LETTER TO THE EDITOR
Response to ‘‘The Life Table Method of Half-Cycle Correction: Getting It Right’’ David M. J. Naimark, MD, MSc, Nader N. Kabboul, MSc, Murray D. Krahn, MD, MSc
W
e thank Dr. Jan Barendregt for his insightful comments regarding our recently published article on the alternatives to the standard half-cycle correction (HCC) used in discrete-time, discretestate transition models.1 He raises two issues concerning Figure 5 in our paper wherein the results of various HCC methods—for a simple model that we depicted in our Figure 4—are compared with a gold standard that consisted of running the model using hourly cycles. First, Dr. Barendregt is surprised by the degree to which the life-table HCC method underestimated the gold standard. As a result, we reexamined our calculations, which had been performed using TreeAge 2009. Indeed, Dr. Barendregt is correct: We had inadvertently placed the utility expression associated with the ‘‘Sick’’ state in the initial utility box associated with the ‘‘Well’’ state for the version of the model that used the life-table HCC. Since uSick < uWell, the resulting quality-adjusted life expectancy was an underestimate. Furthermore, since initial utility values are not discounted, the degree of underestimation increased in direct proportion to the cycle length. Correcting the error in our TreeAge model, and checking the result against a model created in
Received 12 December 2013 from Department of Medicine, University of Toronto, Toronto, ON, Canada (DMJN, MDK); Institute of Health Policy, Management and Evaluation, University of Toronto, Toronto, ON, Canada (DMJN, NNK); and Faculty of Pharmacy, University of Toronto, Toronto, ON, Canada (MDK). Revision accepted for publication 15 December 2013. Supplementary material for this article is available on the Medical Decision Making Web site at http://mdm.sagepub.com/supplemental. Address correspondence to David Naimark, Sunnybrook Health Sciences Centre, Faculty of Pharmacy, Room A139, 2075 Bayview Ave., Toronto, ON M4N 3M5, Canada; e-mail: [email protected]. Ó The Author(s) 2014 Reprints and permission: http://www.sagepub.com/journalsPermissions.nav DOI: 10.1177/0272989X14520719
286
Excel, we find that the quality-adjusted life expectancy from the model using a life-table HCC is closer to Dr. Barendregt’s result. The difference between the latter value and the gold standard is considerably less than we had originally depicted in our paper, as we now show in Figure S1 of the online supplementary material. However, we still find a residual difference between results of our life-table and cycle-tree HCC-based models. In equation 4, within Dr. Barendregt’s letter, he introduces the discounted, quality-adjusted, lifeyears lived, Q, computed using the life-table HCC quantities Wx and Sx, which are the increments in life-years accrued by members of a hypothetical cohort occupying the ‘‘Well’’ and ‘‘Sick’’ states, respectively, during cycle ‘‘x.’’ In the accompanying online supplemental material, we show algebraically that the value of Q is exactly the same as would be found by applying the cycle-tree HCC. We also demonstrate the reason for the residual difference between the results of the life-table and cycle-tree HCC methods shown in Figure S1. The latter is a result of our assumption that in the life-table approach, the fraction of the cohort transitioning from the ‘‘Well’’ to the ‘‘Sick’’ state during a particular cycle would be assigned the utility of the ‘‘Well’’ state. The basis of that assumption was the absence of utility weighting considerations in Dr. Barendregt’s paper introducing the life-table HCC in 2009.2 In his letter, Dr. Barendregt provides alternative calculations for Wx and Sx (Equation 5), which assume a constant hazard over a cycle interval. He notes that the alternative is rarely used because of numerical instability when the proportion of the cohort occupying a state doesn’t change very much over the course of a cycle. The ultimate instability would occur in a microsimulation model where the single hypothetical individual does not transition out of a given state during cycle ‘‘x.’’ In that case, one or other of the Wx or Sx expressions would have zero in both its numerator and denominator. Another
MEDICAL DECISION MAKING/APRIL 2014
Downloaded from mdm.sagepub.com at University of Birmingham on May 29, 2015
RESPONSE TO BARENDREGT
