Partitioned Survival Analysis (PartSA) has been used often in the context of evaluating oncology treatments. PartSA models differ from state progression models in an important way. For detail description of PartSA you can refer to the following article:

The following model built in TreeAgePro (PartSA_2Markov_1DES v1.trex) can be used to see how State Transition models (both Markov and DES) differ from PartSA model. The "PartSA Markov" strategy still uses Markov node but simply iterates through single state for the duration of the analysis. The calulcations of areas under the curves and between the curves is implemented through Event Rewards. The same Overall Survival and Progression Free Survival (Exponential distributions with 2 different lambda parameters) are used in all three models to draw Time - To - Event samples for DES, to calculate Transition Probabilities for Markov State Transition Model and to calculate the partitions of the cohort in Progression Free (PFS) Area, Post Progression (PP) Area under these curves.

The models use 2 different costs and 2 different utilities for PFS and PP areas. These values can be changed to represent different scenario. The model can be evaluated with microsimulation or samping+microsimulation. It is possible to run Cohort Analyses on the individual Markov nodes (DES nodes require microsimulation).

You can change the values of the distribution parameters to see how the results generated by the models change. When Lambda for OS (e.g. 0.04) is small and Lambda for PFS (e.g. 0.7) is order of magnitude or more larger than Lambda OS than the results between State Transition Models (Markov and DES) and the Partitioned Survival Analysis model are relatively close. However, when LambdaPFS and LambdaOS get close to each other, you will get results that are quite different between PartSA model and the State Transition models (DES and Markov). Try LambdaOS of 0.4 and LambdaPFS of 0.7 and you will see that results between PartSA and Transition model are diverging.

In case of Exponential survival curves the exact calculations for the area's under the curve are possible with closed form formulas. Refer to the attached Excel document which calculates the areas under the exponential curves and the corresponding costs and utilities for reference.

ExponentialPartitionedAnalysis.xlsx

For further background on the mathematical formulas behind the PartSA refer to:

PartitionedSurvivalAnalysis v2.pdf

In conclusion the PartSA is a different technique for estimating Costs and Utilities of treatment. It makes different simplifying assumptions about the patient histories in Post Progression state(s). In particular using the same Survival Curves for PartSA and State Transition Models will lead to increasingly different results, where OS and PFS curves converge close to each other.

PartSA models with different survival curves can be easily accommodated in TreeAgePro using either tables (using the empirical Kaplan-Meier curves) or other parametric distributions (e.g. Weibull, etc.). Additional survival curves could be used to partition the cohort into additional groups using analogous area under the curve calculations.

## Comments

Tristan SnowsillExcellent article and example, thank you! Could you please explain why you choose to use event cost and effectiveness as opposed to cycle cost and effectiveness?

Al ChrosnyThe event rewards (cost, effectiveness, etc) are not subject to cycle correction (e.g. traditional HCC or within cycle correction WCC) so they allow for a more explicit (not necessarily simpler) formulation of custom rewards. Since cycle (HCC) and time (WCC) rewards are designed for computing area under reward function (HCC - rectangular, WCC - trapezoidal), there is some more care that would need to be done to express the cycle / time rewards in an appropriate way.

By using the event rewards I focused on the fundamental math behind the partitioned survival analysis. The cycle rewards can certainly be used instead, so the model with event rewards can act as a reference. It is always re-assuring to have several different implementation generating consistent results.

Tristan SnowsillGreat explanation, thanks!

Mohamed Ismail ABDUL AZIZ (MOH)Thanks for the example file! Is it also fair to say that in the Markov state transition model, it is assumed that the probability of death for those who have progressed is similar to those who have not progressed?

Al ChrosnyIn general you should not make the assumption that the probability of death should be similar. However, the Partitioned Survival technique uses Overall Survival to express probability of death for both pre- and post- progression cohort. So in that sense PartSA is using "similar" probabilities of death for both pre and post progression states.

Obviously if you had data that differentiated these survivals, you could create a better informed model and would not need to resort to Partitioned Survival analysis simplifications.

Andrew MunzerPlease note that we will be adding direct support for Partitioned Survival Analysis as a new model type in July, 2019 with TreeAge Pro 2019, R2.

Tina CalentiHow would you incorporate a fixed palliative care cost for every death in a partitioned survival model (i.e., a fixed amount/same amount for every death regardless of length of time spent in PFS or progressed disease)? Thanks

Al ChrosnyThank you for an interesting question. Indeed the topic of Death cost or more generally State Entry or Exit costs within PartSA comes up. Mathematically speaking the Entry or Exit costs are not a direct function of the area under (or between) survival curve(s), they are a function of the derivative of the survival curve. In case of death cost it would be the derivative of the Overall Survival curve that determines how many people die per unit of time.

Surv(@T(n)) - Surv(@T(n+1)) => This gives you the number of people that die between time T(n) and T(n+1). So you could multiply that proportion of people dying in the time period by the fixed Death Cost.

An alternative approach might be to consider that PartSA method was really meant to be strictly an area under the curve method. However, it is morphing rapidly with all sorts of innovative additions, e.g. landmarking, state entry costs, etc.

Just a hypothetical question. For all other states the PartSA uses some sort of average per unit time costs that are then integrated over time. So actual cost is the function of the area under (between) curve(s). Isn't it a methodologically more consistent to treat Death costs on average bases and apply them to the area "above" the OS curve. I do not know the answer as to which approach is better, but it might be a good topic for research.

Finally, TreeAge Pro 2019 R2.0 scheduled for release in July will introduce a completely new support for building PartSA models, new PartSA node type and new PartSA analysis and reports. All of that will be integrated within the other capabilities of sensitivity analysis, etc. You will be able to convert a PartSA model to a Markov model structure. Also the Markov models will be able to generate PartSA like output (the survival curves).

Please reach out on e-mail to support@treeage.com if you would like to learn more about the TreeAge Pro 2019 R2.0 release.

Andrew MunzerWhen our PartSA implementation is released in July, it will have several ways to integrate cost and utility - time-based, interval-based, discrete times, etc.

I don't believe that the first release will include transition cost (i.e., death cost) specifically. However, we have built models with slightly complex formulas that would achieve that end. Likely a subsequent release would handle the transition cost more directly.

Andrew MunzerLooks like Al and I responded at the same time - his response more complete.

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