Modelling Individual Response to Treatment and Its Uncertainty:A Review of Statistical Methods and Challenges for Future Research

Abstract

Clinicians often have to make treatment decisions based on the likelihood that individual patients will receive benefit. This information has to be derived from appropriate studies and provided in a way that easily transforms from the statistical model into clinical practice. This chapter discusses four methodological strategies to provide information for treatment decisions: subgroup analysis, regression models, models for potential outcome, and prediction models. Based on how statistical techniques are able to provide information for stratified, precision, or individualized medicine we present the challenges that exist in analysis, reporting and applying those results. We also identify relevant ethical issues. There is still a substantial gap between formal statistical results on patient treatment interaction and ways in which they can be presented to clinicians as easy to use tools in clinical decision-making. At the moment, most energy is spent in the discovery of biomarkers which help to grasp patient treatment interaction. Less energy is spent in corresponding validation studies and the process of translating statistical results into clinical practice. Furthermore, clinical studies that provide evidence for successful translation of predictive biomarkers into clinical practice are needed.

Statistical Excursus: Cox Regression and Logistic Regression

The Cox proportional Hazards model quantifies the effect of a treatment (G) and a biomarker (X) on an event time T by specifying the survival probability. P[T > t|X, G] represents the probability to survive the time t given biomarker measurement X and treatment G. The proportional hazards model is given by

$$ P\left[T>t\ |\ X,G\right]=exp \left[-{\int}_0^t{\alpha}_0(s)\bullet exp \left\{\ {\upbeta}_{\mathrm{G}}\cdotp \mathrm{G}+{\beta}_X\cdotp X+{\beta}_{GX}\cdotp G\cdotp X\right\}\ ds\right]. $$

Here α₀(s) describes an unspecific baseline hazard and exp{ β _G · G + β _X · X+ β _GX · G · X} quantifies the hazards ratio between subjects treated by IN (G = 1) versus subjects treated by ST (G = 0) and presenting a biomarker measurement of X. In a more general sense the model can also handle nonlinear influence of the biomarker on outcome by

$$ P\left[T>t\ |X,G\right]= exp \left[-{\int}_0^t{\alpha}_0(s)\bullet exp \left\{\ {\beta}_{\mathrm{G}}\cdotp {G}+{f}_X(X)+{f}_{GX}\ (X)\right\}\ ds\right]. $$

Here f _GX(X) = 0 for G = 0 and f _GX(X) = h _X(X) for G = 1. Using the survival function of the baseline group P[T > t |X = 0, G = 0] = P ₀[T > t], the above expression is equivalent to

$$ {P}\left[{T}>{t}\ |{X},{G}\right]={{P}}_0\left[{T}>{t}\right] exp \left\{{\beta}_{{G}}\cdotp {G}+{{f}}_{{X}}\left({X}\right)+{{f}}_{{G}{X}}\ \left({X}\right)\right\}. $$

The term β _G · G + f _X(X) + f _GX (X) or β _G · G + β _X · X+ β _GX · G · X are called linear predictors and written as LP(X,G). Therefore the above formulae translate to

$$ P\left[T>t\ |X,G\right]= exp \left[-{\int}_0^t{\alpha}_0(s)\bullet exp \left\{\ {LP}\left({X},{G}\right)\right\}\ {ds}\right] $$

and

$$ {P}\left[{T}> {t}\ |{X},{G}\right]={P}_0\left[{T}>{t}\right] exp \left({LP}\left({X},{G}\right)\right)\Big\}. $$

The logistic regression quantifies the effect of a biomarker measured by X and treatment defined by G on a binary outcome Y using the model:

$$ {P}\left[{Y}=1\ |{X},{G}\right]= exp \left\{{LP}\left({X},{G}\right)\right\}/\left(1+ exp \left\{{LP}\left({X},{G}\right)\right\}\right). $$

The linear predictor is different from the linear predictor of the Cox regression because it also contains a baseline term β₀:

$$ {LP}\left({X},{G}\right)={\beta}_0+{\beta}_{{G}}\cdotp {G}+{{f}}_{{X}}\left({X}\right)+{{f}}_{{G}{X}}\ \left({X}\right) $$

or

$$ {LP}\left({X,G}\right)={\beta}_0+{\beta}_{{G}}\cdotp {G}+{\beta}_{{X}}\cdotp {X}+{\beta}_{{G}{X}}\cdotp {G}\cdotp {X}. $$

The value exp{LP(X, 1)} quantifies the odds ratio (OR) of a person showing a biomarker measurement of value X and being treated by the innovative treatment versus a person showing a biomarker measurement of value X = 0 and being treated by the standard. The odds ratio for a person with biomarker measurement X = x ₀ between both treatments is

$$ exp \left\{{LP}\left({{x}}_0,1\right)-{LP}\left({{x}}_0,0\right)\right\}= exp \left\{{{h}}_{{X}}\left({X}\right)\right\}. $$