Showing papers in "Biometrics in 1955"

Tests for Linear Trends in Proportions and Frequencies

Abstract: One frequently encounters data consisting of a series of proportions, occurring in groups which fall into some natural order. The question usually asked is then not so much whether the proportions differ significantly, but whether they show a significant trend, upwards or downwards, with the ordering of the groups, In the data shown in Table 1, for, instance, the usual test for a 2 X 3 contingency table yields a x2 equal to 7.89 on 2 degrees of freedom, corresponding to a probability of about 0.02. But this calculation takes no account of the fact that the carrier rate increases with the tonsil size, and it is reasonable to believe that a test specifically designed to detect a trend in the carrier rate as the tonsil size increases would show a much higher degree of significance.

Further contributions to the theory of paired comparisons

Abstract: 1. When a pair of objects is presented for comparison and the two are placed in the relationship preferred: not-preferred, we have what is known as a paired comparison. A set of n objects can be compared, a pair at a time, in some or all of the possible n(n 1)/2 ways of choosing a pair, and the set of paired comparisons so derived gives us a picture of the interrelationships of the objects under preference. A pairedcomparison scheme is more general than a ranking; for with the latter A-preferred-to-B and B-preferred-to-C automatically ensures A-preferred-to-C, whereas with paired comparisons it might happen that C was preferred to A. The existence of these departures from the ranking situation may be due to various reasons, such as the fact that 'preference' is a complicated comparison being made with reference to several factors simultaneously; and one reason for using paired comparisons is to give such effects a chance to show themselves. 2. Situations often occur in which a set of m observers express preferences among n objects and we have to select that object, or perhaps that sub-set of objects, which are, in some sense, "most preferred." The simplest case is the one where there are only two objects, A and B, and every observer votes for either A or B as president of an institution. If 51 per cent of the votes are cast for A and 49 per cent for B we declare A elected. In doing so we have satisfied 51 per cent of the preferences but have had to proceed contrary to 49 per cent; we may say that 49 per cent of the preferences were violated. More generally, when we have to select a subset of the n objects as "elected" we shall in general, in the absence of complete unanimity, violate a inumber of preferences. Circumstances force us to do so to some extent. The problem is to do so to the least possible extent. 3. Consider the case in which 8 members of a body have to elect a committee of three from among themselves. We will suppose that no member votes for himself (though this makes no essential difference) and that there are no abstentions (though this too makes no essential

The Description of Genic Interactions in Continuous Variation

Abstract: The genetical interpretation of the continuous variation (or indeed any variation) shown by a population, family or group of families requires the use of specifications of two distinct kinds. Firstly, it is necessary to specify the genetical structure of the population, family or families. In principle, this requires the specification in suitable terms of the relative frequencies of the various alleles of the genes involved, the distribution of the alleles at a locus between the various possible homozygotes and heterozygotes and the distribution of the alleles of different genes in respect of one another. These specifications will depend on the ancestry of the material, the mating system which has been in force, the selection which has been practised (if any), and the linkage or other relation of the genes in transmission from parent to offspring. Specification of the genotype of every individual, or indeed of any individual, is not essential for most biometrical purposes so long as the relative frequencies of the different possible genotypes can be given, and indeed it is sufficient for many purposes to specify only the average, taken over all genes, of the allele frequencies, homozygosis, linkage relations and so on.

The Exploration and Exploitation of Response Surfaces: An Example of the Link between the Fitted Surface and the Basic Mechanism of the System

Abstract: This is a sequel to an article which recently appeared in this journal [1] and had the same general title. The previous article described a number of applications of newly developed techniques [2] for the study of response surfaces. The present article shows how study of the form of the empirical surface can throw important light on the basic mechanism operating and can thus make possible developments in the fundamental theory of a process. This idea is illustrated in some detail with an example previously discussed only from the empirical standpoint. A theoretical surface, based on reaction kinetics is now derived, rate constants are estimated from the data and the theoretical surface is compared with the empirical surface previously obtained. It is then shown how the canonical variables of the empirical surface can relate to the basic physical laws controlling the system. In this connection the problem of suitable choice of metrics for the variables is discussed. In a final section some general remarks on the process of scientific investigation are appended.

