Journal of Biopharmaceutical Statistics
ISSN: 1054-3406 (Print) 1520-5711 (Online) Journal homepage: http://www.tandfonline.com/loi/lbps20
Hypoglycemic Events Analysis via Recurrent Timeto-Event (HEART) Models Haoda Fu, Junxiang Luo & Yongming Qu To cite this article: Haoda Fu, Junxiang Luo & Yongming Qu (2014): Hypoglycemic Events Analysis via Recurrent Time-to-Event (HEART) Models, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2014.992524 To link to this article: http://dx.doi.org/10.1080/10543406.2014.992524
Accepted online: 01 Dec 2014.Published online: 01 Dec 2014.
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Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lbps20 Download by: [Mount Allison University 0Libraries]
Date: 24 September 2015, At: 08:59
Journal of Biopharmaceutical Statistics, 00: 1–22, 2015 Copyright © Taylor & Francis Group, LLC ISSN: 1054-3406 print/1520-5711 online DOI: 10.1080/10543406.2014.992524
HYPOGLYCEMIC EVENTS ANALYSIS VIA RECURRENT TIME-TO-EVENT (HEART) MODELS
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
Haoda Fu, Junxiang Luo, and Yongming Qu Eli Lilly and Company, Indianapolis, Indiana, USA Diabetes affects an estimated 25.8 million people in the United States and is one of the leading causes of death. A major safety concern in treating diabetes is the occurrence of hypoglycemic events. Despite this concern, the current methods of analyzing hypoglycemic events, including the Wilcoxon rank sum test and negative binomial regression, are not satisfactory. The aim of this paper is to propose a new model to analyze hypoglycemic events with the goal of making this model a standard method in industry. Our method is based on a gamma frailty recurrent event model. To make this method broadly accessible to practitioners, this paper provides many details of how this method works and discusses practical issues with supporting theoretical proofs. In particular, we make efforts to translate conditions and theorems from abstract counting process and martingale theories to intuitive and clinical meaningful explanations. For example, we provide a simple proof and illustration of the coarsening at random condition so that the practitioner can easily verify this condition. Connections and differences with traditional methods are discussed, and we demonstrate that under certain scenarios the widely used Wilcoxon rank sum test and negative binomial regression cannot control type 1 error rates while our proposed method is robust in all these situations. The usefulness of our method is demonstrated through a diabetes dataset which provides new clinical insights on the hypoglycemic data. Key Words: Coarsening at random; Diabetes; Gamma frailty model; Hypoglycemic events; Missing at random; Recurrent events.
1. INTRODUCTION Diabetes mellitus, or simply diabetes, is a disease characterized by elevated blood glucose. It is a major cause of kidney failure, nontraumatic lower-limb amputations, blindness, heart disease, and stroke. As a result, diabetes is one of the leading causes of death. Based on the data from Centers for Disease Control and Prevention (http://www. cdc.gov/diabetes/ pubs), in year 2011, diabetes affects 25.8 million American people which is 8.3% of the US population. The newly diagnosed cases are expected to be at a rate of 1 million people per year. The goal of treating diabetes patients is to lower their blood glucose. However, lowering glucose too much could result in an adverse event, called hypoglycemia. Concern over hypoglycemia has a significant negative impact on diabetes management, Received April 2, 2014; Accepted November 24, 2014 Address correspondence to Haoda Fu, Eli Lilly and Company, Lilly Corporate Center, Indianapolis, IN 46285, USA; E-mail: [email protected] Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lbps.
