Hi Tristan, thank you for sharing knowledge and these great lectures. Can you please also create and share similar tutorials about partitioned survival models - focusing on extrapolating the survival curves, calculating state membership and patient simulation, in Excel and R.
Hi there! I'm having troubles understanding the applicability of these survival functions for Markov modeling. For example, if I'm analyzing the cost-effectiveness of ACALABRUTINIB vs. IBRUTINIB as a second-line treatment, most oncology Markov models would have three states ("Progression-free," "PD," "Death"). Using these survival functions, how can I obtain the time-varying transition probabilities for these three states?
Yes, lots of oncology models are not in fact Markov models but instead use a technique called Partitioned Survival Analysis (PartSA). There is a good report on this technique at www.sheffield.ac.uk/nice-dsu/tsds/partitioned-survival-analysis - it is not necessarily a good technique to use but it has the benefit of being easy to implement. If you have patient-level data, e.g., times and censoring indicators for disease progression and death, then for Markov modelling you would: (1) Fit a parametric model for the time between progression and death, which would give you the transition probability for PD -> Death; (2) Fit a parametric model for the time between initiation of 2nd line treatment and death, censored for progression (so this is just people dying without disease progression) and this gives you the transition probability for Progression-free -> Death; (3) Fit a parametric model for the time between initiation of 2nd line treatment and progression, censored for death, which gives you the transition probability for Progression-free -> PD. If you don't have patient-level data (e.g., you only have Kaplan-Meier curves or somebody has provided you with progression-free survival [PFS] and overall survival [OS] parametric models) then I would probably use PFS to determine the probability of remaining in Progression-free, then have two (or more) parameters to be estimated through calibration, which are the proportion of transitions out of Progression-free which are to Death, plus a parametric model for post-progression survival (the transition probability from PD -> Death).
@TMSnowsill Thank you! These are very helpful videos. Could someone please help me get the source of this formula:- p' = 1-(1-p)^T'/T. Also, I want to understand the mathematics behind 51,809 deaths and 70,971 person-years. Tia.
Hi! You can derive the formula like this: given a risk p over time T, we estimate the rate r. r = -ln(1-p)/T. Now this same rate is assumed to apply over a different period, T', so it should also be true that r = -ln(1-p')/T'. Note that -ln(1-p)/T = -ln((1-p)^(1/T)). So -ln((1-p)^(1/T)) = -ln((1-p')^(1/T')), multiple both sides by -1 and exponentiate both sides, (1-p)^(1/T) = (1-p')^(1/T'), raise both sides to the power T', (1-p)^(T'/T) = (1-p'), now simple addition/subtraction gets you to the desired formula. For the second part, you can get a fairly accurate answer if you just calculate the number of people alive at the beginning of each calendar day and the number of people dying, given that 200/100,000 will die each day. That gives you 51,844 total deaths and 71,020 person-years lived. For full accuracy for the number of deaths we would use the formula risk = 1 - exp(-rate*Time). The rate is 200 per 100,000 person-days, i.e., 365 * 200 per 100,000 person-years, so 0.73. Time is just 1 (year). So risk = 1 - exp(-0.73) = 0.518091, which when multiplied by 100,000 gives us 51,809 deaths. The person-years lived can be calculated as the area under the curve for the number of people alive at a given time, which is exp(-0.73*t). The area under this curve is (1-exp(-0.73))/0.73 = 0.70971, which we multiply by 100,000 (the total cohort at the start) to get 70,971 person-years.
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Hi Tristan, thank you for sharing knowledge and these great lectures. Can you please also create and share similar tutorials about partitioned survival models - focusing on extrapolating the survival curves, calculating state membership and patient simulation, in Excel and R.
Thank you so much!!! I have been confused with these questions for a long time. Look forward to your more videos!!
Glad it was helpful!
Hi there!
I'm having troubles understanding the applicability of these survival functions for Markov modeling.
For example, if I'm analyzing the cost-effectiveness of ACALABRUTINIB vs. IBRUTINIB as a second-line treatment, most oncology Markov models would have three states ("Progression-free," "PD," "Death").
Using these survival functions, how can I obtain the time-varying transition probabilities for these three states?
Yes, lots of oncology models are not in fact Markov models but instead use a technique called Partitioned Survival Analysis (PartSA). There is a good report on this technique at www.sheffield.ac.uk/nice-dsu/tsds/partitioned-survival-analysis - it is not necessarily a good technique to use but it has the benefit of being easy to implement.
If you have patient-level data, e.g., times and censoring indicators for disease progression and death, then for Markov modelling you would: (1) Fit a parametric model for the time between progression and death, which would give you the transition probability for PD -> Death; (2) Fit a parametric model for the time between initiation of 2nd line treatment and death, censored for progression (so this is just people dying without disease progression) and this gives you the transition probability for Progression-free -> Death; (3) Fit a parametric model for the time between initiation of 2nd line treatment and progression, censored for death, which gives you the transition probability for Progression-free -> PD.
If you don't have patient-level data (e.g., you only have Kaplan-Meier curves or somebody has provided you with progression-free survival [PFS] and overall survival [OS] parametric models) then I would probably use PFS to determine the probability of remaining in Progression-free, then have two (or more) parameters to be estimated through calibration, which are the proportion of transitions out of Progression-free which are to Death, plus a parametric model for post-progression survival (the transition probability from PD -> Death).
@TMSnowsill Thank you! These are very helpful videos.
Could someone please help me get the source of this formula:- p' = 1-(1-p)^T'/T. Also, I want to understand the mathematics behind 51,809 deaths and 70,971 person-years. Tia.
Hi! You can derive the formula like this: given a risk p over time T, we estimate the rate r. r = -ln(1-p)/T. Now this same rate is assumed to apply over a different period, T', so it should also be true that r = -ln(1-p')/T'. Note that -ln(1-p)/T = -ln((1-p)^(1/T)). So -ln((1-p)^(1/T)) = -ln((1-p')^(1/T')), multiple both sides by -1 and exponentiate both sides, (1-p)^(1/T) = (1-p')^(1/T'), raise both sides to the power T', (1-p)^(T'/T) = (1-p'), now simple addition/subtraction gets you to the desired formula. For the second part, you can get a fairly accurate answer if you just calculate the number of people alive at the beginning of each calendar day and the number of people dying, given that 200/100,000 will die each day. That gives you 51,844 total deaths and 71,020 person-years lived. For full accuracy for the number of deaths we would use the formula risk = 1 - exp(-rate*Time). The rate is 200 per 100,000 person-days, i.e., 365 * 200 per 100,000 person-years, so 0.73. Time is just 1 (year). So risk = 1 - exp(-0.73) = 0.518091, which when multiplied by 100,000 gives us 51,809 deaths. The person-years lived can be calculated as the area under the curve for the number of people alive at a given time, which is exp(-0.73*t). The area under this curve is (1-exp(-0.73))/0.73 = 0.70971, which we multiply by 100,000 (the total cohort at the start) to get 70,971 person-years.