.21,22 The PEP data we used were collected through in-person interviews at baseline and 18, 36, 54, and 72 months; monthly telephone interviews; and hospital records, up to 7 years after recruitment. We evaluated preexisting (i.e., those reported at the baseline interview) as well as new-onset diseases. Response rates were greater than 94% for the in-person interviews and 99% for the monthly telephone interviews.
Diseases of Interest
We evaluated 4 chronic diseases (heart failure, cancer, chronic lung disease, and dementia) and 1 acute disease (pneumonia). These 5 diseases are common in the elderly population and are frequently reported causes of death. Information on the 4 chronic diseases was ascertained by physician diagnosis. We considered dementia to be present if the participant was taking a cholinesterase inhibitor or memantine.23 We ascertained occurrence of pneumonia by self-report of hospitalization. A review of hospital records for a group of 94 participants showed excellent reliability for self-reported hospitalizations (κ = 0.94).21 We determined deaths from the monthly calls and by review of the local obituaries and confirmed them by obtaining death certificates from national vital statistics data.
We calculated LE-AAFs for preexisting and new onset during follow-up heart failure, cancer, and chronic lung disease. Chronic diseases with reported onset during the first year were regarded as preexisting. Baseline assessment for this cohort did not include dementia, so we considered all cases detected in years 1 to 7 to be new cases; dementia cases diagnosed in the first year could therefore be a mix of new and existing cases. Because we only used the PEP data from the second year on, this probably did not affect the calculation of the contribution from the presence of dementia to death. Pneumonia, the illustrative acute disease, was regarded as present for 6 months after onset. We considered pneumonias reported more than 6 months after a previous episode to be new episodes.
Calculating Longitudinal Extension of Average Attributable Fraction
Pooled logistic regression model for the effects of time-varying diseases on the occurrence of a time-varying outcome.
Because reporting on the diseases in our data was updated monthly, the number of observations equaled the number of months between a participant's second year and either death or the end of the seventh year of follow-up. We used the pooled logistic regression, where each observation represented a person-month and could accommodate time-varying exposures and timed occurrence of or time-to-event outcome.24
We included the time variable, main effects of the diseases, and its significant 2-way interactions as assessed by P values smaller than .05. As in Eide and Gellfeller's method,11 our approach can include interactions. We divided the total follow-up period into single-year intervals and used these as the covariate for time. Thus, the effect of time on the odds of death was updated yearly, and the odds were an approximation of the corresponding piecewise constant baseline hazards, assuming the baseline hazards were constant within an individual year. When a complex functional form of time (e.g., spline) is used and the time intervals are small, the odds ratio of a risk factor obtained from the pooled logistic regression model approximates the hazard ratio from a Cox proportional hazards model, because the hazard in an interval is very small.24
Calculating adjusted attributable fraction of a single disease in the presence of coexisting diseases.
We divided the total follow-up time into J time intervals according to the pooled logistic regression model. We let j = 1, …, J denote the jth interval. The AF for a disease is defined as the proportion by which an adverse outcome is prevented if the population attains the same risk level of the outcome as individuals without the disease. We indexed the disease of interest by k, and s denoted a stratum representing a specific combination for the presence of the other coexisting diseases. Then we calculated an adjusted AF for disease k in the jth time interval, that is, λjk, could be calculated as the sum of stratum s–specific AFs in that time interval over all the strata representing all possible combinations of the coexisting diseases:
 |
where P(Oj) denotes the probability of the occurrence of the adverse outcome (such as death in our example) within jth time interval, and P(Oj |Ejsk) and P(Oj |Ejs0) denote the probabilities of death with and without the presence of the disease k in stratum s, respectively. P(Ejsk) is the prevalence of disease k in stratum s within jth time interval. S is the total number of strata.
For example, in an estimation of the adjusted AF for heart failure (disease k) on death in the presence of 2 coexisting diseases, cancer and pneumonia, a total of 4 strata (S = 4) represented neither cancer nor pneumonia (s = 1), cancer but no pneumonia (s = 2), pneumonia but no cancer (s = 3), and cancer and pneumonia (s = 4). We calculated P(Oj |Ejsk) by using the corresponding regression coefficients for disease k (the heart failure) and for the disease(s) present in stratum s in the inverse of the logit function in the logistic regression as 1/[1+ exp{−(
0 +
1 heart +
2 cancer +
3 pneumonia +
4 heart × cancer +
5 heart × pneumonia)}]. For stratum 2 (s = 2) with only cancer present, the terms
3 pneumonia and
5 heart × pneumonia were zero, and the other terms took the values of their associated s. We calculated P (Oj |Ejs0) similarly, except we regarded disease k as absent. P(Ejsk)/P(Oj) could be estimated as njsk/dj, where dj is the total number of deaths and nsk is the number of participants having disease k in stratum s within interval j. When the follow-up period spanned multiple short intervals for each participant in the data, (adjusted) AF closely approximated the corresponding (adjusted) hazard.
