The minimal clinically significant difference is often used to design phase III trials, and is highly related to the number needed to treat (NNT). This is the number of patients one would be willing to treat with a drug to prevent one bad outcome. For example, a urologist may biopsy 10 men to find one high grade prostate cancer.
There are problems with NNT as it’s primarily calculated on a population level, and ignores heterogeneity in the population. https://discourse.datamethods.org/t/problems-with-nnt/195 If we define a minimum NNT rather than an average NNT, we may use the metric to determine who should be treated from their baseline risk.
Let \(x_i\) = {Risk of outcome for patient i without intervention}, and assume the distribution of risk is given by \(X \sim f_X(x)\).
Now imagine there is an intervention that reduces an individual’s risk by \(1 - \gamma\) in relative terms. Therefore, if patient \(i\) were to receive the intervention, their individual risk would go from \(x_i\) to \(x_i\gamma\).
The risk difference with and without intervention can be represented by \(Z = X - X\gamma = X(1 - \gamma)\). The distribution of \(Z\) is \(f_Z(z)\) given by: \[ \begin{align} f_Z(z) = \frac{\partial}{\partial z}F_Z(z) = \frac{\partial}{\partial z} Pr(Z \le z) = \frac{\partial}{\partial z} Pr(X(1 - \gamma) \le z) &=\\ \frac{\partial}{\partial z} Pr\Big(X \le \frac{z}{1 - \gamma}\Big) = \frac{\partial}{\partial z} F_X\Big(\frac{z}{1 - \gamma}\Big) = f_X\Big(\frac{z}{1 - \gamma}\Big) \frac{1}{1 - \gamma} \end{align} \]
To assess the proportion of patients whose personal risk was reduced by \(t\) percentage point or fewer (absolute reduction), we merely calculate the integral.
\[ \int_{0}^{t} f_Z(z) dz = \int_{0}^{t} f_X\Big(\frac{z}{1 - \gamma}\Big) \frac{1}{1 - \gamma} dz \]
We would only want to include patients in a trial that are expected to experience a risk reduction of at least \(t\) percentage points, which depends on the effectiveness of the treatment.
As we learn more about the effectiveness of the treatment, we can adpatively modify the inclusion criteria of the trial to include/exclude patients who are likely to experience a benefit of at least \(t\) percentage points. By selecting patients to enroll in a trial that will indeed have a meaningful benefit, it’s likely a trial can be completed with fewer patients. The trial, however, may take longer to enroll when being more selective who may enroll.
We will also allow for stopping early for success and for futility.