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By B. J. T. Morgan (auth.)

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27. 2 we shall employ the method-of-scoring approach to obtain the maximum likelihood estimates of (oc, {3). Later in the book we shall consider a range of more complicated models. g. Adby and Dempster, 1974) and computer routines are provided by computer packages and libraries such as: NAG, IMSL, BMDP, SAS, MLP, GLIM and GENST AT. ) The simplex method ofNelder and Mead (1965), available as NAG routine E04CCF, is extremely useful as it does not require any information other than the expression for the log-likelihood, and we shall see several examples of its use later in the book.

If P; denotes the probability that any individual responds at the ith dose, then assuming independence between individuals within doses, and between doses, we obtain the likelihood as the following product of binomial terms: We may model the probabilities of response by assuming a location and scale model, with P; = F(a +/3d;) for the location/scale pair of parameters (a, /3) and (usually) cumulative distribution function F( ) of some continuous random variable. The likelihood, L, then becomes a function, L(a, /3), of the two parameters, a and /3.

Later in the book we shall consider a range of more complicated models. g. Adby and Dempster, 1974) and computer routines are provided by computer packages and libraries such as: NAG, IMSL, BMDP, SAS, MLP, GLIM and GENST AT. ) The simplex method ofNelder and Mead (1965), available as NAG routine E04CCF, is extremely useful as it does not require any information other than the expression for the log-likelihood, and we shall see several examples of its use later in the book. Some numerical procedures use finite differences to approximate derivatives, and this is done in procedure 3R in BMDP for example.

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