Formula 1 is similar to formula 5 by Eide and Gefeller,11 except formula 1 is defined within the time interval j, and formula 5 by Eide and Gefeller is defined for a fixed duration. Thus, the expression within the summation explains the proportional reduction in the occurrence of the adverse outcome in stratum s if the disease of interest were to be removed in that stratum (i.e., the stratum s–specific AF for disease k). When there are no other coexisting diseases, the total number of strata S becomes 1, and this formula equals that of the unadjusted AF for disease k.
Calculating sequential attributable fraction and average attributable hazards for a single disease.
If 2 sets of diseases differed only by the presence of an extra disease k in 1 of the sets, we defined the difference in the adjusted AFs between the 2 sets as a sequential attributable fraction (SAF) for disease k. We classified the common set of diseases in the 2 sets as an overlapping set. We determined the SAF to be the proportional reduction in occurrence of an outcome if the exposure of disease k was eliminated before the coexisting diseases (i.e., the overlapping set). An SAF for disease k therefore depended on the composition of the other coexisting diseases that it adjusted for.
For a single disease k among a total of K coexisting diseases under consideration, the number of the diseases in its overlapping set varied from 0 to (K−1). Diseases could be ordered differently within and outside overlapping sets. Disease k was removed before the diseases in the overlapping set, but after the diseases not in the overlapping set. Hence, a total of K! differently ordered removal sequences were possible for a given disease.
We defined the AAF for the disease as the average of the SAF associated with the K! possible removal orders.11,12 It has been shown that the AAF satisfies the desired properties of additivity and symmetry.11–13 The ordering and the composition of the overlapping sets reflect the multifactorial interrelationships among the coexisting diseases; the causal relationships, however, are often uncertain. We used the R package pARccs to calculate the AAF.25 AAF calculated with the pooled logistic regression closely approximated the corresponding hazards, or average attributable hazards (AAHs). AAF or AAH provides a unique summary of the risk attributable to the disease, accounting for the coexisting diseases.
Calculating the longitudinal extension of the average attributable fraction for a single disease.
We obtained the AAH for a disease k for all the time intervals, that is, {AAHj(k), j = 1, …, J}. Time-dependent AAF (T-AAF) was the whole series AAH for disease k. We calculated the LE-AAF for the disease as the weighted average of all the {AAHj(k), j = 1, …, J} over all the time intervals, with the weights equal to person-month in each time interval.
Because the participant composition in person-months and the estimates from pooled logistic regression changed across the time intervals, AAH for the disease changed across the time intervals. Therefore, the LE-AAFs summarized the proportion of contribution to the outcome from a disease with respect to that of the total contribution from the coexisting diseases across the entire follow-up time and could be interpreted as an average longitudinal measure of the overall risk attributable to the disease during the follow-up period. The LE-AAF also satisfied the properties of additivity and symmetry.
The sampling variability of a LE-AAF comes from 2 sources: (1) the variability in estimating the parameters of the pooled logistic regression model and (2) the calculation of the LE-AAF in the estimation stage. To measure the sampling variability of each LE-AAF, we generated 1000 bootstrap samples and calculated the LE-AAF of each disease in all of the samples. We obtained the lower and upper limits of the 95% confidence interval of the LE-AAF for each disease empirically as the 2.5 and 97.5 percentiles of the 1000 LE-AAFs.
RESULTS
The 713 participants had an average (±SD) age of 78.3 (±5.2) years, 65% were female, and 90% were White. The frequencies of the diseases reported at the baseline interview and during the 84 months of follow-up are listed in Table 1. During the 84-month follow-up period, 231 (32.4%) participants died.
