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### Markov Models II

HS 249T Spring 2008

Brennan Spiegel, MD, MSHS

VA Greater Los Angeles Healthcare System

David Geffen School of Medicine at UCLA

UCLA School of Public Health

CURE Digestive Diseases Research Center

UCLA/VA Center for Outcomes Research and Education (CORE)

Topics

- More on Markov models versus decision trees
- More examples of Markov models
- Calculating annual transition probabilities
- Time independent (Markov chains)
- Time dependent (Markov processes)

- Temporary and tunnel states
- Half-cycle corrections

Disadvantages of Traditional Decision Trees

- Limited to one-way progression without opportunity to “go back”
- Can become unwieldy in short order
- Difficult to capture the dynamic path of moving between health states over time
- Often fails to accurately reflect clinical reality

Markov Models

- Allow dynamic movement between relevant health states
- Allow enhanced flexibility to better emulate clinical reality
- Acknowledge that different people follow different paths through health and disease

Example Markov Model

Inadomi et al. Ann Int Med 2003

Decision Trees and Markov Models may Co-Exist

- Both provide different types of information
- Information from both is not mutually exclusive
- Markov model can be “tacked” onto end of a traditional decision tree

Normal Lifespan

Cirrhosis

Markov Model

Virological Response

Normal Lifespan

No Cirrhosis

Normal Lifespan

No Response

Cirrhosis

Markov Model

Response

Start Adefovir

Resistance

No Response

No Resistance

Con’t Lamivudine

No Therapy

Inteferon

Chronic

HBV

Lamivudine

Adefovir

Adefovir Salvage

Markov Model #1

Uncomplicated

Cirrhosis

Chronic HBV

Virological Resistance

Chronic HBV on Treatment

Virological Response

Virological Relapse

Uncomplicated

Cirrhosis

Complicated

Cirrhosis

Hepatocellular

Carcinoma

Liver

Transplant

Death

Dyspepsia

Myocardial Infarction

GI Bleed

Post Myocardial Infarction

Post

GI Bleed

Death

Sub-Clinical HE

Overt HE

Clinical Response

Hepatocellular Cancer

Non-HE Complication

Liver Transplantation

Death

Annual Probability Estimates

Annual Probability Estimate

Cirrhosis in HBeAg(-) 4.0% Cirrhosis in HBeAg(+) 2.2% Chronic HBV liver cancer 1.0% Cirrhosis liver cancer 2.1% Compensated cirrhosis decompensated 3.3% Decompensated cirrhosis liver transplant 25% Liver cancer liver transplant 30% Death in compensated cirrhosis 4.4% Death in decompensated cirrhosis 30% Death in liver cancer 43%

=

8%

5

Converting Data Into Annual Probability EstimatesCannot simply divide long-term data by number of years

Example:

If 5-year risk of an event is 40%, then annual risk does not amount to:

Converting Data Into Annual Probability Estimates

General rule for converting long-term data into annual probabilities:

1-(1-x)Y = Probability at Y Years

Example of Converting Long Term Data into Annual Probability

If probability of bleed at 5 years = 0.40, then the annual probability = x, as follows:

1- (1-x)5 = 0.40

(1-x)5 = 1 – 0.40

(1-x)5 = 0.60

(1-x)= 0.902

x = 0.097

… or 9.7%

Example of Converting Long Term Data into Annual Probability

Check for errors by back calculating using the inverse equation:

1-(1-annual probability)Y = probability at Y years

1-(1- 0.097)5 = 0.40

1-(0.903)5 = 0.40

0.40 = 0.40

Steps to Combining Time-Independent Transition Probabilities

Step 1 Collect and abstract relevant studies

Step 2 Select common cycle length

Step 3 Convert all studies to common cycle length units

Step 4 Calculate common cycle transition probabilities

Step 5 Combine common cycle probabilities

Mean = 21.3% / 12-month cycle

Many Probabilities are Time Dependent

- Time independence is usually a simplifying assumption
- Progress though many systems in health care (biological, organizational, psychosocial, etc) are erratic and non-linear
- May need to account for time-dependent transitional probabilities using:
- Tables
- Tunnels

Probability

Cycle

Probability

1

0.05

1

0.1

2

0.05

2

0.08

3

0.05

3

0.07

4

0.05

4

0.06

5

0.05

5

0.05

6

0.05

6

0.04

7

0.05

7

0.03

8

0.05

8

0.02

Time Independent

Time Dependent

Using Tables for Time-Dependent Probabilities- Tables allow transition probabilities to vary cycle-by-cycle
- Allow greater precision for processes that are non-linear

Time Independent: Linear Curve

Time Dependent: Non-linear Diminishing Returns

Time Dependent: Non-linear Accelerating Returns

Probability

Cycle

Using Tunnels States

- Some events can interfere with otherwise orderly Markov chains
- Can get “stuck in a rut” that removes subjects from the usual flow of events
- e.g. developing cancer

- Tunnel states add flexibility to Markov models:
- Model getting “stuck in the rut”
- Compartmentalize processes into component states
- Can model various “recovery states” from the “rut”
- Can incorporate time-dependent transitions

Half-Cycle Corrections

- In “real life,” events can occur anytime during a given cycle – it is usually a random event
- The default setting for Markov models is for events to occur at the exact end of each cycle
- Yet the default setting can lead to errors in the calculation of average values
- Will tend to overestimate benefits (e.g. life expectancy) by about half of a cycle

Rationale for Half-Cycle Corrections

“In whatever cycle a ‘member’ of the cohort analysis dies, they have already received a full cycle’s worth of state reward, at the beginning of the cycle. In reality, however, deaths will occur halfway through a cycle on average. So, someone that dies during a cycle should lose half of the reward they received at the beginning of the cycle.”

- TreeAge Pro Manual, p476

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