The Theory of Bacterial Constant Growth Apparatus

Abstract: Recent work in bacterial genetics has emphasised the usefulness of apparatus whose purpose is to maintain a constant population of bacteria in a state of active growth. Several such devices have been described for example by Monod (1950), Novick and Szilard (1950), and Perret (1954). It seems worthwhile to give a short account of the underlying theory as a guide to the problems of design likely to be encountered with organisms of different growth characteristics; or under various conditions of culture. Mathematically the problem may be stated as follows:-Consider an organism growing freely in a limited, conistant volume of nutrient. After some period of growth factors come into play which depress the power of the organism, to divide and eventually stop it growing altogether. These factors may be of several different kinds: for example, exhaustion of nutrient, insufficiency of oxygen, or production of some toxic metabolite, the general effect however, is that the growth rate of the population at any time is a function of its size (n) so that

Determining the Fruit Count on a Tree by Randomized Branch Sampling

Abstract: In crop estimation work and in some areas of biological and pomological research, the problem of determining the total number of fruits on a tree sometimes arises. If an accurate count of all fruits is attempted, this may be quite an onerous and time consuming job,-especially, if the fruits are to be left on the tree undamaged. If the fruits are picked before counting, in order to improve the accuracy of the results, the removal of the fruits may seriously interfere with other aspects of the investigation. A method of obtaining reasonably precise estimates of the total fruits by sampling, so that little time is required, may be of some interest. The purpose of this paper is to describe some possible schemes and compare some aspects of their efficiencies. The object of sampling is to select some portion of a relatively large total which will represent that total reasonably well. In the present case, the object is to select a few of the many smaller branches of a tree in such a manner that counting the fruits on these sample branches will enable us to obtain a reasonably accurate estimate of the total fruits on the tree. At present, we shall consider only those schemes which select the sample branches by a randomizing procedure. Suppose the branching system of a tree is represented as in the following diagram: The trunk, branch number "O", splits into two branches at fork I. Branch 1 of this fork splits into 3 branches at fork II, etc. Suppose all the fruits of the tree are borne on the peripheral branches, the number being indicated by the encircled figures. Thus branch I of fork III has 12 fruits, branch 1 of fork VI has none, etc.; the tree has 64 fruits borne on its 8 "fruiting" branches. Suppose we wish to determine the fruit count of this tree by confining our counts to two fruiting branches selected at random. This could be done by numbering each of the 8 fruiting branches from 1 to

Use of the simplex design in the study of joint action of related hormones

Abstract: In certain toxicological studies the term joint action has come to take a special meaning. If members of a group of related compounds all cause death of an organism when administered separately the simultaneous action of these substances is called their joint action. Bliss (1939) first discussed the analysis of data obtained in this manner. From this time the problem has been examined in terms of tolerance distribution theory and developed in relation to probit analysis (Finney, 1952). Plackett and Hewlett (1951) have extended the tolerance distribution theory to different theoretical forms of joint action and developed a set of mathematical models, each of which is based on many assumptions and is very difficult to fit to experimental data. Fisher (1954) has shown that parameters of the binomial distribution may be estimated without tolerance distribution assumptions. The aim of the present paper is to show that the study of joint action by means of an appropriate experimental design-the simplex designallows ready interpretation of experimental data with no reference to a joint tolerance distribution, and no further assumptions than normally required in quantal analysis. The method is also appropriate without modification to the study of joint action of substances eliciting a graded response simply by applying the standard estimation procedures. Examples will be drawn from the study of the action of oestrogens on the vagina of the ovariectomized mouse. The quantal response in this case is cornification of the vaginal epithelium.

Rules of Thumb for Determining Expectations of Mean Squares in Analysis of Variance

Abstract: Exact procedures for determining the expected values of sample mean squares in terms of population parameters are adequately described in a number of places in statistical literature (1, 3, 7)t. For simple designs with few classifications the processes can be gone through quickly, and with practice, the expectations of such mean squares can be written by inspection. However, when a design involves several classifications, and particularly when the classifications are a mixture of random and fixed variates, the processes become complex and tedious.

