ORIGINAL ARTICLE
Theoretical Foundations and Practical Applications of Within-Cycle Correction Methods Elamin H. Elbasha, PhD, Jagpreet Chhatwal, PhD
Background. Modeling guidelines recommend applying a half-cycle correction (HCC) to outcomes from discretetime state-transition models (DTSTMs). However, there is still no consensus on why and how to perform the correction. The objective was to provide theoretical foundations for HCC and to compare (both mathematically and numerically) the performance of different correction methods in reducing errors in outcomes from DTSTMs. Methods. We defined 7 methods from the field of numerical integration: Riemann sum of rectangles (left, midpoint, right), trapezoids, life-table, and Simpson’s 1/3rd and 3/8th rules. We applied these methods to a standard 3-state disease progression Markov chain to evaluate the costeffectiveness of a hypothetical intervention. We solved the discrete- and continuous-time (our gold standard) versions of the model analytically and derived expressions for various outcomes including discounted qualityadjusted life-years, discounted costs, and incremental cost-effectiveness ratios. Results. The standard HCC
method gave the same results as the trapezoidal rule and life-table method. We found situations where applying the standard HCC can do more harm than good. Compared with the gold standard, all correction methods resulted in approximation errors. Contrary to conventional wisdom, the errors need not cancel each other out or become insignificant when incremental outcomes are calculated. We found that a wrong decision can be made with a less accurate method. The performance of each correction method vastly improved when a shorter cycle length was selected; Simpson’s 1/3rd rule was the fastest method to converge to the gold standard. Conclusion. Cumulative outcomes in DTSTMs are prone to errors that can be reduced with more accurate methods like Simpson’s rules. We clarified several misconceptions and provided recommendations and algorithms for practical implementation of these methods. Key words: state-transition models; discrete time; continuous time; half-cycle correction; numerical integration. (Med Decis Making XXXX;XX:XX–XX)
D
such as life expectancy can easily be estimated from DTSTMs based on the average number of cycles spent in each state. However, the use of discrete steps can introduce error when calculating cumulative outcomes.3 The error arises mainly because DTSTM assumes that state transitions occur only at fixed times, whereas in most biological and healthcare systems, as time runs continuously, state transitions can occur at any time. The ISPOR-SMDM Modeling Good Research Practices Task Force recommends applying a halfcycle correction (HCC) to costs and effectiveness when using DTSTMs.1 The most common approach to correcting the error in total outcomes that would arise from assuming transitions occur at the end of a cycle is to subtract and add half of the outcomes in each state from the first cycle and last cycle, respectively. This method is known as HCC either because it adjusts total outcomes by half-cycle outcomes or because it assumes
iscrete-time, state-transition models (DTSTMs) are arguably the most commonly used approach in health economic evaluation.1,2 In these models, time advances in discrete time-steps (known as cycles), which provides a convenient way of simulating transitions of persons from one state to another at fixed intervals. Cumulative outcomes Received 2 July 2014 from Merck & Co., Inc., Kenilworth, NJ (EHE); and Department of Health Services Research, The University of Texas MD Anderson Cancer Center, Houston TX (JC). Revision accepted for publication 1 April 2015. Address correspondence to Elamin H. Elbasha, Merck Research Laboratories, Merck & Co., Inc., UG1C-60, PO Box 1000, North Wales, PA 19454-1099; telephone: (267) 305-7991; fax: (267) 305-6455; e-mail:
[email protected]. Ó The Author(s) 2015 Reprints and permission: http://www.sagepub.com/journalsPermissions.nav DOI: 10.1177/0272989X15585121
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ELBASHA, CHHATWAL
that transitions occur, on average, halfway through each cycle.3,4 We refer to this approach as the standard HCC method. Others have suggested different correction approaches such as the life-table method and Simpson’s rule.3,5,6 These methods typically apply correction at each cycle, not only at the first or final cycle. For technical accuracy, we refer to all these other methods as within-cycle correction (WCC) methods because some of the methods do not use half-cycle outcomes or assume that transitions can occur within cycle at times other than halfway. Notwithstanding recent publications, there is still confusion in the literature on how to minimize the error created by formulating models in discrete time.7 There is also no consensus on the best workaround method (e.g., HCC, life-table method, or trapezoidal rule) to use.8–10 Although earlier studies compared a few methods for HCC, these studies did not compare the performance of correction methods with the true gold standard—outcomes from a continuous-time state-transition model (CTSTM). Because these studies used a shorter cycle (e.g., hourly cycle) as their gold standard, they may have inadvertently introduced errors by failing to appropriately account for competing risk when converting 1-year transition probabilities into probabilities with a shorter cycle length.11,12 None of the previously published studies compared mathematically the performance of error-correction methods. In addition, a theoretical foundation of HCC and other WCC methods has not been well documented in the medical literature. The objective of our study was to provide a theoretical foundation of WCC using well-established methods from the field of numerical integration,13 compare health outcomes predicted using the discrete- and continuous-time versions of a simple standard disease prevention model, provide mathematical expressions of error using different methods, assess the performance of these correction methods, and provide algorithms for practical implementation of these methods. RATIONALE FOR AND SCOPE OF WITHIN-CYCLE CORRECTION In health economic evaluation, models are routinely used to predict intermediate outcomes over a fixed time horizon (e.g., 20-year risk of cancer) to help with debugging and to enhance validity and transparency of the model results.1 It is also recommended that final outcomes such as discounted qualityadjusted life-years (QALYs) be calculated over a long time horizon.1
2
WCC stems from the need to evaluate the area under a continuous outcome curve when the analysis is based on discrete time. Because closed-form solutions are usually not feasible, the goal is to find an appropriate way of approximating a definite integral of a given function. This is not a new finding. For example, it has long been recognized in the DTSTM literature that calculating life expectancy is equivalent to finding a numerical approximation of the area under a continuous survival curve (see, e.g., Sonnenberg and Beck3). However, it is not well known that the integrand need not be restricted to survival or quality-adjusted survival. WCC methods should be applied to any outcome for which one needs to compute a cumulative quantity over a period of time. This includes intermediate outcomes such as absolute risk of disease over T years as well as final outcomes such as costs and quality-adjusted survival. Outcomes Using Continuous Time (Gold Standard) In a CTSTM, a discounted (at continuous rate r per year) cumulative outcome F over a time horizon of T years is calculated as follows: F5
ðT
f ðt Þdt 5
0
ðT
exprt Oðt Þdt;
0
where exp stands for the base of natural logarithm, and O(t) and f(t) denote undiscounted and discounted outcome (e.g., costs) at time t, respectively. Setting r to zero yields undiscounted outcomes. Outcomes Assuming Discrete Time We start with the premise that DTSTMs can correctly estimate outcomes (e.g., discounted qualityadjusted survival) at discrete time points (i.e., ends of cycles). In other words, with DTSTMs the height of an outcome curve is known only at the ends of cycles and is unknown within cycles. Our point of departure is that the evaluation of the area under a continuous outcome curve when time is discretized requires finding an appropriate way of approximating a definite integral. A plethora of methods in the field of numerical integration can be used for this purpose.13 METHODS OF WITHIN-CYCLE CORRECTION Riemann Sum In mathematics, the Riemann sum formula relates a definite integral (often the area under the curve) of a continuous function f over an interval [0,T] to the
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limit of an infinite series. The formula is typically presented under the assumption that the interval can be divided into subintervals of equal width. To simplify the presentation and the calculations, we assume that each interval consists of 1 year that can be further divided into n cycles of equal length. The total number of subintervals is nT (number of cycles in a year times the time horizon in years). Note that 1=n measures the length of a cycle in years. Thus, tk = 0,1/n, 2/n, . . ., 1,1 1 1/n,. . ., T when k = 0,1,2, 3,. . ., nT; respectively. For example, with monthly cycles, n 5 12 and tk tk1 5 1=n 5 1=12: The definite integral is related to a Riemann sum according to ðT 0
nT nT 1X 1X f ðtk Þ 5 lim ð1 1 rÞtk =n Oðtk Þ; n!‘ n n!‘ n k51 k51
f ðtÞdt 5 lim
where r is the discrete-time discount rate per year. It should be noted that the continuous-time discounting factor is given by exprt whereas the discretetime discounting factor is given by the step function ð1 1 rÞt : The continuous-time discount rate r can be calculated from an annual discount rate r according to r 5 lnð1 1 rÞ: In practice, the sum is calculated by constructing a geometric shape (e.g., a rectangle or trapezoid) at each cycle. The areas of each of these shapes are calculated and added together to form the area under the curve. As the geometric shapes get smaller and smaller (i.e., the length of the cycles gets smaller and n becomes large), the sum approaches the Riemann integral. If a rectangle is formed at left endpoints of the cycles (Figure 1), the method is called a left Riemann sum (L), with the area under the curve given by the following equation: " # nT nT X 1X 1 L5 f ðtk1 Þ 5 f ðt k Þ : f ðt0 Þ f ðtnT Þ 1 n k 51 n k 51
A right Riemann sum (R) is obtained if f is approximated by the value at the right endpoint of the cycles: R5
nT 1X f ðtk Þ: n k51
Using a left or a right Riemann sum is the same as assuming transitions occur at the end or beginning of the cycle, respectively.
