Showing papers in "Biometrics in 1959"

The estimation of environmental and genetic trends from records subject to culling.

Abstract: A very common problem which arises in animal or plant breeding research is that of assessinig the gain which has resulted from a selection program carr'ied oIn ovei a number of years. To be specific, let us suppose that we have a closed dairy herd which has been maintained over a number of years with selection being practiced. The records available for assessing any genetic improvemient consist of production records of cows in the various years and can be represented by a twoway classification, cow by year. At first it might be thought that such a two-way classification could be analyzed by the method of fitting constants [Yates, 1934]. Applications of this technique have, however, led to the apparent conclusion that the environment gradually deteriorated over the period of years, as indicated by the fact that the constants fitted for years tend to decrease year by year. Heinderson [1949] pointed out that a least squares procedure in which the cow effects are regarded as fixed leads to biased estimates. Lush and Shrode [19501 gave a simple explanation of the biases arising in the estimation of age correction factors; similar considerations apply to the estimation of year effects. The present paper is the combined

A multiple comparison rank sum test: Treatments versus control.

Abstract: Problems of applied research have necessitated the investigation of multiple comparison procedures. Such investigations have been carried out almost entirely within the framework of the analysis of variance. Since the assumptions underlying the analysis of variance are not always valid, distribution-free or non-parametric procedures have been developed for data arising from a number of experimental designs. Most such procedures do not provide for multiple comparisons. This paper presents a rank sum multiple comparison test for comparing treatments with a control, when the data are from a one-way classification. Error rate is experimentwise. An experimentwise error rate is, by definition, the ratio of the number of experiments with one or more false significance statements to the total number of experiments. Thus, in computing this error rate, the experiment is the unit; the experiment which leads to a single false significance statement is rated no differently than the one in which all comparisons are falsely declared significant. If we set the error rate at a, then 1 a gives the probability that no false statements of significance will be made, in other words, that all statements will be correct when the null hypothesis is true. In an experiment where k independent comparisons are to be made, it is customary to use a

A Simple Method for Constructing Orthogonal Polynomials When the Independent Variable is Unequally Spaced

Abstract: Orthogonal polynomials are commonly used in the analysis of variance for the construction of orthogonal contrasts among equally spaced levels of a treatment factor. The existence of tables [1; 2] giving the compounding coefficients for these particular contrasts often influences the experimenter to choose an equal spacing. A simple procedure for constructing orthogonal polynomials when the levels are unequally spaced frees the experimenter from this sometimes undesirable restriction.

Experimental Design in the Evaluation of Genetic Parameters

Abstract: Rather surprisingly, there appears to be no discussion in the literature of the best design for the estimation of intraclass correlations and, consequently, in the context of quantitative genetics, of heritabilities. Put rather more specifically, if we have facilities for the measurement of a certain number of objects with a hierarchical classification, what is the most efficient group size for the estimation of intra-group correlations? We shall be dealing here with the situation in which, in any experiment, all groups are of the same size. As more work is now being done on the quantitative genetics of laboratory animals, in which the family structure is rather more under the experimenter's control than in farm animals, it seemed valuable to put these results on paper.

A Unified Theory for Quantal Responses to Mixtures of Drugs: Non-Interactive Action

Abstract: In 1952 we considered how the quantal responses in groups of organisms can be expressed as functions of the doses of two poisons administered together, and what we wrote applies to drugs in general. Our aim was a general theory for the interpretation of data of this kind, a theory properly related to the underlying mechanisms of joint drug action. So far as we know, advance in this field has since been confined to Ashford's [1958] alternative method of deriving an equation for simple similar action. Sampford [1952] made progress in the allied field of response-time distributions for drug mixtures. Despite criticisms (see discussion) we maintain that our approach to the problem was sound, though now think that it was not sustained adequately because, as explained below, the results lacked unity. This paper is the first of a series in which a unified theory is presented. Many of the equations obtained before remain, but as special cases of new ones; the latter clear away difficulties, and should enable a wider variety of data to be interpreted. The biological mechanism of joilnt action can vary according to the pair of drugs, and the relation between the quantal response and the jointly applied doses differs according to this mechanism. Thus in our previous paper (Plackett and Hewlett [1952]) we set up biological models of joint action, and deduced the relations, i.e. mathematical models, from them. The s;et of biological models sprang from a twoway classification. The joint action was defined as similar or dissimilar according as the sites of primary action in the organism were the same

