This article has Open Peer Review reports available.
Additive and multiplicative hazards modeling for recurrent event data analysis
- Hyun J Lim^{1}Email author and
- Xu Zhang^{2}
https://doi.org/10.1186/1471-2288-11-101
© Lim and Zhang; licensee BioMed Central Ltd. 2011
Received: 5 January 2011
Accepted: 27 June 2011
Published: 27 June 2011
Abstract
Background
Sequentially ordered multivariate failure time or recurrent event duration data are commonly observed in biomedical longitudinal studies. In general, standard hazard regression methods cannot be applied because of correlation between recurrent failure times within a subject and induced dependent censoring. Multiplicative and additive hazards models provide the two principal frameworks for studying the association between risk factors and recurrent event durations for the analysis of multivariate failure time data.
Methods
Using emergency department visits data, we illustrated and compared the additive and multiplicative hazards models for analysis of recurrent event durations under (i) a varying baseline with a common coefficient effect and (ii) a varying baseline with an order-specific coefficient effect.
Results
The analysis showed that both additive and multiplicative hazards models, with varying baseline and common coefficient effects, gave similar results with regard to covariates selected to remain in the model of our real dataset. The confidence intervals of the multiplicative hazards model were wider than the additive hazards model for each of the recurrent events. In addition, in both models, the confidence interval gets wider as the revisit order increased because the risk set decreased as the order of visit increased.
Conclusions
Due to the frequency of multiple failure times or recurrent event duration data in clinical and epidemiologic studies, the multiplicative and additive hazards models are widely applicable and present different information. Hence, it seems desirable to use them, not as alternatives to each other, but together as complementary methods, to provide a more comprehensive understanding of data.
1. Background
Sequentially ordered multivariate failure time data or recurrent event time data are commonly observed in biomedical longitudinal studies. Examples include tumor recurrences, epileptic seizures, asthma attacks, migraines, infectious episodes, myocardial infarctions, injuries, and admissions to the hospital.
In general, standard hazard regression methods cannot be applied because of correlations between multivariate failure or recurrent event times within a subject. Adjustment is necessary for existing correlations, and more sophisticated analytic approaches are needed to obtain accurate estimates and efficient inferences. In the presence of the dependence between recurrent event times within a subject and subject-specific susceptibility across subjects, a variety of statistical methods have been proposed for the estimation of the covariate effect. In survival analysis, multiplicative and additive hazards models provide the two principal frameworks for studying the association between risk factors and recurrent event durations for the analysis of multivariate failure time data.
The majority of existing regression methods for analyzing multivariate failure or recurrent event time data assumes multiplicative covariate effects. Various authors have considered multivariate failure time models to be extensions of the Cox proportional hazards model [1]. The multivariate model with a Markov assumption, the conditional approach, the marginal approach, and the random effects approach are among them. Anderson and Gill proposed use of modeling under a Markov assumption [2]. Wei et al and Lee et al proposed use of the marginal approach [3, 4]. Prentice et al proposed use of a semi-parametric model when multivariate failure times are conditionally independent, given the covariates [5]. Others used the random effect frailty model or the conditional frailty model for such recurrent event data analysis [6, 7]. The popularity of these multiplicative models derives not only from their utility and wide applicability, but also from convention and the availability of statistical software. In general, consideration is not given to the possibility that the true underlying covariate effects may add to, rather than multiply, the baseline hazards. The semiparametric additive hazards model proposed by Lin and Ying [8] is the most closely connected analogue of the multiplicative Cox hazards model. Their additive hazards model assumes that covariates act in an additive manner on an unknown baseline hazard rate and that the effect of a covariate is time-invariant. Numerous authors advocated and utilized the additive hazards models for multivariate failure time data [9–14].
In this paper, we applied both multiplicative and additive models to the pediatric firearm victim's emergency department visit data. We considered the gap time model to be a multiplicative hazards model, as recommended for analysis of recurrent event time data by several authors [7, 15]. We considered the Lin and Ying's model [10] to be an additive hazards model in our data analysis. The multiplicative and additive hazards models for analysis of recurrent event data with two scenarios were considered: (i) a varying baseline with a common coefficient effect and (ii) a varying baseline with an order-specific coefficient effect. The proposed models were applied to the emergency department (ED) visits of pediatric firearm victims, and difference between the models was examined.