1
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
2
FU ET AL.
and it is a major factor to prevent patients’ glycemic control from reaching treatment target (Wild et al., 2007). Therefore, it is desired to develop new anti-diabetes agents that lead to less hypoglycemic events while lowering the glucose toward the normal range. As a result, statistical analysis of hypoglycemic events is very important in diabetes clinical research. In clinical trials, hypoglycemic events are generally captured through patients self report in diaries which record the time, date of hypoglycemic events, and the associated attributes. Each patient can experience multiple hypoglycemic events during a period of time. One way to handle the hypoglycemic event data is to treat them as count data, i.e., the number of hypoglycemic events. Then, a natural way to fit the data is to use a Poisson distribution. However, the mean and the variance of observed hypoglycemic events are often quite different. For example, in a 24-week study (Rosenstock et al., 2008), the mean number of hypoglycemic events for patients treated with insulin therapy is 48.70 per patient per year while the corresponding variance is 2343.53. Therefore, Poisson regression is too restrictive for analyzing the number of hypoglycemic events. There are a few methods to deal with the overdispersion under the generalized linear models framework, such as the quasi-likelihood Poisson regression method (McCullagh and Nelder, 1998, Section 9.3), negative binomial regression, quasi-likelihood negative binomial regression, generalized Poisson regression, zero-inflated Poisson/negative binomial regression, and semi-parametric negative binomial regression (McCullagh and Nelder, 1998; Lawless, 1987; Li, 2010). Instead of these parametric models, the rate of hypoglycemic events is often analyzed by nonparametric methods, such as Wilcoxon rank sum test. Among all these methods, the current most popular analysis methods used by pharmaceutical companies and regulatory agents are negative binomial regression and the Wilcoxon rank sum test (Mullins et al., 2007; Bullano et al., 2005; Bulsara et al., 2004; Buse et al., 2011; Gold et al., 1994). There are two gaps in using negative binomial regression or Wilcoxon rank sum test to analyze hypoglycemic events. First, we often have patients who drop out before completing the clinical trial. Both negative binomial regression and the Wilcoxon rank sum test essentially assume the hypoglycemic event rates are the same before and after dropout. The Wilcoxon rank sum test also assumes the distribution of the event rate for those patients with early dropout is the same as those who complete the study. This assumption can be erroneous, as the variance of the event rate generally decreases as the duration of observation increases. As a result, under different drop out rates, the Wilcoxon rank sum test cannot control the type 1 error. Second, the negative binomial regression and the Wilcoxon rank sum test ignore the information of the dates when the hypoglycemic events occur. Therefore, these two methods only evaluate the average hypoglycemic event rate over a period of time. However, it is of clinical interest to evaluate the hypoglycemia rate change over time. The aim of this research paper is to develop a method to analyze hypoglycemic events which is able to account for the shortfalls of the other methods. Our goal is to make the proposed method to be an industry standard method. With this goal in mind, we have the following four principles when developing the model and writing this paper. 1. There should be no ambiguity when applying the proposed model. The model can be well specified in the protocol and statistical analysis plan without controversies. For example, we do not explore models which contain a tuning parameter in this paper, since different ways to choose the tuning parameter may end up with different results. It is critically important for both the sponsor and reviewers to arrive at the same result.
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
HEART MODELS
3
2. The proposed method should not have any more assumptions than the negative binomial regression method. In particular, the negative binomial regression should be a special case of the proposed method. Otherwise, if the proposed method requires more assumptions than the established method, people may disagree on the assumptions and, consequently, on the results. 3. The model should be simple and elegant. It should take care of practical considerations without increasing the model complexities and the risks of numerically instability. For example, missing data are an important practical problem. The proposed method is able to handle a broad range of missing data issues without additional modeling. 