Given Level Of Significance Research Articles

A conservative Sample size for the comparison of several proportions

In many experimental situations we need to test the hypothesis concerning the equality of parameters of two or more binomial populations. Of special interest is the knowledge of the sample sizes needed to detect certain differences among the parameters, for a specified power, and at a given level of significance. Al-Bayyati (1971) derived a rule of thumb for a quick calculation of the sample size needed to compare two binomial parameters. The rule is defined in terms of the difference desired to be detected between the two parameters. In this paper, we introduce a generalization of Al-Bayyatifs rule to several independent proportions. The generalized rule gives a conservative estimate of the sample size needed to achieve a specified power in detecting certain differences among the binomial parameters at a given level of significance. The method is illustrated with an example

An Algorithm for Determining the Distribution Function of the Durbin-Watson Test Statistic

IN REGRESSION ANALYSIS most empirical economists use the well-known Durbin-Watson (DW) procedure [1] to test the hypothesis of no autocorrelation among the disturbances of a linear regression model against the hypothesis of a first-order autocorrelation. The use of this procedure is compromised by the fact that it is a bounds text and, hence, cannot discriminate between the two competing hypotheses for a range of intermediate values of the test statistic. This shortcoming can be eliminated by determining the distribution function [5] for the Durbin-Watson test statistic and enumerating it for a given level of significance and a particular regression matrix. The authors have written a FORTRAN IV program for finding the probability that the DW test statistic is less than the observed value if the null hypothesis of no autocorrelation were true. The program enables the investigator to perform the DW test for either positive or negative correlation by comparing the above probability to a specified level of significance. This procedure provides a conclusive test for first-order autocorrelation. The procedure begins with a transformation of the Durbin-Watson test statistic stated as

Physical comparison of gamma camera and two-headed scanner: figures of merit and times required for threshold detection of spherical sources

Physical comparisons are described between a scintillation camera and a two-headed scanner for their efficiency in detecting radioactive spheres in tanks of water containing radioactivity at different concentrations. For these comparisons, the previously defined figure of merit has to be modified to use “counts recorded per integrating time, i.e. time spent over the abnormal region” instead of “per minute”. A new quantity, “merit-time-product”, is defined and used for this purpose. It is then possible to compare scanners moving at different speeds with each other or with cameras, on the same basis. Results are presented showing the total times required for each device to scan an area, so that a given level of significance is achieved in the threshold detection of active spheres in the presence of background radioactivity. The scintillation camera requires shorter times. The differences are small for 131I, but exceed factors of 20 in overall time for 99mTc. These factors decrease with depth in the scintillation camera, from 20 to 30 at 7 cm to about 10 at 10 cm depth. Possibly the most striking difference between the camera and the two-head scanner is the high efficiency of the camera for relatively shallow tumours and the rapid decrease of efficiency with depth.

On a Locally most Powerful Boundary Randomized Similar Test for the Independence of Two Poisson Variables

By definition $(X, Y)$ is a bivariate Poisson vector if $(X, Y) = (X^\ast + U, Y^\ast + U)$ where $X^\ast, Y^\ast$ and $U$ are three independent Poisson variables with, say, respective expectations $a, b$ and $d$. Let $(X_n, Y_n)n = 1, 2, \cdots, N$ be independent observations on a bivariate Poisson vector $(X, Y)$. It is shown that no test for the independence of $X$ and $Y$ can be both boundary randomized similar in $a$ and $b$ and also uniformly most powerful. However, a test of the form \begin{equation*} \varphi_0(Z\mid s, t) = \begin{cases} 1 \\ \gamma 0 \\ 0 \end{cases} \text{according as} Z = \sum^N_{n = 1} X_nY_n \gtreqqless k_0(N, s, t)\end{equation*} given $S = s$ and $T = t$, where, $\sum^N_{n = 1} X_n = S,\quad\sum^N_{n = 1} Y_n = T$, is boundary randomized similar and locally most powerful. Using a lemma on the convergence to a Normal probability distribution function of the conditional probability distribution function of $\sum^N_{n = 1} (X_n - sN^{-1}) (Y_n - tN^{-1})$ given $S = s$ and $T = t$, asymptotic formulae for the values of the $k_0(N, s, t)$ corresponding to a given level of significance are derived. In addition, it is shown that the asymptotic power of the test can be obtained from an approximation to a Normal probability function and that, in case instead of $k_0(N, s, t)$ its value calculated from the asymptotic formulae is used, the modified test is asymptotically locally most powerful in the sense of Definition 2. To extend the domain of application of this test, we replace $U$ in the definition of the bivariate Poisson vector by another random variable $W$ also taking non-negative integral values according to the probability function $P\{W = w \mid \sigma \geqq 0\} = \int^\infty_0 e^{-\sigma t}\frac{{(\sigma t)}^w} {w!} f(t) dt$ where $f(t)$ is any continuous probability function satisfying $\frac{\partial}{\partial\sigma} \int^\infty_0 e^{-\sigma t} \frac{{(\sigma t)}^w}{w!} f(t) dt \bigg |_{\sigma = 0} = \int^\infty_0 \frac{\partial}{\partial\sigma} \big\{e^{-\sigma t} \frac{{(\sigma t)}^w}{w!}\big\}|_{\sigma = 0} f(t) dt$ $\sigma \geqq 0, w = 0, 1, 2, \cdots$ and $\int^\infty_0 tf(t) dt < \infty$. In this way a class of bivariate probability functions is obtained such that for every member of this class also, regarded as a probability function of the random vector $(X, Y)$, the locally most powerful boundary randomized similar test for the independence of the two random variables $X$ and $Y$ is the same as the one given in the Poisson case.