TABLE 1—
Prevalence and Incidence of Diseases: Precipitating Events Project; New Haven County, CT; 1998–2005
| Diseases | Baseline (n = 713), No. (%) | Year 2 (n = 713), No. (%) | Year 3 (n = 670), No. (%) | Year 4 (n = 628), No. (%) | Year 5 (n = 584), No. (%) | Year 6 (n = 547), No. (%) | Year 7 (n = 500), No. (%) |
| Preexisting chronica | |||||||
| Heart failure | 43 (6.0) | ||||||
| Cancer | 119 (16.7) | ||||||
| Lung | 112 (15.7) | ||||||
| New-onset chronicb | |||||||
| Heart failure | 20 (2.8) | 14 (2.1) | 7 (1.1) | 8 (1.4) | 11 (2.0) | 11 (2.2) | |
| Cancer | 18 (2.5) | 15 (2.2) | 7 (1.1) | 8 (1.4) | 7 (1.3) | 12 (2.4) | |
| Lung | 18 (2.5) | 8 (1.2) | 3 (0.5) | 11 (1.9) | 6 (1.1) | 6 (1.2) | |
| Dementiac | 21 (2.9) | 8 (1.2) | 7 (1.1) | 18 (3.1) | 8 (1.5) | 27 (5.4) | |
| Acuteb: pneumonia | 21 (2.9) | 23 (3.4) | 22 (3.5) | 20 (3.4) | 18 (3.3) | 23 (4.6) |
Note. We excluded the few participants who died during year 1 because limited life expectancy was an exclusion criterion for the parent study and because there were too few deaths to analyze. Two participants without any follow-up data were also excluded.
Onset or occurrence prior to the baseline assessment.
Onset or occurrence during follow-up.
Baseline dementia was not assessed; all dementia occurred during follow-up.
Pooled Logistic Regression Analysis
The parameter estimates ( s) for year, diseases, and interactions from the final pooled logistic regression models are displayed in Table 2. We used these values to calculate adjusted AF, as for formula 1. With a C statistic of approximately 0.80, the pooled logistic model fit the data quite well. The odds ratios of death from the pooled logistic regression closely approximated the corresponding hazard ratios within the corresponding year.
TABLE 2—
Odds and Odds Ratio Estimates from the Final Pooled Logistic Regression Model Predicting Mortality: Precipitating Events Project; New Haven County, CT; 1998–2005
| Model Term | Deaths/Person-Months, No. | b (SE) | P | Oddsa | Marginal ORb | Joint ORc |
| Year 2 | 33/8302 | −6.24 (0.20) | < .001 | 0.0019 | ||
| Year 3 | 33/7784 | −6.20 (0.20) | < .001 | 0.0020 | ||
| Year 4 | 43/7301 | −5.90 (0.18) | < .001 | 0.0027 | ||
| Year 5 | 36/6761 | −6.16 (0.20) | < .001 | 0.0021 | ||
| Year 6 | 44/6323 | −5.82 (0.18) | < .001 | 0.0030 | ||
| Year 7 | 42/5605 | −5.94 (0.19) | < .001 | 0.0026 | ||
| Preexisting chronic diseased | ||||||
| Heart failure | 0.97 (0.22) | < .001 | 2.64 | |||
| Cancer | 0.56 (0.17) | < .001 | 1.75 | |||
| Lung | 0.11 (0.18) | .54 | 1.11 | |||
| New-onset diseasee | ||||||
| Heart failure | 2.14 (0.20) | < .001 | 8.50 | |||
| Cancer | 0.98 (0.26) | < .001 | 2.66 | |||
| Lung | 1.16 (0.25) | < .001 | 3.19 | |||
| Dementiaf | 1.41 (0.20) | < .001 | 4.10 | |||
| Acute disease: pneumonia | 2.58 (0.19) | < .001 | 13.20 | |||
| Interaction | ||||||
| Heart failure × cancer | −1.42 (0.56) | .011 | 5.47 | |||
| Heart failure × pneumonia | −1.49 (0.47) | .002 | 25.48 | |||
| Lung disease × dementia | −1.08 (0.60) | .073 | 4.44 | |||
| Lung disease × pneumonia | −1.21 (0.62) | .05 | 12.55 |
Note. OR = odds ratio. We excluded year 1 data because limited life expectancy was an exclusion criterion for the parent study and because there were too few deaths to analyze.
Odds of death in the year for participants without the presence of any listed disease.
Marginal OR of death for the presence versus absence of the disease without the presence of any other disease.
Joint OR for the presence versus absence of both diseases but without the presence of any other diseases.
Onset or occurrence prior to the baseline assessment.
Onset or occurrence during follow-up.
Baseline dementia was not assessed; all dementia occurred during follow-up.
