n is the number needed per dose. Thus, nK ^ N B/t7 = (/i, - Ho)/o half of the observations at each level is the optimum one for a linear regression E(Y |x) = A + Bx (in the sense that var(b) is minimized). It may, however, be important to have fewer than half of the observations at the placebo or high dose levels (for example, for increasing the chance of receiving a hoped for effective treatment or for ethical reasons such as reducing the risk of side-effects), so the investigator might wish to place some observations at 0, some at 1, and the remainder at intermediate doses. The experimenter may even unbalance the design by making the number of observations at each dose unequal. Similarly, the 0 dose might be increased to x0. This would not be a placebo controlled study, but would still be able to demonstrate effectiveness of x at x = 1 if there were an increasing response for increasing x. In the phase III drug approval context, regulatory agencies expect some of the dose to be at the level for which approval is sought. Thus, a dose-response study with doses at 0, 0-5 and 10 would not suffice for an approval at 0-75. Generally, dose-response studies are done in phase II. We will examine some examples of these when we consider the efficiencies of the designs in later sections.

Assume bivariate observations, (y, x), are taken, where x is the dose given and y is the response to the drug. The dose begins at 0 (placebo) and has a largest value of 1 (this can always be handled by appropriate scaling). With K equally spaced doses, we have values of x at 0, 1 /(K — 1), 2/(K — 1),..., (K — 2)/(K — 1), 1. If we assume a straight-line model, we have where B is the amount of increase in E(Y |x) for a one unit increase in the dose, x. That is, B is still Hi — Ho, the mean difference between the drug at x = 1 and the placebo at x = 0. The variance of Y given x is <r2, and var(b) = o-2/E(xf — x)2.

With an equal number of observations and equal spacing between them, the denominator of var(b) is a2N(K + 1)/[12(K — 1)] where N is the total sample size and N = nK, and n is the sample size per group. This is another example of a uniform design. For testing H0: B = Hi — Ho = 0 against a two-sided alternative Hi. B # 0, the sample size needed to detect a slope of B is given by where a is the significance level, and is the probability of a type II error and Zx is the upper 100 x th percentile of the standard normal distribution.

One can use the above formula to compute the needed sample size. For example, to detect B/it = 1 ((/*! — Ho)/g = 1) with a. = 0-05, and 1 — /? = 0-9, the required sample sizes are given in Table II. The sample sizes are rounded to integer values. In practice the sample size N should be increased so they are divisible by K.

When K > 2, the multiple x values permit us to examine curvature or a polynomial response. The equally spaced doses are not necessarily the optimally spaced x values for such fits. In practice, if there is confidence that the model is not much wrigglier than a quadratic, the value of K should not be much higher than 3 or 4 because of the considerably larger sample size requirements.

Table III. Equivalent maximum two-group dose for K equally spaced doses

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