Comparative Sensitivity of Pair and Triad Flavor Intensity Difference Tests

Abstract: Alternative simple experimental designs for sensory difference tests of flavor intensity lead to the procedures termed "pair", "duo-trio" and "triangular" tests (3). In the first, a unit trial consists in submitting coded aliquots of the two batches in question to a subject in the sequence (X), (Y) or (Y), (X) and requiring him to rank them in the order of appraised flavor strength. In the second, it consists in submitting identified X or Y with the coded sequence (X), (Y) or (Y), (X) and requiring the subject to attempt to match the identified with the like coded aliquot. In the third, it consists in submitting one of the completely coded sequences (X), (X), (Y); (X), (Y), (X); (Y), (X), (X); (Y), (Y), (X); (Y), (X), (Y) or (X), (Y), (Y) and again requiring an attempted matching of like aliquots. Inferences respecting the occurrence or non-occurrence of real discrimination are then made by relating the actual frequency of ranking or matching in repeated trials to percentage points of the binomial distribution expected in the absence of discrimination. It has been suggested (4) that the "triangular" test is "obviously the most efficient" but experimental evidence to the contrary has been reported (1). This note indicates some statistical considerations relevant to efficiency comparisons, and applies them to additional data.

Design and Analysis of Two Phase Experiments

Abstract: It sometimes happens in experimental work that the effects of different treatments cannot be measured directly and a further stage of testing is required in order to evaluate them. Examples of this type of situation are studies of the effect of conditions of growth of parent material on resistance to disease or productivity of progeny; the survival of nodule bacteria under various conditions of storage and appraised by inoculating appropriate legume seedlings; and the effect of various treatments on virus multiplication in leaf tissue, the concentration of virus being ascertained by lesion counts on indicator plants.

Partially Replicated Latin Squares

Abstract: Latin squares have been used for over 30 years in agricultural field trials (1). Often the double restriction imposed by the row and column arrangement brought a welcome reduction in the mean square for error. The row and column mean squares were of no interest. Eventually the Latin square arrangement found application where the rows and columns corresponded to clearly defined physical entities. Youden (3) in studies on tobacco mosaic virus observed that the number of lesions produced on leaves depended upon the position of the leaf on the plant and that plants also differed markedly in susceptibility to lesions. Furthermore, for a given lot of plants, the effect of leaf position was closely the same from one plant to another. Very large reductions in the error mean square resulted from the use of a Latin square. The plants were columns and leaf positions were rows. Here there was some interest in the mean squares for rows and columns though the chief concern was with the treatments applied to the leaves. Later Yates (2) introduced confounding into Latin square designs. Probably it was inevitable that the Latin square arrangement would be tried when the rows and columns were used for factors that not only were likely to interact with the treatments (letters) but also with each other. The form or appearance of a Latin square remains but the substance is lost. The fractional replication of a factorial experiment that results from this practice is a particularly unfortunate one as all

Prediction Equations in Quantitative Genetics

Abstract: One of the fundamental concepts in the application of statistical methods to the analysis of the inheritance of characters showing continuous variation is the additive genetic variance ao , the variance in any character in a population that is due to the average effects of genes. If this is expressed as a fraction of the total variance, oa, we get the related parameter, h2, the heritability (in the narrowest sense) of the character. It can be shown without great labour that the heritability is also equal to the regression coefficient of breeding value on performance or phenotype. This short paper presents an alternative derivation from the point of view of the combination of information from different sources, an approach which may be useful in teaching. Several other important prediction equations in quantitative genetics can be fitted into the same pattern. If we have a measurement P of an individual in a population in which the character measured has a mean P, we may consider ourselves as having two independent pieces of information on the animal's breeding value. They are: (i) that the animal is a member of a population whose mean breeding value is P with variance ao2; (ii) that the animal's own performance is P and that this will have variance ao_ oa2 about the true breeding value. If we knew only (i), we should take P as the best estimate of the individual's breeding and it would have error variance <,, . If we knew only (ii), we should take P as the best estimate with error variance 2 2 O7) U. In combination, the correct weight to give the two estimates is the reciprocal of their respective variances. We then have

