Standard Analysis Of Variance Research Articles

UMAVC: A program for computing comparisons in multifactor designs with unequal cell Ns

While some computer center libraries include programs that perform comparisons for multifactor analysis of variance (ANOVA) designs, these programs are usually limited to designs with equal Ns. Program UMAVC extends these basic capabilities to include factorial designs with unequal Ns (Keppel, 1973; Myers, 1972). The program employs unweighted means techniques to assess comparisons for main effects (example: Acomp) and for interactions (example: Acomp by B by C). Description. This program consists of two major segments. The first segment performs one-, two-, three-, or four-way unweighted means ANOVAs for standard factorial designs (fixed effect only) having either equal or unequal Ns; this program segment is similar in design to Veldman's (1967) AVAR23 program. The second segment performs comparisons requested by the user. These comparisons may be orthogonal or nonorthogonal, or tests for trend depending upon the coefficients that are specified. Input. Three program control cards precede the data: an ANOVA problem identification card, an ANOVA design card, and a data format card. The data is read in accordance with the user-specified format and is arranged with the subscript of the last factor varying the fastest. Cards containing the Ns are placed before each cell of data cards. A card containing the comparison problem identification immediately follows the data. Cards specifying the desired comparisons and their coefficients complete the deck setup. Output. Output from UMAVC consists of (l) a standard unweighted means ANOVA table including the probabilities of the computed F ratios, (2) means and Ns, and (3) a comparison table including sums of squares, degrees of freedom, mean squares and F ratios accompanied by their associated probabilities. Restrictions. The number of levels of each ANOV A factor must not exceed 10. The program can analyze only one comparison per factor in a given run; different comparisons for different factors may be analyzed in a single run. In the case of trend comparisons, for example, to investigate both linear and quadratic trends for factor A would require two runs, while a linear trend for factor A, a quadratic trend for factor B, a cubic trend for factor C, and a quartic trend for factor D could be investigated in a single run. Computer and language. The program is written in FORTRAN IV and was developed on an IBM 370{155 with virtual memory. It executes in approximately 192K of core; reducing the maximum number of levels for the ANOV A factors below the present value of 10 would greatly reduce the program's core requirements. Compilation time is approximately 30 sec. Data sets of average size may be analyzed in 5-10 sec of CPU time. Two direct access devices are required. The program consists of a mainline and 13 subroutines. Availability. A documented listing of UMAVC is available at no cost from the first author at: Southern College of Optometry, 1245 Madison Avenue, Memphis, Tennessee 38104.

Another Line for the Analysis of Variance

A test is developed for hypotheses about the grand mean in the analysis of variance, using the known relationship between the t distribution and the F distribution with 1 df for the numerator. Examples are given to suggest that the test may be of value in some research settings, especially when a known comparison population exists, and collection of data on a control group is expensive. Where appropriate, the test is efficient, as its addition to a standard analysis of variance requires no additional data collection and results in no loss of df from other comparisons in the analysis.

Some effects of planting density and variety on the relationship between tuber size and tuber dry-matter percentage in potatoes

SummaryThree experiments are reported which investigated the effects of planting density and variety on the relationship between tuber dry-matter percentage and tuber size. In the first experiment dry-matter determinations were made on samples of tubers less than 3·8 cm and greater than 3·8 cm from different planting densities obtained at eight fortnightly harvests. Standard analysis of variance was used for each harvest date and apart from Maris Piper giving consistently higher dry-matter percentages than Pentland Crown for the same tuber sizes, effects were small and difficult to interpret. The reasons for the problems in interpretation, which were mainly due to differences in tuber size distribution, are discussed.In the second and third experiments the effects of tuber-size distribution were removed by regression analyses of tuber dry-matter percentage on tuber size in definable grades and curves were fitted to the data. These curves revealed that tuber dry-matter percentage tends to show a quadratic response to increasing tuber size and is affected by variety and stem density. The significance of these results is discussed.

