Introduction to Markov Models (part 1) Henry Glick Epi 550 March 2, 2012
Outline • Introduction to Markov models • 5 steps for developing Markov models • Constructing the model • Analyzing the model – Roll back and sensitivity analysis – First-order Monte Carlo – Second-order Monte Carlo
Decision Trees and Markov Models • Markov models are repetitive decision trees that are used for modeling conditions that have events that may occur repeatedly over time or for modeling predictable events that occur over time (e.g., screening for disease at fixed intervals) – e.g., Cycling among heart failure classes or screening for colerectal cancer • Use of Markov model simplifies the presentation of the tree structure • Markov model explicitly accounts for timing of events, whereas time usually is less explicitly accounted for in decision trees
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"Bushiness" of Repetitive Trees
State Transition or Markov Models • Develop a description of the disease by simplifying it into a series of states – e.g., models of heart failure (HF) might be constructed with five or six health states • Five state model (if everyone in the model begins with HF): HF subdivided into New York Heart Association (NYHA) classes I through 4, and death (either from heart failure or other causes) • Six state model (if the model predicts onset of disease): No disease, HF subdivided into New York Heart Association (NYHA) classes I through 4, and death (either from heart failure or other causes)
State Transition or Markov Models (II) • Disease progression is described probabilistically as a set of transitions among the states in periods, often of fixed duration (e.g., months, years, etc.) • Likelihood of making a transition defined as a set of transition probabilities • Assess outcomes such as resource use, cost, and QALYs based on the resource use, cost and QALY weight experienced: – Method 1: From making a transition from one state to another (e.g., average cost among patients who begin a period in NYHA class 1 and begin the next period in NYHA class 2) (Used here) OR – Method 2: From being in a state for a period (e.g., average cost of being in NYHA class 1 for a year)
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State Transition or Markov Models (III) • Develop a mathematical description of the effects of an intervention as a change in: – The transition probabilities among the states (e.g., by reducing the probability of death) or – The outcomes within the states (e.g., after the intervention, those in NYHA class 1 cost $500 less than do those without the intervention)
State Transition Model, NYHA Class and Death Heart Failure Model
I
III
Death
II IV
5 Steps in Developing Markov Model 1.
Imagine the model, and draw the "tree" 1.A Enumerate the states 1.B Define allowable state transitions
2.
Identify the probabilities 2.A Associate probabilities with the transitions 2.B Identify a cycle length and number of cycles 2.C Identify an initial distribution of patients within the states
3.
Identify the outcome values
4.
Calculate the expected values
5.
Perform sensitivity analyses
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Systemic Lupus Erythematosus (SLE) (I) • The example used here is a Markov model predicting prognosis in SLE * •
The study sample – 98 patients followed from 1950-1966 (the steroid period), 58 of whom were treated with steroids – All patients were seen more than once and were followed at least yearly until death or study termination – No patient was lost to follow-up – Time 0 was time of diagnosis
* Silverstein MD, Albert DA, Hadler NM, Ropes MW. Prognosis in SLE: comparison of Markov model to life table analysis. J Clin Epi. 1988;41:623-33.
SLE (II) • Diagnosis was based on the presence of 3 of 4 criteria: – Skin rash – Nephritis (based on urinary sediment abnormality, with greater than 2+ proteinuria on two or more successive visits) – Serositis – Joint involvement •
All patients would have fulfilled the ARA diagnostic criteria for SLE
•
A set of 11 clinical findings and 9 laboratory values were used to classify patients’ disease into four severity grades, 1 through 4
Step 1.A Enumerate the States • Markov models made up of states • States are all inclusive and mutually exclusive (all patients must be in one and only one state at all times in the model) • Clearly defined, usually according to standard literaturebased notions of disease • Distinguished by their prognosis or transition probabilities • Transition probabilities per unit time estimable from data or the literature • Able to assign costs / outcome weights (e.g., QALYs, etc.)