Fitting the neyman type a (two parameter) contagious distribution

Abstract: One of the difficulties associated with the use of the Neyman contagious distributions (Neyman (1939)) concerns the method of fitting to data. Restricting attention to the two parameter Type A distribution, the method suggested by Neyman (with a remark that its efficiency needed investigation: there are no sufficient estimators) was to equate the corresponding first and second moments of the data and the distribution-this gives two readily soluble equations for the two estimators. Shenton (1949) investigated the efficiency of this moment fit, and outlined a technique for an iterative maximum likelihood fitting process, together with suggestions relating to the circumstances in which the process might be worth applying. Owing to the complicated nature of the recurrence relation for successive probabilities, the distribution is rather tedious to handle in any circumstances, and it is unfortunately the case that the maximum likelihood process suggested by Shenton increases considerably the labour of fitting. Recent papers (e.g., Beall and Rescia (1953)) stress the need for a technique which would reduce the amount of calculation-this paper suggests a method which greatly shortens the labour of obtaining a maximum likelihood fit, and which reduces the calculation necessary for a comparison of observation and expectation whatever the method of fitting used. As with the Shenton technique, the successive approximations for the maximum likelihood fit are based on the NewtonRaphson method. The case where the zero class is unknown is also briefly discussed.

Experimental Design in Industry

Abstract: The design of experiment and analysis of variance are techniques which have been developed mainly in connection with agricultural research. The majority of examples in textbooks on the subject are consequently drawn from this field. These techniques also apply to industrial and technological experimentation. The main purpose of this note is to emphasize that the conditions under which these techniques have to be applied in industry are in several respects essentially different from those prevailing in agriculture; and that these differences imply changes in the method of teaching, of analysis, and of presentation if we are to reap the full benefit and efficiency of these statistical methods in the industrial field.

Sur la Determination de l'Axe d'un Nuage Rectiligne de Points

Abstract: Au cours d'une 6tude de la relation d'allometrie d6velopp6e devant la premiere Conf6rence international de Biom6trie (1948) et parue ici-meme, j'ai 6te conduit a examiner le probleme de la liaison lin6aire de deux variables jouant des roles sym6triques, telles par consequent qu'aucune d'elles ne peut legitimement etre considerde comme independante. Transportant dans le domaine du calcul une technique d'interpolation graphique classique, j 'ai propose de repr6senter la relation cherch6e par la droite qui rend minimum la somme des aires des triangles rectangles ayant cette droite pour hypotenuse commune, deux cotes paralleles aux axes, et leurs sommets aux points X1, X2. La f6rme de l'6quation ainsi obtenue:

Appropriate scores in bio-assays using death- times and survivor symptoms*

Abstract: Death and survival are the most commonly used markers in biological standardization and in evaluation of medical therapy. Numerous methods are available for estimating the parameters of the dosagemortality curve, and the biologist is often in a state of embarrassment of riches in the choice between a dozen well recommended methods to determine his LD50, ED50, TCiD50, or other "D50's" with which he is concerned. The experimental records of good biologists and clinicians usually contain more information than death or survival of the subjects on given treatments, but either the "LD50-fixation" of the investigator prevents these data from entering in the evaluating process, or they are left out because no simple method is available to utilize these observations. In many assays, individual deaths have a quantitative connotation in terms of the time period from exposure to the lethal agent until death occurs. Survival has also quantitative aspects such as time of recovery, severity of symptoms at the time when death is no longer expected, etc. Some data may consist mostly of deaths occurring at different times, in other experiments survivors are in excess. In these cases, group mortality percentages are too large or too small, respectively, to be of biometric use. Attempts are then made to find a transformation of death times or survivor symptoms that can be used as response metameter with approximately linear relationship to dose or with approximately normal distribution, or both. However, in such attempts the problem of truncated or censured distribution will sooner or later present itself, leaving the investigator either in uncomfortable indecision or involved in excessive computation.