Four additional sections comprise this paper. In Section 2, the models and methods for the analysis of recurrent event duration data are reviewed. In Section 3, a description of the ED visit study and the methods applied to this dataset are provided. Section 4 contains a discussion, in which the applicability and appropriateness of each model are discussed.
2. Methods
Within the framework of the multiplicative or additive hazards regression models, a variety of models have been proposed and utilized in real applications. Among the rich selection of different models, the gap time model as an extension of the multiplicative Cox proportional hazards model [5] and the Lin and Ying's additive model (L-Y model) [8, 10] received the greatest attention due to easy interpretation of the covariate effects. These two models assume unspecified baseline hazards and constant covariate effects. In the models, we will assume that all censoring is non-informative and independent.
2.1. Basic Notations
Suppose that there are n subjects and that each subject can experience K failures or recurrent events. Suppose that censoring is non-informative, which means that knowledge of a censoring time for a subject provides no further information about the subject's likelihood of survival at a future time. Let T _{ ik }be the time when the kth failure occurs for the ith subject and C _{ ik }be the corresponding censoring time. T _{ ik }is measured from the subject's study enrollment and the censoring C _{ ik }occurs after the subject has been entered into a study to the right of the last known failure time; thus, it is right censoring. When T _{ ik }is subject to right censoring, the kth failure time X _{ ik }is a minimum of (T _{ ik }, C _{ ik }), i.e., X _{ ik }is equal to T _{ ik }if the event was observed and is equal to C _{ ik }if it is censored. Let δ _{ ik }= I(T _{ ik }≤ C _{ ik }), where I(.) is an indicator function and takes the value 1 when T _{ ik }≤ C _{ ik }and is 0 otherwise. Let Z _{ ik }be a covariate vector of p-dimensions for the ith subject at the kth failure. For each of the K failures, the hazard function for the ith subject with respect to the kth failure,λ _{ ik }(t), is assumed to take additive or multiplicative forms.
2.2. Multiplicative hazards model
where G _{ i,k }= X _{ ik } -X _{ i,k-1 }is the inter-event or gap time interval and Y _{ jk } (t) = I (G _{ ik }≥ t) is a risk set indicator. is a p-vector of regression coefficients of Z _{ i }.
Therefore, it turns out that , which is called the "robust" variance-covariance estimate, and a detailed derivation is given by Wei et al [16] and Lin [17]. To account for the within-subject correlation, we used this robust "sandwich" method in the estimation of standard errors.
2.3. Additive hazards model
Where λ _{0k }(t) is the unknown and unspecified baseline hazard function for the kth gap time and is a p x 1 vector of the regression coefficients. When there is only one failure event (i.e., K = 1), the model (3) reduces to a univariate additive hazards model [8].
Using empirical process theory, U( ) is shown to be a sum of independently, identically distributed random variables and thus follows a zero-mean Gaussian process by the functional central limit theorem; see Pollard [19], page 53, or van der Vaart & Weller [20], Section 2.11. Using Taylor's series expansion and some probability arguments, converges in distribution to a zero-mean normal distribution [11].
3. Results
3.1. Study Description
The pediatric firearm victim's ED visit study was a retrospective cohort study. Data consisted of medical record reviews of follow-ups of firearm victims younger than 19-years-old who were presenting to the Pediatric Emergency Department/Trauma Center at the Children's Hospital of Wisconsin and all other hospitals in the Milwaukee metropolitan area between 1990 and 1997.
Number of revisits to the emergency department.
Number of events | 0 | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|---|
Number of subjects (%) | 248 (48.5) | 130 (25.4) | 52 (10.2) | 30 (5.9) | 25 (4.9) | 9 (1.8) | 17 (3.3) | 511 (100) |
3.2. Analysis of the dataset
Additive and multiplicative hazards models for recurrent event time, with a varying baseline and common coefficient effect.
