For the case with K — 2 and x = 0 and 1, the above produces the sample size for a two-group design with equal number observations in each group, with the required sample size in each group as N/2. This is the usual formula for the two-sample problem.

Given the above, the investigator can consider several issues. First, if the investigator is certain that the response is linear and the maximum dose is the one which will be administered to patients, the optimal two-group design allocates half of the sample to the 0 (placebo) dose and half to 1 and no multiple dosage design (K > 2) is more efficient in the sense that no other design can have a smaller variance of b. However, the high dose may have some toxicity which the investigators want to avoid. The alternatives are to reduce the high dose to a lower dose at x < 1 or to conduct a study with multiple doses so that fewer patients receive the highest dose. If we compare a dose-response study that has equal allocation to equally spaced doses between 0 and 1 (a uniform design) to a two-group design at 0 and x, can the dose-response design have a smaller variance of b than the two-group design where the slope B is still — /¿0 and the total sample size N does not exceed that of the two-group design? (Answer: Yes). Where does the dose-response design begin to be better? (Answer: it depends on x). How does the number of doses relate to the maximum level (x) in the two-group case? (Partial answer: if x < 1/^/3 any multiple point design wins). How do the variances of b and c compare when the doses are evenly spaced versus optimal design placement? (Answer: see below). It is sometimes proposed to have fewer observations at dose = 0 and dose = 1 for ethical reasons (fewer patients at dose = 0 to have more patients receiving something, fewer at dose = 1 to have less potential toxicity). What is the effect of reducing the number of observations at the extreme doses on the variances of the parameter estimates? (Answer: the increase in variance can be pretty bad).

If the corresponding maximum dose given in the two-group design is x = 0-5 (for example, 500 mg in a 1000 mg maximum dose study), the change in mean response would be half that of the maximum. The sample size required (Table II for K = 2) would be multiplied by 4, and 172 patients (86 per group) would be required. This is larger than a dose-response design with 10 levels of drug, and one would clearly prefer a dose-response design. If the maximum dose for two-group dose design is 0-75, then the number of patients required would be 76 (38 per group) and the two group design would require about as many subjects as a dose-response design with four dosages (at 0, 0-33,0-67 and 1). A two-group study and a dose-response study will have the same sample size if the formulae for N are equal. Some algebra shows that a two-group study where the drug is given at dose x will have the same sample size requirement as a dose-response study with K doses equally spaced from 0 to 1 if x2 = (K + 1)/[3(K - 1)]. This leads to Table III as a table of equivalent sample size studies.

This can be interpreted to mean that if the maximum dose in the two-group study is 0-745, a four level study with doses at 0, 0-33, 0-67 and 1 will provide estimate of B with the same precision. Assuming equal spacing, we can show that the equivalent dose is never less than 0-577 (1/^/3). These results suggest that 3 to 5 levels in a dose-response study will provide most of the gain when the two group dose is less than the maximum dose in the dose-response study. When

a the dose to be studied in the two-group study is the maximum dose that would be administered in a dose-response study, the dose-response study does not provide a gain in power or precision of estimate. If the maximum dose in a two-group study is less than 0-8, an increase in power can usually be realized with a dose-response study.

Dose-response (regression) designs enable us to compare the response to different drug levels and evaluate the responses for possible curvature. If we assume a quadratic response, E(Y\x) = A + Bx + Cx2, we can find optimal designs which minimize var(c), var(y(x)) for a given x, or the generalized variance (the determinant of the covariance matrix of the estimates). Here, y(x) is the predicted value of y given x. These designs will provide estimates of the curvature (that is, the quadratic coefficient) and also allow us to estimate the linear dose-response. We next discuss these concepts.

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