Revisiting the relationship between baseline risk and risk under treatment
© Wang et al; licensee BioMed Central Ltd. 2009
Received: 07 August 2008
Accepted: 17 February 2009
Published: 17 February 2009
In medical practice, it is generally accepted that the 'effect model' describing the relationship between baseline risk and risk under treatment is linear, i.e. 'relative risk' is constant. Absolute benefit is then proportional to a patient's baseline risk and the treatment is most effective among high-risk patients. Alternatively, the 'effect model' becomes curvilinear when 'odds ratio' is considered to be constant. However these two models are based on purely empirical considerations, and there is still no theoretical approach to support either the linear or the non-linear relation.
Presentation of the hypothesis
From logistic and sigmoidal Emax (Hill) models, we derived a phenomenological model which includes the possibility of integrating both beneficial and harmful effects. Instead of a linear relation, our model suggests that the relationship is curvilinear i.e. the moderate-risk patients gain most from the treatment in opposition to those with low or high risk.
Testing the hypothesis
Two approaches can be proposed to investigate in practice such a model. The retrospective one is to perform a meta-analysis of clinical trials with subgroups of patients including a great range of baseline risks. The prospective one is to perform a large clinical trial in which patients are recruited according to several prestratified diverse and high risk groups.
Implications of the hypothesis
For the quantification of the treatment effect and considering such a model, the discrepancy between odds ratio and relative risk may be related not only to the level of risk under control conditions, but also to the characteristics of the dose-effect relation and the amount of dose administered. In the proposed approach, OR may be considered as constant in the whole range of Rc, and depending only on the intrinsic characteristics of the treatment. Therefore, OR should be preferred rather than RR to summarize information on treatment efficacy.
A simulation approach  based on numerical models of drug action has explored the whole range of Rc and suggested a curvilinearity at least in some settings. However, there is still no theoretical approach to support such a non-linear relation. Consequently, we propose an alternative approach to revisit the relationship between Rc and Rt, based on a phenomenological model that includes the possibility of integrating both beneficial and harmful therapeutic effects.
Presentation of the hypothesis
Where R corresponds to the probability of the outcome, β0 and β1 are the intercept and slope of linear function respectively, and E is the pharmacodynamic effect (or similarly a risk factor of the spontaneous outcome).
where E is the pharmacodynamic effect, E0 represents baseline value of E, Emax is the maximum theoretical effect, D is the dose, ED50 is the dose at which 50% of the maximum effect is achieved, and γ is the sigmoidicity parameter.
a model with Rc as independent variable, Rt as dependent variable, and β1, Emax, ED50, D, and γ as parameters.
Therefore, OR which represents the size of treatment efficacy, does not vary with Rc. OR is here a constant only determined by pharmacotherapeutic parameters. This remains also true in case of an active control therapy (D ≠ 0).
where i = 1 for benefit and i = 2 for harm.
Then from models (7) and (8), for a treatment with one or two effects respectively, Rt and other measures of treatment effect, such as RR and ARR, can be easily translated and the relationships between baseline risk and treatment effect can be graphically simulated over the whole range of Rc (Figure 2C). The U-shape relation between Rc and Rt indicates that the patients with moderate risk obtain most from the treatment while those at highest risk benefit less or not at all. For a treatment involving two independent mechanisms, a harmful effect is observed in low-risk groups, and treatment effects under different doses show the potential difficulties of choosing an optimal dose for a patient, especially for those in low baseline risk.
Testing the hypothesis
Two approaches (retrospective and prospective) can be proposed in order to investigate in practice such a non-linear 'effect model'. The retrospective one is to perform a meta-analysis of clinical trials with subgroups of patients with a great range of baseline risks. An alternative is to conduct a meta-analysis of individual patient data, a practical way of detecting differential treatment effects among various risk groups.  The prospective approach is to perform a large clinical trial ('megatrial') or several trials, in which patients are recruited according to several diverse risk groups that are prestratified depending on the information taken from previous small trials . Severe patients, who are usually excluded from clinical trials, will be then preferentially included to detect the real treatment effect in high-risk conditions.
Implications of the hypothesis
These results are however strongly dependent on the underlying models considered. Logistic regression, one of a class of models known as generalized linear models, is a type of predictive model that can be used with two types of target variables : a categorical one that has exactly two categories (i.e., a binary or dichotomous variable), or a continuous one that has values in the range 0 to 1 representing probability values or proportions; explanatory variables can be either categorical or quantitative. The log odds is then a linear function of the explanatory variable(s), leading to an OR constant. This model is now used extensively in the medical science, where a clinical outcome is in most cases binary (e.g. death/alive, event/no event). The dose-concentration-effect model (e.g. Hill model) is based on the law of mass-action and on receptor occupancy theory, supposing a reversible drug pharmacodynamic effect which is related to the clinical (binary) outcome. Depending on the biological effects of drugs and their mechanisms of action, such assumptions may not be always valid, particularly in case of irreversible mechanisms, tolerance/rebound phenomena, and synergistic/antagonist effects.
For most patients (e.g. those in low or moderate risk), one can assume a linear relation with an absolute benefit proportional to the baseline risk. However for high-risk patients, appropriate data are required to show up whether it remains proportional or decreases to zero. The values of OR and RR may consequently differ according not only to the level of risk under control conditions (Rc), but also to the characteristics of the dose-effect relation and the amount of the dose chosen. In the proposed approach, OR may be considered as constant in the whole range of Rc, and depending only on the intrinsic characteristics of the treatment. Therefore, OR should be preferred to summarize information on treatment efficacy.
We are grateful to Michel Cucherat, and Michel Lièvre, for helpful discussions.
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