Covariance Analysis as an Alternative to Stratification in the Control of Gradients

Abstract: Federer and Schlottfeldt discussed measurements of the heights of tobacco plants in an experiment on seven treatments arranged in eight randomized blocks. A fertility gradient within the blocks was suspected and they therefore calculated a quadratic covariance analysis on a measure of distance in this direction. For the study of a regression trend of higher degree, the computations are simplified by using standard orthogonal polynomial values from Fisher and Yates's Statistical Tables (2) based upon distance from the centre of the experiment, this modification involving no difference in principle. Table I reproduces the yields from (1) and also the covariates, xI to xs for the corresponding orthogonal polynomials. The analysis of squares and products up to the third degree is shown in Table II, which includes the quantities required for subsequent covariance adjustments and agrees with Tables III and IV of (1) with respect to xI , x2 and y. To estimate the regression coefficients in the cubic analysis, the following set of equations must be solved.

An Analysis of Perennial Crop Data

Abstract: In many fields of study, multiple observations are made on each individual. Treatment of such multivariate data differs. A trait-bytrait analysis may be made. Methods which consider several characters simultaneously include variance, covariance, components of variance and regression analysis. One of these may adequately answer the questions raised or test the hypotheses stated by the research worker when designing the experiment or survey. However, cases arise where none of these procedures is wholly adequate or appropriate. A multivariate analysis may be both appropriate and adequate in such cases. The term multivariate analysis will be applied to analyses of data where several variables are considered jointly with none relegated to the position of an independent variable. If such multiple observations are analyzed on the basis of separate variables, the combination of the results of univariate tests and the assignment of a measure of credibility to any inference drawn present problems. Thus if the observations are perfectly correlated, the same conclusions are drawn from each variable; if the observations are completely independent and it is agreed to claim a difference at the 5% level if at least one variable shows significance, then one falsely claims significance with probability 1 (.95)n with n variables; if the rule is to claim a difference only if all variables show significance, then the

The Bradley-Terry Probability Model and Preference Tasting

Abstract: Broadly speaking, two methods of appraising sensory differences may be distinguished: scoring and ranking. There are also two sources of sensory difference: that among the intensities of several stimuli identical in kind, and that among preferences in a group of stimuli that may or may not be of the same kind. This paper is concerned with the applicability of the simplest form of ranking, namely, pair comparisons, to testing in general and taste preferences in particular. In organoleptic work it is usually rewarding to postulate a sensory continuum whose points, S, are monotonically related to the concentration, C, of a given stimulus in a given medium. Some controversy has centred on the meaningfulness of this notion, and on its right to come within the ambit of metrology at all; here, without discussion, the view will be adopted that the concept is operationally valid, and that the practical problem is to refine the measuring technique. A common further assumption is that the relation approximates to the Fechnerian form S = a + ES log (C/CO), where a and ES are constants and CO is the threshold detectable concentration, over a certain critical range. We shall not at the moment perpend this relation, but may observe that, whatever the true equation, its parameters are likely to be biological variants over the universe of tasters. A preference continuum is a more nebulous concept. In simplest form, it may be thought of as a series of points forming an ordinate of preference, P, to an abscissa of concentration, C, of a given stimulus; and it is not difficult to visualize a curve that maximizes P at some particular concentration. But a preference continuum must also be applicable to a series of stimuli that are at least partially different in kind. This necessitates a more complex model in which P is a function of an n-dimensional vector quantity. Can such a continuum be validated? In other words, can a subject's preference statements, in suitably selected and controlled situations, concerning a set of flavors, be

On the Analysis of Variance of a Two-Way Classification with Unequal Sub-Class Numbers