The Regression Analysis of Causal Paths

Abstract: The purpose of this presentation is to acquaint biologists and biometricians with an important tool, path analysis. This tool can be of help in dealing with complex causal networks. These often, though not always, prove amenable to common regression technics. Path analysis, originated by Sewall Wright [1918], is a convenient approach to regression problems involving two or more regression equations. For those unskilled in statistics, path analysis provides one method of depicting regression problems by simple diagram. The path diagram, commonly representing the flow of cause and effect, often permits one to write estimators of parameters immediately upon inspection. Path analysis thus facilitates the process of abstraction for both mathematician and biologist. The analytic process is here explained, two computational algorithms (rules-of-thumb) are given, and an example involving feedback is detailed. Inclusion of feedback, and thus homeostasis, is an important feature of this presentation. Since Wright's early work [1918, 1921, 1924, 1934, and others] the treatment of multiple equations has been extensively developed in econometrics (see especially Hood and Koopmans, [1953]) but generally without use of the standardized regression coefficients used by Wright or of the path diagrams and algorithms which characterize Wright's technic. Wright himself [1921] used unstandardized coefficients and the term path regression, but in general [1954] has favored the standardized form. Tukey [1954] in a critical review pointed out advantages in working with unstandardized regression coefficients. Recently Kemp-

An Approach to the Analysis of Data for Semi-Quantal Responses in Biological Assay

Abstract: In a typical biological assay a population of living organisms is exposed to varying quantities of the material under investigation and the reaction of the various individual members of the population is subsequently correlated with the applied dose. The methods of statistical analysis used to handle data of this kind may be divided into two broad classes, depending on whether the reaction of the individual organism is expressed in quantitative or quantal terms. In certain circumstances it is apparent that the underlying reaction is essentially continuous, although owing to difficulties of observation it can only be measured in terms of a quantal response. When these conditions apply, the observed quantal response may be defined by one or more subdivisions of the underlying reaction scale. A single subdivision corresponds to a "binomial" response, the subject being classified as "responding" or "not responding" according to whether or not the underlying reaction is in excess of some specified value. Two or more subdivisions correspond to a "multinomial" or "semi-quantal" response. The problem of relating an observed quantal response to an underlying quantitative reaction has been considered briefly by Finney [1952] and, in more detail, by Hewlett and Plackett [1956] who show that the quantal dosage-response relationship may be derived from the corresponding quantitative dosage-response relationship. The ideas put forward by Hewlett and Plackett, though plausible, have not, however, been verified experimentally in any particular bioassay situation. Aitchison and Silvey [1957] describe a method of analysis for multiple response data, but these authors do not consider the problem from the point of view of an underlying quantitative reaction. The purpose of this paper is to examine the application in reverse of some of the results given by Hewlett and Plackett, first to determine