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States for Modeling Systemic Lupus • Four disease states – State 1: Remission • No disease activity – State 2: Active • Severity grades 1 through 3 – State 3: Flare • Severity grade 4 – State 4: Death (from any cause)
States for Modeling Systemic Lupus (II) • Each patient year was classified by the greatest severity of disease activity during the year, even if severity was only present during a portion of the year – e.g., a patient whose disease activity was severity grade 4 during any visit in a calendar year was considered to have a flare year – No patient was observed to have more than 1 flare per year and all patients were seen at least once a year
Step 1.B Define Allowable State Transitions • Nonabsorbing states: once in the state, one can move out of it •
Absorbing states: once in the state, one cannot move out of it (e.g., death)
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SYSTEMIC LUPUS
Active
Remis-
Death
sion
Flare
Developing The Treeage Lupus Model
Usual Care Lupus Intervention
Add the 4 States
Remission Active Usual Care Flare Lupus
Death Intervention
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Add the Transitions from Remission Remission Active
Remission
Flare Death Usual Care
Active Flare
Lupus
Death Intervention
Add the Remaining Usual Care Transitions Remission Remission
Active Flare Death Remission
Active
Active Flare
Usual Care Death Remission Flare
Lupus
Active Flare Death
Death Intervention
Step 2.a Associate Probabilities with the Transitions • Suppose you had data from a lupus registry that was following 98 patients – Observations were made at the beginning and end of each year – During the period of observation, you had 1115 patient years of observation – Pooling across the years of observation, you identified • 100 patient years classified as remission • 935 patient years classified as active disease • 80 patient years classified as flare
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Remission Transition Probabilities • Suppose that among the 100 who spent a year in remission – 59 spent the following year in remission – 41 spent the following year with active disease – None spent the following year with flare or or died • What are the annual transition probabilities?
Active Transition Probabilities • Suppose that among the 935 who spent a year with active disease – 66 spent the following year in remission – 806 spent the following year with active disease – 56 spent the following year with flare – 9 died • Probabilities?
Flare Transition Probabilities • Suppose that among the 80 spent a year with a flare – 0 spent the following year in remission – 22 spent the following year with active disease – 18 spent the following year with a flare – 40 died • Probabilities?
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Transition
Data *
Prob
95% CI
Remission ö Remission
59 / 100
Remission ö Active
41 / 100
0.41 (0.31 to 0.51)
Remission ö Flare
0 / 100
0.00 (0.00 to 0.03)
Remission ö Death
0 / 100
0.00 (0.00 to 0.03)
Active ö Remission
66 / 937
0.07 (0.06 to 0.09)
Active ö Active
806 / 937 0.86 (0.83 to 0.88)
Active ö Flare
56 / 937
0.06 (0.05 to 0.08)
Active ö Death
9 / 937
0.01 (0.00 to 0.02)
Flare ö Remission
0.59 (0.49 to 0.69)
0 / 80
0.00 (0.00 to 0.06)
Flare ö Active
22 / 80
0.27 (0.18 to 0.39)
Flare ö Flare
18 / 80
0.23 (0.14 to 0.33)
Flare ö Death
40 / 80
0.50 (0.38 to 0.62)
* Counts are approximations of actual data (not provided in article)
SYSTEMIC LUPUS 0.86 Active
0.41
0.01 0.07
0.59
Remission
Death
0.06 0.27
1.00
0.50 Flare
0.23
Probability Estimation • Large number of methods exist for estimating transition probabilities – Simple methods as suggested in Lupus example – If available data are hazard rates (i.e., instantaneous failure rates) per unit of time (Rij[t]), they can be translated into probabilities as follows:
Pij (t) = 1- e
-Rij t
where Pij(t) equals the probability of moving from state i at the beginning of period t to state j at the beginning of period t+1; Rij equals the instantaneous hazard rate per period (e.g., per year); and t equals the length of the period
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Step 2.B Identify a Cycle Length and Number of Cycles (Markov Termination) • Currently accepted practice for cycle length: – Strategy 1: Have the cycle length approximate clinical follow-up – Strategy 2: Allow the cycle length to be determined by the study question or available data; ignore differences that don’t make a difference • Current probabilities are for annual cycles • Markov Termination :
_stage > 1999
Step 2.C Identify an Initial Distribution of Patients Within the States • Use a population approach: e.g., one might want to use the distribution in which patients present to the registry Remis
Active
Flare
0.10
0.85
0.05
Step 2.C Identify an Initial Distribution of Patients Within the States (II) • Alternatively, start everyone in one state, (e.g., to determine what will happen to patients who begin in remission, make the probability of being in remission 1.0) Remis
Active
Flare
Start in Remission
1.0
0.0
0.0
Start in Active
0.0
1.0
0.0
Start in Flare
0.0
0.0
1.0
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Hypothetical Lupus Initial Distribution
Remission:
0.10
Active:
0.85
Flare:
0.05
Insert Initial Distribution, Probabilities, and Number of Cycles in Tree 0.59 Remission 0.1
# 0 0 0.07
Active 0.85
0.86 0.06
Usual Care
# 0 Flare
Lupus
0.27
0.05 0.23 #
Remission Active Flare Death Remission Active Flare Death Remission Active Flare Death
Death # Intervention
Step 3. Identify the Outcome Values • Basic result of model calculation is cycles of survival in the different states • Also should identify: – Costs of making a transition from one state to another state or of being in a state – Health outcomes other than survival (e.g., qualityadjusted life expectancy)
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Outcomes for Transitions • For the current analysis, outcomes are modeled as a function of making a transition from one state to another – e.g., number of hospitalizations (and cost) or QALYs experienced by patients who at the beginning of time t are in state i and at the beginning of time t+1 are in state j (e.g., the transition from remission to active disease)
Lupus Outcome Variables • Hypothetical Cost Data – Costs modeled as # of hospitalizations × $ • cHosp assumed to equal 10,000 * – Suppose that our hospitalization data were derived from the observation of subjects for a year • We recorded their disease status at the beginning and end of the year and measured the number of times they were hospitalized during the year – We use these data to estimate the (hypothetical) mean number of hospitalizations for those who begin in state i and end in state j: * Krishnan, Hospitalization and mortality of patients with systemic lupus erthematosus. J Rheumatol. 2006;33:1770-4.
Numbers of Hospitalizations
Remission
Remis.
Active
Flare
Death
0.05
0.25
0.00
0.00
Active
0.10
0.20
1.00
0.50
Flare
0.00
0.25
1.25
0.75
• e.g., Patients who begin in remission and remain in remission will have 0.05 hospitalizations during the year; those who begin with active disease and develop a flare will have 1 hospitalization during the year
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Hypothetical QALY Data (I) • Suppose you found a study that reported preference weights from the cross sectional observation of subjects (i.e., the authors assessed preference for current health among cohorts of patients who were in remission, active disease or flare) •
We observed the following (hypothetical) QALY weights (NYHA class weights provided for reference): QALY Weight
SLE Stage
NYHA Class
QALY Weight
Remission
0.90
--
--
Active
0.70
1
0.71
Flare
0.50
3
0.52
Hypothetical QALY Data (II) • The hypothetical preference weights can be used to estimate QALYs for those who begin in state i and end in state j: – For the transition between remission and active disease, we know that people in remission experience 0.9 QALYs and those in active disease experience 0.7 – If we assume that the transition between remission and active disease occurs at the mid-interval, the mean QALYs among those who begin the period in remission and end it in active disease are: (0.5 x 0.9) + (0.5 x 0.7)
Hypothetical QALY Transition Rewards Transition R to R
Preference Score 0.9
R to A A to R A to A A to F A to D
(0.9+0.7)/2 (0.7+0.9)/2 0.7 (0.7+0.5)/2 0.7/2
F to A F to F F to D
(0.5+0.7)/2 0.5 0.5/2
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Other Outcomes • Years of life – 1 for every transition other than the transition to death – 0.5 for every transition to death • Discounted years of life – Years of life rewards that include discounting • Number of discounted hospitalizations – Calculated by setting cHosp = 1
Discounting • Rewards experienced over time, and thus must be discounted • Can write out the discounting equation as part of the reward – e.g., for annual transition from REM to REM (cHosp * 0.05) / ((1+r)^_stage) – where r = the discount rate (e.g., 0.03) and _stage represents Treeage’s cycle counter (first cycle = 0) • OR Can use Treeage’s discounting function Discount(payoff; rate; time) = payoff / ((1 + rate)time ) – e.g., Discount(cHosp * 0.05;0.03;_stage)
Remission Transition Rewards --- Markov Information Trans Cost: (cHosp*.05)/((1+r)^_stage) Trans Eff: 0.90/((1+r)^_stage)
Remission
0.59
Remission --- Markov Information Init Cost: 0 Incr Cost: 0 Final Cost: 0 Init Eff: 0 Incr Eff: 0 Final Eff: 0
--- Markov Information Trans Cost: (cHosp*.25)/((1+r)^_stage) Trans Eff: 0.80/((1+r)^_stage)
Active
# --- Markov Information Trans Cost: 0 Trans Eff: 0
Flare
0
0.1 --- Markov Information Trans Cost: 0 Trans Eff: 0
Death
0 Remission
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Trans Cost: 0 Trans Eff: 0
0 Active Transition Rewards --- Markov Information Trans Cost: (cHosp*.10)/((1+r)^_stage) Trans Eff: 0.80/((1+r)^_stage)
Remission
0.07
Active --- Markov Information Init Cost: 0 Incr Cost: 0 Final Cost: 0 Init Eff: 0 Incr Eff: 0 Final Eff: 0
ual Care Markov Information rm : _stage>2000
--- Markov Information Trans Cost: (cHosp*.20)/((1+r)^_stage) Trans Eff: 0.70/((1+r)^_stage)
Active
0.86 --- Markov Information Trans Cost: (cHosp*1.0)/((1+r)^_stage) Trans Eff: 0.60/((1+r)^_stage)
Flare
0.06
0.85 --- Markov Information Trans Cost: (cHosp*.50)/((1+r)^_stage) Trans Eff: 0.35/((1+r)^_stage)
Death
# Remission
--- Markov Information
Trans Cost: (cHosp .50)/((1+r) _stage) Trans Eff: 0.35/((1+r)^_stage)
#
Flare Transition Rewards
Remission
--- Markov Information Trans Cost: 0 Trans Eff: 0
0 Active
--- Markov Information Trans Cost: (cHosp*.25)/((1+r)^_stage) Trans Eff: 0.60/((1+r)^_stage)
Flare --- Markov Information Init Cost: 0 Incr Cost: 0 Final Cost: 0 Init Eff: 0 Incr Eff: 0 Final Eff: 0
0.27 Flare
--- Markov Information Trans Cost: (cHosp*1.25)/((1+r)^_stage) Trans Eff: 0.5/((1+r)^_stage)
0.23
0.05
Death
--- Markov Information Trans Cost: (cHosp*.75)/((1+r)^_stage) Trans Eff: 0.25/((1+r)^_stage)
#
Death --- Markov Information I it C t 0
Systemic Lupus 0.41 0.25 0.80 0.59 0.05 0.90
Remission
Active
0.07 0.10 0.80
0.23 1.25 0.50
0.06 1.00 0.60
0.86 0.20 0.70
0.01 0.50 0.35
0.27 0.25 0.60
Flare
Death
1.00 0.00 0.00
0.50 0.75 0.25
