Deriving Transition 

Probabilities for Decision 

Models 

Risha Gidwani, DrPH 

June 2014 
Transition probabilities drive the 

decision model

 Probability of moving from one health 

state to another (state-transition model)

 Probability of experiencing an event 

(discrete-event simulations)

2 
Goal

 (Transition) probabilities are the engine 

to a decision model

 You will often derive these probabilities 

from literature-based inputs

 Learn when and how you can do this 

3 
Acknowledgements 

 Rita Popat, PhD 

Clinical Associate Professor

Division of Epidemiology

Dept. of Health Research and Policy

Stanford University School of Medicine

4 
Probabilities in a Decision Model

 You have a cost-effective model, now you need inputs for 

your transition probabilities
Probabilities in a Decision Model

 Does not have to be 2 drugs – can be any strategies
Probability Inputs
Ways to derive model inputs

 Obtain existing data from a single study

 Synthesizing existing data from multiple studies

– Meta-Analysis 

– Mixed Treatment Comparisons

– Meta-Regression
USING EXISTING DATA FROM A 

SINGLE STUDY
Plucking inputs from the literature

 

 If you are extremely lucky, you will read a 

journal article that will have exactly the type of 

information you need.

– The vast majority of people are not extremely 

lucky.

 Modify existing literature to derive your model 

inputs
Using inputs from the literature

 Many types of inputs are available from the literature

– Probability (risk)

– Rate (mortality)

– Relative Risk 

– Odds Ratio

– Risk Difference

– Mean

– Median

– Mean Difference

– Standardized Mean Difference

 We need data in the form of probabilities for use in a 

model
What do these inputs mean? 

Statistic Evaluates Range 

Probability/Risk

(aka Incidence Proportion)

# of events that occurred in a time period

# of people followed for that time period

0-1 

Rate # of events that occurred in a time period

Total time period experienced by all subjects followed

0 to ∞

Relative Risk (aka Risk 

Ratio)

Probability of outcome in exposed

Probability of outcome in unexposed

0 to ∞

Odds Probability of outcome

1 − Probability of outcome

0 to ∞

Odds Ratio Odds of outcome in exposed

Odds of outcome in unexposed

0 to ∞

Risk Difference Difference in risk (probability) of event amongst exposed 

and unexposed

-1 to 1

Survival Curve Point = 

# of people who are alive at time t | being alive at time t - 1 

0 to n

Mean Sum of all observations

Total # of observations

- ∞ to ∞ 

Mean Difference Mean of Group 1 – Mean of Group 2 0 to ∞

Standardized Mean 

Difference

Mean of Group 1 − Mean of Group 2

Pooled Standard Deviation

- ∞ to ∞ 
Comparative, Non-Comparative Data

Transform to 

Non-Comparative 

Data

Statistic Evaluates Type of Data

Probability/Risk # of events that occurred in a time period

# of people followed for that time period

Non-Comparative 

Rate # of events

Total time period experienced by all subjects followed

Non-Comparative 

Odds Probability of outcome

1 − Probability of outcome

Non-Comparative 

Odds Ratio Odds of outcome in exposed

Odds of outcome in unexposed

Comparative 

Relative Risk (aka 

Risk Ratio)

Probability of outcome in exposed

Probability of outcome in unexposed

Comparative 

Risk Difference Difference in risk (probability) of event amongst 

exposed and unexposed

Comparative 

Survival Curve Point = # of people who are alive at time t | being alive at 

time t - 1 

Non-Comparative 

Mean Sum of all observations

Total # of observations

Non-Comparative 

Mean Difference Mean of Group 1 – Mean of Group 2 Comparative 

Standardized Mean 

Difference

Mean of Group 1 − Mean of Group 2

Pooled Standard Deviation

Comparative 

Inputs for a decision model require non-comparative data:

Ex. 1) Probability of controlled diabetes with Drug A as the first input

 2) Probability of controlled diabetes with Drug B as the second input

 
Using probabilities from the 

literature

 Literature-based probability may not exist for your time 

frame of interest

 Transform this probability to a time frame relevant for 

your model

 Example: 

– 6-month probability of controlled diabetes is reported in the literature

– Your model has a 3-month cycle length

– You need a 3-month probability
Probabilities cannot be 

manipulated easily

 Cannot multiply or divide probabilities

– 100% probability at 5 years does NOT mean a 

20% probability at 1 year

– 30% probability at 1 year does NOT mean a 120% 

probability at 4 years
Probabilities and Rates

 Rates can be mathematically manipulated -- added, 

multiplied, etc.

– Probabilities cannot

 To change time frame of probability:

 Probability  Rate  Probability

 Note: Assumes the event occurs at a constant rate over a particular time period 
Rates versus probabilities

 In a rate, you care when the event happened – this changes the rate 

(but not the probability)

3 4 

 The rate of death is 3/(3+4+1+2) = 3/10: 

 3 per 10 person-years, 0.3 per person-year

 The probability of death is 3⁄4:

 75%

=dead

=alive at end

0 1 2 3 4

Time
=dead

=alive at end

0 1 2 3 4

Time

Rate of death = 0.3 per person-year 

3/(3+4+1+2) = 3/10 

Probability of death = 75%

= 3⁄4 

=dead

=alive at end

0 1 2 3 4

Time

Rate of death = 0.5 per person-year 

3/(0.5+4+1+0.5) = 3/6

Probability of death = 75%

= 3⁄4 

Rates versus probabilities, 2
Probability-Rate Conversions

 Probability to rate

RRRRRRRR = −llll (11−pp)

tt

 Rate to probability

 PPPPPPPP PPPP PP PP = 11 − eeee (−rr )

p = probability

t = time

r = rate
Example

 3-year probability of controlled diabetes is 60%

– What is the 1-year probability of controlled diabetes?

 Assume incidence rate is constant over 3 years: 

– Rate = −ln (1−pp)

tt 

– = −ln (1−0.6)

3

= 0.3054

– Probability = 1 − eeeeee(−rr )

 = 1 − ee(−0.3054×1) = 00. 22222222

 = 26% 
Question 

 30% of people have controlled diabetes at 5 years. 

 What is the 1-year probability of controlled diabetes?
Answer 

 A 5-year probability of 30% is a 1-year probability of 6.89%. 

 Equations: 

 Rate : −ln 1−0.30

5

= 0.0713

 Probability : 1 − ee −0.0713×1 = 6.89%
Converting to Probabilities? 

Statistic Convert to probability?

 Probability/Risk

(aka Incidence Proportion)

Yes (it is already one, but use rates to convert the time period to which 

they apply) 

 Rate Yes

Relative Risk (aka Risk Ratio)

Odds

Odds Ratio

Risk Difference (x-y=z)

Survival Curve 

Mean

Mean Difference Potentially

Standardized Mean Difference Potentially

Beyond the scope of this 

seminar 
In the beginning...

there were 2 by 2 tables:

 

 Probability of outcome in exposed =

aa

aa+ bb

OOOOOOOO RRRRRRRRRR: 

aa

bb

cc

dd

=

aa

bb

×

dd

cc

=

aa

bb

RR RRRRRRRRRR RRRR RR: 

( aa

aa+bb)

( cc

cc+dd)

Outcome – Yes Outcome - No

Exposed a b 

Unexposed c d 
In the beginning...

there were 2 by 2 tables:

 

 Probability of controlled diabetes with Drug A =

aa

aa+ bb

OOOOOOOO RRRRRRRRRR: 

aa

bb

cc

dd

=

aa

bb

×

dd

cc

=

aa

bb

RR RRRRRRRRRR RRRR RR: 

( aa

aa+bb)

( cc

cc+dd)