4. This paper is to target a broad audience. The contents of this paper are carefully selected to make this paper self-contained to a large degree. We include enough mathematical derivations which enable the practitioner to understand the details while we do not require any background knowledge of counting process, martingale, or survival analysis. In addition, we provide sufficient additional references to people who would like to understand this topic more in depth. Since hypoglycemic event times are recorded on clinical report forms, we can consider the data as recurrent time to events. There is a large amount of literature on modeling recurrent event data (see Cook and Lawless, 2007; Kalbfleisch and Prentice, 2002; Fleming and Harrington 2005 among others). Based on the four principles of this paper, we discuss why we choose our model step-by-step. We introduce the basic concepts of the counting process and recurrent event models in Section 2.1. This also provides an overview of different approaches modeling recurrent events data and we propose to model hypoglycemic events data through event counts. In Section 2.2 we review the key theoretical results to develop the likelihood function. This will serve as a refresher and also as a way of collecting a few results that will often be useful. In Section 2.3, we propose to use a gamma frailty recurrent event model to analyze hypoglycemic events. Based on the four principles, we propose to use a piecewise constant baseline rate function and draw statistical inference based on a full likelihood function. This allows us to overcome the shortcomings of some current estimation methods, and it also allows us to develop a few tests with clinical meaning in this section. For example, S-plus and R (function coxph) use partial penalized likelihood method which do not provide standard error estimate of the frailty variance, and, as a consequence, it cannot provide the variability of the mean cumulative function as well (Therneau and Grambsch, 2000). In Section 2.4, we discuss the missing data issue. The missing or censoring has been discussed by other authors, and a coarsening at random condition was proposed (Heitjan and Rubin, 1991; Robins and Rotnitzky, 1992; Miloslavsky et al., 2004). However, as critiqued by Sinha et al. (2008), the coarsening at random condition is difficult to understand and hard to verify. We provide an intuitive proof, and translate the abstract mathematical definition to clinical settings. Connections among gamma frailty model and other approaches are discussed in Section 2.5. We evaluate the performance of our model by simulations. In Section 3.2, we evaluate its ability to estimate the parameters when there are no missing data and when data are missing at random (MAR). In Section 3.3, we evaluate its ability for statistical inference. In particular, we compare the proposed method with negative binomial regression and Wilcoxon rank sum test, and we point out that the traditional methods are not able to control type 1 error under certain conditions. We analyze a dataset from a recent diabetes study in Section 4. In Section 5, we provide a discussion on this method for analyzing hypoglycemic events.
4
FU ET AL.
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
2. MODELING RECURRENT EVENT DATA In this section, we review the basic results to model the recurrent event data and propose a gamma frailty model to fit hypoglycemic events. We start from basic concepts of the counting process and derive general likelihood functions. Then, we introduce a gamma frailty model with a piecewise constant baseline rate function. In the end, we discuss the conditions to handle missing data and connect our model with negative binomial regression and the Wilcoxon rank sum test. The concepts and derivations introduced in this section are self contained and we try to make the results easily understood without using advanced probability theory. More rigorous treatment on this topic can be found in many classical books including Cook and Lawless (2007), Andersen (1993), and Fleming and Harrington (2005).
2.1. Basic Concepts To illustrate the concepts, we use a dataset from a diabetes trial (Rosenstock et al., 2008). In this trial, some patients experienced multiple hypoglycemic events. The event times from some randomly selected patients are listed in Table 1. Clinical report forms collect each hypoglycemic event time which enables us to treat the hypoglycemic events as recurrent events. The concepts of counting processes and intensity functions are especially useful for both modeling and statistical analysis of recurrent event data. The hypoglycemic event times for each patient are considered as a single event process. Let 0 T1 < T2 < Tn denote the event times during a study period ½0; . The associated counting process fN ðtÞ; 0 Pt1 g records the cumulative number of events generated by this process, i.e., N ðtÞ ¼ i¼1 I ðTi t Þ, where IðÞ is the indicator function. Therefore, N ðtÞ is the number of events occurring over the time interval ½0; t. Let ΔN ðtÞ ¼ N ðt þ Δt Þ N ðt Þ denote the number of events in the interval ½t; t þ ΔtÞ, and HðtÞ ¼ fN ðsÞ : 0 s < t g denote the event history of the process before time t. By assuming two events cannot occur simultaneously, the intensity function of this process is defined as PrfΔN ðtÞ ¼ 1jHðtÞg : Δt!0 Δt
λftjHðtÞg ¼ lim
(2:1)
We illustrate the counting process representation of a single process from a patient in the panel (a) of Fig. 1. In panel (b) and (c), for illustration, we simulated the data based on different intensify functions (details on how to simulate event times are provided in Table 1 Selected hypoglycemic events data from the Rosenstock study Paitent ID
Hypoglycemic event times (days)
1
14
38
66
87
101
133
163
2
102
110
117
3
50
70
74
75
87
91
113
4
28
29
5
99
37
57
66
152
Censored time (days) 169 173
114
122
126
149
157
161 172 168
5
Figure 1 Counting process representation of recurrent event data. (a) Counting process representation of the first patient in Table 1. Panels (b) and (c) are the simulated data based on different intensity functions.