The mortality rates across the years were similar. The risk of death in any year in the absence of exposure to the listed diseases, corresponded to the piecewise baseline hazards, and the risk was low, as shown in the fourth column of Table 2. In this cohort, 2 preexisting diseases, cancer and heart failure, achieved statistical significance at 0.05, and the hazard ratio for death exceeded 2 for the other new-onset chronic diseases. The hazard ratio of pneumonia exceeded 13. Each of the 2-way interactions had negative coefficients, indicating that the contributions to death were less than additive for any 2 coexisting diseases. Nevertheless, the hazard ratios for the simultaneous presence of 2 diseases were quite large (all exceeding 4).
Longitudinal Extension of the Average Attributable Fraction
We used the estimated coefficients from Table 2 to calculate the (piecewise) year-specific AAH for each disease over each year, as shown in Table 3. A corresponding LE-AAF for a disease was the weighted average of the 6 year-specific T-AAFs, with the person-months in each year serving as weights.
TABLE 3—
Longitudinal Extension of Average Attributable Fractions: Precipitating Events Project, New Haven County, CT
| LE-AAF, % (95% CI) | Year 2 T-AAF, % (95% CI) | Year 3 T-AAF, % (95% CI) | Year 4 T-AAF, % (95% CI) | Year 5 T-AAF, % (95% CI) | Year 6 T-AAF, % (95% CI) | Year 7 T-AAF, % (95% CI) | |
| Preexisting chronic diseasea | |||||||
| Heart failure | 5.2 (2.4, 8.7) | 7.4 (3.0, 13.4) | 5.9 (2.5, 10.8) | 4.5 (1.9, 7.7) | 4.0 (1.9, 6.7) | 4.1 (1.9, 6.5) | 4.6 (2.2, 7.1) |
| Cancer | 7.1 (2.0, 12.0) | 8.3 (2.3, 14.9) | 7.5 (2.0, 13.4) | 7.0 (2.0, 12.4) | 6.4 (1.9, 10.9) | 7.0 (2.1, 11.6) | 5.5 (1.5, 9.3) |
| Lung | 1.3 (−3.5, 5.4) | 1.4 (−3.8, 6.3) | 1.6 (−4.3, 6.4) | 1.3 (−3.4, 5.4) | 1.2 (−3.3, 5.1) | 1.3 (−3.5, 5.1) | 1.1 (−3.2, 4.4) |
| New-onset chronic diseaseb | |||||||
| Heart failure | 13.0 (8.8, 18.1) | 7.7 (3.5, 12.5) | 11.0 (5.9, 17.0) | 13.8 (8.8, 20.2) | 15.1 (10.1, 21.0) | 16.1 (10.7, 22.7) | 16.5 (10.2, 24.1) |
| Cancer | 3.5 (0.2, 7.1) | 1.1 (−0.03, 2.8) | 2.6 (−0.02, 6.3) | 3.7 (−0.02, 7.6) | 4.4 (0.5, 8.7) | 4.7 (0.4, 9.5) | 6.0 (0.9, 11.9) |
| Lung | 4.5 (1.4, 7.9) | 1.9 (0.5, 4.3) | 3.4 (1.0, 7.0) | 4.9 (1.3, 9.2) | 6.4 (2.3, 11.1) | 5.7 (0.6,10.3) | 5.5 (1.3, 10.4) |
| Dementiac | 9.4 (5.6, 13.5) | 6.4 (3.1, 10.1) | 6.9 (3.6, 11.1) | 8.1 (4.8, 12.1) | 12.2 (7.1, 18.1) | 11.5 (6.4, 17.8) | 13.3 (7.6, 20.2) |
| Acute disease: pneumonia | 12.1 (8.0, 17.1) | 17.0 (10.1, 25.3) | 13.5 (7.3, 21.5) | 10.5 (5.6, 17.4) | 10.6 (5.2, 17.2) | 7.0 (3.2, 13.3) | 12.4 (6.8, 20.0) |
| Total | 56.1 (49.3, 63.6) | 51.2 (41.5, 61.7) | 52.4 (44.4, 61.4) | 53.7 (46.3, 61.8) | 60.3 (52.1, 68.1) | 57.4 (50.6, 66.0) | 64.9 (57.6, 73.8) |
Note. CI = confidence interval; LE-AAF = longitudinal extension of average attributable fraction; T-AAF = time-dependent average attributable fraction. Values were calculated from 1000 bootstrap samples.
Onset or occurrence prior to the baseline assessment.
Onset or occurrence during follow-up.
