Showing papers in "Biometrika in 1948"

A table for the calculation of working probits and weights in probit analysis

Abstract: The estimation of the parameters of a distribution of individual tolerances, from data relating to numbers of subjects manifesting a characteristic quantal response at different levels of a stimulus, is a problem frequently encountered in the application of statistical science to dose-mortality studies, biological assay, detonation of explosives, and other problems. A typical situation is that of exposing batches of insects to various doses of an insecticide, recording the proportion killed at 'each level of dose and then requiring to estimate the mean tolerance (or median lethal dose) of individual insects and the variance of the tolerance distribution. Gaddum (1933) and Bliss (1935a, 6; 1938) have been instrumental in developing a method, that of the probit transformation, which greatly simplifies the calculations necessary to the estimation. The exact statistical analysis appropriate to the transformation was first shown by Fisher (1935), and the theory and uses of the method have been discussed fully in many subsequent publications (Finney, 1947a, b). Tables required in the practice of the method, in sufficient detail for most purposes, have been given by various writers (Fisher & Yates, 1943; Finney, 1947 a). Occasionally, however, the statistician needs values of the various functions at finer intervals of the argument, and for his benefit the following Table has been prepared. A brief account of the tabulated functions will suffice for all who are familiar with the probit method; those who require fuller information on the theory and analysis should consult the list of References. Given a proportion P, and its complement Q = 1 — P, the probit of P is, to all intents and purposes, the deviate from the mean which divides the normal curve of unit variance in the ratio P: Q. In the formal definition, however, 5 is added to the deviate in order to avoid the necessity of computing with negative numbers. The advantage of this modification may be questioned, but it is now well established and will be adopted here. The probit, Y, of the proportion P is thus defined by

The precision of observed values of small frequencies

Abstract: In recent genetical work numerous observers have recorded the frequencies of rare events, notably mutations. It has been realized that it is misleading to state the observed frequencies with their standard errors, since the distribution is decidedly skew. Various devices have been suggested to avoid this difficulty. But so far as I know it has not been pointed out that, when the frequency is small, its cube root is almost normally distributed. This will be proved and applied to actual observations. Let a rare event be observed in a out of n trials, where n is much greater than a2. Let x be the true value of the frequency, whose observed value is p = a/n. Let the a priori distribution of x be dF = 0q(x)dx.

The studentized form of the extreme mean square test in the analysis of variance.

Abstract: t= (= I) n/s. As was to be expected, this distribution was independent of o-. The knowledge of the distribution of t made it possible to draw inferences, with the help of evidence entirely supplied by the sample, about the location-parameter 4a of a normal population, without making any assumptions about the generally unknown scale-parameter, oc. Neyman & Pearson (1928) extended this notion to an analogous problem connected with the rectangular and exponential populations. Fisher (1924) obtained the distribution of s'/s where s' and s are two independent estimates of oc, calculated by the root-mean-square method. This distribution is also independent of o. Fisher also showed that 'Student's' t-distribution could be extended to a wide range of sampling problems. Sukhatme (1937) has shown that analogues to t and s'/s of 'normal theory' can be developed for an exponential population, again eliminating the unknown scale parameter. This notion of eliminating unknown scale parameters from the distribution laws of statistics has come to be known as 'studentization'.

Testing the significance of correlation between time series

Abstract: Verification that two things are related to each other is achieved by showing empirically that there is a correspondence between their behaviours greater than could be expected by chance. Where continued experiment is not possible the data available are usually severely limited in both quantity and range of variation and it therefore becomes essential to have precise ideas whether the agreement with a hypothesis is actually a verification of it or whether it is more reasonable to consider the agreement as merely the result of a chance correspondence. This paper discusses the problem of deciding when a correlation between two economic time series is great enough to make it unreasonable to assume that the series are unrelated. The testing of the significance of a correlation involves a comparison with what would have been obtained between non-related series thought to be analogous to the observed series. And, of course, the significance found for the correlation will depend upon the analogy deemed to be appropriate. The choice of an analogy depends upon experience as to which aspects of the real series being correlated are vital, in the sense that they affect the probability of obtaining chance correlations between non-related series. We can never be certain that some important aspect has not been overlooked, but, as our experience is broadened and we learn to take into account more and more factors, the chances of our running into a situation in which the analogy we choose is actually misleading becomes less and less. If we were forced to base our choice of an appropriate analogy to use in any individual situation on the data of that situation alone, the uncertainty of our tests of significance would be very great. Usually, however, the situation is one which we have learned to be similar to some larger class of experiences, and it is this larger class of experiences that furnishes a greater measure of information as to the analogy appropriate to all members of the class. The most commonly used sampling model for generating series of independent terms is to draw them at random from a normal population of values; and in applying tests based upon this sampling model one must take account of the length of the series being dealt with. Fortunately, there is some evidence that tests of significance based on this sampling model are insensitive to variation of the frequency distribution of the population of values from which the random sampling is done (Pearson, 1931), and this makes it reasonable to apply such tests even when little is known of the frequency distributions from which the items of our real series have been drawn. There is, however, one obvious point at which the analogy underlying such tests of significance may break down when one js concerned with economic time series, namely, if the consecutive terms are really correlated. In economic time series, in meteorological time or spatial series, or, for that matter, in biological time series, autocorrelation usually exists. Production, or employment, or price-level series never go directly from

The power function of the test for the difference between two proportions in a 2×2 table

Abstract: Neyman & Pearson's (1933) conception of the power of a test of a statistical hypothesis, Ho, was developed, in the first instance as a means of guiding the choice between alternative tests. This, it was shown, could be done by comparing the effectiveness of the tests in discriminating between Ho and a set of admissible alternative hypotheses regarded as most relevant to the question under test. Where there is no doubt about the most appropriate test and no sequential scheme of sampling is possible, the power function may play a useful part in indicating, before the data are collected, how large the samples should be to avoid an inconclusive result. If this procedure is to be easily applied, a ready means must be available of calculating the power of the test for a given significance level and sample size. The tables of the power function of the t-test (Neyman & Tokarska, 1936) and Tang's Tables (1938) applicable in the Analysis of Variance, are examples of such aids. The present paper aims at providing in simple, if approximate, form a means of determining the power function of the test for the difference between two proportions. The test may be briefly outlined as follows. In two 'infinite' populations the proportions of individuals possessing a character A are pl(A) and p2(A) respectively. Random samples of m and n are drawn from the two populations and the result is represented in a 2 x 2 table, thus: