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Effects of many medical procedures appear after a time lag, when a significant change occurs in subjects’ failure rate. This paper focuses on the detection and estimation of such changes which is important for the evaluation and comparison of treatments and prediction of their effects. Unlike the classical change-point model, measurements may still be identically distributed, and the change point is a parameter of their common survival function. Some of the classical change-point detection techniques can still be used but the results are different. Contrary to the classical model, the maximum likelihood estimator of a change point appears consistent, even in presence of nuisance parameters. However, a more efficient procedure can be derived from Kaplan-Meier estimation of the survival function followed by the least-squares estimation of the change point. Strong consistency of these estimation schemes is proved. The finite-sample properties are examined by a Monte Carlo study. Proposed methods are applied to a recent clinical trial of the treatment program for strong drug dependence.

Change-point models studied in clinical research usually refer to changes in the failure rate. Many articles and clinical reports describe situations when after a certain survival period, the failure rate is expected to change due to the treatment or during the after-treatment recovery. Detection of such changes, their estimation, and their comparison between different groups of patients (the treatment arm and the placebo arm is the classical example) is important understanding the treatment’s effect and for the evaluation of the treatment’s success. For example, during the zoster pain resolution trial [

Survival data with a change point are described by two models for the failure rate, namely, one model before the change point and the other model after the change point. When a subject passes the change point, the failure rate typically reduces, and the probability of the overall survival increases.

This situation is conceptually and mathematically different from the classical change-point model, see e.g. [

Despite the fundamental deviation from the classical change-point model, we will show that classical methods for the standard change-point analysis can be to a certain extent applied to the survival data. Developing these methods, we can also account for the right censoring that is typical for survival data.

The goal of this paper is to find efficient change-point detection methods for the piecewise constant failure rate models [

Developed methodologies are applied to the recent clinical trial of the treatment program for methamphetamine dependence conducted by Research Across America in Dallas TX [

The rest of the paper is organized as follows. The failure rate change-point model is introduced in Section 2. In Section 3, we give a brief review of maximum likelihood estimate and its properties. We propose an alternative least square estimator, find its convergence rate, and prove its strong consistency in Section 4. In Section 5, we extend the strong consistency of the least square estimator to a more general model, Cox proportional hazard model with a change point. We compare the two estimation procedures by means of a simulation study in Section 6. Section 7 shows application of these methods to the Prometa clinical trial. Conclusion is given in Section 8. Proofs of theorems, lemmas, and corollaries are in the Appendix section.

We assume a constant failure rate function

where

Consider a sample of

is observed instead of

The indicator variable

will show whether the

Throughout the paper,

Under model (1), the likelihood function of

which yields the log-likelihood ratio

where

When

When

where

The effect of random censorship has been studied by many authors. [

In this section, we introduce a different change-point estimation procedure which is based on Kaplan-Meier estimator of the survival function. Since the Kaplan-Meier method is nonparametric, the change-point estimation scheme proposed here can be easily extended to a wide variety of survival models with change points arising in clinical trials and other applications.

Kaplan and Meier (1958) proposed a famous estimator for the survival function

This is a step function with jumps at observations

1) It is the nonparametric maximum likelihood estimator of the true survival function

2) It has an asymptotically normal distribution for any

3) It converges almost surely to

4) If no censoring occurs or all variables are censored at the same time, then the Kaplan-Meier estimator reduces to the usual empirical distribution function.

Under the piecewise constant failure rate model (1) with a change point

Let

where

Lemma 1. At

The proof can be found in the Appendix.

To prove the strong consistency of the vector of least squares estimators

where

The uniform convergence of

Since we assume that there is indeed a change-point, it is reasonable to make the following assumption.

Assumption (A): There exist known

Assumption (A) is a classical assumption in the case when a change point is estimated in presence of nuisance parameters, and it ensures that samples of a sufficient size are used to estimate the nuisance parameters.

Under Assumption (A), the least squares estimator

Theorem 1.

The proof can be found in the Appendix.

Theorem 2.

Proof. 1) We will prove

We prove by contradiction. Suppose for any

From Theorem 1 and (12), we get

From (13), we have

for all

Also,

Hence, for sufficiently large

which contradicts (14).

2) We will prove

We also prove this by contradiction. Suppose for any

for all

Then

for all

Also,

Hence,

From (11) and Theorem 1, we can get

Hence

whereas

Hence

for sufficiently large

Combining 1) and 2) gives

Theorem 3.