The reliability of a point biserial coefficient of correlation

An attempt is made to elaborate upon Lev's results concerning the reliability of the point biserial coefficient of correlation in a manner that will be helpful to the psychological statistician. Procedures required in the use of the non-centralt tables prepared by Johnson and Welch are described as they relate to the determination of the fiducial limits for a point biserial coefficient. A normal approximation technique for the estimation of fiducial limits is also suggested. Numerical evidence is presented which shows that relative to a given level of significance the width of the fiducial interval estimated from a point biserial coefficient of any size is smaller than that of the fiducial interval corresponding to an ordinary Pearsonian coefficient of the same magnitude.

Most Powerful Tests of Composite Hypotheses. I. Normal Distributions

For testing a composite hypothesis, critical regions are determined which are most powerful against a particular alternative at a given level of significance. Here a region is said to have level of significance $\epsilon$ if the probability of the region under the hypothesis tested is bounded above by $\epsilon$. These problems have been considered by Neyman, Pearson and others, subject to the condition that the critical region be similar. In testing the hypothesis specifying the value of the variance of a normal distribution with unknown mean against an alternative with larger variance, and in some other problems, the best similar region is also most powerful in the sense of this paper. However, in the analogous problem when the variance under the alternative hypothesis is less than that under the hypothesis tested, in the case of Student's hypothesis when the level of significance is less than $\frac{1}{2}$, and in some other cases, the best similar region is not most powerful in the sense of this paper. There exist most powerful tests which are quite good against certain alternatives in some cases where no proper similar region exists. These results indicate that in some practical cases the standard test is not best if the class of alternatives is sufficiently restricted.

ACCURACY OF THE LOCAL‐LESION METHOD FOR MEASURING VIRUS ACTIVITY. IV. SOUTHERN BEAN MOSAIC VIRUS

DETAILED PROCEDURES for estimating the relative potency of drugs from the response induced in test animals by injection of graded doses were worked out by Bliss and Marks (1939a and 1939b). These procedures make it possible to calculate not only the most probable relative activity of the drug in terms of a standard, but also the limits within which the true activity will lie for any given level of significance. In recent work (Spencer and Price, 1943; Price and Spencer, 1943a, 1943b), the procedures of Bliss and Marks were applied in determining the accuracy of measuring the relative activities of certain plant viruses. From the results it was found that the relative activities of tobacco mosaic, tobacco necrosis, and tobacco ringspot viruses could be estimated with errors not exceeding 10, 11, and 14 per cent, respectively, under the prevailing experimental conditions and that the standard error of estimate was a reliable measure of the experimental error. The results with alfalfa mosaic virus were less satisfactory; the standard error of estimate could not be counted upon to give a reliable estimate of the true error. The present paper describes the results obtained when essentially the same procedures were applied in measuring the relative activity of southern bean mosaic virus (Marmor laesiofaciens Zaumeyer and Harter). THE VIRUS.-Southern bean mosaic virus was first described by Zaumeyer and Harter (1942, 1943) who found all tested varieties of bean (Phaseolus vulgaris L.) to be susceptible. In some varieties, reddish necrotic lesions about 1 mm. or less in diameter were produced on inoculated leaves, and the virus was localized in these lesions. In other varieties no symptoms appeared on inoculated leaves, but the virus became systemic and caused a mottling disease in the young leaves. More recently, a strain of the Blue Lake variety was found to develop a systemic necrosis when infected with the virus (Zaumeyer and Harter, 1944). Mottling type varieties serve as a means of propagating the virus and of obtaining it in relatively large quantities. Local lesion type varieties offer a means of determining the relative activity of virus samples used in experimental work. In the present work, the virus was propagated in the Bountiful variety of garden bean and its activity was determined by inoculation of the Early Golden Cluster variety.2 The virus samples used in the rela-

Drift of Net Assimilation Rate in Plants

IN reply to the letter from R. F. Williams1, may I say that I much regret not having mentioned his paper2 and that of Ballard and Petrie3 in my note on the effect of age on net assimilation and relative growth-rates in the cotton plant4, but I have just seen them for the first time. In stating that the absence of general trend of net assimilation rate up to time of flowering in my experiments was not proved, I was merely mentioning an inherent characteristic of the statistical test of significance. In testing the significance of a linear regression coefficient, the hypothesis is made that its true value is zero. This hypothesis may be disproved, at a given level of significance, in which case a significant general trend is shown, but it can never be finally proved. My results were clearly consistent with the hypothesis that the linear regression of net assimilation rate on time up to first flowering was zero, that is, that there was no general rise or fall, and it would seem, therefore, that to claim that my data confirmed the findings of Gregory5 for barley was justifiable. Furthermore, mzy results agreed with those obtained for cotton by Crowther6 in the Sudan, although no rigid statistical test was in the latter case applied.