• Row 1, Transition probabilities; Row 2, Number of hospitalizations; Row 3, QALYs
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The Hypothetical Intervention • Hypothetical intervention must be taken for life, but affects only the transition from remission to active disease – Relative risk = 0.8537 (0.35 / 0.41) R to A: 0.8537 * 0.41 – Where does 0.1463 * 0.41 go? • In this case it remains in remission. You’ve got to make sure the residual of the changed probability goes to the correct state • Cost of hypothetical intervention per year:
365
Construct Intervention Subtree • Change “Intervention” node to a Markov node • Place cursor on “Usual Care” node • \Subtree\Select Subtree OR Right click: Select Subtree • \Edit\Copy • Place cursor on intervention node • \Edit or Right click \ Paste
Construct Intervention Subtree (2) • Everything should have copied EXCEPT Markov termination – If pay-offs aren’t copied, check to make sure that you changed the “Intervention node to a Markov node • Double click Markov Information box under Intervention; _stage>1999 • Revise Remission probabilities • Add intervention cost (cInterv = 365)
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Final Rwd: 0
#
Remission Transition Probabilities Remission
--- Markov Information Trans: ((cHosp*.05)+cInterv)/((1+r)^_stage)
# Active
--- Markov Information Trans: ((cHosp*.25)+cInterv)/((1+r)^_stage)
Remission --- Markov Information Init Rwd: 0 Incr Rwd: 0 Final Rwd: 0
(rr*0.41) Flare
--- Markov Information Trans: 0
0.1
0 Death
--- Markov Information Trans: 0
0 Remission
--- Markov Information Trans: ((cHosp* 10)+cInterv)/((1+r)^ stage)
Systemic Lupus, Hypthetical Intervention 0.35 0.25 0.80 0.65 0.05 0.90
Remission
Active
0.07 0.10 0.80
0.23 1.25 0.50
0.06 1.00 0.60
0.86 0.20 0.70
0.01 0.50 0.35
0.27 0.25 0.60
Flare
Death
1.00 0.00 0.00
0.50 0.75 0.25 Intervention cost, 365 / cycle
• Row 1, Transition probabilities; Row 2, Number of hospitalizations; Row 3, QALYs
Calculate the Expected Values • The principal analysis can be performed in 1 of 3 ways: – “Iterate” the model – Monte Carlo simulation – Matrix algebra solution (Not discussed)
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Iterate the Model • Use the data on the initial distribution and the transition probabilities to estimate the distribution of patients in later periods (e.g., years) of the model •
Initial Distribution:
•
Disease Transition Probabilities:
Remission: 0.10; Active: 0.85; Flare: 0.05
Time t+1 Time t
Remis.
Active
Flare
Remission
0.59
0.41
0.00
Active
0.07
0.86
0.06
Flare
0.00
0.27
0.23
Transition to Remission • Assuming that the probability that patient is in the three states at the beginning of the model is 0.1, 0.85, and 0.05, what is the probability a patient will be in remission next year? State i,t
Pi,t
Pi,Rem
P t+1
Remission
0.10
0.59
0.059
Active
0.85
0.07
0.0595
Flare
0.05
0.00
PRem,t+1
0.00 0.1185
(i.e., multiply the initial distribution times the first column of the transition matrix)
Transition to Active • Will have Active disease? State i,t
Pi,t
P i,Act
P t+1
Remission
0.10
0.41
0.041
Active
0.85
0.86
0.731
Flare
0.05
0.27
0.0135
PAct,t+1
0.7855
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Transition to Flare • Will experience a Flare? State i,t
Pi,t
P i,Flr
P t+1
Remission
0.10
0.00
0.00
Active
0.85
0.06
0.051
Flare
0.05
0.23
0.0115
PFlr,t+1
0.0625
Transition to Death • Will die? State i,t
Pi,t
Pi,Dth
P t+1
Remission
0.10
0.00
0.00
Active
0.85
0.01
0.0085
Flare
0.05
0.50
0.0250
PDth,t+1
0.0335
Expected Cost of Hospitalization • Use data on the initial distribution, the transition probabilities, and the number of hospitalizations per transition/period to estimate the expected number of hospitalizations in each period of the model •
Number of Hospitalizations
Remission
Remis.
