Showing papers in "Biometrics in 1951"

Testing a linear relation among variances.

Abstract: hold among the Oi , where the ci1 are known numbers. This problem, usually involving a single linear relation, has been encountered occasionally in the applications of statistical methods to data. With the current increased interest in the "components of variance" technique, it is not unlikely that the problem will appear more frequently in the future. Two examples will be described briefly. An experiment is repeated at r different places on each of c different occasions, because it is expected that the performance of the treatments (t) will differ from one place or one time to another. The places and times are assumed to be a random sample of the population of places and times in which the results will be used. If the usual "components of variance" model is set up, the expectations of the four principal items in the analysis of variance of the results are as shown in Table 1.

Components in regression.

Abstract: WE SHALL BE principally concerned with simple linear regression where both variates are subject to "error", although the same problems and methods apply to problems involving three, four, or many variates. By saying that we deal with "both variates subject to error" we mean that two measured quantities, which we may as well call x and y, are thought of as made up of two independent (or at least orthogonal) parts as follows

The present status of variance component analysis.

Abstract: rHE ELEMENTARY THEORY of variance component analysis has been 'discussed in recent papers, by Daniels (10), Crump (8, 9) and Eisenhart (11). Since the appearance of the last of these the theory has been extended in a number of directions. It seems worthwhile at this time then to present a unified discussion of the theory in its present state which indicates the available results in order that further work on the theory may be directed at the important unsolved problems. Although an effort will be made to avoid excessive discussion of elementary well known results, enough of these will be included to make the general content of the paper essentially self contained and coherent. It seems pertinent to begin by defining rather carefully what problems are included in the province of variance component analysis. This is most easily done by considering a particular type of multiple classification which contains all of the essential elements of the situation. It must be remembered however that the general features of the next few paragraphs are common to any set of data arranged in a multiple classification and described by a linear model. Data arranged in a two-way classification with the same number of observations in each cell will be used to illustrate. Let the classes in the two criteria of classification be A, , A2 , B e, A, and B1 , B2 , * , Rb respectively. Then if yhii denotes the j-th datum in subclass AhBi the complete linear model for these data is included in the following equations:

Evaluation of a Class of Diagnostic Tests

Abstract: MEDICAL diagnostic tests, such as stool examination for specific infections, or the NIH swab for oxyuriasis, for which a positive diagnosis is based on definite identification of the organism, or its cysts or trophozoites in the stool, or of the pinworm egg in the swab, constitute a class of diagnostic tests which, under certain control conditions, yield no false positives. How good any one such diagnostic test is, is measured by the probability that an infected person will be found positive by a single application of the test. If we assume that this probability is the same for all infected individuals, we may term this probability the efficiency of the diagnostic test. This assumption implies that we are considering only infections diagnosable by the test used, rather than cases for which the locus or form of infection is such that the test cannot diagnose it. Given a group of known infected persons, the estimation of efficiency is simple; it is given by the ratio of number of positive results obtained to the total number of examinations performed on the known infected persons. In general, however, we do not have such a group of known infected persons, but have instead a group of unknown persons on each of wvhom one or more examinations are made. Also, in addition to estimating the efficiency of the test, we may be required to estimate the prevalence of the infection in the population for which our group is considered to be a representative sample. These problems, estimation of efficiency and prevalence, will be considered here.

Consistency of estimates of variance components.

Abstract: THE ESTIMATION OF VARIANCE COMPONENTS has important applications in genetic research, and geneticists were among the first to use analysis of variance extensively for that purpose. Early examples of component estimation in genetics are contained in papers by Lush et al. (1934), Bywaters (1937), and Stonaker and Lush (1942). Numerous later papers could also be cited. The earlier component estimates in the genetic literature were not accompanied by measures of reliability or confidence limits of any sort. The first attempt to place confidence limits were made by Knapp and Nordskog (1946) in connection with estimation of the ratio of genetic to total variance in beef cattle. Research on the genetics of quantitative characters of economic plants now in progress at the North Carolina Experiment Station relies heavily on variance component estimation. In designing our experiments, prediction of the sampling variance of component estimates has been predicated on normal distribution theory. Whether this is satisfactory depends on conformity of observed distributions of component estimates with assumptions of normality and homogeneity of variances throughout the experimental material. A preliminary investigation of the problem has been made using data collected for estimation of genetic variance components in corn.