An Examination of Some Methods of Comparing Several Rates or Proportions

Abstract: The investigation of such problems as the association between smoking and lung cancer, between congenital malformations and exposure to radiation, or between coronary disease and obesity, usually requires quantitative estimates of the increase in the "risk" of the condition under study (Sheps [1958]). Several rather different methods of measuring changed risks have been used by investigators. In the field trials of poliomyelitis vaccine (Francis [19551), for example, the incidence of paralytic poliomyelitis per 100,000 was 16 (12) among vaccinated children and 57 (Ii) among those who had received placebo injections. The effectiveness of the vaccine was estimated as 100(1 12/I1) = 100(1 16/57) = 71.9 per cent protection. Hammond and Horn [1954] compared death rates of smokers (D18) with the mortality rates observed in their sample of nonsmokers (DO), by deriving the ratio D8/Do . Cornfield developed a method (Cornfield [1956], Neyman [1955]) for estimating an analogous "relative risk" from retrospective data on the smoking history of men with and without cancer. He defined relative risk as C8/Co where C. and C0 represent the prevalence of lung cancer among smokers and nonsmokers respectively. On the other hand, Berkson [1958] estimated the effect of smoking on mortality as the difference between two mortality rates, i.e., D. Do. For example, the standardized mortality rates from lung cancer observed by Doll and Hill [1956] were 1.66 and 0.07 per 1000 for heavy smokers and nonsmokers respectively. Rather than consider this a "relative risk" of 1.66/0.07 = 23.7, Berkson calculated the additional rate of 1.66 0.07 = 1.59 per 1000 smokers. He contended that his comparison was more appropriate since it treated the increased risk as proportional to those who would have survived if they had not smoked, rather than treating the risk as proportional to the deaths. As will be shown below, the meaning of the described statistics

Confidence Limits for the LD 50 using the Moving Average-Angle Method

Abstract: The use of moving averages as a smoothing procedure is well known to statisticians. Moving average interpolation for LD50 estimates was proposed by Thompson 11947] and recently extended by Bennett [1952] to include the angular transformation. The purpose of this transformation is, of course, to make the variance of an observed proportion largely independent of the unknown true proportion. Thompson and Weil [19521 and Weil [1952] have published formulas and tables useful in the calculation of approximate confidence limits for the LD50 without reference to the angular transformation. The following sections present "exact"* confidence limits and significance tests under the moving average-angle procedure, when the doses are separated by a constant interval (almost always a log interval in bioassay) and an equal number of animals has been exposed to each dose. Under the moving average-angle method, formulas to handle differing intervals and/or numbers of animals may be easily derived when needed (cf. Bennett, [1952]). The observed proportional response at each dose is transfoimecd to an angle, 0(p) = aresin -Vp, using, for example, Table XII in Fisher and Yates [1948]. When p = 0 or 1, Bartlett's adjustments (cf. Eisenhart, [1947]) should be applied, namely 0(0) = arcsin V/1/4n and 0(1) = 90 0(0), where n is the number of animals at each dose. Then simple averages of A; successive angles are computed, where A; is usually 3 or 5. Each average angle is associated with the middle dose of the respective set of A; doses. The LD50 is estimated by linear interpolation between the two

Use of Contingency Tables in the Analysis of Consumer Preference Studies

Abstract: A consumer preference study involving three varieties of snap beans was conducted at Mississippi State College. One lot of each of the three varieties (V1 , V2 , and V3) was displayed in retail stores, and each of n consumers was asked to rank the beans according to first, second, alnd third choices. The data obtained in one store on one day are presented in Table 1. I was asked if the usual x2-test with four degrees of freedom could be used to test for independence of varieties and ranks, i.e., that each variety had the same chance (4) of receiving a given rank, regardless of rank.

Some Recent Results in Chi-Square Goodness-of-Fit Tests

Abstract: The aim of this paper is to relate and extend some recent work on chi-square goodness-of-fit tests. There is no discussion of any problems which are specifically associated with more than one categorical variable. The main topics are the effect of estimation on chi-square and its partitions and their relation to Neyman's smooth goodness-of-fit tests, and the effect of grouping a univariate distribution according to the disposition of the sample on the distribution of the chi-sqcuare statistic and on the smooth test statistic.

Calculation of chi-square to test the no three-factor interaction hypothesis

Abstract: A technique is demonstrated which is particularly well suited for modern high-speed computers. The technique involves the superimposition of Newton's method of functional interaction onto the procedure suggested by Horton. (W.D.M.)