Model | Covariate | Estimate | S.E | Chi-square | p-value |
---|---|---|---|---|---|
Additive Hazards Model | Gender ^{#} | - 0.152 | 0.034 | 19.7 | <0.0001 |
Race/ethnicity ^{&} | 0.058 | 0.027 | 4.71 | 0.030 | |
Age | 0.009 | 0.003 | 6.34 | 0.012 | |
Parent ^{@} | 0.033 | 0.024 | 1.91 | 0.167 | |
------------------ | ---------------- | --------- | -------- | -------- | -------- |
Multiplicative Hazards Model | Gender | -0.654 | 0.139 | 35.4 | <0.0001 |
Race/ethnicity | 0.335 | 0.148 | 6.31 | 0.012 | |
Age | 0.043 | 0.016 | 8.22 | 0.004 | |
Parent | 0.152 | 0.117 | 2.37 | 0.124 |
Recurrent event time models with varying baseline and order-specific coefficient effects, from the additive and multiplicative hazards models.
Model | Order | Covariate | Estimate | S.E | Chi-square | p-value |
---|---|---|---|---|---|---|
Additive Hazards Model | 1 | Gender ^{#} | -0.103 | 0.06 | 2.96 | 0.085 |
Race/ethnicity ^{&} | 0.068 | 0.036 | 3.66 | 0.056 | ||
Age | 0.004 | 0.005 | 0.60 | 0.439 | ||
Parent ^{@} | 0.063 | 0.034 | 6.70 | 0.065 | ||
2 | Gender | -0.202 | 0.067 | 9.13 | 0.003 | |
Race/ethnicity | 0.038 | 0.051 | 0.55 | 0.457 | ||
Age | 0.015 | 0.005 | 7.94 | 0.005 | ||
Parent | 0.028 | 0.049 | 0.33 | 0.567 | ||
3 | Gender | -0.181 | 0.053 | 11.7 | 0.001 | |
Race/ethnicity | 0.003 | 0.078 | 0.002 | 0.967 | ||
Age | 0.011 | 0.008 | 1.92 | 0.165 | ||
Parent | -0.025 | 0.046 | 0.30 | 0.583 | ||
4 | Gender | -0.158 | 0.013 | 2.65 | 0.103 | |
Race/ethnicity | 0.068 | 0.121 | 0.32 | 0.571 | ||
Age | 0.021 | 0.013 | 2.65 | 0.104 | ||
Parent | -0.075 | 0.088 | 0.73 | 0.394 | ||
--------------------- | ------- | --------------- | ------------ | ------------ | ----------- | ---------- |
Multiplicative Hazards Model | 1 | Gender | -0.464 | 0.179 | 7.36 | 0.007 |
Race/ethnicity | 0.391 | 0.175 | 5.05 | 0.025 | ||
Age | 0.021 | 0.021 | 1.15 | 0.285 | ||
Parent | 0.347 | 0.144 | 5.99 | 0.014 | ||
2 | Gender | -0.864 | 0.235 | 15.7 | < 0.0001 | |
Race/ethnicity | 0.185 | 0.262 | 0.45 | 0.505 | ||
Age | 0.078 | 0.029 | 6.33 | 0.012 | ||
Parent | 0.112 | 0.215 | 0.31 | 0.577 | ||
3 | Gender | -0.747 | 0.25 | 8.21 | 0.004 | |
Race/ethnicity | 0.085 | 0.445 | 0.04 | 0.842 | ||
Age | 0.051 | 0.047 | 1.03 | 0.309 | ||
Parent | -0.138 | 0.248 | 0.29 | 0.589 | ||
4 | Gender | -0.732 | 0.28 | 5.44 | 0.02 | |
Race/ethnicity | 0.357 | 0.469 | 0.48 | 0.49 | ||
Age | 0.103 | 0.073 | 1.71 | 0.191 | ||
Parent | -0.31 | 0.323 | 0.89 | 0.345 |
The estimates from the additive and multiplicative hazards models had the same signs, indicating the same directions of the covariate effects. The standard errors from the additive hazard model were smaller than those of the multiplicative model. However, while the p-values for the two models differ, the inferences were consistent. Almost all of the models from both the common and order-specific covariate effects showed that gender was the significant risk factor for ED revisits.