consideration is that using equation 5 in models built using decision-analytic software, such as TreeAge or DecisionMaker, would require relatively cumbersome coding similar to that needed for implementation of the Simpson Rule HCC as illustrated in our article. Thus, it is gratifying that our group and Dr. Barendregt have converged to the same recommendation regarding the HCC: We are essentially using the same technique but we merely refer to it by different names. It would also be reasonable to conclude that the calculation of Q using the life-table approach would be most suitable for discrete-time, discretestate transition models built in spreadsheets, while the cycle-tree HCC method lends itself to models constructed in decision-analytic software. The second issue Dr. Barendregt raises has to do with our calculation of the gold standard value of the quality-adjusted life expectancy using hour-long cycle lengths. In particular he was concerned about our method of converting annual probabilities of becoming ill or dying to hourly probabilities. Recent best-practice guidelines from ISPOR3 suggest converting probabilities from one time scale to another via rates but do not offer specific guidance regarding how to actually perform the calculations. We chose a simple method: Within the ‘‘Well’’ state we defined pS(1) and pS(t) as the per cycle and annual probabilities, respectively, of becoming ill, and pD(1) and pD(t) as the per cycle and annual probabilities, respectively, of dying, where ‘‘t’’ is the number of cycles per year. Note that in Figure 4 of our paper we refer to pD(1) as ‘‘pDie.’’ For hypothetical subjects who become ill during a cycle, pD(1) was modified by a relative risk term, R (referred to as ‘‘RRsick’’ in our paper), which was assumed to be time-insensitive. Our time-scale conversions were then lnð1 pSðtÞÞ 5 1 ½1 pSðtÞ1=t pSð1Þ 5 1 exp t : ð1Þ lnð1 pDðtÞÞ 5 1 ½1 pDðtÞ1=t pDð1Þ 5 1 exp t
The per cycle probabilities were then used within the cycle tree so that the transition probabilities became pðW ! WÞ 5 ½1 pSð1Þ½1 pDð1Þ pðW ! SÞ 5 pSð1Þ½1 ðR pDð1ÞÞ : pðW ! DÞ 5 ½pSð1Þ R pDð1Þ 1 ½ð1 pSð1ÞÞ pDð1Þ ð2Þ
Similar calculations were used for determining the transition probabilities in the ‘‘Sick’’ state. The transition probabilities above are different than those suggested by Dr. Barendregt in Equation 7 of his letter and, for a given model, these two different ways of calculating the gold standard value of the quality-adjusted life expectancy may change the estimated magnitude of over- or underestimation produced by various HCC methods. However, across models, another factor comes into play: the convexity of the true state membership function. We define the latter as the proportion of a theoretical cohort that exists in a given state as a function of time since the initiation of a continuous-time, random-transition, discrete-state model. As we show in supplementary Figure S2, trapezoidal HCC corrections still overestimate the increment in quality-adjusted life-years accrued during a cycle when the true state membership function is convex-upward and underestimate the value when the function is convex-downward. The shapes of these curves for the states in a model are dependent on its structure and transition probability values. The overall magnitude and direction of the under- or overestimation of quality-adjusted life expectancy for a discrete-time model that uses an HCC compared with the continuous-time version are a complex mix of the under- and overcorrections of the component health states. This overall behavior is difficult to predict a priori. Despite our error, and despite minor differences, Dr. Barendregt and our group agree on the fundamentals: that the standard approach to the HCC is flawed and should be abandoned, that the simplest remedy is based on a trapezoidal half-cycle correction method (which Dr. Barendregt refers to as a life-table HCC and we refer to as the cycle-tree HCC), and that this correction is not perfect and becomes less so as the cycle length increases for a given model. REFERENCES 1. Naimark DM, Kabboul NN, Krahn MD. The half-cycle correction revisited: redemption of a kludge. Med Decis Making. 2013;33(7): 961–70. 2. Barendregt JJ. The half-cycle correction: banish rather than explain it. Med Decis Making. 2009;29:500–2. 3. Siebert U, Alagoz O, Bayoumi AM, et al. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Med Decis Making. 2012;32(5):690–700.