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
6
FU ET AL.
Section 3.1). The plots illustrate that it is more likely to observe an event when the underlying intensity function has a high value. Our statistical inference for hypoglycemic events is based on the intensity function. The two fundamental ways of modeling recurrent events are through gaps between successive events and through event counts. Modeling based upon gaps are often useful when the events are rare and when prediction of the next event is of interest. Some examples can be found in modeling system failures where repairs are made at each failure time. The renewal processes are the canonical models in this situation. Some references can be found in Gail et al. (1980), Prentice et al. (1981), Kalbfleisch and Prentice (2002), and Lawless (2003) among many others. Models based on event counts are often useful when individuals frequently experience the events, and when the trend of the intensity function is of interest (Aalen, 1978; Andersen and Gill, 1982; Therneau and Grambsch, 2000; Cook and Lawless, 2007). Depending upon the antidiabetic drugs, hypoglycemic events can frequently occur and comparing the intensity functions from different treatment groups are of interest. Therefore, in this paper, we model the hypoglycemic events through event counts.
2.2. Likelihood Function In this subsection, we provide the key techniques to develop the likelihood function. We assume that the events happen in a fixed study period ½0; , i.e., is a fixed constant. In real clinical trials, patients can dropout earlier due to different reasons. In Section 2.4, we extend our results to handle the situations where is a random variable, and we provide conditions under which the results developed in this section are still valid. The following theorem lays a foundation to develop likelihood functions which are used for parameter estimation and statistical inferences in this paper. Theorem 1 By assuming two events cannot occur simultaneously, the intensity function for an event process is defined as in equation (2.1). Over a specified interval ½0 ; , conditional on H ð0 Þ, the probability density of outcome n events ðt1 ; . . . ; tn Þ, at times 0 t1 < < tn , where n 1, for a process with intensity function as λft jH ðtÞg, is, n Y i¼1
ð λfti jH ðti Þg exp λft jH ðt Þgdt ; 0
(2:2)
and
" ð # b PrfN ða; bÞ ¼ 0jH ðaþ Þg ¼ exp λft jH ðt Þgdt ;
(2:3)
a
where N ða; bÞ ¼ N ðbÞ N ðaÞ denotes the number of events over a time period ða; b.
The proof of this theorem is Appendix A. The Poisson process provides an important framework to analyze the recurrent event data, and it is a foundation to develop the gamma frailty model in the later section.
HEART MODELS
7
Definition 1 (Poisson process) Poisson processes are Markov with an intensity function of the form as
λft jH ðt Þg ¼ lim
Δt!0
PrfΔN ðt Þ ¼ 1g ¼ ρðt Þ; Δt
(2:4)
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
where ρðtÞ is the rate function giving the marginal (i.e., unconditional) instantaneous probability of an event at time t.
The Poisson process is independent of its history. Hence, equation (2.4) says that for two non-overlap time intervals, the number of event occurrences in these two intervals are independent. By the definition of the Poisson process, we have the following results which are useful when we develop the gamma frailty model, and when we show the connections between different models. The mean cumulative function is defined as ðt ρðsÞds:
μð t Þ ¼ E fN ð t Þ g ¼
(2:5)
0
If n events are from a Poisson process, by Theorem 1 and equation (2.4), the probability density function of these n events occur at time t1 < < tn in ½0; is (
n Y ρ tj
) expfμð Þg;
(2:6)
j¼1
where n 1. The rationale for referring to this as a Poisson process is because it is connected with a Poisson distribution. This connection can be seen as below: "ð Prðn events in½0; Þ ¼
(
ð
0t1 ...