Both additive and multiplicative hazards models with varying baseline and common coefficient effects gave similar results with regard to covariates selected to remain in the model (Tables 2). Three covariates showed significant impact on ED revisits in both hazards models: gender, race/ethnicity, and age. The result obtained under the additive model with a varying baseline and common coefficient effect in Table 2 suggested that females tended to have more-delayed ED revisits, compared to males, and being older and black were associated with significantly shorter gap times. On the other hand, the gap time did not seem to be related to having a parent as a guardian; the p-values were 0.167 and 0.124 for the additive and multiplicative hazards model, respectively. Estimates of the ED revisit order-specific covariate effects for the models with varying baselines are shown in Table 3. For all orders, k = 1, .., 4, gender was again the only significant risk factor for an ED revisit in the multiplicative hazards model, but this was not true for the additive hazards model. In both models, age was significant for the 2^{nd} revisit but not for any other revisit. Ethnicity and having a parent as a guardian were not significant in the additive model for all k.
4. Discussion
Among recurrent event data, correlation between event durations within a subject exists. For an example, one can suppose that more frequently a subject experiences episodes of injury, the sooner the next injury is likely to occur. In this study, the additive and multiplicative hazards regression models for the recurrent event duration analysis were examined and illustrated with a real dataset. Differences in estimates from the models under (i) a varying baseline with a common covariate effect and (ii) a varying baseline with an order-specific covariate effect were compared using the pediatric firearm victim's ED visit data. The additive and multiplicative models revealed similar results with regard to covariates selected to remain in the model: gender, race/ethnicity, and age. The estimated survival functions, based on the additive and multiplicative hazards models from our data, were similar. Our example showed that the goodness-of-fit of both multiplicative and additive hazards models was satisfactory.
The standard errors increase as the order increases because the size of the risk set for the models decreases after each revisit. If the risk set decreases rapidly, then it yields estimates that are less reliable with a small risk set size. However, in our study, the standard errors increased moderately. The coefficients of the models cannot be compared directly because the coefficients of the former act in a multiplicative way on an unknown baseline hazard, whereas the coefficients of the latter act in an additive way on unknown baseline hazard or represent coefficient function for added risks. A naïve way of comparing the models would be comparing p-values, which would indicate the power of rejecting the null hypothesis for selected covariate in the models. In order to detect any difference between models in terms of prediction, a comparison between two model-based survival curves with the nonparametric estimate of the survival function could be performed. In our study, the survival curves of these two models were larger than the Kaplan-Meier estimate for all order, but the differences were negligible.
The additive model is plausible for many applications and is often attractive in epidemiologic applications, for example, a study of diabetic patients [23]. In such a study, λ_{0} is taken to be the baseline mortality of the standard population and β measures excess risk for the patients under study. The excess mortality is more appealing than the relative mortality to provide an inference on how the study population' mortality differs from that of the standard population.
The additive and multiplicative hazards models can capture the risk process for patients with average comorbidity profiles equally well. In cases where both the additive and multiplicative models fit the data fairly well, an additive specification may be preferred, due to the interpretation of the regression parameters. One of the major advantages of using the additive hazards model over the multiplicative hazards model is that the resulting regression parameter estimator has a closed form [13]. Regression coefficients from the additive model are more interpretable in public health research since they represent differences in event rates, as opposed to ratio [24]. A practical drawback of using the additive models is that the current standard procedure for fitting additive models is still limited, whereas statistical software for the multiplicative model is available and easy to use.
In the presence of the dependence between recurrent events in multivariate failure time data, frailty model have been proposed for the estimation of the covariate effect by incorporation of additional unobserved random frailty effects into standard survival models [6]. The covariate estimates in the frailty model are estimated conditionally on the unobservable frailty, and because of this, their interpretation is often ambiguous [7]. When the primary interest of investigation is a measurement of the dependence of correlated repeated events within a subject, the frailty model approach is adequate [25]. However, our study does not focus on the dependence measurement of recurrent events.
