William G. Cochran

The Combination of Estimates from Different Experiments

When we are trying to make the best estimate of some quantity A that is available from the research conducted to date, the problem of combining results from different experiments is encountered. The problem is often troublesome, particularly if the individual estimates were made by different workers using different procedures. This paper discusses one of the simpler aspects of the problem, in which there is sufficient uniformity of experimental methods so that the ith experiment provides an estimate xi of u, and an estimate si of the standard error of xi . The experiments may be, for example, determinations of a physical or astronomical constant by different scientists, or bioassays carried out in different laboratories, or agricultural field experiments laid out in different parts of a region. The quantity xi may be a simple mean of the observations, as in a physical determination, or the difference between the means of two treatments, as in a comparative experiment, or a median lethal dose, or a regression coefficient. The problem of making a combined estimate has been discussed previously by Cochran (1937) and Yates and Cochran (1938) for agricultural experiments, and by Bliss (1952) for bioassays in different laboratories. The last two papers give recommendations for the practical worker. My purposes in treating the subject again are to discuss it in more general terms, to take account of some recent theoretical research, and, I hope, to bring the practical recommendations to the attention of some biologists who are not acquainted with the previous papers. The basic issue with which this paper deals is as follows. The simplest method of combining estimates made in a number of different experiments is to take the arithmetic mean of the estimates. If, however, the experiments vary in size, or appear to be of different precision, the investigator may wonder whether some kind of weighted meani would be more precise. This paper gives recommendations about the kinds of weighted mean that are appropriate, the situations in which they

The analysis of groups of experiments

When a set of experiments involving the same or similar treatments is carried out at a number of places, or in a number of years, the results usually require comprehensive examination and summary. In general, each set of results must be considered on its merits, and it is not possible to lay down rules of procedure that will be applicable in all cases, but there are certain preliminary steps in the analysis which can be dealt with in general terms. These are discussed in the present paper and illustrated by actual examples. It is pointed out that the ordinary analysis of variance procedure suitable for dealing with the results of a single experiment may require modification, owing to lack of equality in the errors of the different experiments, and owing to non-homogeneity of the components of the interaction of treatments with places and times.

Errors of Measurement in Statistics

In this review of some of the recent work in the study of errora of measurement, attention is centered on the type of mathematical model used to represent errom of measurement, on the extent to which standard techniques of analysis become erroneous and misleading if certain types of errors are present (and the possible remedial procedures), and the techniques that are available for the numerical study of errors of measurement.

Planning and Analysis of Observational Studies.

Treats studies, primarily in human populations, that show casual effects of certain agents, procedures, treatment or programs. Deals with the difficulties that comparative observational studies have because of bias in their design and analysis. Systematically considers the many sources of bias and discusses how care in matching or adjustment of results can reduce the effects of bias in these investigations.

Some Classification Problems with Multivariate Qualitative Data

Since 1935, when Fishers discriminant function appeared in the literature, methods for classifying specimens into one of a set of universes, given a series of measurements made on each specimen, have been extensively developed for the case in which the measurements are continuous variates. This paper considers some aspects of the classification problem when the data are qualitative, each measurement taking only a finite (and usually small) number of distinct values, which we shall call states. Our interest in the problem arose from discussions about the possible use of discriminant analysis in medical diagnosis. Some diagnostic measurements, particularly those from laboratory tests, give results of the form: -, + (2 states); or -, doubtful, + (3 states); or (with a liquid), clear, milky, brownish, dark (4 states). With qualitative data of this type an optimum rule for classification can be obtained as a particular case of the general rule (Rao, [1952], Anderson, [1958]). The rule is exceedingly simple to apply (Section 2). In practice, qualititative data are frequently ordered, as with -, doubtful, +. The classification rule discussed in this paper takes no explicit advantage of the ordering, as might be done, for instance, by assigning scores to the different states so as to produce quasi-continuous data. The best method of handling ordered qualitative data is a subject worth future investigation.

