An Application of Markov Models in Estimating Transition Probabilities for Postmenopausal Women with Osteoporosis
Li, Zhengqing
Vertebral fracture is a common consequence of osteoporosis for postmenopausal women. Modeling the progression of the disease in vertebral fractures has been of great interest to physicians as well as patients. In this paper, we propose a simple Markov chain model to study the progression of the vertebral fractures for untreated postmenopausal women with osteoporosis. In this model, the state space consists of {0, 1, 2, . . . 13}, which represents the possible number of vertebral fractures for a patient, as commonly assessed via spinal x-ray. Based on the model, a simple estimator of the one-year transition probability matrix is proposed. An estimate for the m-year transition probability matrix is then derived following an application of the ChapmanKolmogorov equations. To estimate the confidence intervals for the m-year transition probabilities, a bootstrap procedure is proposed. The proposed bootsIrap procedure can also be employed to construct confidence intervals for differences between two treatment groups. The methods are illustrated with data from clinical trials for an anti-resorptive therapy.
Key Words
Bootstrap; Markov chains; Osteoporosis; Stochastic processes; Transition probabilities
INTRODUCTION
Vertebral fractures, that is, severe deformation of spinal vertebra, are a well-recognized consequence of postmenopausal bone loss and are the most common osteoporotic fracture (1). All vertebral fractures are associated with increased mortality and morbidity, including back pain and decreased activity. Previous data have indicated that prevalent vertebral fractures substantially increase the risk of future vertebral fractures (2). However, the speed of progression of the disease has not been well characterized and is usually addressed via epidemiological studies. Data from clinical trials of osteoporosis therapies provide unique data that facilitate alternative modeling techniques.
As a useful mathematical tool, Markov models have been well developed in the mathematical and statistical literature (3) and have been used in many areas (4). For example, for aggregated data (data that consist of the number of individuals in each state at specified observation times), statistical methods have been developed in estimating the transition probabilities (5).
In the therapeutic area of postmenopausal osteoporosis, vertebral fractures have been assessed in clinical trials of up to three years duration using annual spinal x-rays to see whether a patient sustains a fracture and the time period when the fracture occurs (6, 7, 8). It is important to note that patients may experience multiple vertebral fractures in any time period. Typically, assessment is made on 13 vertebrae in the thoracic and lumbar regions of the spine.
In this paper, we demonstrate how to model the progression of vertebral fractures using a simple Markov model for this type of data. Based on this Markov model, we estimate the one-year transition probability matrix using a simple estimate. The long-term transition probability matrix is derived using matrix multiplication with the Chapman-Kolmogorov equations. To construct confidence intervals for m-year long-term transition probabilities, we propose using a bootstrap method that preserves the correlation structure among the transition probabilities for a Markov chain. The proposed bootstrap method is also adopted to construct confidence intervals for between-group differences.
ESTIMATES OF THE TRANSITION PROBABILITIES
We consider the following data collected from a clinical study with a duration of three years. Suppose vertebral fracture assessment is made for each patient at baseline, end of year 1, end of year 2, and end of year 3. Denote the cumulative number of vertebral fractures at the four time points by n^sub 0^, n^sub 1^, n^sub 2^, and n^sub 3^. The fracture status path for a patient during the study can be represented by the configuration (n^sub 0^, n^sub 1^, n^sub 2^, n^sub 3^). Clearly, the numbers of vertebral fractures satisfies 0
To calculate the one-year transition probabilities, we need to determine the paths for each transition. A requirement for the transition probabilities from state i is [Sigma].^sub i^ P^sub ij^ = 1. To fulfill this requirement, the configurations for transitions from i to k and i to l must be mutually exclusive for k[not =]l. For all transitions starting from zero fracture, the configuration space can be represented by [Omega]0 = {(0, n^sub 1^, n^sub 2^, n^sub 3^): 0
In the configuralions above, we use * to denote any possible nonnegative integer that is greater than or equal to the integer in the previous year. Since the transition from zero to zero corresponds to a path of no change in fracture status, the only configuration for this transition is (0, 0, 0, 0). The configurations such as (0, 0, 0, 1) and (0, 0, 1, 1) correspond to paths in which the fracture status has changed. Therefore, these configurations do not belong to the transition from zero to zero. Including these configurations in the transition from zero to zero also violates the mutual exclusivity in the transition probability calculation. For the transitions starting from one fracture, the configuration space can be written as [Omega]0=[(*,*,*,*,),(*,1,*,*)’ (*, *, 1, *)]. Since the transition from one to one corresponds to paths in which a patient keeps her fracture status at one after she sustains the first fracture, the configurations associated with these paths can be written as [(*, *,1, 1)] = [(0, 0, 1, 1), (0, 1, 1, 1),(1, 1, 1, 1)]. A configuration such as (0, 1, 1, 2) does not belong to the transition from one to one since the fracture status is changed after the first fracture occurs. Specifically, we can write the configurations explicitly for the transition from one to j.