A midpoint Riemann sum computes the area of a collection of rectangles (Mk) whose heights are determined by the values of the function at a midpoint (tk21 1 tk)/2 (Figure 1): Mk 5
1 tk 1 tk1 : f 2 n
Note that the frequently made statement that transitions occur halfway through the cycle is equivalent to using a midpoint Riemann sum. However, this method typically cannot be implemented in practice because the function f cannot be evaluated at a midpoint (tk21 1 tk)/2.3 Trapezoidal Rule If on each subinterval the function f is approximated with a straight line, the region under the graph of the function will consist of trapezoids. The sum is obtained by calculating and adding up the areas of these of trapezoids (Figure 1). Thus, with right endpoint tk and left endpoint tk21, the area of the of the trapezoid over the interval [tk21,tk] is Zk 5
1 ½f ðtk1 Þ 1 f ðtk Þ: 2n
The general formula for the trapezoidal rule is nT X f ðtk1 Þ 1 f ðtk Þ
Z5
k51
2n
5
nT f ðt0 Þ f ðtnT Þ 1X 1 f ðtk Þ: 2n n k 51
The trapezoidal approximation is the average of the left Riemann sum and the right Riemann sum: Z5
L1R f ðt0 Þ f ðtnT Þ f ðt0 Þ f ðtnT Þ 5L 5R1 : 2 2n 2n
Therefore, to arrive at a trapezoidal approximation when transitions occur at the end (beginning) of the cycle, one would need to subtract (add) half of the difference between the initial outcome and final outcome. It is logical to deduce that the basis for the standard HCC can be found in the trapezoidal rule for approximating a definite integral. Contrary to the frequently made statement that HCC is based on the assumption that transitions occur halfway through the cycle, the trapezoidal rule stipulates that the function between the beginning and end of the cycle is linear (i.e., assuming a uniformly, instead of exponentially, distributed waiting time within a cycle).
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Figure 1 Geometric illustration of the Riemann sums, trapezoidal rule, and composite Simpson’s rules.
Life-Table Method
rule, Simpson’s composite 1/3rd rule, and Simpson’s 3/8th rule.
In demography, life expectancy, Ek, over the age interval [k21, k] is calculated from a complete life table using the following equation14,15: 1 Ek 5 ½lk1 1 lk ; 2
where lk is the number of persons alive at age k in years. Barendregt5 suggested the life-table method as a better alternative to the standard HCC method. By comparing Ek with Zk, it is obvious that the lifetable method is a special application of the trapezoidal rule (i.e., f is given by the survival function).10 Thus, if the function f and the units of time are defined the same way, the standard HCC method, the trapezoidal rule, and the life-table method will give the same results.
The simple Simpson’s rule assumes that the function f can be evaluated at a midpoint (tk21 1 tk)/ 2 and a quadratic curve passes through the 3 points {tk21, (tk21 1 tk)/2, tk}. The area of the region formed by the quadratic curve over the interval [tk21,tk] is Ak 5
1 tk1 1 tk 1 f ðtk Þ : f ðtk1 Þ 1 4f 2 6n
Note that that the simple Simpson’s rule is the weighted average of the trapezoidal rule and the midpoint Riemann sum (Mk): Ak 5
Simpson’s Rules Simpson’s rules approximate the definite integral of function f using a higher order polynomial rather than straight line segments as in the trapezoidal rule. We describe 3 types of rules: Simpson’s simple
4
Simple Simpson’s Rule
2 1 Mk 1 Z k : 3 3
Because both the midpoint Riemann sum and the simple Simpson’s rule require breaking up the cycle length into 2 subintervals, we will not analyze them any further. Instead, we will analyze separately the
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impact of shortening the cycle length on the results of all other methods.
Thus, the area under the curve over the interval [tk,tk 1 3] is given by
Composite Simpson’s 1/3rd Rule
CRk 5
The area under the quadratic curve passing through the 3 points {tk21, tk, tk 1 1} over the interval [tk21,tk 1 1] is
Because it uses 3 segments, Simpson’s 3/8th rule requires the time horizon to be a multiple of 3. The general Simpson’s 3/8th rule (skipping 2 cycles) is given by
ðtk 1 1 tk1 Þ ½f ðtk1 Þ 1 4f ðtk Þ 1 f ðtk 1 1 Þ 6 1 ½f ðtk1 Þ 1 4f ðtk Þ 1 f ðtk 1 1 Þ: 5 3n
Ck 5
CR 5
Tn1 X
Ck 5
k 5 1; 3; 5;
Note that the coefficient of the value function in the Simpson’s 1/3rd rule changes from 4 to 2 as the cycle number changes from odd to even. Likewise, the Simpson’s 3/8th rule has the coefficient of the value function changing from 3 to 2 as the cycle number becomes a multiple of 3. These patterns are very useful for the implementation of composite Simpson’s rules. Implementation of WCC Methods
3
7 Tn1 Tn2 P P 1 6 7 6 f ðtk Þ þ 2 f ðtk Þ þ f ðtnT Þ7: 6f ðt0 Þ þ 4 5 3n 4 k ¼ 1; 3; 5; k ¼ 2; 4; 6; |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
Appendix A shows how to implement the trapezoidal rule, Simpson’s 1/3rd rule, and Simpson’s 3/ 8th rule within commonly used software such as Microsoft Excel (Microsoft Corp., Redmond, WA) and TreeAge (TreeAge Software, Williamstown, MA).
even
odd
3 ½f ðt0 Þ 1 3f ðt1 Þ 1 3f ðt2 Þ 1 2f ðt3 Þ 1 3f ðt4 Þ 8n
1 3f ðtnT1 Þ 1 f ðtnT Þ:
1 ½f ðt0 Þ1 4f ðt1 Þ1 2f ðt2 Þ 1 4f ðt3 Þ1 . . . 14f ðtnT1 Þ1 f ðtnT Þ 5 3n ¼
CRk 5
1 3f ðt5 Þ 1 2f ðt6 Þ 1 . . . 1 2f ðtnT3 Þ 1 3f ðtnT2 Þ
Tn1 X 1 ½f ðtk1 Þ 1 4f ðtk Þ 1 f ðtk 1 1 Þ 3n k 5 1; 3; 5;
2
Tn3 X k 5 0; 3; 6;
Because the formula contains ‘‘1/3,’’ this method is sometimes referred to as Simpson’s 1/3rd rule. As the formation of each curve requires 2 subintervals (tk 1 1 – tk21 = 2/n), the summation over k is performed every other cycle (skipping a cycle) ending at time Tn21: C5
3 ½f ðtk Þ 1 3f ðtk 1 1 Þ 1 3f ðtk 1 2 Þ 1 f ðtk 1 3 Þ: 8n
The composite Simpson’s 1/3rd rule requires that the total number of subintervals or time horizon to be even.