Abstract: In many avenues of research it is necessary to analyse the variance of data which are classified in two ways with unequal numbers of observations falling into each sub-class of the classification. For data of this kind special. methods of analysis are required because the inequality of the sub-class numbers causes lack of orthogonality among the main effects and interaction comparisons. Table I below gives the basic notation for dealing with an analysis of a two-way classification with unequal sub-class numbers. Several writers have dealt with the analysis of data of this form and various methods have been put forward. Some of the more prominent articles and discussions are cited below [1-13]. A simple preliminary step common to all methods is to separate the variance within sub-classes from the variance between sub-classes. Table II gives the analysis of variance for this preliminary step. The problem of extending the analysis to the main effects and to the interaction between the main effects now arises. The (pq 1) degrees of freedom for between sub-classes can be partitioned in the usual way into (p 1) degrees of freedom for between A classes, (q 1) degrees of freedom for between B classes and (p 1) (q 1) degrees of freedom for the interaction between the two classifications. The main difficulties arise in determining the correct sums of squares to be associated with each of these. One difficulty is that the addition theorem for sums of squares does not apply unless the sub-class numbers are proportional, and thus the interaction sum of squares cannot be computed by the usual method of differences. In fact, situations may occur where this procedure would give a negative result for the sum of squares for interaction. Frequently we assume, from the nature of the data or from previous information or experience, that interaction is absent or if present, negligible. Making this assumption, we are interested in testing if there are any significant differences between the A-classes and between the B-classes.

Quantitative studies in diphtheria prophylaxis: an attempt to derive a mathematical character- ization of the antigenicity of diphtheria prophylactic*

Abstract: Examined quantitatively, the antibody responses of animals and children to inoculatioiis of different forms of diphtheria prophylactic vary greatly. The dose-response curves from such materials, however, do show a similar pattern, and there is a great variation between different forms of prophylactic in the dosage required to induce some arbitrary level of response (Jerne & Maal0e, 1949). As Jerne & Wood (1949) point out, an assay of a test preparation (T.P.) in terms of a standard preparation (S.P.) is strictly valid only if "the less potent preparation behaves as though it were a dilution of the other in a completely inert dilutent. The relative potency of the T.P. in terms of S.P., defined as the ratio of doses required to produce a given response, is then independent of the dose level of response at which it is measured. They continue ". . . this is the only definition of relative potency that would normally be regarded by the bio-assayist as satisfactory . . . . An instance of current interest in which this assumption does not hold is the assay of diphtheria and tetanus toxoids in commercial products containing aluminium hydroxide, using as S.P. a reference sample of highly purified toxoid . . . the doseresponse curves of the two preparations have different upper asymptotes and cannot be described by the same form .... The assay is thus invalid." Here then we have the problem; two preparations have a property in common, viz. the ability to cause the development of antitoxin when injected, but the one cannot be expressed quantitatively in terms of the other in the usual way. The present communication is concerned with (a) an attempt to overcome this difficulty by finding antigenicity equations applicable to all types of diphtheria prophylactic (and probably other antigeiis)

Matrices in Quantal Analysis

Abstract: Fisher (1954) has recently discussed the various transformations of probability used in the analysis of binomial data, and in that paper a full account of the statistical theory is given. While the assumption is often made that a distribution of thresholds must be postulated before efficient analysis may be made of binomial data (Finney, 1952a), Fisher (1954) has clearly demonstrated that this is unnecessary. Transformations of the expected proportion responding may thus be simply regarded as a different scale for the measurement of response. In this paper practical methods of relating the binomial variable to the coordinates of experimental designs are given, and matrix methods are employed so that the results may immediately be applied to any experiment with known design matrix. A considerable time has been devoted in the past to methods supposed to give quick estimation of parameters in quantal analysis. These graphical or semigraphical methods are usually employed in order to avoid efficient but tedious probit analysis in routine work. In this paper it will be shown that quick efficient solution is afforded by use of the angular transformation. Recently Berkson (1953) has advanced a "simplified and quick" method for the estimation of parameters of binomial data by means of a modified logit technique. The method is still tedious when compared with the method exemplified here, which gives estimates equivalent to those derived by Berkson's method.