The proof can be found in the Appendix.

Now let us investigate the convergence rate of

Theorem 4. For any

for sufficiently large

The proof can be found in the Appendix.

Corollary 1. The change-point estimator

for sufficiently large

Proof. According to Theorem 4, for any arbitrary sequence

Since the sum of probabilities converges, by the Borel-Cantelli lemma, with probability one,

It remains to let

Generalizing the previous results, in this section we develop change-point estimation techniques for a more general model, Cox proportional hazard model with a change point. Under this model, the hazard rate function has the form,

where

It is well known that Cox proportional hazard model is semiparametric. Indeed, it puts no assumptions on the form of baseline hazard rates

Introduce the following notations:

・

・

・

・

・

・

・

Under model (16), the survival function is expressed as

so that

The least squares estimator

of the error sum of squares

where components

Extention of the results of Section 4 on the strong consistency of the change point estimator and estimators of the nuisance parameters to Cox proportional hazard model is straightforward. Indeed, the uniform strong consistency of the Kaplan-Meier estimator holds for any type of the underlying distribution of survival times. Therefore, the error sum of squares can be split into four parts as in (8), with almost sure convergence holding for each part.

Along the same lines as in the constant hazard rate model, we obtain the following results.

Lemma 2. At

Theorem 5. With known

for some

Proof. The proof is similar to the proof of Theorem 4.5 and Corollary 4.6 of Section 4.2. □

The following results show that the strong consistency of

Theorem 6. The estimated slopes

Theorem 7. Under unknown slope parameters

Strong consistency of

In classical cases, under the usual regularity assumptions, the maximum likelihood estimator is asymptotically the uniformly minimum variance unbiased estimator. Change-point models violate the regularity conditions because of the discontinuity of the likelihood function at the change-point parameter. As a result, the maximum likelihood estimator may no longer be optimal. In this section, we compare the maximum likelihood estimator and the least squares estimator by means of the following Monte Carlo simulation study.

Generating samples from model (1) is quite simple. We generate an

Samples are generated with the change point

1) Both MLE and LSE of

2) Both MLE and LSE become more accurate when the difference between

Sample size | MLE | LSE | ||||||
---|---|---|---|---|---|---|---|---|

0.3 | 0.1 | 100 | 2.8 | 0.33 | 0.150 | 3.925 | 0.239 | 0.159 |

200 | 2.701 | 0.315 | 0.156 | 5.117 | 0.233 | 0.157 | ||

300 | 2.979 | 0.312 | 0.147 | 5.917 | 0.222 | 0.155 | ||

0.25 | 0.15 | 100 | 2.809 | 0.271 | 0.173 | 3.860 | 0.234 | 0.188 |

200 | 2.93 | 0.263 | 0.176 | 3.808 | 0.254 | 0.184 | ||

300 | 3.146 | 0.262 | 0.171 | 4.232 | 0.251 | 0.182 | ||

0.2 | 0.15 | 100 | 3.44 | 0.208 | 0.161 | 4.136 | 0.212 | 0.169 |

200 | 3.403 | 0.208 | 0.159 | 4.72 | 0.225 | 0.166 | ||

300 | 3.261 | 0.208 | 0.158 | 5.111 | 0.242 | 0.164 |

Sample size | MSE for MLE | MSE for LSE | ||||||
---|---|---|---|---|---|---|---|---|

0.3 | 0.1 | 100 | 10.005 | 0.112 | 0.025 | 15.919 | 0.059 | 0.026 |

200 | 7.98 | 0.101 | 0.025 | 29.864 | 0.059 | 0.025 | ||

300 | 9.7615 | 0.098 | 0.022 | 38.455 | 0.055 | 0.024 | ||

0.25 | 0.15 | 100 | 10.239 | 0.076 | 0.031 | 16.177 | 0.057 | 0.036 |

200 | 9.549 | 0.07 | 0.032 | 20.361 | 0.077 | 0.034 | ||

300 | 11.238 | 0.069 | 0.03 | 25.67 | 0.071 | 0.033 | ||

0.2 | 0.15 | 100 | 12.609 | 0.044 | 0.028 | 19.848 | 0.055 | 0.03 |

200 | 12.799 | 0.044 | 0.026 | 29.5 | 0.064 | 0.028 | ||

300 | 12.161 | 0.044 | 0.025 | 34.978 | 0.038 | 0.027 |

3) The LSE of

In this section, we apply both the maximum likelihood method and the least squares method to a recent clinical trial for treating methamphetamine-dependent patients conducted by Research Across America, an outpatient clinical research center in Dallas, Texas [

Fifty patients participated in an open-label study over the time frame of 84 days. In this study, all of the participants were long-term users of methamphetamine. After the screening visit on day 0, patients received five infusions during the first three weeks and conducted 14 follow-up visits.