ISSN: 1054-3406 (Print) 1520-5711 (Online) Journal homepage: http://www.tandfonline.com/loi/lbps20
Hypoglycemic Events Analysis via Recurrent Timeto-Event (HEART) Models Haoda Fu, Junxiang Luo & Yongming Qu To cite this article: Haoda Fu, Junxiang Luo & Yongming Qu (2014): Hypoglycemic Events Analysis via Recurrent Time-to-Event (HEART) Models, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2014.992524 To link to this article: http://dx.doi.org/10.1080/10543406.2014.992524
Accepted online: 01 Dec 2014.Published online: 01 Dec 2014.
Submit your article to this journal
Article views: 66
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lbps20 Download by: [Mount Allison University 0Libraries]
Date: 24 September 2015, At: 08:59
Journal of Biopharmaceutical Statistics, 00: 1–22, 2015 Copyright © Taylor & Francis Group, LLC ISSN: 1054-3406 print/1520-5711 online DOI: 10.1080/10543406.2014.992524
HYPOGLYCEMIC EVENTS ANALYSIS VIA RECURRENT TIME-TO-EVENT (HEART) MODELS
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
Haoda Fu, Junxiang Luo, and Yongming Qu Eli Lilly and Company, Indianapolis, Indiana, USA Diabetes affects an estimated 25.8 million people in the United States and is one of the leading causes of death. A major safety concern in treating diabetes is the occurrence of hypoglycemic events. Despite this concern, the current methods of analyzing hypoglycemic events, including the Wilcoxon rank sum test and negative binomial regression, are not satisfactory. The aim of this paper is to propose a new model to analyze hypoglycemic events with the goal of making this model a standard method in industry. Our method is based on a gamma frailty recurrent event model. To make this method broadly accessible to practitioners, this paper provides many details of how this method works and discusses practical issues with supporting theoretical proofs. In particular, we make efforts to translate conditions and theorems from abstract counting process and martingale theories to intuitive and clinical meaningful explanations. For example, we provide a simple proof and illustration of the coarsening at random condition so that the practitioner can easily verify this condition. Connections and differences with traditional methods are discussed, and we demonstrate that under certain scenarios the widely used Wilcoxon rank sum test and negative binomial regression cannot control type 1 error rates while our proposed method is robust in all these situations. The usefulness of our method is demonstrated through a diabetes dataset which provides new clinical insights on the hypoglycemic data. Key Words: Coarsening at random; Diabetes; Gamma frailty model; Hypoglycemic events; Missing at random; Recurrent events.
1. INTRODUCTION Diabetes mellitus, or simply diabetes, is a disease characterized by elevated blood glucose. It is a major cause of kidney failure, nontraumatic lower-limb amputations, blindness, heart disease, and stroke. As a result, diabetes is one of the leading causes of death. Based on the data from Centers for Disease Control and Prevention (http://www. cdc.gov/diabetes/ pubs), in year 2011, diabetes affects 25.8 million American people which is 8.3% of the US population. The newly diagnosed cases are expected to be at a rate of 1 million people per year. The goal of treating diabetes patients is to lower their blood glucose. However, lowering glucose too much could result in an adverse event, called hypoglycemia. Concern over hypoglycemia has a significant negative impact on diabetes management, Received April 2, 2014; Accepted November 24, 2014 Address correspondence to Haoda Fu, Eli Lilly and Company, Lilly Corporate Center, Indianapolis, IN 46285, USA; E-mail: [email protected] Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lbps.