APPLICATION OF WCC METHODS TO A 3-STATE MODEL
Composite Simpson’s 3/8th Rule We illustrate the application of the above numerical integration methods and their performance using a standard 3-state DTSTM—Well, Disease, and Dead (Figure 2). The objective of the model was to estimate
Simpson’s 3/8th rule uses 4 points {tk, tk 1 1, tk 1 2, tk 1 3} and derives the rule by adding the areas of the cubic curve passing through these four points.
Well (W) Initial cost: $I Recurring costs: $0 QoL: u
b(1 −e)
Disease (S) Initial cost: $0 Recurring costs: $c QoL: q
d
Dead (D) Initial cost: $0 Recurring costs: $0 QoL: 0
m Figure 2 Transition diagram of the discrete-time, state-transition model of disease progression—Well, Disease, and Dead. b = transition probability to the Disease state; d = transition probability to the Dead state from the Disease state; m = probability of all-cause death for Well persons; e = probability of intervention reducing disease progression. The transfer diagram for the continuous-time state-transition model is similar (not shown here). However, the transition rates are denoted with Greek letters.
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Table 1 Description
Parameters of the Discrete-Time, State-Transition Model Symbol
Probability or Input per Cycle (n Cycles per Year)
n o 1 1 bð1 eÞ ð1 dÞn ½1 ð1 eÞb mn
Disease progression
ð1 eÞb
p^12 5
All-cause mortality
m
p^13 5
Disease-specific death Discount rate Disease cost
d r c
p^23 5 1 ð1 dÞn 1 r^ 5 ð1 1 rÞn 1 c n
h bð1 eÞ 1 imd n o 1 1 ð1 eÞb 1 ð1 dÞn ðd mÞ 1 ½1 ð1 eÞb mn ð1 eÞb 1 m d 1
Note: b = transition probability to the Disease state; d = transition probability to the Dead state from the Disease state; e = probability of intervention reducing disease progression; m = probability of all-cause death for well persons; r = discount rate.
the clinical benefits and cost-effectiveness of a hypothetical intervention to prevent disease progression.1,16,17 In the absence of interventions, Well persons progress to disease with probability b per year. Disease results in death with probability d, costs $c per year, and degrades quality of life of a Well person from u to q. A Well person dies with probability m per year. The intervention has efficacy e and a onetime cost of $I. To simplify, we assumed that transition probabilities are constant over time. Several chronic diseases can be described by this model. For example, progression of chronic hepatitis C virus infection to liver complications can be halted by antiviral therapy.18 Also, 3-state transition models are frequently used in the economic evaluation of oncology products (e.g., Ja¨kel and others19). The 1-step transition probabilities matrix pij is given by 2 P 54
1 ð1 eÞb m 0 0
ð1 eÞb 1d 0
3
m d 5: 1
With constant transition probabilities, the DTSTM model can be solved using matrix algebra showing the distribution of persons in each state and the overall expected value of each outcome.17 The probability of being in a given health state after t years is obtained by raising the matrix P to the power t. Assuming that a fraction w0 of the cohort of size 1 starts in the Well state, the distribution of persons in a given health state t years from now is obtained by evaluating ðw0 ; 1 w0 ; 0Þ:Pt ; where ‘‘’’ denotes the dot product of a vector and a matrix. With time (t=k/n) measured in cycles, the distribution of persons in a given health k state at cycle k is ðw0 ; 1 w0 ; 0Þ:Pn (Appendix B).
6
A 3-State Continuous-Time Model We next define the corresponding 3-state CTSTM, which serves the purpose of a gold standard for comparing different WCC methods. A CTSTM is formulated using instantaneous transition rates in contrast to per-cycle transition probabilities as in a DTSTM. We denote the rate of moving from the Well state to the Disease state by (1 2 E)b and from Disease to Dead by d. The hazard rate of death among well persons is given by m. In the presence of intervention, the transition intensity matrix qij is given by20,21 2
½ð1 eÞb 1 m Q54 0 0
3 ð1 eÞb m d d 5: 0 0
Similarly, the CTSTM can also be solved using matrix algebra by evaluating the matrix exponential exp (Appendix B): P ðtÞ 5 expQt :
To link the two models, we relate the transition probabilities per cycle (denoted by p^ij ) and the instantaneous transition rates per year qij to the transition probabilities pij per year (Tables 1 and 2 and Appendix B). Because we wanted to express quantities in terms of transition probabilities per year (a standard reference discrete-time model), we replaced instantaneous rates by their equivalent formula from Table 2. Because the standard HCC method, the trapezoidal rule, and the life-table method yield the same results, we present only cumulative outcomes using the left and right Riemann sums, trapezoidal rule, Simpson’s 1/3rd rule, and Simpson’s 3/8th rule.
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Table 2 Parameters of the Continuous-Time, State-Transition Model Description
Symbol
Instantaneous Rate per Year
Disease progression
(12e)b
ð1 eÞb 5
All-cause mortality
m
Disease-specific death Discount rate
d r
m5 d 5 lnð1 dÞ r 5 lnð1 1 rÞ
bð1eÞflnð1dÞln½1ð1eÞbmg ð1eÞb 1 md ðdmÞ ln½1ð1eÞbmbð1eÞ lnð1dÞ ð1eÞb 1 md
Note: b = transition probability to the Disease state; d = transition probability to the Dead state from the Disease state; e = probability of intervention reducing disease progression; m = probability of all-cause death for well persons; r = discount rate; b = disease progression rate; E = efficacy; d = death rate of diseased persons; m = all-cause mortality rate; r = discount rate.
Table 3
Formulas for State Membership and Cumulative Outcomes over T Years as Functions of Parameters for Continuous-Time, State-Transition Model (Gold Standard)
Outcomea
Persons well Persons with disease Persons dead Risk of disease over T years Life expectancy Discounted QALYs Discounted disease costs Net monetary benefits
Formula
W ðtÞ 5 w0 ½1 ð1 eÞb mt ½bð1eÞðdmÞð1w0 Þð1dÞt w0 bð1eÞ½1ð1eÞbmt bð1eÞ 1 md ½bð1eÞðdmÞð1w0 Þ½1ð1dÞt w0 ðdmÞf1½1ð1eÞbmt g DðtÞ 5 bð1eÞ 1 md bð1eÞf1½1ð1eÞbmT gw0 fln½1ð1eÞbmlnð1dÞg RISKG 5 ½bð1eÞ 1 md ln½1ð1eÞbm ½bð1eÞðdmÞð1w0 Þ½1ð1dÞT w0 ðdmÞf1½1ð1eÞbmT g LIFEG 5 ½bð1eÞ 1 md ln½1ð1eÞbm ½bð1eÞ 1 md lnð1dÞ
SðtÞ 5
QALYG ðeÞ 5
COSTG ðeÞ 5
1ð1eÞbm T
w0 ½bð1eÞðuqÞðdmÞu 1½ 1 1 r ½bð1eÞ 1 mdfln½1ð1eÞbmlnð1 1 rÞg
1ð1eÞbm T
cw0 bð1eÞ 1ð 1 1 r Þ ½ð1eÞ 1 mdfln½1ð1eÞbmlnð1 1 rÞg
T
q½bð1eÞðdmÞð1w0 Þ 1ð11d 1 rÞ ½bð1eÞ 1 mdflnð1dÞlnð1 1 rÞg
T
c½bð1eÞðdmÞð1w0 Þ 1ð11d 1 rÞ ½ð1eÞ 1 md½lnð1dÞlnð1 1 rÞ
NMB 5 l½QALYG ðeÞ QALYG ð0Þ ½COSTG ðeÞ 1 I COSTG ð0Þ
Note: b = transition probability to the Disease state; c = disease cost per period; COST = discounted cost of disease; d = transition probability to the Dead state from the Disease state; D(t) = persons in the Dead state; e = probability of intervention reducing disease progression; G = gold standard; I = intervention cost; LIFE = life expectancy; m = probability of all-cause death for well persons; NMB = net monetary benefits; q = quality of life loss; QALY = quality-adjusted life-years; r = discount rate; RISK = cumulative risk of disease; S(t) = persons in the Disease state; T = time horizon; w0 = proportion of the cohort initially in the Well state; W(t) = persons in the Well state; l = willingness-to-pay for a QALY. a. Outcomes in the absence of the intervention are obtained by setting e = I = 0.