Sources of variation in the number of lambs born in a grazing experiment

Considerable variation was observed in the number of lambs born to fine-woolled Merino ewes during a 4-year grazing experiment in which the factors were four pasture management systems x three stocking rates. Attempts at interpretation by standard analysis of variance were ineffectual and accordingly the data were subjected to further examination by using non-orthogonal and contingency models and pattern analysis. The strongest influence discerned was the tendency of particular ewes to consistently produce 0, 1, or 2 lambs. The majority of births (72%) were singles; twinning was relatively uncommon (<10 %) and was most prevalent in the third year; barrenness was relatively high (18 %) particularly in the first year. Significant associations were found between the number of lambs born and the ewe's liveweight and liveweight change before and after mating. Pattern analysis suggested that part of this variation was associated with differences in the initial pasture composition between plots. There was a highly significant year effect in all the relevant animal variables examined but it was not possible to separate the contributions of weight and age to this effect. It is suggested that some of the variation in reproduction rate between plots was due to non-uniform allocation of the inherently barren and twinning ewes.

Analysis of Variance Applied to Measures of Central Tendency and Dispersion in Sediments

ABSTRACT Two problems that frequently confront researchers' performing routine sediment analysis involve: (1) the choice of which mathematical technique to use in order to evaluate the size frequency distribution and (2) the number of sieves needed to insure sufficient accuracy for the size analysis. To investigate this, samples of recent beach and river sediments were collected and were sieved using a 0.5 and 1.0 phi sieve interval. Mean diameters, standard deviation, skewness and kurtosis were calculated using (1) the central moment measure equations, (2) Folk and Ward's equations and (3) Inman's equations. The results obtained with the central-moment measure equations, and the 0.5 phi sieve data, were assumed to provide the most accurate estimates of the various parameters. These values were then compared with those obtained with Folk and Ward's and Inman's equations for both 0.5 and 1.0 phi sieve data to determine what differences resulted. Comparisons were made by both graphical plots and standard analysis of variance F and T tests to determine if statistically different resu ts were obtained using the three different methods. The results of this study indicated that: (1) no significant differences result when mean diameters and standard deviations are calculated by the three methods, if a 0.5 phi sieve interval is used, (2) only minor differences are found between the three methods if the mean diameter of beach and river sediments are derived using 1.0 phi sieve data, (3) noticeable loss of accuracy results if the standard deviation of either beach or river sediments is computed, using 1.0 phi data, (4) significant differences from the central-moment estimates occur if skewness and kurtosis are calculated for well sorted sediments using Folk and Ward and Inman's equations, especially if 1.0 phi sieve data is used, (5) neither Folk and Ward nor Inman's equation can be used to reliably determine kurtosis, re ardless of whether 0.5 or 1.0 phi sieve data is used.

A Monte-Carlo trial on the behaviour of the non-additivity test with nonnormal data

SUMMARY The behaviour of the t test based on Tukey's one degree of freedom for non-additivity is investigated by a small Monte-Carlo trial on two types of nonnormal data. As is to be expected, the levels of significance are considerably exaggerated when there are no real additive effects, but there is no serious disturbance when additive effects are introduced. The implication of these findings in the various contexts in which the test may be used are discussed. The standard analysis of variance of a p x q two-factor table gives a partition into the average (additive) effects of the two factors with p - 1 and q - 1 degrees of freedom, respec- tively and their interaction with (p - 1) (q - 1) degrees of freedom. The interaction term gives a general measure of the observed deviations from the additive model. It is to be expected that these deviations, in so far as they represent real departures from the additive model, will be orderly. It is therefore often informative to subdivide the (p - 1) (q - 1) degrees of freedom into subsets, so that components likely to be of importance are separated from those which are expected to be small in relation to error. Often appropriate subdivisions can be specified in terms of a linear model. Thus if both factors are quantitative and give responses that are approximately linear the 'linear x lin- ear' component may well adequately represent the actual interactions. The coefficients of this linear function with 1 degree of freedom are given by the products of the coefficients which give the linear regressions for each factor separately. If the responses are not linear, but each response curve can be expressed as a function of a single unknown parameter, at least approximately, linear functions can be chosen which give the most accurate estimates of these parameters. Here again the 1 degree of freedom given by the products of these coefficients may well isolate the greater part of the real interaction. This may be termed a 'generalized linear x linear interaction' with parameters chosen a priori. If the factors are not quantitative, or if the form of the response curves is not known, the marginal means may themselves be used as a basis for subdivision. An early example of this is given by Yates & Cochran (1938). In the analysis of a set of variety trials on five varieties of barley at six centres in two consecutive years, the mean yields of all the varieties at each centre were taken as a measure of the fertility at that centre. Regressions of the separate varieties on these mean yields were found to be approximately linear; the mean regression is of course strictly linear with unit slope. This gave the subdivision of the 20 degrees of