Active
Flare
Death
0.05
0.25
0.00
0.00
Active
0.10
0.20
1.00
0.50
Flare
0.00
0.25
1.25
0.75
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Expected Cost of Hospitalization for Usual Care Patients who Begin in Remission, Period 1? • What is the expected cost of hospitalization for Usual Care patients who begin in remission? State i,t+1
Pi
Pij
Hij
Nhosp
Remission
0.10
0.59
0.05
.00295
29.50
Active
0.10
0.41
0.25
.01025
102.50
Flare
0.10
0.00
0.00
.00
Death
0.10
0.00
0.00
.00
Total
0.10
--
--
.0132
x 10,000
0.00 0.00 132.00
Expected Cost of Hospitalization for Usual Care Patients who Begin with Active Disease? • Begin with Active disease? State i,t+1
Pi
Pij
Hij
Remission
0.85
0.07
0.10
Nhosp .00595
x 10,000
Active
0.85
0.86
0.20
.1462
1462.00
Flare
0.85
0.06
1.00
.051
510.00
Death
0.85
0.01
0.50
.00425
Total
0.85
--
--
.2074
59.50
42.50 2074.00
Expected Cost of Hospitalization for Usual Care Patients who Begin in Flare? • Begin with Flare? State i,t+1
Pi
Pij
Hij
Remission
0.05
0.00
0.00
Nhosp .00
x 10,000
Active
0.05
0.27
0.25
.003375
33.75
Flare
0.05
0.23
1.25
.014375
143.75
Death
0.05
0.50
0.75
.01875
187.50
Total
0.05
--
--
.0365
365.00
0.00
• Total cost of hospitalization, Usual Care: 132 + 2074 + 365 = 2571
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Expected QALYs • Use initial distribution, transition probabilities, and QALY weights to estimate the expected QALYS / period Transition
Preference Score
R to R
0.9
R to A
(0.9+0.7)/2 = 0.8
A to R
(0.7+0.9)/2 =0.8
A to A
0.7
A to F
(0.7+0.5)/2 = 0.6
A to D
0.7/2 = 0.35
F to A
(0.5+0.7)/2 = 0.6
F to F
0.5
F to D
0.5/2 = 0.25
Expected QALYs, Period 1 • What are the expected QALYs for patients who begin in remission? State i,t+1
Pi
P ij
Qij
Remission
0.10
0.59
0.90
.0522
QALY i
Active
0.10
0.41
0.80
.0328
Flare
0.10
0.00
0.00
.00
Death
0.10
0.00
0.00
.00
Total
0.10
--
--
.0850
Expected QALYs, Period 1 (cont.) • And so on... •
Total QALYS: 0.085 + 0.592875 + 0.0201 = 0.697975
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Roll Back Results • For a patient who initially has a 0.1, 0.85, and 0.05 probability of being in the three states, respectively Nat Hist
Interv
Life expectancy (undisc)
24.48
25.10
Life expectancy (disc)
14.44
14.63
QALYs (disc)
10.34
10.53
Cost (disc)
38,188
43300
3.82
3.80
Hospitalization, N (disc)
Roll Back, Patients Beginning in Remission Nat Hist
Interv
Life expectancy (undisc)
27.44
28.46
Life expectancy (disc)
16.08
16.45
QALYs (disc)
11.83
12.21
Cost (disc)
39,398
44,953
3.94
3.89
Hospitalization, N (disc)
Roll Back, Patients Beginning with Active Disease Nat Hist
Interv
Life expectancy (undisc)
27.44
25.60
Life expectancy (disc)
16.08
14.92
QALYs (disc)
10.53
10.71
Cost (disc)
38,965
44,201
3.90
3.88
Hospitalization, N (disc)
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Roll Back, Patients Beginning with Flare Nat Hist
Interv
Life expectancy (undisc)
9.74
9.95
Life expectancy (disc)
5.94
6.00
QALYs (disc)
4.07
4.13
22,549
24,669
2.25
2.25
Cost (disc) Hospitalization, N (disc)
(CEA)/Analysis/Cost-Effectiveness/Text report Strat
Cost
UC
38188
Int
43300
Incr Cst
Eff
Incr Eff Incr C/E
10.3388 5112
10.5342 0.1955
C/E 3694
26155
4110
One-Way Sensitivity Analysis, cInterv
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One-Way Sensitivity Analysis, RR (.65-.95)
2-Way Sensitivity Analysis, cInterv and rr
First-Order Monte Carlo Simulation *
Mean SD * Min
Usual Care Cost QALYs 10.3494 38202 6.7608 17146 0.25 5000
Intervention Cost QALYs 43296 10.5330 19951 6.8808 5183 0.25