ANALYTICAL RESULTS We provide the solutions of DTSTM and CTSTM and cumulative outcomes in terms of mathematical expressions relating model outcomes (e.g., qualityadjusted life expectancy) to model inputs (e.g., probabilities) and time horizon (Tables 3 and 4; Appendix B). Such analytic solutions provide a relationship between inputs and outputs under different conditions which otherwise cannot be easily obtained from the numerical results. First, we show that there is no need to apply WCC to estimate state membership (i.e., persons in Well, Disease, and Dead states in a given cycle) in this 3state DTSTM. Substitution of the inputs in Table 2 into the equations for W(t), S(t), and D(t) shows that the number of persons in each health state predicted by CTSTM and DTSTM are identical (Tables 3 and 4;
Appendix B). This implies that state membership results of CTSM and DTSM are not dependent on discretization of time, and this is true for all input parameters. In addition, there is no need to apply WCC to estimate the cumulative incidence of death because death is specified as an absorbing health state. Next, we show that unless cumulative outcomes are specified as absorbing states, all correction methods lead to errors in cumulative outcomes from DTSTM. This is because the standard HCC method can be derived from a comparison between a method of rectangles and trapezoidal rule rather than a comparison with the gold standard of an exact formula for the definite integral, and WCC methods provide only approximation of such integral. The formulas in Tables 3 and 4 can be used to compare outcomes analytically and gauge the magnitude of the error of each WCC method under different conditions,
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1
n
Þ
1ð1eÞbm T 11r
NMBx 5 lQALYx COSTx I
cw0 bð1eÞ 1ð
o1 1
c½bð1eÞðdmÞð1w0 Þ 1ð11d 1 rÞ
q½bð1eÞðdmÞð1w0 Þ 1ð11d 1 rÞ
T h i 1 n 1ð1eÞbm n n½bð1eÞ 1 md 1½ 1 1 r n½bð1eÞ 1 md ð11d 1 rÞ 1 1ð1eÞbm T
T cw0 bð1eÞ 1ð 1 1 r Þ c½bð1eÞðdmÞð1w0 Þ 1ð11d 1 rÞ 1 COSTZ 5 COSTR 2n½bð1eÞ 1 md 2n½bð1eÞ 1 md h i h i
1 2 1 2 1bð1eÞm n 1bð1eÞm n 1ð1eÞbm T 1d n 1d n 1d T c½bð1eÞðdmÞð1w0 Þ 1 1 4ð1 1 rÞ 1 ð1 1 rÞ 1ð1 1 rÞ cw0 bð1eÞ 1 1 4½ 1 1 r 1 ½ 1 1 r 1½ 1 1 r h i n o COSTC 5 2 2 1bð1eÞm n n 3n½bð1eÞ 1 md 1ð11d 3n½bð1eÞ 1 md 1½ 1 1 r 1 rÞ h i3 h i
1 3 1 T 1bð1eÞm n 1ð1eÞbm T n 3c½bð1eÞðdmÞð1w0 Þ 1 1 ð11d 1ð11d 3cw0 bð1eÞ 1 1 ½ 1 1 r 1½ 1 1 r 1 rÞ 1 rÞ h i n o COSTCR 5 3 3 1bð1eÞm n n 8n½bð1eÞ 1 md 1ð11d 8n½bð1eÞ 1 md 1½ 1 1 r 1 rÞ COSTR 5
T h i QALYR 5 1 1 n n½bð1eÞ 1 md ½ n½bð1eÞ 1 md ð11d 1 rÞ 1 1ð1eÞbm T
T w0 ½bð1eÞðuqÞðdmÞu 1½ 1 1 r q½bð1eÞðdmÞð1w0 Þ 1ð11d 1 rÞ QALYZ 5 QALYR ðeÞ 1 1 2n½bð1eÞ 1 md 2n½bð1eÞ 1 md h i h i
1 2 1 2 1bð1eÞm n 1bð1eÞm n 1ð1eÞbm T n 1d n 1d T q½bð1eÞðdmÞð1w0 Þ 1 1 4ð11d 1 1 w ½ b ð 1e Þ ð uq ÞðdmÞu 1 1 4½ 1 1 r 1 ½ 1 1 r 1½ 1 1 r Þ ð Þ ð Þ 0 1r 11r 11r h i n o 1 QALYC 5 2 2 1bð1eÞm n n 3n½bð1eÞ 1 md 1ð11d 3n½bð1eÞ 1 md 1½ 1 1 r 1 rÞ h i3 h i
1 3 1 T 1bð1eÞm n 1ð1eÞbm T n 3q½bð1eÞðdmÞð1w0 Þ 1 1 ð11d 1ð11d 3w0 ½bð1eÞðuqÞðdmÞu 1 1 ½ 1 1 r 1½ 1 1 r 1 rÞ 1 rÞ h i n o QALYCR 5 1 3 3 1bð1eÞm n n 8n½bð1eÞ 1 md 1ð11d 8n½bð1eÞ 1 md 1½ 1 1 r 1 rÞ o
1ð1eÞbm T 11r 1 1ð1eÞbm n 1 11r
w0 ½bð1eÞðuqÞðdmÞu 1½
n
8½bð1eÞ 1 md 1½1bð1eÞmn
bð1eÞf1½1ð1eÞbmT gw0 ð1dÞn ½1ð1eÞbmn
1 1 ½bð1eÞ 1 md 1½1ð1eÞbmn 1 1 T bð1eÞf1½1ð1eÞbm gw0 ð1dÞn ½1ð1eÞbmn RISKZ 5 RISKR 1 2½bð1eÞ 1 md n o 1 1 4½1bð1eÞmn1 1 ½1bð1eÞmn2 1½1ð1eÞbmT 1 1 f g RISKC 5 bð1 eÞw0 ð1 dÞn ½1 ð1 eÞb mn 3 2 3½bð1eÞ 1 md 1½1bð1eÞmn
1 1 1 3 3bð1eÞw0 ð1dÞn ½1ð1eÞbmn 1 1 ½1bð1eÞmn f1½1ð1eÞbmT g RISKCR 5 3 RISKR 5
Dk 5
k
Note: b = transition probability to the Disease state; c = disease cost per period; C = Simpson’s 1/3rd rule; COST = discounted cost of disease; CR = Simpson’s 3/8th rule; d = transition probability to the Dead state from the Disease state; Dk = persons in the Dead state; e = probability of intervention reducing disease progression; I = intervention cost; m = probability of all-cause death for well persons; NMB = net monetary benefits; q = quality of life loss; QALY = quality-adjusted life-years; r = discount rate; R = right Riemann; RISK = cumulative risk of disease; Sk = persons in the Disease state; T = time horizon; w0 = proportion of the cohort initially in the Well state; Wk = persons in the Well state; Z = trapezoidal rule; l = willingness-to-pay for a QALY. a. Outcomes in the absence of the intervention are obtained by setting e = I = 0.