1
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
2
FU ET AL.
and it is a major factor to prevent patients’ glycemic control from reaching treatment target (Wild et al., 2007). Therefore, it is desired to develop new anti-diabetes agents that lead to less hypoglycemic events while lowering the glucose toward the normal range. As a result, statistical analysis of hypoglycemic events is very important in diabetes clinical research. In clinical trials, hypoglycemic events are generally captured through patients self report in diaries which record the time, date of hypoglycemic events, and the associated attributes. Each patient can experience multiple hypoglycemic events during a period of time. One way to handle the hypoglycemic event data is to treat them as count data, i.e., the number of hypoglycemic events. Then, a natural way to fit the data is to use a Poisson distribution. However, the mean and the variance of observed hypoglycemic events are often quite different. For example, in a 24-week study (Rosenstock et al., 2008), the mean number of hypoglycemic events for patients treated with insulin therapy is 48.70 per patient per year while the corresponding variance is 2343.53. Therefore, Poisson regression is too restrictive for analyzing the number of hypoglycemic events. There are a few methods to deal with the overdispersion under the generalized linear models framework, such as the quasi-likelihood Poisson regression method (McCullagh and Nelder, 1998, Section 9.3), negative binomial regression, quasi-likelihood negative binomial regression, generalized Poisson regression, zero-inflated Poisson/negative binomial regression, and semi-parametric negative binomial regression (McCullagh and Nelder, 1998; Lawless, 1987; Li, 2010). Instead of these parametric models, the rate of hypoglycemic events is often analyzed by nonparametric methods, such as Wilcoxon rank sum test. Among all these methods, the current most popular analysis methods used by pharmaceutical companies and regulatory agents are negative binomial regression and the Wilcoxon rank sum test (Mullins et al., 2007; Bullano et al., 2005; Bulsara et al., 2004; Buse et al., 2011; Gold et al., 1994). There are two gaps in using negative binomial regression or Wilcoxon rank sum test to analyze hypoglycemic events. First, we often have patients who drop out before completing the clinical trial. Both negative binomial regression and the Wilcoxon rank sum test essentially assume the hypoglycemic event rates are the same before and after dropout. The Wilcoxon rank sum test also assumes the distribution of the event rate for those patients with early dropout is the same as those who complete the study. This assumption can be erroneous, as the variance of the event rate generally decreases as the duration of observation increases. As a result, under different drop out rates, the Wilcoxon rank sum test cannot control the type 1 error. Second, the negative binomial regression and the Wilcoxon rank sum test ignore the information of the dates when the hypoglycemic events occur. Therefore, these two methods only evaluate the average hypoglycemic event rate over a period of time. However, it is of clinical interest to evaluate the hypoglycemia rate change over time. The aim of this research paper is to develop a method to analyze hypoglycemic events which is able to account for the shortfalls of the other methods. Our goal is to make the proposed method to be an industry standard method. With this goal in mind, we have the following four principles when developing the model and writing this paper. 1. There should be no ambiguity when applying the proposed model. The model can be well specified in the protocol and statistical analysis plan without controversies. For example, we do not explore models which contain a tuning parameter in this paper, since different ways to choose the tuning parameter may end up with different results. It is critically important for both the sponsor and reviewers to arrive at the same result.
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
HEART MODELS
3
2. The proposed method should not have any more assumptions than the negative binomial regression method. In particular, the negative binomial regression should be a special case of the proposed method. Otherwise, if the proposed method requires more assumptions than the established method, people may disagree on the assumptions and, consequently, on the results. 3. The model should be simple and elegant. It should take care of practical considerations without increasing the model complexities and the risks of numerically instability. For example, missing data are an important practical problem. The proposed method is able to handle a broad range of missing data issues without additional modeling. 