Following the same method, we can list the configurations for the transition from i fractures to j fractures. Note it is impossible Lo make the transition from i to j for j
AN EXAMPLE
Risedronate is a pyridinyl bisphosphonate with high affinity for hydroxyapatile crystals in bone. The efficacy of risedronate in reducing the risk of vertebral fractures compared to placebo in postmenopausal women with osteoporosis has been demonstrated in two randomized, double blind, placebo-controlled studies (6,7). In these two studies (VERT NA and VERT MN), the primary efficacy endpoint was the incidence of vertebral fractures over a three-year treatment period. In addition, risedronate has also been shown to reduce the risk of hip fractures in elderly women in a large randomized and placebo-controlled study (8). In all these studies, vertebral fracture assessments were made at baseline and also annually over the three-year treatment period.
In this analysis, we estimated the long-term transition probabilities in vertebral fractures for the placebo patients who had osteoporosis from the three studies (VERT NA, VERT MN, and HIP) using the methods described in previous sections. The placebo patients in all studies received calcium or vitamin D supplementation. The population that we were interested in was those osteoporotic patients as assessed by femoral neck t-scores i + 3. Based on the 1-year transition matrix we derived the 5-year and 10-year transition matrices. For illustration purposes, we were interested in the following transition probabilities:
* The probabilities that a patient will sustain 1 or more or 2 or more verLebral fractures in the next 5 or 10 years given that she has no prevalent vertebral fracture,
* The probabilities that a patient will sustain 1 or more or 2 or more vertebral fractures in the next 5 or 10 years given that she has 1 prevalent vertebral fracture,
* The probabilities that a patient will sustain 1 or more or 2 or more vertebral fractures in the next 5 or 10 years given that she has 2 prevalent vertebral fractures.
These probabilities provide an assessment of how the disease progressed when patients were not treated with active therapies. In the analysis, all placebo patients with postmenopausal osteoporosis who had known fracture status at baseline and had postbaseline vertebral fracture assessments were included. When calculating the 95% confidence intervals, 500 bootstrap samples were generated. The estimated probabilities and the corresponding 95% confidence intervals are included in Table 1. For an untreated woman who had no vertebral fractures, the probability that she will sustain at least 1 vertebral fracture was estimated to be approximately 33% in 5 years and 55% in 10 years. For those patients who have sustained fractures, the risk that they will sustain more fractures in 5 or 10 years increased dramatically. This analysis clearly indicates the importance of treating patients early and the necessity of preventing the first vertebral fracture.
To illustrate how one can compare the transition probabilities between the two treatment groups, we also calculated the transition matrix for the risedronate 5 mg group. For a risedronalc 5 mg-treated woman with no prevalent vertebral fractures, the probability that she will sustain at least one vertebral fracture in five years was estimated to be 19.9%. This probability was 13.2% (95% CI: 3.3%, 23.1%), lower than a similar placebo-treated patient and associated with a relative risk of 0.60. The probability of sustaining at least two new fractures in five years for a risedronate-treated patient was estimated to be 5.5%, 5.9% (95% CI: 1.1 %, 10.7%) lower than a similar placebo-treated patient. The relative risk associated with this difference was 0.48. These differences were statistically significant since zero was not included in the 95% CIs for the differences in transition probabilities.
CONCLUSION AND DISCUSSION
Although the application of Markov models has been investigated in many different areas (4), to our knowledge they have not been used in trials of therapies for osteoporosis. When fracture assessments are made periodically, a Markov model provides a simple approach to modeling the progression of disease. The estimates of the long-term transition probabilities in fractures are not only instrumental to researchers for understanding the diseases but they also provide guidance to physicians when treating patients and to health policymakers and regulatory practitioners in public health policymaking. We proposed using the bootstrap method to calculate the corresponding confidence intervals for the long-term transition probabilities. With the current advancements in computing, this method is computationally straightforward and can be easily implemented. Furthermore, this method preserves the correlation structure of the transition matrix.