A re-examination of rahn's data on the number of genes in bacteria

Abstract: A CCORDING to current cytogenetic theory, each chromosome in every I cell of a living organism comprises a number of genes. Inter alia, these genes have the following properties:(a) They are the carriers and transmitters of hereditary characters; (b) They control the specific enzymatic activity of the cell; (c) They play an important part in the initiation of cell division. This paper is concerned only with the third of these properties, and with the light it can throw on the numbers of genes in a cell. The natural process of cell division must be preceded by duplication of the constituent chromosomes, which in turn requires the duplication of every gene and is the culmination of an elaborate synthetic process. Luria (1950) has pointed out that 'There is something peculiar to homologous replication that sets it aside from other types of synthetic reactions. The replication of specific biological units must involve the building of complex specific molecules or molecular aggregates, the only permissible limitation to identity of model and replica being the production of "mutated" structures the production, that is, of modified elements replicated in the modified form'. Claude (1949) has emphasized the difficulties in this 'template theory, elaborated to account for the supposedly unique process of gene reproduction', which 'assumes that the gene or the chromosome serves as a mold for the systematic apposition of a new substance'. He goes on to state that 'in the light of biochemical processes already known, it is conceivable that the duplication of essential and characteristic cell substances is the end

Analysis of variance with unequal but proportionate numbers of observations in the sub-classes of a two-way classification.

Abstract: A NALYSIS OF VARIANCE is used to provide the solution to two more or less distinct problems: (1) to detect and estimate components of variance in a composite population, (2) to detect and evaluate the significance of differences among means of sub-sets (Eisenhart, 1947). Attention has been drawn to complexities and some unsolved problems when there are disproportionate numbers of observations in each subclass of a multiple classification. But the case of proportionate subclass frequencies is usually passed over in a way which may lead the unwary to suppose that it has the same simplicity as when all sub-class means have equal weight. The purpose of this note is to call attention to the condition that such supposition is incorrect. When the sub-class numbers, although unequal, are proportionate to their marginal totals, it is well known that the additive property of sums of squares still holds good, and the analysis of variance can be carried through in the usual way. Beyond this it is generally implied although never explicitly stated, that interpretation as well as arithmetical procedure follows the usual lines; for example, Snedecor (1946), Sec. 11.9, writes, ". . . causing no injury to the analysis of variance". The implication however requires qualification (1) when the problem is to estimate components of variance, and (2) for tests of significance in which the within class variance fails to provide the appropriate estimate for error. Suppose that there are p classes A, A, 2 * , with numbers of observations proportional to a, , a2 ,. a, ; and q classes B1, B2 ...

Some Comments on Ëstimates of the LD 50 : A Critique"

Abstract: IN THE December 1950 issue of Biometrics [1] Bross presented a comparative study of three methods of estimating the LD50 , Kairber's method, the Reed-Muench method and the method of maximum likelihood. Dr. Bross' general conclusion on the comparative advantages of the first and third of these methods is "The Cornfield-Mantel iterative approximations to the maximum likelihood estimate do not improve the accuracy of the Spearman-Karber's method. The additional work seems to be wasted insofar as the LD50 is concerned." Dr. Bross' results and conclusions are based upon successive samples drawn from known populations. The only source of practical interest in the problem, however, arises when samples are drawn from unknown populations. In most studies in experimental sampling, the conclusions drawn for known populations can be applied without modification to unknown populations. In the present case, because of the special characteristics of Kairber's method, this unfortunately cannot be done. In fact, the situation appears to be:

On the errors of biological assays with graded responses, and their graphical derivation.