Later, a double-blind, placebo-controlled study was conducted to better evaluate the effect of treatment. In the double-blind study, neither the participants nor the clinicians knew which patients belong to which treatment arm. The reason for blinding and placebo controls is to determine (as much as possible) whether the effects observed in the study are due to the treatment itself and not other factors. For each participant, the survival time is the time to relapse, which is the duration of time without the use of drugs.

Our goal here is to detect the after-treatment effect of Prometa, which results in a significant reduction of failure rate some time after the first three infusions. We detect such changes with both the maximum likelihood method and the least squares method. Results are listed in

First, we estimate the change point for the 50-subject open-label study.

1) Using the maximum likelihood method, day 13 maximizes the log-likelihood ratio in

2) Using the least squares method, the estimate for change point is 14.2373 and the failure rate drops from 0.1281 to 0.0142, which are very close to the results from maximum likelihood estimate. The graph of error sum of squares is in

Change points for the female and male groups are compared to see whether occurrence of a change point depends on gender.

1) Using the method of maximum likelihood, the estimated change points for males and females are 8 and 17 from

2) Using the least squares method, the change-point estimator for males is about day 14 and the failure rate reduces from 0.1494 to almost 0, while the change-point estimator for females is 13 and the failure rate reduces from 0.1495 to almost 0. We can see that there is almost no difference between male group and female group in change-point estimators from graph 3, right.

Open-label study | Male group | Female group | |||||||
---|---|---|---|---|---|---|---|---|---|

MLE | 13 | 0.1402 | 0.0105 | 8 | 0.1649 | 0.0201 | 17 | 0.1387 | 0 |

LSE | 14.2 | 0.1281 | 0.0142 | 14 | 0.1494 | 0 | 13 | 0.1495 | 0 |

Prometa group | Placebo group | |||||
---|---|---|---|---|---|---|

MLE | 13 | 0.0781 | 0.0139 | 18 | 0.1145 | 0.0532 |

LSE | 17 | 0.0720 | 0 | 14 | 0.1255 | 0.0016 |

Finally, we estimated the change points for the randomized double-blind placebo-controlled study. Change points are estimated separately for the active treatment group and for the placebo group.

1) The graph of log-likelihood ratios is in

2) With the least squares method, the change-point estimator for the treatment group is around day 17 and the failure rate reduces from 0.0720 to almost 0, while the change-point estimator for Placebo is around 14 and the failure rate reduces from 0.1255 to 0.0016. The graph for error sum of squares is in 4, right.

As a result, besides statistical significance, existence of change-points in the survival curves for both treatment groups has an important clinical significance. It shows a drop in the risk of relapse after a certain period of abstinence. Although the MLE and LSE methods slightly disagree on the exact location of change-points in the two treatment groups, both methods show that the after-change failure rate is significantly lower for the active treatment groups. Essentially, a patient has to abstain from methamphetamine for two weeks after receiving the treatment, and then the failure rate reduces significantly.

Detection of change-points in survival curves and estimation of their location finds important application in clinical research. This problem is conceptually different from the standard change-point analysis, where the distribution of data changes at unknown times. Nevertheless, similar statistical techniques can be used. The maximum likelihood approach yields a tractable change-point estimator, however, a more efficient procedure can be obtained by the Kaplan-Meier estimator of the survival function coupled with the method of least squares. Unlike the standard change-point problems, here both methods result in strongly consistent estimators.

We thank the Editor and the referee for their comments. Research of M. Baron is funded by the National Science Foundation grant DMS 1322353. This support is greatly appreciated.

Proof. Express

Define

and for any

Since

Hence

Proof. From (9), we have

On the other hand,

Hence we have

Proof. From Theorems 1, 2, and (10), we obtain

On the other hand,

Hence

Proof. First, express

and

Hence

Let

and the theorem is proved.