Recurrent event duration data are the archetypical example of series data, which differ from parallel multivariate failure time data. Because the study period is typically less than the first failure time, the marginal distribution of the second gap time is not identifiable unless within-subject failure times are independent. Even if the total times are censored independently, the subsequent failure times will be subject to induced dependent censoring [17, 18]. To analyze such recurrent event duration data, the non-informative censoring assumption is required for the validity of the statistical analysis. However, when the recurrence is influencing a censoring mechanism such as dropout or death, censoring is informative about the event process; therefore, the non-informative censoring assumption is violated, and subjects in the risk set do not form a representative sample from the target population. An important assumption of the models examined in this paper is that the recurrent event process is independent of the censoring process. Suitable modification of the methodologies needs to be further studied to adjust for such informative censoring mechanisms related to terminal events in the recurrent event analysis [26]. In addition, there is relatively little information in the literature on the goodness-of-fit for multiple failure time models.
5. Conclusion
In this study, we illustrated and compared the additive and multiplicative hazards models for analysis of recurrent event durations. In summary, the choice between the additive and multiplicative models will typically be an empirical matter. Due to the frequency of recurrent event duration data in clinical and epidemiologic studies, the proposed additive and multiplicative methods are widely applicable. The two modeling approaches have sound biological bases, providing complementary information about the association between risk factors and death. An overall conclusion is that the additive and multiplicative hazards models present different aspects of the association between risk factors and the event durations. Hence, two hazards models give different information and it seems desirable to use them -- not as alternatives to each other, but as complementary methods -- together to gain a more comprehensive understanding of the data. Practitioners may benefit from the use of these statistical models, which help in predicting the effect of one or more variables and in verifying their influence on study outcomes.
Appendix
Suppose a subject has four recurrent events, say at t _{ 1 }, t _{ 2 }, t _{ 3 }, and t _{ 4 }. For the model analysis the subject is represented as a set of rows with time intervals of (0, t _{ 1 }], (0, t _{ 2 }- t _{ 1 }], (0, t _{ 3 }- t _{ 2 }], and (0, t _{ 4 }- t _{ 3 }] for the gap time. Letting gap be equal to t _{ k }- t _{ k-1 }for k = 1, .. 4, then the following programs specify the models.
A1. Data Managementfor recurrent events
data counting;
set firearm;
gap = tstop-tstart;
bguardn1 = 0;
if visit = 1 then bguardn1 = bguardn;
bguardn2 = 0;
if visit = 2 then bguardn2 = bguardn;
bguardn3 = 0;
if visit = 3 then bguardn3 = bguardn;
bguardn4 = 0;
if visit = 4 then bguardn4 = bguardn;
bracen1 = 0;
if visit = 1 then bracen1 = bracen;
bracen2 = 0;
if visit = 2 then bracen2 = bracen;
bracen3 = 0;
if visit = 3 then bracen3 = bracen;
bracen4 = 0;
if visit = 4 then bracen4 = bracen;
bage1 = 0;
if visit = 1 then bage1 = bage;
bage2 = 0;
if visit = 2 then bage2 = bage;
bage3 = 0;
if visit = 3 then bage3 = bage;
bage4 = 0;
if visit = 4 then bage4 = bage;
sexn1 = 0;
if visit = 1 then sexn1 = sexn;
sexn2 = 0;
if visit = 2 then sexn2 = sexn;
sexn3 = 0;
if visit = 3 then sexn3 = sexn;
sexn4 = 0;
if visit = 4 then sexn4 = sexn;
run;
A2. SAS PHREGprocedure for multiplicative model