Baseline dementia was not assessed; all dementia occurred during follow-up.
According to our LE-AAF assessment, 5.2% of deaths over the 6 years were attributable to preexisting heart failure. The T-AAFs for preexisting heart failure varied over the years. The LE-AAF assessment attributed 7.1% of deaths over the 6 years to preexisting cancers. With the exception of the sixth year, the T-AAFs displayed a downward trend in the contribution of preexisting cancer to risk of death. The T-AAFs were consistent through each year for preexisting chronic lung disease, which in this cohort accounted for a nonsignificant 1.3% of overall deaths over the 6 years. As assessed by the confidence intervals of their LE-AAFs, significant proportions of death were attributable to diseases with onset after baseline. For all the LE-AAFs, we found that the bootstrap confidence intervals stabilized around 800 samples.
The point estimates of the LE-AAFs from the 5 diseases, both preexisting and arising during follow-up, equaled the sum of the individual disease-specific LE-AAFs. Because the lower limit of the 95% confidence interval for the LE-AAF was 49.3%, over the 6-year period the contribution of the 5 diseases together likely exceeded 49%.
DISCUSSION
Estimation of the contribution of multiple risk factors or diseases to a health outcome could help guide clinical and policy decisions. We developed a method for apportioning the combined contribution of multiple time-varying risk factors to timed occurrence of or time-to-event outcomes. We avoided the nonadditivity of the (adjusted) AF by extending the AAF method of Eide and Gefeller11 to the longitudinal situation that accommodates time-varying exposures. We followed Eide and Gefeller in not specifying causal orderings of diseases and likewise agreed that averaging across all possible orderings provides the best analytical alternative for considering a large number of coexisting risk factors with diverse relationships.
We obtained the 95% confidence intervals for the LE-AAFs through the bootstrap method. Although Eide and Gefeller11 provide a formula for the asymptotic variance estimate of AAF, they do not calculate the standard errors. Furthermore, their formula does not account for the variability in estimating the parameters from the pooled logistic regression model. We provided a bootstrap variance estimate for LE-AAF because the calculation of its asymptotic variance is very complicated.
The LE-AAFs depend on the actual risk factors included in the pooled logistic model. The risk fraction attributable to a risk factor may be sensitive to the changes in the effects of other coexisting risk factors, and such changes are likely to induce changes in the correlation among the multiple risk factors. However, risk factors other than those included in the logistic model may contribute to death (or other outcome of interest), which is why the total from all the included risk factors is less than 1.
Important assumptions underlie the AF and AAF–LE-AAF methods. A given removal order of diseases is used only for the purpose of calculation and not intended to represent a clinical or public health scenario. AAF and LE-AAF assume that the removal orders are equally probable. In some situations, the probability of one removal order may be different from another. For example, because peripheral vascular disease is linked to diabetes, it would not be possible to remove peripheral vascular disease independently of diabetes. Under such circumstances, the AAF–LE-AAF is regarded as the effect of absence of the disease and is still the best way to apportion the combined effects of multiple, co-occurring exposures for which the causal relationships may not be monotone. When investigators are sure about the causal order of the diseases, methods that sequentially partition attributable effects can be adopted.8
In addition to its efficacy in analyzing prospective longitudinal data, the LE-AAF method can be used in panel data, multiple-wave survey data, and registry data where the status of the risk factors and the health outcome may vary over time within or across different individuals. The LE-AAF of a risk factor depends on the population in which the data are collected and the variety of coexisting risk factors taken into account. Because PEP represents a small cohort of older adults in New Haven County, Connecticut, and the self-reported ascertainment of diseases may not have been complete or accurate (as in many other studies), our results are not intended to infer relative contributions of the listed diseases to death in the general population.
Our LE-AAF method particularly applies to longitudinal situations where the risk factors are not independent and are time varying with no monotonic causal pathway and where the outcome is also time varying. Future work on AAF–LE-AAF should directly adjust for continuous covariates such as age, high dimensionality (i.e., many more risk factors), and extension to multivariate outcomes such as recurrent events.26
Acknowledgments
This work was supported by the National Institutes on Aging (grants R01 AG031850-01, R21AG030109, R37AG17560, and R01AG022993 and Midcareer Investigator Award in Patient-Oriented Research K24AG021507) and by the Yale Claude D. Pepper Older Americans Independence Center (grant P30AG21342).
Human Participant Protection
Protocol approval was waived by the Yale University human investigation committee because the data were previously collected.
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