Abstract: CALCULATION of the fiducial limits of error of a bioassay is intricate. A technical assistant, computing by rote, is unlikely to be able to decide whether an abnormal result is due to miscalculation or is inherent in the data. Nomography and the use of error curves can reduce his labors, promote his confidence, and advance his insight into the factors contributing to the assay error. Several graphical aids for bioassays and errors are already available, e.g. those of de Beer [1], Knudsen & Randall [2], Bliss [3], Sherwood [4], Goyan & Dufrenoy [5], Koch [6], and Harte [7]. Lees's specific calculating machine [8] should also be mentioned in this connexion. The most recent aid, and perhaps the one most nearly suited to the requirements of the writer's laboratory, is that of Healy [9], whose nomogram gives errors for assays of given size and structure. The present requirements, however, were error charts covering 4and 6point assays of given design but greatly varying size. It was not required to calculate graphically the mean estimate itself, the writer being unable to appreciate the advantages of so replacing the simple arithmetic operations involved. The devisal of appropriate error charts, the main subject of this paper, has thrown into relief some aspects of assay design and error factors that may be worthy of concomitant record.

The effect of sulfamerazine on the erythrocyte and hemoglobin content of trout blood.

Abstract: THE DISCOVERY, at Leetown in 1945, of the effectiveness of sulfamerazine in the treatment of furunculosis in trout (1, 2), and the consequent widespread use of this drug for this purpose, made it important to learn the effects of this sulfonamide on trout. Accordingly an experiment was run to test the effects of various dosage rates, from approximately the minimum effective rate to one greatly in excess of the highest rate recommended. The daily dosage rate originally recommended (2) was 8 grams per 100 pounds of fish. Later it was found that a lighter rate was sufficient and that a 4-gram rate was about as effective as the 6-gram rate subsequently recommended (3, 4, 5). The dosage rates used in this experiment were 0, 5, 10 and 15 grams of sulfamerazine per day per 100 pounds of fish. The duration of treatment originally recommended was three weeks. In this experiment sulfamerazine was administered through more than twice this period. In the original experiment (1, 2) the species of trout used was the Eastern brook trout, Salvelinus fontinalis, a species very susceptible to furunculosis. This experiment, described in part by Gutsell and Snieszko (12), included two other species: the brown trout, Salmo trutta or Salmo fario, generally less susceptible; and the rainbow trout, Salmo gairdnerii, which only occasionally suffers noticeable loss from this disease. The test using brook and rainbow trout was run in 1946; that using brown trout in 1947. Erythrocyte counts and hemoglobin measurements were made only in 1947 and therefore could include only the brown trout.

Estimating precision of textile instruments.

Abstract: ONE OF THE MOST consistently difficult problems in the textile industry is the measurement of moisture for the assessment of value of material bought or sold on a dry basis. Complete solution of this problem involves a more complete knowledge of the relation of the water and textile fiber than is available. Complications exist in the many forms of water known to be associated with natural and synthetic fibers. Adsorbed, absorbed, and chemically bound portions contribute differently to different methods of estimation of total moisture content. The method of assessing moisture which is the oldest and best known is by loss-in-weight during drying, here designated as the oven method. This method is not only time-consuming but is also unsatisfactory in many instances, and particularly with fabrics because of the necessity of cutting samples. Another is measurement of some electrical property, such as resistance, impedance, capacitance, or dielectric constant, which shows an appreciable change with change in moisture content of the sample. The oven has more apparent validity in its direct measurement of amount of water volatilized, but it is only an arbitrary standard. Calibration of another method against oven moisture should, therefore, provide an alternate arbitrary standard which could be considered an equally good measurement. The problem of measurement of a quantity whose exact relation to a desired property is not clearly defined may seem, at first glance, to be unusual. However, further consideration will undoubtedly provide, in the mind of any experimentalist, many similar instances. This phase of the problem will therefore be dismissed without additional discussion. Calibrations of several types of electrical meters intended for use in measurement of moisture in textiles have previously been discussed (3, 4, 6, 7). Variables which produce a major change in the dependence of meter readings upon the moisture, as measured by the oven method, have been emphasized, particularly in the second and fourth papers of

Relationships among characters observed in urea clearance tests.

