The regressions computed from the three designs are given in Tables 1(c) to 1(h). The first response model, E(Y\x) = 5x, is the correct model for the computer output in Tables 1(c), (d) and (e). The second response model, E(7|x) = llx - 6x2 is the correct model for the output in Tables !(/). (9) and (h). The output is taken from ST ATA.2 Most standard statistical software packages

From Table 1(c), the coefficient B is estimated as 5-58 (118) where (118) is the standard error. From 1(d), the estimate of B is 2-97 (1-25) and from 1(e), the estimate is 5-29 (1-41). These are close to the correct value of 5 (none is significantly different from 5). From Tables I(/) and 1(h), where the correct model is E(Y\x) = llx - 6x2, we note that the quadratic coefficient C cannot be estimated since there are only two doses which are given to the subjects. To fit a quadratic model at least three distinct values of x are needed. The estimates fit a straight line between dose x = 0 and dose x = 1 (or x = 075). Since E(Y\x = 1) = 5, and E(Y\x = 0) = 0, the slope is again 5, and the estimates 3-61 (0-94) and 6-74 (118) reflect that (from Tables 1(f) and 1(h)). The only design of the three which allows us to estimate C, (Table 1(0)) gives 13-30 (5-76) as the estimate for B and - 811 (5-52) as the estimate for C. Neither estimate is significantly different from this parameter, which are B = 11 and C = - 6, at the 0 05 significance level. The first and third designs do not permit estimation of some parameters. We note that the N/2 at 0 and N/2 at 1 is optimal for a simple linear regression with an intercept. It is not optimal for the model E(Y |x) = 5x when the intercept is known to be 0. We see in these examples that the choice of design for a particular

Determining patterns of dose responses is an important part of new drug evaluation. This may consist of a simple comparison of two levels (placebo and drug at some level), or may consist of placebo (dose x = 0) and multiple levels of the drug. We describe some of the design options the researcher has, and provide some guidance on choices. We consider first the equivalence of the two-group design and a two-dose design, and note some properties when the higher dose in the two-group design is less than the maximum dose in a dose-response design. We then review optimal designs and give the efficiencies of several candidate designs for the simple linear and quadratic regression models. For regression notation, we use capital letters (A, B, C) to denote the parameters, and lower case letters (a, b, c) to indicate their estimates. We assume normal errors with mean zero and variance a2 throughout. In all cases, we can use standard multiple regression software packages to estimate the parameters A, B, C and a.

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