Pair Comparison, with and without Ties

Abstract: A probabilistic model for pair comparison, in various experimental settings, and with and without admission of ties, is described. When discrimination is the objective, admission of tied decisions theoretically increases the power of the test of the null hypothesis, but in practice this may be offset by a decrease in the subject's efficiency of decision, and in these circumstances it is better to prohibit ties. When preference is the objective, ties should be admitted as they add information.

The comparison of the sensitivities of similar experiments. model ii of the analysis of variance

Abstract: The appropriate scale of measurement may not be apparent in many experimental situations and this is particularly true when subjective appraisals of samples are required. In this paper we consider the possibility that two scales of measurement, or two experimental techniques, are under consideration. Preliminary experimentation may be required to decide on the better of the two scales of measurement. The better scale of measurement, or the more sensitive one, is by definition the one that better demonstrates treatment effects (Model I of the analysis of variance) or the existence of a between-treatments component of variance (Model II), depending on the model used. We assume preliminary experimentation in the form of two parallel exoeriments, appropriate for use of analysis of variance, that are distinct and independent in probability but which use similar experimental designs with separate samples from the same treatments. Cochran [1943] and Carlin, Kempthorne, and Gordon [1956] have discussed this situation earlier as have the present authors. This paper is not directly concerned with the relationships that may exist between two scales. Various such relationships are discussed by Cochran and, when the form of a functional association between scales is known, special techniques for comparing the sensitivities of the scales may be devised. The usual situation appears to be the one in which scales are monotonically related but in which the relationship is otherwise unknown. In order to have meaningful experiments where

The analysis of experiments on growth rate.

Abstract: comparable groups of growing subjects over a period of time. The attribute of interest to the experimenter-e.g., liveweight-is measured initially and at regular intervals during the experimental period. The results of such an experiment are most commonly considered in terms of average growth rate-e.g., gain in liveweight per week-and this quantity is most simply obtained by dividing the difference of the initial and final measurements by the duration of the experiment. Experiments of this type were discussed by Wishart [1938, 1939]. He pointed out that this method of assessing growth rate ignored any information provided by the intermediate measurements, and that other features of the growth curve besides the average slope might merit consideration. Difficulties arise, however, because successive measurements on the same subject cannot be considered independent. Wishart accordingly suggested that the growth curve for each subject should be broken down into its mean and linear, quadratic, etc., components, each of these being subjected to separate analysis. The effects of treatments on the average growth rate could be seen from the analysis of the linear components, and analyses of the further components would show to what extent the treatments were affecting the shapes of the growth curves. In an experiment of short duration, or when the treatment effects are small, a summary solely in terms of average growth rate will often be adequate. When a single treatment is compared with a control, this is equivalent to assuming that the treatment effect-the average difference between treated and control subjects-increases linearly with time. It is natural to estimate this effect from the difference between the linear components of the growth curves. However, in many types of

Extra-Period Change-Over Designs

Abstract: Designs involving cyclical arrangements of treatments are frequently used in experiments extending over several periods in order to reduce errors due to variation between experimental units. These designs are known as change-over designs. In the simplest type of change-over design the sequences of treatments are determined by the columns of one or more Latin squares. Any Latin square can be used when the effects of treatments do not persist beyond the period of application. Frequently, however, it is desirable to take residual effects into account. Change-over designs which provide for the estimation of first residual effects, i.e., residual effects in the period immediately after application, have been devised by Cochran, Autrey, and Cannon [1941] and Williams [1949] using special Latin squares. The residual effects are estimated by introducing additional constants into the ordinary analysis of Latin squares. For details of these designs and their analysis reference can be made to the review of the subject by Cochran and Cox [1957, p. 133]. Designs which permit the number of periods to be less than the number of treatments have been investigated by Patterson [1951, 1952]. These designs consist of series of incomplete Latin squares and are analysed accordingly with provision for first residual effects. Yates [1951], Lucas [1957], and Cochran and Cox [1957] have pointed out that in the analysis of change-over designs (a) residual effects are less accurately determined than direct effects