Abstract: I N LABORATORIES of clinical pathology the observation usually made to measure efficiency of kidney action is that known as the van Slyke urea clearance test. Failure of renal function results eventually in increased concentration of urea in the blood; the objective of the test is to provide a measure of partial failure before the onset of that consequence. It has been supposed that this can be obtained by observing the rate of urea excretion in urine. Following observations (under somewhat limited and special conditions) that urea excretion is approximately proportional to concentration in the blood, the measure adopted for test purposes has been the ratio of urea excreted per minute/urea concentration in blood usually described as measuring the volume of blood "cleared" of urea per minute, although of course no blood is actually cleared. A second consideration which the van Slyke test was devised to meet is that rate of urea excretion depends also on the rate of water excretion, particularly when this is slow. It was observed that rate of urea excretion is maintained nearly constant when rate of urine excretion exceeds about 2 ml. per minute. Consequently "maximum urea clearance" was defined as

Statistical Treatment of the Data of Blood Concentration of a Drug

Abstract: 1. Summary. Considering the case of a single dosage administered enterally' to an individual, a mathematical model is derived for the blood concentration of a drug in terms of absorption and excretion. A transformation of the experimental data is described which allows the practical evaluation of this model by the method of least squares. The coefficients of the model so estimated are combined to give statistics which have a geometrical, statistical, and physiological meaning. It is shown that these statistics allow a valid comparison of two or more concentration curves, in terms of an efficiency of treatment scale. This scale is a segment of the standardized normal distribution.

Notes on the estimation of gross and net reproduction rates by methods of statistical sampling.

Abstract: GROSS AND NET reproduction rates are usually computed from data on births and deaths relating to the total population. But if the data are faulty the degree to which such figures are correct cannot be assessed. The purpose of these notes is to point out that these rates may also be estimated from data, possible to obtain, by methods of statistical sampling which also supply methods of estimating their accuracy. In the case of the latter index, supplementary actuarial information in the form of female survival rates, l , for specific ages are necessary. If, at a given time period, in any geographical region the entire population of women is considered, the gross reproduction rate G, according to Kuczynski, is simply the sum of the specific fertility rates for female births p, , where the subscript x denotes, the age-group (x 2, x + 2) of the mother. The net reproduction rate R invented by B6ckh (D.V.G., 1947), will be the sum of the products of the agespecific fertility rates and each corresponding female survival rate. We need only be concerned with those groups included in the reproductive period which is usually between 15 and 50 years. In symbols

Statistical Aspects of the Simultaneous Detection of Thyroid and Thyrotrophic Hormones in Human Sera, Based on the Data of D'Angelo and Gordon

Abstract: rFHE DATA presented by D'Angelo and Gordon (1950) and D'Angelo, tGordon and Charipper (1942) indicate that the stasis tadpole is a suitable and sensitive test animal for the bioassay of thyrotrophic hormone. It was also reported by D'Angelo and Gordon (1950) that it is possible to determine in a relatively quantitative manner, both thyroid and thyrotrophic hormone levels simultaneously in the same sample of serum by an analysis of the effects of the injected sample on developmental advance and thyroid follicular cell height. Such a method should be extremely valuable in advancing our knowledge of the thyroidanterior pituitary gland interactions. It was felt, however, because of the interactions between these glands, that a further analysis of the original data of D'Angelo and Gordon was necessary for complete understanding of the method. It is fortunately possible to examine the reported data by means of a factorial analysis which has previously been applied to problems of endocrinology by Snedecor and Breneman (1945) and Breneman (1951). A factorial analysis is a convenient statistical method for evaluating the effects of a given group of treatments, both individually and with all possible interactions. The fact that this method permits evaluation of interactions between the treatments is particularly useful in this case inasmuch as it is necessary to evaluate the possible interactions of two hormones known to have physiological interactions if they are to be quantitatively determined.

An Analysis of Rectangular Lattices with Unequal Block Sizes, Using Inter-Block Information

Abstract: THE square lattice design (2) and the three-dimensional cubic lattice design (7) are available for variety trials and experiments with large numbers of treatments; both designs permit recovery of inter-block information. Yates (6) has also given the design and analysis of a rectangular lattice for pq varieties, using blocks of unequal sizes, but without recovery of inter-block information. The present paper extends the analysis to recover this information, using the assumption that the inter-block and intra-block components of variance are little affected by the unequal block sizes. In practice p and q need seldom differ by more than one, and when q = p + 1 the rectangular lattice design with blocks of equal size, developed by Harshbarger (4), is preferable, using the method of analysis given by Cochran and Cox (1), (cf. also Grundy, 3). The unequal block design may, however, be useful in some cases where the difference between p and q exceeds unity, but is neither greater than 5 nor greater than 10% of pq.