Net monetary benefits Method x (R, Z, C, CR)
Simpson’s 3/8th rule
Simpson’s 1/3rd rule
Trapezoidal rule
Discounted disease costs Right Riemann sum
Simpson’s 3/8th rule
Simpson’s 1/3rd rule
Trapezoidal rule
Discounted QALYs (years) Right Riemann sum
Simpson’s 3/8th rule
Simpson’s 1/3rd rule
Trapezoidal rule
Risk of disease over T years Right Riemann sum
Cases of death
k
½bð1eÞðdmÞð1w0 Þð1dÞn w0 bð1eÞ½1ð1eÞbmn bð1eÞ 1 md ½bð1eÞðdmÞð1w0 Þ½1ð1dÞk w0 ðdmÞf1½1ð1eÞbmk g bð1eÞ 1 md
Persons with disease Sk 5
k
Wk 5 w0 ½1 ð1 eÞb mn
Formula
Formulas for State Membership per Cycle and Cumulative Outcomes with Different Methods for Discrete-Time, State-Transition Model
Persons well
Outcomea
Table 4
FOUNDATIONS OF WITHIN-CYCLE CORRECTION METHODS
Table 5
Error Analysis of Risk of Disease over T Years with Various Correction Methods Relative Errora
Within-Cycle Correction Method
Left Riemann
RISKL 51 ERRORL 5 1 RISK G
Right Riemann
ERRORR 5 1
Trapezoidal rule
RISKR RISKG
1
1
lnð1dÞ
1ln½1ð1eÞbm 1
o
1
ð1dÞn ½1ð1eÞbmn ½1ð1eÞbmn
51
RISKC 51 ERRORC 5 1 RISK G
Simpson’s 3/8th rule
1 1½1ð1eÞbmn
RISKZ 51 ERRORZ 5 1 RISK G
Simpson’s 1/3rd rule
1
ð1dÞn ½1ð1e n Þbmn
1
1½1ð1eÞbmn
n
lnð1dÞ
1ln½1ð1eÞbm
o
1 1 1 ½1ð1eÞbmn o n
1 1 ð1dÞn ½1ð1eÞbmn
1
2 1½1ð1eÞbmn
lnð1dÞ
1
1
ð1dÞn ½1ð1eÞbmn
CR ERRORCR 5 1 RISK RISKG 5 1
1ln½1ð1eÞbm
1
n 2
3 1½1bð1eÞmn
3
2
1 1 4½1bð1eÞmn 1 ½1bð1eÞmn lnð1dÞ
o
1ln½1ð1eÞbm
1 3 1 1 ½1bð1eÞmn o n
1 1 ð1dÞn ½1ð1eÞbmn
3
8 1½1bð1eÞmn
lnð1dÞ
1ln½1ð1eÞbm
Note: b = transition probability to the Disease state; d = transition probability to the Dead state from the Disease state; e = probability of intervention reducing disease progression; ERROR = relative error; m = probability of all-cause death for well persons; RISK = cumulative risk of disease.
including types of disease (probability of progression, costs, mortality, and effects on quality of life) and properties of intervention (efficacy and cost). We also show analytically that the performance of each of these methods depends on the outcome function that needs to be estimated. For example, Table 5 shows the results of the error analysis of risk of disease over T years with various WCC methods. The error is the relative difference between the risk of disease using DTSTM with n cycles per year and the exact value of the definite integral. We found that all methods produced a positive error, indicating underestimation of T years risk. The ranking (from best to worst) of methods in terms of accuracy is left Riemann sum, trapezoidal rule, Simpson’s 3/8th rule, Simpson’s 1/3rd rule, and right Riemann sum (Appendix C). The size of the error decreases monotonically with a shorter length of a cycle, reaching zero when the length of a cycle is infinitesimally small (i.e., approaching CTSTM). NUMERICAL RESULTS We assigned numerical values to the mathematical expressions in Tables 325 to compare outcomes across WCC methods. Does Within-Cycle Correction Matter? We found situations where a wrong decision can be made if the more accurate method is not applied (Table 6). The gold standard yielded an incremental
cost-effectiveness ratio (ICER) of $47,790/QALY, indicating a cost-effective intervention at a threshold of $50,000/QALY. Using an annual cycle, all withincycle correction methods indicated lack of cost-effectiveness, clearly a wrong recommendation to follow. With a semiannual cycle, both Simpson’s 3/8th rule and Simpson’s 1/3rd rule gave the correct decision, whereas all other methods provided a wrong recommendation. With a shorter cycle length, all methods converged. Simpson’s 1/3rd rule was the fastest method to converge to the gold standard. Also, Simpson 1/3rd rule demonstrated best accuracy compared with other WCC methods. This example also illustrates the fault in the conventional wisdom that the approximation errors cancel each other out or become insignificant when incremental outcomes are calculated. When Does Within-Cycle Correction Matter? It is difficult to derive general conditions where WCC matters. However, it is instructive to focus on some important questions for illustration purposes only. It is important to ask how often the decision changes depending on WCC method used. Because the cost-effectiveness of an intervention is determined in relation to a threshold, choice of a correction method may not always affect the decision even though the ICER may vary greatly across methods but remain bounded away from the threshold. However, we found that when the ICER is near a threshold, the choice of method determines whether an intervention is cost-effective (Table 6).
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Table 6
Incremental Cost-Effectiveness Ratio (ICER) with Different Methods and Cycle Length
Intervention
QALYs
Cost
ICER
Semiannual (n =2) ICER
No Yes No Yes No Yes No Yes No Yes
7.4059 7.6613 7.9059 8.1613 7.8831 8.1469 7.8853 8.1472 7.8799 8.1467
255,369 270,776 255,368 270,776 258,950 272,466 258,344 272,378 259,795 272,546
— 60,333 — 60,333 — 51,224 — 53,594 — 47,790
— 51,042 — 51,042 — 48,075 — 48,371 — 47,790
Annual (n =1) Method
Right Riemann Trapezoidal rule Simpson’s 1/3rd rule Simpson’s 3/8th rule Gold standard
Monthly (n =12) ICER
Weekly (n =52) ICER
Daily (n= 365) ICER
— 47,881 — 47,881 — 47,790 — 47,791 — 47,790
— 47,795 — 47,795 — 47,790 — 47,790 — 47,790
— 47,790 — 47,790 — 47,790 — 47,790 — 47,790
Note: b = 0.80; c = $30,000; d = 0.08; e = 0.4; I = $30,000; ICER = incremental cost-effectiveness ratio; m = 0.05; q = 0.85; QALYs = quality-adjusted life-years; r = 0.03; T = 100; u = 1.0; l = 50,000. A dash indicates that the ‘‘No Intervention’’ is the reference, and it is not applicable to compute ICER. a. The error of approximation is the relative difference between the outcome using discrete-time, state-transition model with n cycles per year and the exact value of the definite integral.