The primary interest of this research was to provide a statistical approach to modeling the progression of ostcoporosis for an untreated population. Under the framework of bootstrap sampling, one can also make between-group comparisons for transition probabilities. However, the between-group comparisons outlined should be viewed as hypothesis generating rather than as hypothesis testing. An appropriately designed randomized controlled study is the definitive tool for generating efficacy data for a new agent. The between-group differences that we presented in the example based on Markov models with bootstrap sampling should be interpreted with this caution. The risk reductions for patients without prevalent vertebral fractures based on our modeling, however, appear to be consistent with the risk reductions observed in randomized clinical trials (6,7). This provides further support to the validity of our methods in modeling the progression of the disease.
One assumption that we have made in the model is that all past history and other factors (eg, bone mineral density) of a patient affect the risk of future vertebral fractures only through the present number of vertebral fractures that the patient sustains, that is, given the present number of vertebral fractures that a patient sustains, the risk of future vertebral fractures is independent of all the past history. While the other factors such as bone mineral density may have an impact on the risk of fractures, analyses have shown that the number of prevalent vertebral fractures is the most important factor in determining the risk of future fractures (2). Although similar statistical approaches could be developed to incorporate other important factors in the model, the calculation and interpretation of the model will be more complicated. This is an analysis that deserves further investigation.
Acknowledgments-The authors would like to express their gratitude to Michael Bramley for his help with the SAS programming needed for the example. The authors are also thankful to Drs. Michael Meredith, Eilcen King, and Mike Manhart for their valuable comments and suggestions.
REFERENCES
1. Ross PD. Clinical consequence of vertebral fractures. Am ; Med. 1997;10 (suppl):30s-43s.
2. Lindsay R, Silverman SL, Cooper C, Hanley DA, Barton I, Broy SB, Licata A, Benhamou L, Geusens P, Flowers K, Stracke H, Seeman E. Risk of new vertebral fracture in the year following a fracture. JAMA. 2001;285:320-323.
3. Cox DR, Miller HD. The Theory of Stochastic Processes. London, United Kingdom: Methuen; 1965.
4. Chiang CL. An Introduction to Stochastic Processes and their Applications. New York, NY: Krieger; 1980.
5. Kalbfleisch JD, Lawless JF, Vollmer WM. Estimation in Markov models from aggregate data. Biometrics. 1983;39:907-919.
6. Reginster JY, Minne HW, Sorensen OH, Hooper M, Roux C, Brandi ML, Lund B, Ethgen D, Pack S, Roumagnac I, Eastell R. Randomized trial of the effects of risedronate on vertebral fractures in women with established poslmenopausal osteoporosis. Osteoporosis Int. 2000; 11:83-91.
7. Harris ST, Watts NB, Genant HK, McKeever CD, Hangartner T, Keller M, Chesnut C, Brown J, Eriksen EF, Hoseyni MS, Axelrod DW, Miller PD. Effects of risedronate treatment on vertebral and nonvertebral fractures in women with postmenopausal osteoporosis. JAMA. 1999;282: 1344-1352.
8. McClung MR, Geusens P, Miller PD, et al. Effect of risedronate on the risk of hip fracture in elderly women. New Engl J Med. 2001:344:333-340.
9. Ross SM. Introduction to Probability Models. San Diego, CA: Academic Press; 1993.
10. Athreya KB, Fuh CD. Bootstrap Markov chains: countable case. J Stat Plan Inference. 1992; 33:311331.
Zhengqing Li, PhD
Section Head, Biometrics and Statistical Sciences, Procter & Gamble Pharmaceuticals, Mason, Ohio
Simon Pack, PhD
Associate Director, New Drug Development, Europe, Procter & Gamble Pharmaceuticals, United Kingdom
Correspondence Address
Zhengqing Li, Biometrics and Statistical Sciences, The Procter & Gamble Company, 8700 Mason-Montgomery Rd., Box 2199, Mason, OH
45040 (e-mail: li.z@pg.com)
Copyright Drug Information Association 2004
Provided by ProQuest Information and Learning Company. All rights Reserved
