-Discuss pros and cons of decision modeling using decision trees vs. a formal deterministic model.
-Understand the components and structure of discrete time Markov models.
-Discuss how to structure and parameterize a transition probability matrix.
-Understand how to construct a Markov trace using a transition probability matrix and state occupancy vector.
The ministry of health is considering five preventive care strategies that reduce the risk of becoming sick:
Strategy | Description | Cost |
---|---|---|
A | Standard of Care | $25/year |
B | Additional 4% reduction in risk of becoming sick | $1,000/year |
C | 12% reduction in risk | $3,100/year |
D | 8% reduction in risk | $1,550/year |
E | 8% reduction in risk | $5,000/year |
Decision tree for two full cycles.
Strategy A decision tree for 5 cycles.
Pros | Cons |
---|---|
Simple, rapid & can provide insights |
Pros | Cons |
---|---|
Simple, rapid & can provide insights | |
Easy to describe & understand |
Pros | Cons |
---|---|
Simple, rapid & can provide insights | |
Easy to describe & understand | |
Works well with limited time horizon |
Pros | Cons |
---|---|
Simple, rapid & can provide insights | Difficult to include clinical detail |
Easy to describe & understand | |
Works well with limited time horizon |
Pros | Cons |
---|---|
Simple, rapid & can provide insights | Difficult to include clinical detail |
Easy to describe & understand | Elapse of time is not readily evident. |
Works well with limited time horizon |
Pros | Cons |
---|---|
Simple, rapid & can provide insights | Difficult to include clinical detail |
Easy to describe & understand | Elapse of time is not readily evident. |
Works well with limited time horizon | Difficult to model longer (>1 cycle) time horizons |
Common approach in decision analyses that adds additional flexibility.
Pros | Cons |
---|---|
Can model repeated events | |
\(\quad \quad \quad \quad \quad \quad\) |
Common approach in decision analyses that adds additional flexibility.
Pros | Cons |
---|---|
Can model repeated events | |
Can model more complex + longitudinal clinical events |
Common approach in decision analyses that adds additional flexibility.
Pros | Cons |
---|---|
Can model repeated events | |
Can model more complex + longitudinal clinical events | |
Not computationally intensive; efficient to model and debug |
The advantages of Markov models derive from the fact that they are structured around mutually exclusive disease states.
These disease states represent the possible states or consequences of strategies or options under consideration.
Because there are a fixed number of disease states the population can be in, there is no need to model complex pathways, as we saw in the decision tree “explosion” a few slides back.
It is also common to pair a Markov model with a decision tree.1
It is also common to pair a Markov model with a decision tree.1
A simple decision tree is implicit in nearly every decision analysis.
Treatment A:
We defined the decision problem earlier in this lecture, so we’ll repeat the basic objectives briefly here.
Goal: model the cost-effectiveness of alternative strategies to prevent a disease from occurring.
Strategy | Description | Cost |
---|---|---|
A | Standard of Care | $25/year |
B | Additional 4% reduction in risk of becoming sick | $1,000/year |
C | 12% reduction in risk | $3,100/year |
D | 8% reduction in risk | $1,550/year |
E | 8% reduction in risk | $5,000/year |
Two major steps:
Diagram constructed using the Graphviz Visual Editor
Basic steps
Basic steps
The challenge of selecting an appropriate cycle length boils down to how we deal with competing risks.
The challenge of selecting an appropriate cycle length boils down to how we deal with competing risks.
Pros | Cons |
---|---|
Can model repeated events | Competing risks are a challenge |
Can model more complex + longitudinal clinical events | |
Not computationally intensive; efficient to model and debug |
Pros | Cons |
---|---|
Can model repeated events | Can only transition once in a given cycle |
Can model more complex + longitudinal clinical events | Shortening the cycle can create computational challenges. |
Not computationally intensive; efficient to model and debug |
More challenges …
Pros | Cons |
---|---|
Can model repeated events | Can only transition once in a given cycle |
Can model more complex + longitudinal clinical events | Shortening the cycle can create computational challenges. |
Not computationally intensive; efficient to model and debug | Shortening cycle can cause “state explosion” if tunnel states are used |
3b.i. Source and define the base case values.
3b.ii. Source and define sources of uncertainty.
Please note that future lectures will give you specific further guidance on sources and strategies for 3b.i. and 3b.ii!!
We defined many of the underlying parameters earlier in this lecture, so we’ll repeat them briefly here.
Each strategy has a different cost and impact on the likelihood of becoming sick.
Strategy | Description | Cost |
---|---|---|
A | Standard of Care | $25/year |
B | Additional 4% reduction in risk of becoming sick | $1,000/year |
C | 12% reduction in risk | $3,100/year |
D | 8% reduction in risk | $1,550/year |
E | 8% reduction in risk | $5,000/year |
It is critical to follow a formal process for parameterizing your model.
It is critical to follow a formal process for parameterizing your model.
It is critical to follow a formal process for parameterizing your model.
It is critical to follow a formal process for parameterizing your model.
All of the above highlight the importance of adopting a formal process for naming and tracking the value, source, and uncertainty distribution of all model parameters in one place.