Controlled 

Diabetes 

Uncontrolled 

Diabetes 

Drug A a b 

Placebo c d 
OR versus RR

 RR is easier to interpret than OR

 But, the OR has better statistical properties

– OR of harm is inverse of OR of benefit

– RR of harm is not the inverse of RR of benefit

 Much data in the literature is reported as OR
Probability from RR

RRRR =

pp pp pp pp ii eeeeee ee

pp pp pp pp ii uu

prob (exposed) = RR * prob(unexposed)

 

This requires that you are able to find the probability of unexposed in the journal article
Probability from RR, example

29 

 Example: 

– RR = 2.37

– Probability in unexposed = 0.17

– Probability in exposed = 2.37 * 0.17 

 = 0.403 

 = 40.3%

 over the entire study period
Probability from RR, caveat

 If the RR is the result of a regression, it has been adjusted for 

covariates

 But, the probability in the unexposed will be unadjusted 

 

 Therefore, your derived probability estimate will have some bias 

– so make sure you vary this extensively in sensitivity analyses!! 

30 
RR are nice, but...

 OR are more likely to be reported in the 

literature than RR

31 
Probability from OR

 IF the outcome is rare (≤ 10%), then you 

can assume that the OR approximates the 

RR 

 If the outcome is not rare  advanced 

topic, do not proceed without consulting 

a statistician

32 
OR versus RR

33 

 OR is a good measure of RR when the outcome is rare
Calculating Probability from OR

 If outcome is rare, assume OR approximates RR

 Prob (exposed) = RR * prob(unexposed)

Example:

OR = 1.57

Prob. of outcome in unexposed: 8%  rare

prob (exposed) = 1.57 * 0.08 = 12.56% 

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OR

Outcome 

– Yes

Outcome 

- No

Exposed 12 88

Unexposed 8 92
ProbUnexposed

 Whether you can assume the OR 

approximates RR depends on the 

probability of outcome in the unexposed 

 Should be available in the paper 

 If it is not – try going to the literature to 

find this value for a similar group of 

patients 

35 
Probability from Odds

PP =

1

7�

8

7�

=

0.143

1.143 = 0.125

=

pp pp pp pp

1 − pp pp pp pp

Odds of 1/7 = 

Probability of 0.125

pp pp pp pp =

1 +
Probability from Stata, directly

 “Margins” command

 logistic y i.x 

 margins i.x 

 Will give you predicted probabilities of x, 

given y = 1

37 
Converting to Probabilities?

Statistic Convert to probability?

 Probability/Risk

(aka Incidence Proportion)

Yes (it is already one, but use rates to convert the 

time period to which they apply) 

 Rate Yes

 Odds Yes

 Odds Ratio Yes, if the outcome is rare (≤ 10%) & you have 

probability in unexposed

 Relative Risk (aka Risk Ratio) Yes, if you have the probability in the unexposed

Risk Difference (x-y=z)

Survival Curve 

Mean
Risk Difference

 Risk in one group minus risk in the other 

 Risk = probability 

 ProbDrug A- Probplacebo = Risk Difference 

 0.84-0.17 = 0.67 

39 
Risk Difference, con’t 

 Probtreatment- Probcontrol = Risk Difference

 If the article gives you Risk Difference, it will often 

give you Probtreatment or Probcontrol 

 If article gives you Probtreatment, use that directly

 If article Probcontrol, use that and the Risk Difference 

to derive Probtreatment

40 
Converting to Probabilities?

Statistic Convert to probability?