title 'Gap Time Multiplicative Model with different betas and different baselines;
proc phreg data = counting;
model gap*status(0) = guard1 guard2 guard3 guard4 race1 race2 race3 race4
sex1 sex2 sex3 sex4;
strata visit;
id IDnumber;
run;
title 'Gap Time Multiplicative Model: common beta and different baselines';
proc phreg data = counting covs(aggregate);
model gap*status(0) = guard race sex;
strata visit;
id IDnumber;
run;
A3. SAS Macrofor additive model
%est(counting, gap, fail, visit, 4, id, bguardn bracen sexn bage);
proc transpose data = best out = best;
proc transpose data = se out = se;
proc transpose data = wlwse out = wlwse; run;
data item;
Variable = 'bguardn'; output;
Variable = 'bracen'; output;
Variable = 'sexn'; output;
Variable = 'bage'; output;
run;
data all;
merge item best (rename = (col1 = Estimate))
se (rename = (col1 = Naive_SE))
wlwse (rename = (col1 = Sandwich_SE));
Chisq = (Estimate/Sandwich_SE)**2;
Prob = 1-probchi(Chisq,1);
drop _name_;
run;
title 'Additive model - gap time';
title2 'common beta, different baselines';
proc print data = all; run;
%est(jing2, gap, fail, visit, 4, id,
bguardn1 bracen1 sexn1 bage1 bguardn2 bracen2 sexn2 bage2 bguardn3 bracen3 sexn3 bage3 bguardn4 bracen4 sexn4 bage4);
proc transpose data = best out = best;
proc transpose data = se out = se;
proc transpose data = wlwse out = wlwse; run;
data item;
Variable = 'bguardn1'; output;
Variable = 'bracen1'; output;
Variable = 'sexn1'; output;
Variable = 'bage1'; output;
Variable = 'bguardn2'; output;
Variable = 'bracen2'; output;
Variable = 'sexn2'; output;
Variable = 'bage2'; output;
Variable = 'bguardn3'; output;
Variable = 'bracen3'; output;
Variable = 'sexn3'; output;
Variable = 'bage3'; output;
Variable = 'bguardn4'; output;
Variable = 'bracen4'; output;
Variable = 'sexn4'; output;
Variable = 'bage4'; output;
run;
data all;
merge item best (rename = (col1 = Estimate))
se (rename = (col1 = Naive_SE))
wlwse (rename = (col1 = Sandwich_SE));
Chisq = (Estimate/Sandwich_SE)**2;
Prob = 1-probchi(Chisq,1);
drop _name_;
run;
title 'Additive model - gap time';
title2 'different betas, different baselines';
proc print data = all; run;
Declarations
Acknowledgements
The authors would like to thank Dr. Meltzer-Lange for the use of data. This study is supported by a research grant from the College of Medicine at the University of Saskatchewan. The authors would like to thank the reviewers for their constructive and valuable comments.
Authors’ Affiliations
References
- Cox DR: Regression models and life-tables (with discussion). Journal of the Royal Statistical Society B. 1972, 34: 187-220.Google Scholar
- Anderson PK, Gill RD: Cox's regression model for counting processes: A large sample study. The Annals of Statistics. 1982, 10: 1100-1120. 10.1214/aos/1176345976.View ArticleGoogle Scholar
- Wei LJ, Lin DY, Weissfeld L: Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. Journal of American Statistical Association. 1989, 84: 1065-1073. 10.2307/2290084.View ArticleGoogle Scholar
- Lee EW, Wei LJ, Amato DA: Cox-type regression analysis for large number of small groups of correlated failure time observations. Survival Analysis: State of the Art. Edited by: JP Klein, PK Goel. 1992, Dordrecht: Kluwer Academic PublisherGoogle Scholar
- Prentice RL, Williams BJ, Peterson AV: On the regression analysis of multivariate failure time data. Biometrika. 1981, 68: 373-379. 10.1093/biomet/68.2.373.View ArticleGoogle Scholar
- Oakes D: Frailty models for multiple event times. Survival Analysis: State of the Art. Edited by: JP Klein, PK Goel. 1992, Dordrecht: Kluwer Academic PublisherGoogle Scholar
- Box-Steffensmeier JM, Suzanna DB: Repeated events survival models: The conditional frailty model. Statistics in Medicine. 2006, 25: 3518-33. 10.1002/sim.2434.View ArticlePubMedGoogle Scholar