4. This paper is to target a broad audience. The contents of this paper are carefully selected to make this paper self-contained to a large degree. We include enough mathematical derivations which enable the practitioner to understand the details while we do not require any background knowledge of counting process, martingale, or survival analysis. In addition, we provide sufficient additional references to people who would like to understand this topic more in depth. Since hypoglycemic event times are recorded on clinical report forms, we can consider the data as recurrent time to events. There is a large amount of literature on modeling recurrent event data (see Cook and Lawless, 2007; Kalbfleisch and Prentice, 2002; Fleming and Harrington 2005 among others). Based on the four principles of this paper, we discuss why we choose our model step-by-step. We introduce the basic concepts of the counting process and recurrent event models in Section 2.1. This also provides an overview of different approaches modeling recurrent events data and we propose to model hypoglycemic events data through event counts. In Section 2.2 we review the key theoretical results to develop the likelihood function. This will serve as a refresher and also as a way of collecting a few results that will often be useful. In Section 2.3, we propose to use a gamma frailty recurrent event model to analyze hypoglycemic events. Based on the four principles, we propose to use a piecewise constant baseline rate function and draw statistical inference based on a full likelihood function. This allows us to overcome the shortcomings of some current estimation methods, and it also allows us to develop a few tests with clinical meaning in this section. For example, S-plus and R (function coxph) use partial penalized likelihood method which do not provide standard error estimate of the frailty variance, and, as a consequence, it cannot provide the variability of the mean cumulative function as well (Therneau and Grambsch, 2000). In Section 2.4, we discuss the missing data issue. The missing or censoring has been discussed by other authors, and a coarsening at random condition was proposed (Heitjan and Rubin, 1991; Robins and Rotnitzky, 1992; Miloslavsky et al., 2004). However, as critiqued by Sinha et al. (2008), the coarsening at random condition is difficult to understand and hard to verify. We provide an intuitive proof, and translate the abstract mathematical definition to clinical settings. Connections among gamma frailty model and other approaches are discussed in Section 2.5. We evaluate the performance of our model by simulations. In Section 3.2, we evaluate its ability to estimate the parameters when there are no missing data and when data are missing at random (MAR). In Section 3.3, we evaluate its ability for statistical inference. In particular, we compare the proposed method with negative binomial regression and Wilcoxon rank sum test, and we point out that the traditional methods are not able to control type 1 error under certain conditions. We analyze a dataset from a recent diabetes study in Section 4. In Section 5, we provide a discussion on this method for analyzing hypoglycemic events.
4
FU ET AL.
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
2. MODELING RECURRENT EVENT DATA In this section, we review the basic results to model the recurrent event data and propose a gamma frailty model to fit hypoglycemic events. We start from basic concepts of the counting process and derive general likelihood functions. Then, we introduce a gamma frailty model with a piecewise constant baseline rate function. In the end, we discuss the conditions to handle missing data and connect our model with negative binomial regression and the Wilcoxon rank sum test. The concepts and derivations introduced in this section are self contained and we try to make the results easily understood without using advanced probability theory. More rigorous treatment on this topic can be found in many classical books including Cook and Lawless (2007), Andersen (1993), and Fleming and Harrington (2005).
2.1. Basic Concepts To illustrate the concepts, we use a dataset from a diabetes trial (Rosenstock et al., 2008). In this trial, some patients experienced multiple hypoglycemic events. The event times from some randomly selected patients are listed in Table 1. Clinical report forms collect each hypoglycemic event time which enables us to treat the hypoglycemic events as recurrent events. The concepts of counting processes and intensity functions are especially useful for both modeling and statistical analysis of recurrent event data. The hypoglycemic event times for each patient are considered as a single event process. Let 0 T1 < T2 < Tn denote the event times during a study period ½0; . The associated counting process fN ðtÞ; 0 Pt1 g records the cumulative number of events generated by this process, i.e., N ðtÞ ¼ i¼1 I ðTi t Þ, where IðÞ is the indicator function. Therefore, N ðtÞ is the number of events occurring over the time interval ½0; t. Let ΔN ðtÞ ¼ N ðt þ Δt Þ N ðt Þ denote the number of events in the interval ½t; t þ ΔtÞ, and HðtÞ ¼ fN ðsÞ : 0 s < t g denote the event history of the process before time t. By assuming two events cannot occur simultaneously, the intensity function of this process is defined as PrfΔN ðtÞ ¼ 1jHðtÞg : Δt!0 Δt
λftjHðtÞg ¼ lim
(2:1)
We illustrate the counting process representation of a single process from a patient in the panel (a) of Fig. 1. In panel (b) and (c), for illustration, we simulated the data based on different intensify functions (details on how to simulate event times are provided in Table 1 Selected hypoglycemic events data from the Rosenstock study Paitent ID
Hypoglycemic event times (days)
1
14
38
66
87
101
133
163
2
102
110
117
3
50
70
74
75
87
91
113
4
28
29
5
99
37
57
66
152
Censored time (days) 169 173
114
122
126
149
157
161 172 168
5
Figure 1 Counting process representation of recurrent event data. (a) Counting process representation of the first patient in Table 1. Panels (b) and (c) are the simulated data based on different intensity functions.