An Inverted Matrix Approach for Determining Crop-Weather Regression Equations

Abstract: We would like to know whether year-to-year changes, or month-tomonth changes in crop yields or prospects are consistent with observed weather data. Generally, historical weather records extend back farther than records of crop yields. We wish to make use of weather data for the entire period of record even though yield data may be available for a much shorter period. This paper reports on an exploratory inverted matrix approach used in one phase of a crop-weather study. The application of multiple regression methods in the study of relationships between crop yields and weather factors is, of course, not new, but the large amount of computational labor involved has discouraged many workers and our people from attempting correlations studies on a very extensive scale. As pointed out by R. A. Fisher, the use of the inverse matrix solution of a set of normal equations greatly reduces the amount of computations when the same set of independent variables is used repeatedly; in addition, it serves to simplify the calculation of sampling errors of the regression coefficients. However, a large amount of computational work is still required when the various dependent variables are available for only relatively few years, and these periods vary from crop to crop because of the fact that the data or series were started at different points in time. We would like some way of utilizing all the weather and crop yield data available. Therefore, we would like to devise what might be called "generalized inverse matrix solution" for a given State or area which could be used whenever the given set of weather factors were appropriate. However, the sampling errors of the regression coefficients cannot be computed using the elements of this generalized solution where the dependent variables are used for only a subperiod. The inverse matrix solution is obtained for a given set of independent variables (i.e., weather factors) for the entire period of the weather

Statistical Analysis of Multiple Slope Ratio Assays

Abstract: The statistical analysis of slope ratio assays for one test solution has been described in detail by Finney (1952), and routine methods of computation fully outlined, including tests for statistical and fundamental validity. Clarke (1952) has given a method for assays involving any number of test preparations, and this paper describes an adaptation of Clark's procedure using response totals directly to compute the various slopes, and gives a further analysis of the sum of squares associaated with the test for fundamental validity. The method of analysis given here applies only to assays in which the response for each preparation is a linear function of the dose. The assay design must be completely symmetrical, i.e. there must be equal spacing between the dose levels for each preparation, the same number of dose levels for each preparation, and equal replication for all treatments. It is generally preferable to run a test at the zero dose level since this gives improved tests for validity, Finney (1952), Wood and Finney (1946); but the suggested method is developed to cover assays with, and without, tests at the zero dose level.

Brief of Presidential Address: The 1954 Trial of the Poliomyelitis Vaccine in the United States

Abstract: (1) Poliomyelitis is a relatively rare disease. From past experience, the rate of paralytic polio in the study areas might be anticipated to be about 30 cases per 100,000 children aged 6-9 years. Given this attack rate, table I shows the probability of obtaining a statistically significant result (5% level) for various numbers of children and for various degrees of true effectiveness of the vaccine. With a vaccine that actually was 50% effective, about half a million children would be needed to make the risk of an inconclusive result small. Table II shows

A Further Note on Missing Data

Abstract: whereas Nelder has (rt 1) in the numerator. It should prove helpful to some to point out that inspection suffices to show that Nelder's formula is incorrect. Remembering the mathematical model, it is obvious that the general mean, the constant for the affected block and that for the affected treatment can all be estimated with any desired accuracy, simply by increasing the numbers of blocks and of treatments. Hence so can their sum, which is the estimate of the missing value. Nelder's formula is not conformable with this observation, having a lower limit of o2 as r and t become large. OIn the other hand, his formula for the r X r Latin square is correct, and is of the order of 3o-2/r as the square becomes large. In referring to Query 96, which raised a question about "impossible" estimated values, another error has occurred in Nelder's paper. The missing value, estimated to be -6.64, has a sampling error of 8.23 on 32 degrees of freedom. The 95% confidence interval is therefore -6.64 ? 16.76 (rather than Nelder's value of 8.10), thus giving no appreciable indication whether the estimated value is based oIn an erroneous model. There is some interest in the fact that not only missing values may have "impossible" estimated values. In the example of Query 96 the model leads to estimates of -3.23 and -1.48 for bait A for replications 4 and 11, respectively, but these are small compared with the sampling error of 8.23. While tests of "possibility" of estimated values may occasionally prove useful, it is probably always better to test for additivity, as discussed for this example by Tukey (1954).