We recommend a structured approach based on parameter naming conventions and parameter tables.
Naming conventions:
type | prefix |
---|---|
Probability | p_ |
Rate | r_ |
Matrix | m_ |
Cost | c_ |
Utility | u_ |
Hazard Ratio | hr_ |
Note: Only a subset of model parameters are shown in table.
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
param column is the short name of the parameter
Note: Only a subset of model parameters are shown in table.
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
base_case is the parameter value for the base case.
Note: Only a subset of model parameters are shown in table.
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
formula defines model parameter formulas for parameters that are functions of other model parameters.
Note: Only a subset of model parameters are shown in table.
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
description provides a text description of the parameter.
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
Note: Only a subset of model parameters are shown in table.
notes is an optional column where you add additional notes or context for the parameter.
Note: Only a subset of model parameters are shown in table.
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
distribution specifies the uncertainty distribution for the parameter. It is used for probabilistic sensitivity analyses, which we will cover in a future lecture.
Note: Only a subset of model parameters are shown in table.
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
Note: Only a subset of model parameters are shown in table.
source provides the source for the parameter. It could be a published research article, an assumption, or just simply an unsourced modeling parameter (e.g., the starting age of the modeled cohort).
Parameter Table | ||||||
param | base_case | formula | description | notes | distribution | source |
---|---|---|---|---|---|---|
n_age_init | 25.00 | Age at baseline | Modeling Parameter | |||
n_age_max | 100.00 | Maximum age of followup | Modeling Parameter | |||
u_H | 1.00 | Utility weight of healthy (H) | beta(shape1 = 200, shape2 = 3) | Leech et al. (2022) | ||
u_S | 0.75 | Utility weight of sick (S) | beta(shape1 = 130, shape2 = 45) | Leech et al. (2022) | ||
c_S | 1000.00 | Annual cost of sick (S) | gamma(shape = 44.4, scale = 22.5) | Graves et al. (2022) | ||
c_trtA | 25.00 | Cost of treatment A | gamma(shape = 12.5, scale = 2) | Martin et al. (2022) | ||
c_trtB | 1000.00 | Cost of treatment B | gamma(shape = 12, scale = 83.3) | Assumption | ||
c_trtC | 3100.00 | Cost of treatment C | gamma(shape = 36.144, scale = 83) | Assumption | ||
n_cycles | 75.00 | (n_age_max - n_age_init) | Time horizon |
Healthy | Sick | Dead | |
---|---|---|---|
Healthy | 0.856 | 0.138 | 0.007 |
Sick | 0 | 0.982 | 0.02 |
Dead | 0 | 0 | 1 |
Health State Occupancy at Beginning of Cycle
Transition Probability Matrix
Health State Occupancy at Beginning of Cycle
Transition Probability Matrix
Health State Occupancy at Beginning of Cycle
Transition Probability Matrix
\(\quad \quad \quad \quad \quad \quad \quad \quad\)
\(s =\)
H S D
1 0 0
\(P =\)
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
\(\quad \quad \quad \quad\)
Health State Occupancy at Beginning of Cycle
Transition Probability Matrix
Health State Occupancy at End of Cycle
\(s =\)
H S D
1 0 0
\(P =\)
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
\(s \cdot P=\)
H S D
0.856 0.138 0.007
Health State Occupancy at Beginning of Cycle
Transition Probability Matrix
Health State Occupancy at End of Cycle
\(s =\)
H S D
1 0 0
\(P =\)
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
\(s \cdot P=\)
H S D
0.856 0.138 0.007
Health State Occupancy at Beginning of Cycle
Transition Probability Matrix
Health State Occupancy at End of Cycle
\(s =\)
H S D
1 0 0
H S D
0.856 0.138 0.007
\(P =\)
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
\(s \cdot P=\)
H S D
0.856 0.138 0.007
H S D
0.733 0.254 0.015
Health State Occupancy at Beginning of Cycle
Transition Probability Matrix
Health State Occupancy at End of Cycle
\(s =\)
H S D
1 0 0
H S D
0.856 0.138 0.007
H S D
0.733 0.254 0.015
\(P =\)
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
H S D
H 0.856 0.138 0.007
S 0.000 0.982 0.018
D 0.000 0.000 1.000
\(s \cdot P=\)
H S D
0.856 0.138 0.007
H S D
0.733 0.254 0.015
H S D
0.627 0.35 0.025
Health State Occupancy at End of Cycle
H S D
0.856 0.138 0.007
H S D
0.73274 0.25364 0.015476
H S D
0.62722 0.3502 0.025171
Health State Occupancy Over Ten Cycles
cycle H S D
0 1.00000 0.00000 0.000000
1 0.85600 0.13800 0.007000
2 0.73274 0.25364 0.015476
3 0.62722 0.35020 0.025171
4 0.53690 0.43045 0.035865
5 0.45959 0.49679 0.047371
6 0.39341 0.55127 0.059531
7 0.33676 0.59564 0.072207
8 0.28826 0.63139 0.085286
9 0.24675 0.65981 0.098669
10 0.21122 0.68198 0.112273