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
Downloaded by [Mount Allison University 0Libraries] at 08:59 24 September 2015
6
FU ET AL.
Section 3.1). The plots illustrate that it is more likely to observe an event when the underlying intensity function has a high value. Our statistical inference for hypoglycemic events is based on the intensity function. The two fundamental ways of modeling recurrent events are through gaps between successive events and through event counts. Modeling based upon gaps are often useful when the events are rare and when prediction of the next event is of interest. Some examples can be found in modeling system failures where repairs are made at each failure time. The renewal processes are the canonical models in this situation. Some references can be found in Gail et al. (1980), Prentice et al. (1981), Kalbfleisch and Prentice (2002), and Lawless (2003) among many others. Models based on event counts are often useful when individuals frequently experience the events, and when the trend of the intensity function is of interest (Aalen, 1978; Andersen and Gill, 1982; Therneau and Grambsch, 2000; Cook and Lawless, 2007). Depending upon the antidiabetic drugs, hypoglycemic events can frequently occur and comparing the intensity functions from different treatment groups are of interest. Therefore, in this paper, we model the hypoglycemic events through event counts.
2.2. Likelihood Function In this subsection, we provide the key techniques to develop the likelihood function. We assume that the events happen in a fixed study period ½0; , i.e., is a fixed constant. In real clinical trials, patients can dropout earlier due to different reasons. In Section 2.4, we extend our results to handle the situations where is a random variable, and we provide conditions under which the results developed in this section are still valid. The following theorem lays a foundation to develop likelihood functions which are used for parameter estimation and statistical inferences in this paper. Theorem 1 By assuming two events cannot occur simultaneously, the intensity function for an event process is defined as in equation (2.1). Over a specified interval ½0 ; , conditional on H ð0 Þ, the probability density of outcome n events ðt1 ; . . . ; tn Þ, at times 0 t1 < < tn , where n 1, for a process with intensity function as λft jH ðtÞg, is, n Y i¼1
ð λfti jH ðti Þg exp λft jH ðt Þgdt ; 0
(2:2)
and
" ð # b PrfN ða; bÞ ¼ 0jH ðaþ Þg ¼ exp λft jH ðt Þgdt ;
(2:3)
a
where N ða; bÞ ¼ N ðbÞ N ðaÞ denotes the number of events over a time period ða; b.
The proof of this theorem is Appendix A. The Poisson process provides an important framework to analyze the recurrent event data, and it is a foundation to develop the gamma frailty model in the later section.
HEART MODELS
7
Definition 1 (Poisson process) Poisson processes are Markov with an intensity function of the form as
λft jH ðt Þg ¼ lim
Δt!0
PrfΔN ðt Þ ¼ 1g ¼ ρðt Þ; Δt
(2:4)
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where ρðtÞ is the rate function giving the marginal (i.e., unconditional) instantaneous probability of an event at time t.
The Poisson process is independent of its history. Hence, equation (2.4) says that for two non-overlap time intervals, the number of event occurrences in these two intervals are independent. By the definition of the Poisson process, we have the following results which are useful when we develop the gamma frailty model, and when we show the connections between different models. The mean cumulative function is defined as ðt ρðsÞds:
μð t Þ ¼ E fN ð t Þ g ¼
(2:5)
0
If n events are from a Poisson process, by Theorem 1 and equation (2.4), the probability density function of these n events occur at time t1 < < tn in ½0; is (
n Y ρ tj
) expfμð Þg;
(2:6)
j¼1
where n 1. The rationale for referring to this as a Poisson process is because it is connected with a Poisson distribution. This connection can be seen as below: "ð Prðn events in½0; Þ ¼
(
ð
0t1 ...
