Choosing inclusion criteria that minimize the time and cost of clinical trials

Figure 1 Separation of patient response phenotypes to a tested treatment according to an aggregate predictive variable, x. The fraction of type 1 responders to the right of the cutoff is the true positive fraction. The fraction of type 2 non-responders to the right of the cutoff is the false positive fraction. In this general example the units of x are arbitrary.

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Figure 2 Calculation of power from probability density distributions for the null hypothesis (H0) and for an alternative hypothesis (H1). The dashed line shows critical value for significance (1.96 for two-tailed P < 0.05). The area under the thick curve to the right of the critical value is the statistical power of the test of H0.

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Figure 3 A sample receiver operating characteristic curve for a hypothetical screening test. In this example type 1 patients had screening scores, x, with a mean of 5.5 and a standard deviation of 1; type 2 patients had screening scores, x, with a mean of 4 and a standard deviation of 1. As the cutoff value xc is swept from 1.0 toward zero, a family of true positive and false positive fractions is created to generate the receiver operating characteristic (ROC) curve.

Figure 4 Numbers of patients screened and enrolled in a model study of heterogeneous responders having a statistically significant positive result. For this model the proportion of type 1, good responders q = 0.2, the response probability for type 1 patients, π₁ = 1.0, the response probability for type 2, poor responders, π₂ = 0. The response probabilities for both phenotypes to the control treatment, π₃ and π₄ both equal 0.2. The mean value of the z statistic for the alternative hypothesis is 2.96 (84% power for the trial). The proportion of patients, α, assigned to the experimental group is 0.5.

Figure 5 Cost estimates in a model study of heterogeneous responders. Cost constants in thousands of dollars are as follows: screening cost per case c₁ = 1, treatment cost c₂ = 10, opportunity cost c₃ = 100/yr, case rate r = 50/yr, follow up time t =1 yr. Other details as in Figure 4.

Figure 6 Cost estimates in a scenario with good responsiveness to the control treatment in patients who are non-responsive to the experimental treatment. π₁ = 1.0, π₂ = 0, π₃ = π₄ = 0.4. Other details as in Figure 5. Dashed line divides the x-domain into regions of a significant negative effect (to the left) vs a significant positive effect (right). Near x_c = 4.4 the cost of disproving the null hypothesis when it is exactly true becomes infinite.

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Figure 7 Fraction of patients. A: Separation of observed responders and non-responders to the experimental treatment along the x-domain in this reconstructed preliminary study. The fraction of patients with each x-value is show on the vertical axis. Patients with x-scores over 60% have a much greater likelihood of responding; B: ROC curve for the screening procedure.

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Figure 8 Regression analysis on the last two columns of Table 3. A plot of the hybrid variable, y = pC(u)(π + u), vs u can be used to evaluate model parameters π₃ and π₄. The slope of the regression line is π₄, and the intercept divided by π is π₃.

Figure 9 Cost estimates in a realistic test data set for targeted drug therapy of lung cancer, presented in Tables 2 and 3. Cost constants in thousands of dollars are as follows: screening cost per case c1 = 1, treatment cost c₂ = 10, opportunity cost c₃ = 100/year, case rate r = 50/year, follow up time t =1 year. Cost to the right of the dashed vertical asymptote are for a significant positive result (experimental treatment better than control). Costs to the left of the dashed vertical asymptote are for a significant negative result (experimental treatment worse than control).

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