- Lin DY, Ying Z: Semiparametric analysis of the additive risk model. Biometrika. 1994, 81: 61-71. 10.1093/biomet/81.1.61.View ArticleGoogle Scholar
- McKeague IW: Asymptotic theory for weighted least squares estimators in Aalen's additive risk model. Contemporary Mathematics. 1988, 80: 139-152.View ArticleGoogle Scholar
- Lin DY, Ying Z: Additive regression models for survival data. Proceedings of the First Seattle Symposium in Biostatistics: Survival Analysis. Edited by: Lin DY, Fleming TR. 1997, Springer: New York, 185-198.View ArticleGoogle Scholar
- Yin G, Cai J: Additive hazards model with multivariate failure time data. Biometrika. 2004, 91: 801-818. 10.1093/biomet/91.4.801.View ArticleGoogle Scholar
- Schaubel DE, Cai J: Analysis of clustered recurrent event data with application to hospitalization rates among renal failure patients. Biostatistics. 2005, 6: 404-419. 10.1093/biostatistics/kxi018.View ArticlePubMedGoogle Scholar
- Schaubel DE, Zeng D, Cai J: A semiparametric additive rate model for recurrent event data. Lifetime Data Anal. 2006, 12: 389-406. 10.1007/s10985-006-9017-x.View ArticlePubMedGoogle Scholar
- Sun L, Park D, Sun G: The additive hazards model for recurrent gap times. Statistica Sinica. 2006, 16: 919-932.Google Scholar
- Kelly PJ, Lim LY: Survival analysis for recurrent event data: an application to childhood infectious diseases. Statistics in Medicine. 2000, 19: 13-33. 10.1002/(SICI)1097-0258(20000115)19:1<13::AID-SIM279>3.0.CO;2-5.View ArticlePubMedGoogle Scholar
- Wei LJ, Lin DY, Weissfeld L: Regression analysis of multivariate incomplete failure time data by modeling marginal. Journal of the American Statistical Association. 1989, 84: 1065-1073. 10.2307/2290084.View ArticleGoogle Scholar
- Lin DY: Cox regression analysis of multivariate failure time data: the marginal approach. Statistics in Medicine. 1994, 13: 2233-2247. 10.1002/sim.4780132105.View ArticlePubMedGoogle Scholar
- Andersen PK, Borgan Ø, Gill RD, Keiding N: Statistical Models Based on Counting Processes. 1993, Springer: New YorkView ArticleGoogle Scholar
- Pollard D: Empirical Processes: Theory and Applications. Regional Conference Series in Probability and Statistics V.2. 1990, Institute of Mathematical Statistics, Howard, CaliforniaGoogle Scholar
- Van Der Vaart AW, Weller JA: Weak convergence and empirical processes. 1996, New York: SpringerView ArticleGoogle Scholar
- Marcelle DR, Melzer-Lange MD: Project UJIMA: Working together to make things right. Wisconsin Medical Journal. 2001, 100: 222-5.Google Scholar
- Lim HJ, Liu J, Melzer-Lange M: Comparison of methods for analyzing recurrent event data: application to emergency department visits of pediatric firearm victims. Accident Analysis & Prevention. 2007, 39: 290-299. 10.1016/j.aap.2006.07.009.View ArticleGoogle Scholar
- Rosato R, Ciccone G, Pagano FG, Gregori D: Evaluating cadiovascular mortality in type 2 diabetes patient: an analysis based on competing risks Makov chains and additive regression models. J of Eval in Clinical Practice. 2007, 13: 422-428. 10.1111/j.1365-2753.2006.00732.x.View ArticleGoogle Scholar
- Lim HJ, Zhang X: Semi-parametric additive risk models: application to injury duration study. Accident Analysis & Prevention. 2009, 41: 211-216. 10.1016/j.aap.2008.07.015.View ArticleGoogle Scholar
- Clayton D, Cuzick J: Multivariate generalizations of the proportional hazards model. Journal of the Royal Statistical Society A. 1985, 148: 82-117. 10.2307/2981943.View ArticleGoogle Scholar
- Wang MC, Chiang CT: Nonparametric method for recurrent event data with informative and non- censorings. Statistics in Medicine. 2002, 21: 445-456. 10.1002/sim.1029.View ArticlePubMedGoogle Scholar
- The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1471-2288/11/101/prepub
Pre-publication history
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.