WCC also matters when assessing model’s validity in predicting risk of disease over a short period of time. Within-Cycle Corrections Methods Are Not the Gold Standard The gold standard for calculating a cumulative outcome is an exact expression of a definite integral of the outcome function derived from CTSTM. WCC methods provide approximations to this integral and can over- or underestimate the value of the cumulative outcome. Therefore, it would be erroneous to state that WCC methods will always produce more accurate results compared with not performing a correction at all. In fact, we found situations in which applying WCC correction can do more harm than good. For example, all methods showed underestimation of 10year risk of disease when the following parameters were used: e = 0, b = 0.1, m = 0.05, d = 0.25, w0 = 1. Compared with the gold standard result of 61.85%, the results of other methods ranged from 45.51% (R) to 53.54% (L), with a relative error range of 13.44% (L) to 26.42% (R). Thus, the left Riemann sum L (i.e., no correction was applied) performed the best. The size of the relative error was vastly reduced when shorter cycle length was selected. For example, with a monthly cycle (n = 12) and using the same parameter values, the range for the relative error was 1.19% to 2.52%. For all methods, the relationship between the relative error and the cycle length is given by a convex function, indicating that the error decreases at a diminishing rate as the cycle length gets shorter.
10
The following example also illustrates the case where, without WCC correction, the results of an incremental analysis are the same as those obtained with a CTSTM. Let b = 0.7, m = 0.05, d = 0.055, w0 = 1, r = 0.03, q = 0.99, u = 1.0, e = 0.93888, and T = 50. Total QALYs without the intervention using CTSTM, right Riemann sum, and trapezoidal rule are 11.423, 10.9371, and 11.4304 years, respectively. The respective total QALYs with the intervention using the 3 methods are 11.8883, 11.4024, and 11.8950 years. So, the right Riemann sum gives the same incremental QALYs of 0.4653 years as the CTSTM, whereas the application of the trapezoidal rule results in 0.4646 years, an underestimate of 0.0007. Therefore, applying WCC could lead to inaccurate estimates of incremental outcomes compared with the gold standard or without correction at all. DISCUSSION The existing modeling literature on WCC methods applies numerical adjustments to cumulative outcomes from DTSTM that are not based on a careful treatment of the underlying theory. We show that WCC methods in DTSTM should be derived from an analysis designed to find the most appropriate way of approximating a definite integral of a given function over a specified period of time. Thus, we reviewed correction methods from the field of numerical integration and applied these methods to a standard 3-state disease progression model to evaluate the cost-effectiveness of a hypothetical intervention. To our knowledge, this is the first
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study to provide a complete analytic solution to both the discrete- and continuous-time versions of the model and derive analytically the formulas for cumulative outcomes and relative error for each method compared with the true gold standard. Others have criticized the standard application of HCC of adjusting total cumulative costs and outcomes using specific terms (for its alleged incompatibility with discounting and varying utilities and costs by cycle) and have suggested alternatives such as life-table method5 or cycle-tree method.7,8 Our recommendation regarding the standard application of HCC is similar, albeit for a different reason. We show that it is compatible with discounting and applicable to utilities and cost functions that vary by cycle. With careful definition of the outcome functions, standard HCC resulted in the same cumulative outcome as the life-table method and the trapezoidal rule. However, even if it is done properly, at best HCC is just an approximation based on the trapezoidal rule of integration, not a gold standard. We showed that WCC methods are needed to accurately estimate cumulative outcomes. The demand for accuracy should trump the desire for simple calculations. One clear support for this principle can be found in the recent recommendations that stress the importance of a model’s ability to accurately predict outcomes.22 Because a model’s validity is frequently tested using intermediate outcomes over a short period of time, applying the type of method for calculating those outcomes can affect validation results and ultimately confidence in the model’s results.
3.
4.
5.
Misconceptions A number of misconceptions are common in the literature concerning the need for and application of HCC specifically or WCC methods in general. We discuss a few of these myths and facts. 1. Myth: WCC is needed to correctly estimate state membership in DTSTM. Fact: If the transition probability matrix is constructed correctly, the state membership in a DTSTM will be the same as that in a CTSTM. Although the DTSTM calculates state membership only at discrete time points, these calculations require no assumptions about when the transitions occur within a cycle. Such assumptions, and hence WCC methods, are needed only when calculating cumulative outcomes. 2. Myth: If one applies WCC to state membership, all cumulative outcomes are calculated correctly. Fact: This is true only for special outcomes (e.g.,
6.
survival), and it leads to erroneous results if applied for most other outcomes (e.g., discounted qualityadjusted survival). This is because some cumulative outcomes depend not only on state membership but also on other time-varying factors (e.g., discounting factor). Myth: If one assumes that transitions occur at the end (beginning) of a cycle, not applying HCC overestimates (underestimates) cumulative outcomes. Fact: This is true only if the outcome function (e.g., survival function) is monotonically decreasing with respect to time. In fact, the opposite is true if the function in question is monotonically increasing (e.g., cost function): Not applying HCC underestimates cumulative outcomes when transitions occur at the end of a cycle. Myth: HCC is not compatible with discounting. This claim was made frequently in the recent debate on HCC.7–9 For example, Barendregt9 arrived at different results using the life-table method compared with the HCC. We think the inconsistency in results arose mainly from the way discounting was applied. Instead of applying the life-table method to the discounted quality-adjusted survival function, Barendregt applied it to state membership first and then calculated QALYs (by applying discounting and quality of life adjustments) afterward. Naimark and others7 stated that deriving a properly discounted expression for HCC ‘‘would be extremely cumbersome.’’ Fact: With the right definition of discounted outcome functions (i.e., apply WCC after discounting), HCC is compatible with discounting. We were able to correctly calculate the present value of several cumulative outcomes using a constant discount rate r per year and showed that the results of HCC, life-table method, and trapezoidal rule are equivalent (Table 4). Myth: HCC can lead to erroneous results when unit costs and utilities vary per cycle. Fact: By definition, HCC is designed to approximate the integral of an outcome that vary over time. When unit costs and utilities vary per cycle, the shape of costs and quality-adjusted survival will be different from those when unit costs and utilities are constant, but the methods of how to approximate their integral do not change. Myth: Even without a correction, either the approximation errors cancel each other out or the remaining errors are small when incremental outcomes are calculated. Fact: This may be true in some situations (especially when the time horizon is long). However, this should not be used as evidence that WCC or more accurate integration methods are not needed. To do so is akin to accepting the logical fallacy of ‘‘two wrongs make a right.’’ We presented an example to illustrate that the results of DTSTM without any correction could lead to a different decision in comparison with the gold standard.
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7. Myth: If one applies HCC appropriately or uses another WCC, cumulative outcomes are calculated correctly. Fact: Regardless of how good the method is, there will always be an approximation error as long as time is assumed to be discrete.
Limitations Following precedents,3,7,8,10,20 we considered continuous-time models as the gold standard for representing the process governing diseases and other biological and healthcare phenomena. However, there may be situations where the underlying process is discrete in nature. The use of discrete-time steps many not introduce errors when calculating cumulative outcomes in such situations. We made several simplifying assumptions to obtain analytical results. First, we did not allow transition probabilities to vary by time. This may not be realistic in several situations. For example, all-cause mortality changes according to the age of the person. Second, we used only a 3-state model and assumed that the disease is progressive without the possibility of recovery. Considering a more flexible model structure that allows recovery from disease, transition probabilities to change with time, and inclusion of additional health states is a logical extension of this framework. We expect the general conclusions of this paper to remain valid despite these changes. Moreover, the WCC methods presented in this paper are general and can be applied to any DTSTM. We focused only on some classic methods of integration that were common before the modern
computer age. All the methods used in this paper are special cases of Newton-Cotes formulas that approximate the integrand by an nth order polynomial (we went as far as a third-degree polynomial). However, these represent a small strand in a vast literature on numerical integration techniques. Perhaps other methods such as Simpson’s 6-points rule, Romberg’s quadrature rule, Boole’s rule, or Gauss’ quadrature formula may prove more accurate than the trapezoidal rule or the two Simpson’s rules reviewed in this paper.13 We hope that our work will motivate others to develop the theory of WCC further and provide additional practical algorithms for easy implementation in commonly used software. CONCLUSION We showed that cumulative outcomes in DTSTMs are prone to errors and that use of WCC methods can reduce the errors. We clarified several misconceptions regarding the application and scope of WCC methods. We found that the trapezoidal rule has less accuracy than the Simpson’s 1/3rd or 3/8th rule when compared with the gold standard. Although WCC methods require the analyst to perform the correction at each cycle, we showed that it is straightforward to implement all these methods within commonly used software such as Microsoft Excel and TreeAge. We recommend using one method (preferably Simpson’s 1/3rd rule) for the base-case analysis and showing the results with other methods in the sensitivity analysis.
APPENDIX A: IMPLEMENTATION OF WCC IN COMMONLY USED SOFTWARE PROGRAMS Programming various WCC methods in spreadsheet-based tool (e.g. Microsoft Excel) and commercial software such as TreeAge Pro is not a difficult task. We illustrate the application of these methods using QALY calculations for a model with an annual cycle. We represent discounted quality-adjusted survival at cycle k by Qk. In our 3-state model this is given by Qk 5 ðuWk 1 qSk Þð1 1 r^Þk : Implementation in Spreadsheet-Based Tool Compute Qk at each cycle (starting from initial time where k = 0). Trapezoidal rule: Starting at first cycle (where k = 1), compute the average at the beginning and end of a cycle, fx= (Qk–1 1 Qk)/2, at each cycle and add up to obtain total QALYs. Simpson’s 1/3rd rule: Compute initial QALYs Q0/3 and final QALYs QnT/3. Using mod function, compute intermediate QALYs fx= if(mod(k,2)=0,2,4)*(1/3)*Qk at each intermediate cycle (for k = 1, 2, 3, . . .., nT–1). Add up initial, intermediate, and final QALYs to obtain total QALYs. Simpson’s 3/8th rule: Compute initial QALYs Q0*(3/8) and final QALYs QnT*(3/8). Using mod function, compute intermediate QALYs fx= if(mod(k,3)=0,2,3)*(3/8)*Qk at each cycle (for k = 1, 2, 3, . . .., nT–1). Add up initial, intermediate, and final QALYs to obtain total QALYs.
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FOUNDATIONS OF WITHIN-CYCLE CORRECTION METHODS
Implementation in TreeAge Trapezoidal rule: Using _stage variable, set Init Rwd = utility/2, Final Rwd = discount(utility; r; nT)/2; and Incr Rwd = discount(utility; r; _stage), where ‘‘utility’’ is the utility associated with a given state, nT is the terminal stage, and r is the discount rate. Simpson’s 1/3rd rule: Using function discount, modulo, and _stage, set set Init Rwd = utility/3; Final Rwd = discount(utility; r; nT)/3; and Incr Rwd = if(modulo(_stage;2)=0; 2; 4)*discount(utility; r; _stage)/3. Simpson’s 3/8th rule: Define Init Rwd = utility *(3/8); Final Rwd = discount(utility; r; nT)*(3/8); and Incr Rwd = if(modulo(_stage;3)=0; 2; 3)*discount(utility; r; _stage)*(3/8).
There are minor complications in implementing Simpson’s 1/3rd rule when the time horizon is odd. When time horizon T is odd, implement Simpson’s 1/3rd rule over interval [0,T – 1] and trapezoidal rule in the final cycle. Likewise, Simpson’s 3/8th rule requires the time horizon to be divisible by 3. To implement Simpson’s 3/8th rule when T is not divisible by 3, use Simpson’s 1/3rd or trapezoidal rule in the final 2 cycles if the remainder is 2 and trapezoidal rule if the remainder of the division by 3 is 1. It should be noted that the above algorithm is based on a model with an annual cycle length. If a different cycle length is used, one would need to multiply total outcomes by the length of the cycle in years in order to compute outcomes on annual basis. APPENDIX B: MODELS’ SOLUTIONS AND CALCULATIONS OF OUTCOMES The probability of being in a given health state after t years from now is obtained by raising the matrix P to the power t. Because we assumed that each year can be divided in n cycles, the transition probability at cycle k (where k = n t) is given by taking the power of the matrix to (t =) k/n. The matrix power can be achieved by diagonalizing P using Eigen decomposition and expressing it as k
k
P n 5 V An V 1 ;
where A is a diagonal matrix of the eigenvalues {1, 12d,12(12e)b2m} and the matrix of eigenvector V and its inverse are given by: 2
3 Þb 1 ð1eð1e Þb 1 md 1 V 541 1 0 5; 1 0 0 2 3 0 0 1 5: 1 1 V 1 5 4 0 Þb dm 1 ð1eð1e Þb 1 md ð1eÞb 1 md
The Eigen decomposition makes calculations of the matrix power feasible because raising A to the power k/n is the same as raising the diagonal elements of A to the power k/n. Thus, k
k
P n 5 V An V 1
2
k
6 ½1 ð1 eÞb mn 56 4 0 0
k
k
ð1eÞb ð1dÞn ½1ð1eÞbmn ð1eÞb 1 md k
ð1 dÞn 0
3
p^13
7 k 7; 1 ð1 dÞn 5 1
Where h i n o k k ð1 eÞb 1 ð1 dÞn ðd mÞ 1 ½1 ð1 eÞb mn
^13 5 p
ð1 eÞb 1 m d
:
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We assume throughout the paper that ð1 eÞb 1 m d 6¼ 0: It can easily be verified that at year 1 (i.e., when k = n) the above transition matrix reduces to the one-step transition probability matrix P. 1 1 A word of caution regarding the resulting matrix Pn is necessary at this juncture. Because Pn is a probability matrix it has to be stochastic: a) all rows add up to 1 and b) all cells are constrained to be between 0 and 1. It is 1 straightforward to verify that matrix Pn satisfies the first condition. However, there are situations when the second condition can be violated. Assuming that a fraction w0 of the cohort of size 1 starts in the well health state, the distribution of persons in k a given health state at cycle k is obtained by evaluating ðw0 ; 1 w0 ; 0Þ: Pn : Thus, k
Wk 5 w0 ½1 ð1 eÞb mn ; k
k
½bð1 eÞ ðd mÞð1 w0 Þð1 dÞn w0 bð1 eÞ½1 ð1 eÞb mn : Sk 5 bð1 eÞ 1 m d
Because they are assumed to be constant over time, the one-step transition probabilities matrix pij, the cyclespecific transition probabilities matrix p^ij , and the intensity rates qij are related according to 1 P 5 expQ ; Q 5 log P; P^ 5 P n ;
where log P denotes the matrix logarithm of a matrix P. These sets of equations were solved to obtain the values of p^ij and qij shown in Table 1. It should be noted that in the limit when the cycle length is infinitesimally small (i.e., n approaches infinity) P^ 5 Q: For example, it can be shown (using l’Hoˆpital’s rule) that 1
^23 5 lim lim p
n!‘
1 ð1 d Þn
n!‘
1 n
5 lnð1 dÞ:
The matrix exponential of Q can be obtained in a similar manner as the matrix power: expQ 5 EexptL E1 ;
