1. Logical Inference on Treatment Efficacy When Subgroups Exist
Ying Ding
Department of Biostatistics
University of Pittsburgh
Joint work with Yue Wei and Xinjun Wang

2. Ding Y*, Wei Y, Wang X. Logical Inference on Treatment Efficacy When Subgroups Exist. Book Chapter In: Ting N, Cappelleri JC, Ho S, Chen DG. Design and Analysis of Subgroups with Biopharmaceutical Applications. New York: Springer, 2019 Forthcoming.

3. Targeted Therapies
Paradigm of medicine shifts from “one-fits-all" to “targeted therapies”.
Two aspects of targeted therapies:
Tailoring existing therapies to individual patients so that each patient can get the most “suitable” treatment; aims at identifying Optimal Treatment Regimes (OTR).
Develop new treatments that target a subgroup of patients; aims at identifying subgroups of patients with enhanced treatment benefit.

4. Targeted Therapy Development
Patient population is thought of as a mixture of two or more subgroups that may derive differential treatment efficacy.
In RCT, a new treatment Rx is compared to a control treatment C.
The “relative” treatment effect between Rx and C measures the treatment efficacy.
This process involves the measurement of efficacy in subgroups and combination of subgroups.
Subgroup 1
Subgroup 2
Subgroup 3
Subgroup 4
Marker -
Marker +

5. Which Group(s) Need to be Assessed?
Most of the GWAS are doing the case control studies using the binary outcomes
Until last year the first GWAS using the time-to-event outcome was published (in my project, I will focus on the bilateral time-to-progression outcomes).
Conclusion: The efficacy in all three groups (g+, g- and combined) are important in the targeted treatment development.

6. Logic-respecting Efficacy Measures
Definition: if the treatment efficacy for g+ is a, and for g- is b, then the efficacy for the combined group should be within [a, b] (assume a<b).
Intuitive but not fully recognized.
if a treatment is truly efficacious in g+ and in g-, then it should be truly efficacious in their mixture {g+, g-}.
The efficacy measures depend on outcome types.
Binary
Continuous
Survival (Time-to-event)

7. Efficacy Measures for Binary Outcomes
OR{g+, g-} ≠ OR g+ (= OR g- )
Conclusion: OR (Odds Ratio) is NOT logic-respecting.

8. Efficacy Measures for Binary Outcomes (Cont)
RR: Relative Response (ratio of response probability between Rx and C)
As shown in Lin et al. (2019) Biometrical Journal:
Conclusion: RR (Relative Response) is logic-respecting.

9. Efficacy Measures for Survival Outcomes
HR: (Hazard Ratio) between Rx and C
Even if both g+ and g- have constant HRs, the overall population typically does not have a constant HR
Conclusion: HR (Hazard Ratio) is NOT logic-respecting.

10. Efficacy Measures for Survival Outcomes (Cont.)
In Ding et al (2016) Stats In Med, it has been shown that the ratio (or difference) of median (or mean) survival (between Rx and C) are suitable efficacy measures for survival outcomes.
Logic-respecting
Direct clinical interpretation
For example, if the median survival time for patients randomized to Rx is 36 months and is 24 months for patients randomized to C, then the interpretation is that the Rx extends the median survival time for 12 (= 36-24) months or for 1.5 (36=24) times as compared with C.

11. Prognostic or Predictive?
In targeted treatment development process, researchers aim to identify “predictive” markers, rather than “prognostic” markers.
This “M&M” plot illustrates whether a biomarker is prognostic or predictive depends on the efficacy measure.

12. Inference on Mixture Populations
One common approach is to ignore the subgroup labels and use the marginal means.
Another popular approach is to apply the Least Squares means (LSmeans) technique indiscriminately for any type of outcome.
For example, applying LSmeans on the log RR from a log linear model to estimate the combination group's RR.
Unfortunately none of these approaches is correct.
The correct estimation procedure has to respect the logical relationship among efficacy parameters.

13. Subgroup Mixable Estimation (SME)
Originally proposed in Ding et al. (2016).
Key of the SME principle: if an efficacy measure is logic-respecting, then its estimation should be logic-respecting as well.
The principle is general, independent of the model and efficacy measure (so long as the efficacy measure is suitable and logic- respecting for mixture populations).

14. Marginal Means Can Produce Paradoxical Result

15. SME Principal
Key: instead of “mixing” the subgroup's efficacy g+ and g- to directly compute the mixture's efficacy {g+, g-} , one has to estimate the clinical response within each treatment (Rx and C) first, for each subgroup and their combinations.

16. Simultaneous Confidence Intervals
In targeted treatment development, clinical effect size matters and confidence intervals (CIs) are a lot more informative than mere p- values.
For example, a reduction in HbA1c between 0.8 and 1.2 is much more clinically meaningful than a reduction between 0.4 and 0.6. Yet the confidence intervals (0.8, 1.2) and (0.4, 0.6) can have identical p-values.
Simultaneous CIs to infer efficacy in g+, g- and {g+, g-}.
1-α simultaneous CIs I+, I-, I±:
This also respect the logic relationships among the true efficacy in g+, g- and {g+, g-}.

17. References
Ding Y*, Lin HM, Hsu JC. (2016). Subgroup Mixable Inference on Treatment Efficacy in Mixture Populations, with an Application to Time-to-Event Outcomes. Statistics in Medicine. 35(10): PMID:
Ding Y*, Li GY, Liu Y, Ruberg SJ, Hsu JC. (2018). Confident Inference For SNP Effects On Treatment Efficacy. Annals of Applied Statistics. 12(3):
Lin HM, Xu H, Ding Y, Hsu JC. (2019). Correct and Logical Inference on Efficacy in Subgroups and Their Mixture for Binary Outcomes. Biometrical Journal. 61(2): PMID:
Ding Y*, Wei Y, Wang X. Logical Inference on Treatment Efficacy When Subgroups Exist. Book Chapter In: Ting N, Cappelleri JC, Ho S, Chen DG. Design and Analysis of Subgroups with Biopharmaceutical Applications. New York: Springer, Forthcoming.

18. Thank you! Questions? yingding@pitt.edu