2.5% 10% Median 90%
7500 17282 36997 61358
0.35 1.5182 9.5957 20.2777
7683 18568 42043 70401
0.35 1.5183 9.8013 20.6087
97.5% Max
72335 118874
23.3180 26.2454
82038 129978
23.6433 26.8382
* 20,000 trials; Seed set to 1
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Introduction to Markov Models (part 2) Henry Glick Epi 550 March 14, 2012
Steps in Performing Probabalistic Sensitivity Analysis Step 1. Construct your tree Step 2. Define your probability distributions Step 3. Define your payoff distributions Step 4. Analyze the "stochastic" tree Step 5. Calculate a significance test or confidence interval
Dirichlet Distribution • The Dirichlet Distribution is the multinomial (more than 2 categories) extension of the binomial Beta distribution • Defined by counts for each of the outcomes – e.g., For transitions from Remission (tRemiss) List(59;41;0;0) OR List(59;41) OR Beta distribution – e.g., For the transitions from Active (tActive) List(66;806;56;9) – e.g., For transitions from Flare (tFlare) List(0;22;18;40) OR List(22;18;40) – e.g., For initial distribution List(100;937;80) (Don’t include count for death)
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Assigning Dirichlet Distribution to Nodes • In my tree, tActive is the second distribution • One adds this distribution to the tree as follows: – Active to Remission: Dist(2;1) – Active to Active: Dist(2;2) – Active to Flair Dist(2;3) – Active to Dead Either Dist(2;4) or #
Relative Risk, Remission to Active • Hypothetical experimental data Intervention
Usual Care
Rem to Act
35 (a)
41 (b)
Rem to Rem
65 (c)
59 (d)
100 (a+c)
100 (b+d)
• Relative risk:
0.35 / 0.41 = 0.8537
Log Relative Risk • Log(RR) and SE Log(RR) ln(RR) = ln(a) + ln(b+d) - ln(b) - ln(a+c)
se[ln(RR)] =
1 1 1 1 + a b a+c b+d
• RR distributed log normal (2 parameters) – µ (ln RR): ln(35)+ln(100)-(ln(41)+ln(100)) = -.1582 – sigma (se ln(RR)): ((1/35)+(1/41)-((1/100)+(1/100)))^.5 = .1816 • NOTE: The mean of this distribution (0.8679) does not equal the point estimate for the RR (0.8537)
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Cost Distributions • Number of hospitalizations – Single parameter Poisson distributions (lambda = point estimate); separate distribution for each possible transition e.g., hdAtoA, poisson, 0.2; hdAtoF, poisson 1.0 • Cost per hospitalization – Normal distribution (mean, SE) – Assume mean = 10,000; SE = 100 • Cost of intervention – Normal distribution (mean, SE) – Assume mean = 365; SE = 50
Gamma Cost Distributions • Cost per hospitalization – α = 10,0002 / 1002= 10,000 – λ = 10,000 / 1002 = 1 • Cost of intervention – α = 3652 / 502 = 53.29 – λ = 365 / 502 = 0.146
QALY Distributions • Assume normal distribution (mean, SE) • Assume SD = 0.1 • Assume the QALY scores were measured in 100, 100, and 70 patients in remission, active, and flare, respectively Mean
SD/N0.5
SE
Remis
0.9
.1/1000.5
0.01
Active
0.7
.1/1000.5
0.01
Flare
0.5
.1/500.5
0.0141
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Editing Distributions in Excel (1) • Create the desired distributions in the Treeage distributions window – Open the distributions window and create each of the distributions needed for the tree (e.g. 4 normal, 4 Dirichlet, etc.). Don’t worry about defining the parameters for the distribution • Highlight (click on) one of the specific distributions for which you want to enter/edit the parameter values. Click the "Open in new Excel Spreadsheet" button (third button from the right in the row of icons above "Index | Type....)