LETTER TO THE EDITOR
287
Downloaded from mdm.sagepub.com at University of Birmingham on May 29, 2015
Response to ‘‘The Life Table Method of Half-Cycle Correction: Getting It Right’’ David M. J. Naimark, MD, MSc, Nader N. Kabboul, MSc, Murray D. Krahn, MD, MSc
W
e thank Dr. Jan Barendregt for his insightful comments regarding our recently published article on the alternatives to the standard half-cycle correction (HCC) used in discrete-time, discretestate transition models.1 He raises two issues concerning Figure 5 in our paper wherein the results of various HCC methods—for a simple model that we depicted in our Figure 4—are compared with a gold standard that consisted of running the model using hourly cycles. First, Dr. Barendregt is surprised by the degree to which the life-table HCC method underestimated the gold standard. As a result, we reexamined our calculations, which had been performed using TreeAge 2009. Indeed, Dr. Barendregt is correct: We had inadvertently placed the utility expression associated with the ‘‘Sick’’ state in the initial utility box associated with the ‘‘Well’’ state for the version of the model that used the life-table HCC. Since uSick < uWell, the resulting quality-adjusted life expectancy was an underestimate. Furthermore, since initial utility values are not discounted, the degree of underestimation increased in direct proportion to the cycle length. Correcting the error in our TreeAge model, and checking the result against a model created in
Received 12 December 2013 from Department of Medicine, University of Toronto, Toronto, ON, Canada (DMJN, MDK); Institute of Health Policy, Management and Evaluation, University of Toronto, Toronto, ON, Canada (DMJN, NNK); and Faculty of Pharmacy, University of Toronto, Toronto, ON, Canada (MDK). Revision accepted for publication 15 December 2013. Supplementary material for this article is available on the Medical Decision Making Web site at http://mdm.sagepub.com/supplemental. Address correspondence to David Naimark, Sunnybrook Health Sciences Centre, Faculty of Pharmacy, Room A139, 2075 Bayview Ave., Toronto, ON M4N 3M5, Canada; e-mail: [email protected]. Ó The Author(s) 2014 Reprints and permission: http://www.sagepub.com/journalsPermissions.nav DOI: 10.1177/0272989X14520719
286
Excel, we find that the quality-adjusted life expectancy from the model using a life-table HCC is closer to Dr. Barendregt’s result. The difference between the latter value and the gold standard is considerably less than we had originally depicted in our paper, as we now show in Figure S1 of the online supplementary material. However, we still find a residual difference between results of our life-table and cycle-tree HCC-based models. In equation 4, within Dr. Barendregt’s letter, he introduces the discounted, quality-adjusted, lifeyears lived, Q, computed using the life-table HCC quantities Wx and Sx, which are the increments in life-years accrued by members of a hypothetical cohort occupying the ‘‘Well’’ and ‘‘Sick’’ states, respectively, during cycle ‘‘x.’’ In the accompanying online supplemental material, we show algebraically that the value of Q is exactly the same as would be found by applying the cycle-tree HCC. We also demonstrate the reason for the residual difference between the results of the life-table and cycle-tree HCC methods shown in Figure S1. The latter is a result of our assumption that in the life-table approach, the fraction of the cohort transitioning from the ‘‘Well’’ to the ‘‘Sick’’ state during a particular cycle would be assigned the utility of the ‘‘Well’’ state. The basis of that assumption was the absence of utility weighting considerations in Dr. Barendregt’s paper introducing the life-table HCC in 2009.2 In his letter, Dr. Barendregt provides alternative calculations for Wx and Sx (Equation 5), which assume a constant hazard over a cycle interval. He notes that the alternative is rarely used because of numerical instability when the proportion of the cohort occupying a state doesn’t change very much over the course of a cycle. The ultimate instability would occur in a microsimulation model where the single hypothetical individual does not transition out of a given state during cycle ‘‘x.’’ In that case, one or other of the Wx or Sx expressions would have zero in both its numerator and denominator. Another
MEDICAL DECISION MAKING/APRIL 2014
Downloaded from mdm.sagepub.com at University of Birmingham on May 29, 2015
RESPONSE TO BARENDREGT