Optimum Group Size in Half-Sib Family Selection

Abstract: When indirect methods of selecting breeding stock have to be used there is often a limit to the number of offspring or other relatives that can be examined. Alan Robertson in a recent paper [1957] discussed the problem of deciding how many sires should be tested on how many offspring when selecting by progeny testing with a given size of test population. He shows that a general solution can be arrived at if the number of offspring tested for each sire is expressed in terms of the fraction of sires kept for breeding. If there are S sires kept for breeding and N animals tested all told, n for each sire, the proportion of sires selected, p, is nS/N since N/n is the number of sires tested. N/S, the number of tested animals for each sire kept, is Robertson's testing ratio, K. It turns out that the genotypic superiority of selected sires is proportional to V1 + (alpK) Z/p where a is (4 h2)/h2, h2 is the fraction of variation that is genetic, and Z/p the superiority of selected sires in standard measure. This expression can be differentiated to give a value of K in terms of p, which maximises the genetic superiority of selected sires. Robertson states that exactly the same formulae will hold in half-sib family selection. This is no doubt true enough for most practical purposes but is not strictly true. In progeny testing when a sire is chosen on the performance of his offspring, each being by a different dam, he is chosen to be used again on other dams, or if he is not to be used again his offspring, by dams other than those in the test, are to be kept. It is the assessment of the sire's genotype which is decisive. The dams of his progeny are irrelevant, except as a possible source of error or bias in his evaluation; they cannot contribute to his genotype. On the other hand, in the half-sib family selection system the dams are not irrelevant since it is members of the half-sibship that are used again or kept and, provided the dams

The Use of the Power Function to Determine an Adequate Number of Progeny per Sire in a Genetic Experiment Involving Half-Sibs

Abstract: Genetic experiments which involve large animals are often limited in scope by factors outside the experiment. Due to a lack of sufficient time, manpower, and space, in addition to the expense of such undertakings, there are few extensive genetic experiments using large animals. Instead, the experimenter must use whatever data or animals are available, and he needs to know when he has an adequate number of observations per genetic group to undertake a genetic study concerning an estimate of heritability of a magnitude of practical importance. A common genetic group used in studies of large animals consists of paternal half-sibs. Such is the case in most progeny tests, and one of the most common methods of estimating heritability is a function of the paternal half-sib correlation. However., the reported estimates of the heritability of a given quantitative trait are often as divergent as the bounds (zero to one). Consequently the geneticist is led to wonder what constitutes an adequate sample of a sire's genes. This paper proposes a method for determining the number of observations required per genetic group in order to detect significant differences among groups when the heritability of the trait is at least as great as some predetermined, minimum magnitude.

Host Variability in Dilution Experiments

Abstract: Moran [1954a, b] and Armitage and Spicer [1956] have discussed the theory of experiments in which doses containing different numbers of infectious particles are administered to groups of host organisms, and the proportion of hosts infected at each dose is recorded. Suppose that, for any one host, a proportion p of the particles will initiate a fatal infection on any one occasion, and let p be a random variable with density function f(p). The problems commonly discussed are those of estimating certain characteristics of f(p), and of testing the null hypothesis, Ho , that p is constant. Armitage and Spicer [1956] stated (pp. 408, 413) that a maximum likelihood approach to the estimation problem was inappropriate, in particular because the expected values, on Ho , of the second derivatives of the likelihood function do not all converge. This view appears now to be incorrect, and the purpose of this paper is to discuss the maximum likelihood approach to this problem, and to describe tests of Ho appropriate in various circumstances.

Sensory Item Sorting

Abstract: A probabilistic model for dichotomous sensory sorting is described; it covers many experimental designs including the simplest, pair comparison. In a theoretical analysis of the probabilities involved, distinction is drawn between the perception and the characterization of an objective difference. In some circumstances a faculty of matching is postulated. Furthermore, if the contrasted stimuli differ in magnitude, the probability of correct ranking has to be invoked. Finally, there is the concept of a probability of preference, of interest for its own sake or as a means of testing discriminability.