Editing Distributions in Excel (2) • Enter the requested parameters (you can edit the index, the name, the type, or the parameter values) • In the "TreeAge 2012" menu in Excel, click on "Add or Update Distributions" • You can, but needn't save the resulting treeage file
Usual Care Remission Transition Rewards
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Intervention Remission Transition Rewards
Defined Variables
CE Analysis, Numbers vs Distributions 2012 N
Cost
Incr Cost
UC
38188
Int
43300
5112
Cost
Incr Cost
2012 D* UC
38035
Int
43247
Eff
Incr Eff
10.3388 10.5342 .1955
Eff
Incr Eff
IC/IE
C/E
0
3964
26,155
4110
IC/IE
C/E
10.2890 5212
10.4603 .1713
3697 30,425
4134
* lupus.final.2012.1dis
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Running the PCEA: Sampling • To analyze both therapies simultaneously, place cursor on root node • \Analysis\Monte Carlo Simulation\Sampling (Probabilistic Sensitivity...) – Set number of samples – Ensure that you are sampling from all distributions • \Distributions\Sample all – Set seed (optional) • \Seeding\Seed random number generator\[#] – Begin
Second-Order Monte Carlo Simulation * Usual Care Cost QALYs
Intervention Cost QALYs
Mean
38,490
10.3680
43,688
10.5847
S.E. Min
48,050 0
0.8603 8.1318
47,534 4170
0.9093 8.1327
2.5% 10% Median 90% 97.5%
1935 6148 19,415 117,958 152,625
8.7377 9.2533 10.3707 11.4443 12.1082
7225 11,396 24,884 122,650 159,823
8.8666 9.4165 10.5695 11.7606 12.3421
Max
299,348
13.2062
302,771
13.8813
* lupus.final.2012.1dis; 1,000 trials; Seed set to 2
Cost-Effectiveness Analysis vs Sampling * CEA
Cost
Incr Cost
UC
38035
Int
43247
5212
Cost
Incr Cost
2012 D* UC
38490
Int
43688
Eff
Incr Eff
IC/IE
10.2890
3697
10.4603 .1713
Eff
Incr Eff
30,425
4134
IC/IE
C/E
10.3680 5198
C/E
10.5847 .2167
3712 23,987
4127
† lupus.final.2012.1dis; 1,000 trials; Seed set to 2
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Normal vs Gamma Cost Distributions Usual Care
Intervention
Normal distribution † Mean
38,490
43,688
SE
48,050
47,534
Min
0
4170
Mean
38,468
43,629
SE
48,044
47,542
Min
0
4066
Gamma distribution †
† lupus.final.2012.1dis; 1,000 trials; Seed set to 2
\Graph\Distribution of Incrementals
TreeAge Pro 2012 Stats Report (Incrementals)
Mean: SD *:
Cost 5198 1754
QALYs 0.2167 0.2443
Minimum 2.5% Median 97.5%
-3158 1993 5142 8478
-0.4567 -0.1950 0.1954 0.7588
Maximum: * Represents standard error
17655
1.1666
† \Charts\Output Distributions\Incremental\Intervention v. Usual Care\#bars\Stats Report lupus.final.2012.1dis; 1,000 trials; Seed set to 2
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Parametric Tests of Significance *
Mean: Std Dev †: T-statistic P-value ‡ P-value, z score * By assumption: dof = 1100
Cost 5198
QALYs 0.2167
1754 2.9635 0.003 0.003
0.2443 .8867 0.38 0.38
† Represents standard error ‡ 2*ttail(1100,(5198/1754)) | 2*(1-norm(5198/1754)) 2*ttail(1100,(.2167/.2444)) | 2*(1-norm(.2167/.2444))
Cost-Effectiveness Plane
Incremental CE Plot Report QUADRANT
INCR
INCR
EFF
COST
FREQ
PROPORTION
C1
IV
IE>0
IC0
IC>0 ICER0 ICER>50k
169
0.169
C5
III
IE