consideration is that using equation 5 in models built using decision-analytic software, such as TreeAge or DecisionMaker, would require relatively cumbersome coding similar to that needed for implementation of the Simpson Rule HCC as illustrated in our article. Thus, it is gratifying that our group and Dr. Barendregt have converged to the same recommendation regarding the HCC: We are essentially using the same technique but we merely refer to it by different names. It would also be reasonable to conclude that the calculation of Q using the life-table approach would be most suitable for discrete-time, discretestate transition models built in spreadsheets, while the cycle-tree HCC method lends itself to models constructed in decision-analytic software. The second issue Dr. Barendregt raises has to do with our calculation of the gold standard value of the quality-adjusted life expectancy using hour-long cycle lengths. In particular he was concerned about our method of converting annual probabilities of becoming ill or dying to hourly probabilities. Recent best-practice guidelines from ISPOR3 suggest converting probabilities from one time scale to another via rates but do not offer specific guidance regarding how to actually perform the calculations. We chose a simple method: Within the ‘‘Well’’ state we defined pS(1) and pS(t) as the per cycle and annual probabilities, respectively, of becoming ill, and pD(1) and pD(t) as the per cycle and annual probabilities, respectively, of dying, where ‘‘t’’ is the number of cycles per year. Note that in Figure 4 of our paper we refer to pD(1) as ‘‘pDie.’’ For hypothetical subjects who become ill during a cycle, pD(1) was modified by a relative risk term, R (referred to as ‘‘RRsick’’ in our paper), which was assumed to be time-insensitive. Our time-scale conversions were then lnð1 pSðtÞÞ 5 1 ½1 pSðtÞ1=t pSð1Þ 5 1 exp t : ð1Þ lnð1 pDðtÞÞ 5 1 ½1 pDðtÞ1=t pDð1Þ 5 1 exp t
The per cycle probabilities were then used within the cycle tree so that the transition probabilities became pðW ! WÞ 5 ½1 pSð1Þ½1 pDð1Þ pðW ! SÞ 5 pSð1Þ½1 ðR pDð1ÞÞ : pðW ! DÞ 5 ½pSð1Þ R pDð1Þ 1 ½ð1 pSð1ÞÞ pDð1Þ ð2Þ
Similar calculations were used for determining the transition probabilities in the ‘‘Sick’’ state. The transition probabilities above are different than those suggested by Dr. Barendregt in Equation 7 of his letter and, for a given model, these two different ways of calculating the gold standard value of the quality-adjusted life expectancy may change the estimated magnitude of over- or underestimation produced by various HCC methods. However, across models, another factor comes into play: the convexity of the true state membership function. We define the latter as the proportion of a theoretical cohort that exists in a given state as a function of time since the initiation of a continuous-time, random-transition, discrete-state model. As we show in supplementary Figure S2, trapezoidal HCC corrections still overestimate the increment in quality-adjusted life-years accrued during a cycle when the true state membership function is convex-upward and underestimate the value when the function is convex-downward. The shapes of these curves for the states in a model are dependent on its structure and transition probability values. The overall magnitude and direction of the under- or overestimation of quality-adjusted life expectancy for a discrete-time model that uses an HCC compared with the continuous-time version are a complex mix of the under- and overcorrections of the component health states. This overall behavior is difficult to predict a priori. Despite our error, and despite minor differences, Dr. Barendregt and our group agree on the fundamentals: that the standard approach to the HCC is flawed and should be abandoned, that the simplest remedy is based on a trapezoidal half-cycle correction method (which Dr. Barendregt refers to as a life-table HCC and we refer to as the cycle-tree HCC), and that this correction is not perfect and becomes less so as the cycle length increases for a given model. REFERENCES 1. Naimark DM, Kabboul NN, Krahn MD. The half-cycle correction revisited: redemption of a kludge. Med Decis Making. 2013;33(7): 961–70. 2. Barendregt JJ. The half-cycle correction: banish rather than explain it. Med Decis Making. 2009;29:500–2. 3. Siebert U, Alagoz O, Bayoumi AM, et al. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Med Decis Making. 2012;32(5):690–700.
LETTER TO THE EDITOR
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