By D. Collett (auth.)
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Additional info for Modelling Binary Data
13) could still be used here, the normal approximation is more accurate when the value of the standard error of Pl - P2 under the null hypothesis is used in the denominator of z. To obtain this standard error, suppose that the common value of the success probability under Ho is p. Each of the two sets of binary data provide information about this common success probability, and so it is natural to combine the data from each data set to give a pooled estimate of p. 12), and will only differ appreciably when the numbers of observations in the two samples are very different, or when Pl and P2 differ considerably.
This is just one, since the sequence of observations is 0000 .... Now, ( n) o n! 1 = O! x n! = ill and for this to be equal to one, O! must be defined to be equal to one. This extends the definition of the factorial function to all non-negative integers. The total probability of obtaining a sequence of y ones and n - y zeros, independent of the ordering of the binary observations, will be (;) multiplied by the probability of any particular arrangement of the y ones and n - y zeros. The probability of y successes in n observations is therefore given by for y = 0, 1 ...
The vertical axis exhibits the binomial probabilities for each value of y on the horizontal axis. has an approximate normal distribution with zero mean and unit variance, known as the standard normal distribution. This distribution will be denoted by N(O,1), and is the distribution used in many of the approximate methods described below. The adequacy of this approximation depends on the degree of asymmetry in the binomial distribution, and hence on the value of both nand p. McCullagh and NeIder (1989) state that the approximation will be satisfactory over most of the range of the distribution of Y when np(1 - p) ~ 2.
Modelling Binary Data by D. Collett (auth.)