On the Performance of the Linear Discriminant Function

This paper considers the question: can the probability of misclassification given by a discriminant function, when used to classify specimens into one of two populations, be predicted accurately from the probabilities given by the variates when used individually? Theoretical consideration of the role of correlations between variates shows that (i) there is no mathematical reason why such predictions should be accurate and (ii) positive correlations (in the sense defined) are generally harmful and negative correlations helpful. Examination of 12 well-known numerical examples from the literature suggests that in practice (i) most correlations are positive (ii) it is usually safe to exclude from a discriminant, before computing it, a group of variates whose individual discriminatory powers are poor, except for any such variate that has negative correlations with most of the good discriminators (iii) the performance of the discriminant function can be predicted satisfactorily from a knowledge of individual po...

Estimators for the One-Way Random Effects Model with Unequal Error Variances

Abstract Using the random effects model, yij = μ + α i + ∈ ij , (i = 1, …, k; j = 1, …, ni ), where α i and ∈ ij are normal with means zero and variances σα 2 and σ i 2, this article considers eight methods of estimating σ i 2, σα 2, and thirteen corresponding procedures of estimating μ. Biases and mean squared errors (MSEs) of these procedures are examined for variations in the magnitudes of the unknown parameters, the sample sizes, and the number of groups.

The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce.

In a number of cereal experiments, three on wheat, three on barley and one on oats, the yields of grain and straw per plot were estimated by weighing the total produce on each plot and taking samples, usually from the sheaves, to estimate the ratio of grain to total produce. This paper discusses the sampling errors of this method. The method proved considerably less accurate than was anticipated from previous calculations made by Yates & Zacopanay. Amongst the reasons which are suggested to account for this are the larger sizes of plot and sampling unit used in these experiments and the additional variability introduced by the presence of weeds, undergrowth and moisture. Nevertheless, the method appears to be substantially superior to the older method of cutting small areas from the standing crop, without weighing total produce, only about one-quarter of the number of samples being required to obtain results of equal precision. The samples were taken both by an approximately random process and by grabbing a few shoots haphazardly from each of several sheaves. The grab samples gave on the whole a slightly higher yield of grain, the greatest positive bias being 6%, but were otherwise just as accurate as the random samples. Since the grab samples can be selected and bagged in about one-third of the time required for random samples, their use is recommended for the majority of the samples required in any experiment. The validity of an approximate formula for calculating the variance of a ratio (in the present instance the ratio of grain to total produce) is discussed briefly in an appendix.

Statistical Problems of the Kinsey Report

Abstract * This article consists of the main text, but not the appendices, of the report of a committee appointed in 1950 by S. S. Wilks as President of the American Statistical Association, to review the statistical methods used by Alfred C. Kinsey, Wardell B. Pomeroy, and Clyde E. Martin in their Sexual Behavior in the Human Male (Philadelphia, W. B. Saunders Co., 1948). For further details on the appointment of the committee and its charge, see Section 1, p. 676 below. For an outline of the appendices, as well as of this paper, see Section 3, pp. 678–81, Appendix G, “Principles of Sampling,” will appear as an article in the March issue of this Journal. The full report, including both the text given here and the appendices, will be published as a monograph by the American Statistical Association in 1954.

Sampling Theory When the Sampling-Units are of Unequal Sizes

JN SAMPLING, the sampling-units are usually chosen so as to be similar in size and structure. With some types of population, however, it is convenient or necessary to use sampling-units that differ in size. Thus the farm is often the sampling-unit for collecting agricultural data, though farms in the same county may vary in land acreage from a few acres to over 1,000 acres. Similarly, when obtaining information about sales or prices, the sampling-unit may be a dealer or store, these ranging from small to large concerns. In such cases the question arises: Should differences between the sizes of the sampling-units be ignored or taken into account in selecting the sample and in making estimates from the results of the sample? This paper contains a preliminary discussion of the problem, though further research is needed, many of the results given below being only large-sample approximations. It is convenient to consider first the problem of estimation, since it appears that the best method of distributing the sample depends on the process of estimation that is to be used.