where L is a diagonal matrix of the eigenvalues {0,2d,2(12e)b2m} of matrix Q and the matrix of eigenvector E and its inverse are given by: 2
1 E541 1
ð1eÞb ð1eÞb 1 md
1 0
2
0 0 1 E1 5 4 0 Þb 1 ð1eð1e Þb 1 md
3 1 0 5; 0 3
1 1 dm ð1dÞb 1 md
5:
The Eigen decomposition makes calculations of the matrix exponential feasible because finding the exponential of tL is the same as exponentiating the diagonal elements of tL. Thus, 2
exp½bð1eÞ 1 mt 6 expQt 5 EexptL E1 5 4 0 0
bð1eÞfexpdt exp½bð1eÞ 1 mt g bð1eÞ 1 md dt
exp 0
3
bð1eÞð1expdt Þ 1 ðmdÞf1exp½bð1eÞ 1 mt g bð1eÞ 1 md 7 dt 5:
1 exp 1
The distribution of persons in a given health state at time t is obtained by evaluating ðw0 ; 1 w0 ; 0Þ:expQt : The closed-form solution for the number of persons in the Well state and Disease state at year t is
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FOUNDATIONS OF WITHIN-CYCLE CORRECTION METHODS
W ðtÞ 5 w0 exp½bð1eÞ 1 mt ;
SðtÞ 5
½bð1 eÞ ðd mÞð1 w0 Þexpdt w0 bð1 eÞexp½bð1eÞ 1 mt : bð1 eÞ 1 m d
Using the relationships ð1 eÞb 1 m 5 ln½1 ð1 eÞb m and d 5 lnð1 dÞ in the above equations, we find that Wk 5 W ðk=nÞ and Sk 5 Sðk=nÞ: OUTCOMES USING CTSTM AND DTSTM We need to specify the functional form of f for each outcome O. Risk of disease over a given period consists of two components: the rate of development of disease and population at risk at time k/n. The latter is given by W ðk=nÞ and is identical in both models. In CTSTM the former is based on a hazard rate whereas in DTSTM ^ 12 : Unless the cycle length is very the risk of disease is calculated using a probability of disease per cycle: p Þ 6¼ p^12 : In fact, small, the hazard within a cycle will be different from a probability per cycle bð1e n bð1 eÞflnð1 dÞ ln½1 ð1 eÞb mg n½ð1 eÞb 1 m d n o 1 1 bð1 eÞ ð1 dÞn ½1 ð1 eÞb mn ^12 : 5p bð1 eÞ 1 m d
Therefore, the functional form for the risk of disease will not be the same across the two models irrespective of whether a within-cycle correction is applied or not. For both models, the definitions for life expectancy, discounted QALYs, and discounted disease costs are: W ðtÞ 1 SðtÞ; ½uW ðtÞ 1 qSðtÞð1 1 rÞt ; and cSðtÞð1 1 rÞt ; respectively. We will illustrate the calculations with QALYs assuming a lifetime horizon (T ! ‘). For CTSTM (gold standard), QALYG 5
Z‘
exprt ½uW ðt Þ 1 qSðt Þdt 5
1 w0 ½bð1 eÞðu qÞ ðd mÞu q½bð1 eÞ ðd mÞð1 w0 Þ : ½bð1 eÞ 1 m d ln½1 ð1 eÞb m lnð1 1 r Þ lnð1 dÞ lnð1 1 r Þ
0
Applying a right Riemann sum, remaining QALYs using DTSTM are
QALYR 5
‘ 1X ðuWk 1 qSk Þ
n k 5 1 ð1 1 rÞk=n
9 8 > > < 1 w0 ½bð1 eÞðu qÞ ðd mÞu q½bð1 eÞ ðd mÞð1 w0 Þ= : 5 1 h i1
1d n1 > > n½bð1 eÞ 1 m d : 1bð1eÞm n ; 1 1 1 1 r 11r
If the cycle length is chosen to be infinitesimally small (i.e., number of cycles in a year becomes infinite), QALYs using the DTSTM will be identical to those from the CTSTM. Thus, lim QALYR 5
n!‘
1 w0 ½bð1 eÞðu qÞ ðd mÞu q½bð1 eÞ ðd mÞð1 w0 Þ 5 QALYG : ½bð1 eÞ 1 m d ln½1 ð1 eÞb m lnð1 1 r Þ lnð1 dÞ lnð1 1 r Þ
The latter is obtained by observing that 1
lim
n!‘
x n 1 1 n
5 ln x:
QALYs calculated using the Trapezoidal rule are
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ELBASHA, CHHATWAL
QALYZ 5
‘ 1X 1 uWk 1 qSk uWk1 1 qSk1 1 k k1 n k51 2 n ð1 1 rÞ ð1 1 rÞ n
! 5
‘ w0 u 1 qð1 w0 Þ 1X ðuWk 1 qSk Þ w0 u 1 qð1 w0 Þ 1 1 QALYR : 5 2n n k 5 1 ð1 1 rÞk=n 2n
It should be noted that the above calculations show that HCC requires adding a half cycle worth of QALYs to total QALYs without correction (applying a right Riemann sum). Discounted QALYs in the absence of the intervention are obtained by evaluating the above expression when e = 0. When there is no adjustment to survival (i.e., u = q = 1), one obtains remaining life expectancy. Discounted QALYs using the composite Simpson’s rule 1/3rd are 8 h i1 h i2 9 h
1d n1 1d n2 i Þm n Þm n > > > > w0 ½bð1 eÞðu qÞ ðd mÞu 1 1 4 1bð11e 1 1bð11e = < q ½ b ð 1 e Þ ðd mÞð1 w Þ 1 1 4 1 1 r 1 r 0 11r 11r 1 : QALYC 5 1 2 2 h i
> > 3n½bð1 eÞ 1 m d > n 1d n 1b ð 1e Þm > 1 11r ; : 1 11r
Again, it can be shown that lim QALYC 5 QALYG : n!‘
Discounted QALYs using the Simpson’s 3/8th rule are 8 h i 1 3 9 h >
1d n1 i3 1bð1eÞm n > > > > > w0 ½bð1 eÞðu qÞ ðd mÞu 1 1 > 8n½bð1 eÞ 1 m d > n n > 1 11d > > 1 1bð1eÞm 1r : ; 11r
Similarly, it can be shown that lim QALYCR 5 QALYG : The evaluation of the limit becomes straightforward n!‘ 1 1 2 3 3 once one uses lim ð1 1 xn Þ3 5 lim 1 1 3xn 1 3xn 1 xn 5 8: Also, lim n 1 xn 5 3 ln x: n!‘
n!‘
n!‘
APPENDIX C: ERROR IN ESTIMATES OF RISK OF DISEASE WITH VARIOUS CORRECTION METHODS The relative error terms, ERROR, in estimates of risk of disease with various correction methods shown in Table 5 can be rewritten as follows: ERRORL 5 1 H; ERRORR 5 1 Hy; ERRORZ 5 1 H
11y 1 1 4y 1 y 2 3ð1 1 yÞ3 ; ERRORC 5 1 H ; ERRORCR 5 1 H ; 2 3ð 1 1 y Þ 8ð1 1 y 1 y 2 Þ
1
1
xy n n where H 5 ð1yÞð1ln x= ln yÞ ; x 5 ð1 dÞ ; y 5 ½1 ð1 eÞb m . Note that 0 x 1 and 0 y 1: If y . x, then ln y . ln x and H . 0. Also, H is a monotonically increasing function in x and y, reaching a maximum value of 1 when x = y =1 (i.e., when cycle length 1/n approaches zero). Thus, 0\H 1: Therefore, 0\ERRORL 1: It can be verified that
y
1 1 4y 1 y 2 3ð1 1 y Þ3 11y 1: 2 3ð1 1 y Þ 8 ð1 1 y 1 y 2 Þ
Therefore, ERRORL ERRORZ ERRORCR ERRORC ERRORR :
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FOUNDATIONS OF WITHIN-CYCLE CORRECTION METHODS
ACKNOWLEDGMENTS We thank Scott Cantor and John R. Cook for helpful comments and discussion. We also acknowledge the helpful comments and suggestions made by Pelham Barton.
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