 Probability/Risk

(aka Incidence Proportion)

Yes (it is already one, but use rates to convert the 

time period to which they apply) 

 Rate Yes

 Odds Yes

 Odds Ratio Yes, if the outcome is rare (≤ 10%) & you have 

probability in the unexposed

 Relative Risk (aka Risk Ratio) Yes, if you have the probability in the unexposed

 Risk Difference (x-y=z) Yes, if the paper reports x or y in addition to z

Survival Curve

Mean
Survival data and probabilities

 All previous probabilities were assumed to be 

constant throughout the model 

(Don’t have to be, but can be)

 Survival should NOT be assumed to be constant over 

time

 So, you will have multiple probabilities for “death” in 

your model – one for each time period of interest. 
Sources of survival data

 All-cause mortality (CDC)

– age- and sex-adjusted rates

 Disease-specific/Treatment-specific 

literature 

– Probability of death at t 

– Survival Curves

43 
Reported survival rates

Transform rate to a probability
Survival Rate  Probability 

Probability = 1 − ee(−rr )

Cycle Age Prob. of death

0 75 .0439 

1 76 .0439 

2 77 .0439 

3 78 .0439 

4 79 .0439 

5 80 .0709 

6 81 .0709 

7 82 .0709 

8 83 .0709 

9 84 .0709 

10 85 .1429 

11 86 .1429 

. . . 

. . . 

Rate Prob. of death

4493.7/100,000 =0.44937 4.39% 

7358.2/100,000 =.073582 7.09% 

15414.3/100,000 =0.154143 14.29% 

Age range CDC 

numbers

75-79 4493.7 

80-84 7358.2 

85+ 15414.3 
Disease-Specific Survival Data

 Kaplan-Meier Curve

– Unadjusted 

 Use with RCT data

 Cox Proportional Hazards Curve 

– Adjusted

 Use with observational data

46 
Survival Data from Curves

47 

Prob. of living at 

Month 18 ≈ 54% 

for control group
Deriving probability from mean 

(continuous distribution)

 1) Need a validated way to generate a binary variable 

from a continuous distribution -- threshold

– HbA1c < 7 = controlled diabetes

 2) Need an estimate of variation around the mean 

(SD, variance) or median (IQR, range)

– Involve a statistician!
Converting to Probabilities?

Outcome Convert to probability?

 Probability/Risk

(aka Incidence Proportion)

Yes (it is already one, but use rates to convert the time 

period to which they apply) 

 Rate Yes

 Odds Yes

 Odds Ratio Yes, if the outcome is rare (≤ 10%) & you have prob. in 

unexposed

 Relative Risk (aka Risk 

Ratio)

Yes, if you have the probability in the unexposed

 Risk Difference (x-y= z) Most likely, because they will give you x or y in addition 

to z

 Survival Curve Yes, but remember they are conditional and may change 

with each time period

 Mean Yes, if you have estimate of variation
Estimates of variation

 You will still need to derive estimates of 

variation around your derived point estimate of 

probability 

 Necessary for sensitivity analyses

 Advanced Topic

50 
Quality of the literature

 THE QUALITY OF THE LITERATURE MATTERS

GREATLY!!

 Preferences for literature-based inputs:

1. These two treatments studied in a head-to-head RCT

2. a) Drug compared to placebo in RCT, and 

 b) Diet/Exercise/Telehealth compared to 

 placebo in another RCT, and 

 c) these two RCTs enrolled similar patients

51 
Summary 

 Need to transform reported data to 

probabilities for use in a decision model – Easiest: Rate, OR if outcome <10%, RR, survival data

– More difficult, but possible: Continuous data with estimate of variation

– Advanced topics: OR when outcome > 10%, mean difference, standardized mean 

difference 

 Probs. apply to particular length of time

– To change the length of time to which a 

probability applies: 

 Probability  rate  probability

52 
References

 Miller DK and Homan SM. Determining Transition 

Probabilities: Confusion and Suggestions. Medical Decision 

Making 1994 14: 52.

 Naglie G, Krahn MD, Naimark D, Redelmeier DA, Detsky

AS. Primer on Medical Decision Analysis: Part 3 --

Estimating Probabilities and Utilities. Medical Decision 

Making 1997 17: 136.
Questions?

risha.gidwani@va.gov

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