A unifying computational method for the analysis of complete factorial experiments

A computational method which may be used for the calculation of sums of squares in the analysis of variance of complete factorial experiments and in the computation of main effect or interaction means is described. The method is elucidated as unifying since one method can be used for a variety of purposes each previously requiring different methods. The programming advantages of such a method are obvious. The following variants are discussed: (1) the standard analysis of variance; (2) analyses omitting certain levels of one or more factors; (3) separate analyses for some levels of a factor or for combinations of levels of more than one factor. These are performed simultaneously; (4) the calculation of main effect or interaction means. The method expects the data in standard order and it leaves the data in that order so that many analyses of the same data can be performed without rearrangement. The total sum of squares, excluding a replication sum of squares, is partitioned into all polynomial partitions and their interactions each with one degree of freedom. This is so even if factors have unequally spaced factor levels.

The estimation of residual variance in quadratically balanced least-squares problems and the robustness of the F-test

In the first part of the paper a new simple proof is given of a special case of a result due to Hsu (1938) concerning the estimation of residual variance in a linear least-squares model. In the second part of the paper similar methods are applied to examine the effect of non-normality in the standard analysis of variance F -test. A condition is found for the test to be insensitive to non-normality; the condition is satisfied in the standard set-ups, such as balanced two-or more-way arrangements, Latin squares, balanced incomplete block designs and equi-replicated nested designs. It is shown that non-normality has little effect on inference made by intrablock analysis in a wider class of incomplete block designs with a particular property. Expressions for judging the effect of non-normality in other practical situations are given.

The Analysis of Variance and Derivation of Standard Errors for Incomplete Data

In a previous paper [10] based on the fundamental paper by Yates [11], the writer has discussed the estimation of missing observations in incomplete data. When the data are completed with these estimates, standard calculations give the correct estimates of treatment effects, etc., and a standard analysis of variance yields the correct residual sum of squares for the estimation of error. However, as Yates [11] observed, other component sums of squares in the standard analysis are incorrect (though perhaps not seriously so). For the randomized blocks design, and for a Latin square with one missing value, Yates derived formulae for adjusting the treatment sum of squares to its correct value. Cornish [2, 3] gave the corresponding results for incomplete block designs, and for designs such as lattice squares, with one missing value. The present paper, following on from the previous paper [10], deals with correcting the standard analysis of variance, and derives a general formula for the necessary corrections. Specific formulae are given which, in particular, provide the necessary correction of the treatment sum of squares when several observations are missing, for the designs with two-way restriction. The standard formulae for determining variances and covariances of treatment comparisons, etc., will also need to be adjusted when the comparisons involve missing values. Tocher [8] gave a general formula for this purpose, which is here extended to cover the singular cases. The principles discussed are illustrated by application to the analysis of a Latin square experiment (numerical example), and in the derivation of an analysis for B.I.B. designs with missing blocks.

Estimation of Missing Values for the Analysis of Incomplete Data

When only a few observations are missing from data that otherwise conform to a planned experimental design, the fitting of a linear model to the data by the principle of least squares is most simply carried out using estimated values for the missing observations, as in this way most effective use is made of the remaining symmetry in the experimental results. A comprehensive account of this procedure was given by Yates [12] in 1933. Firstly, the data are completed by inserting estimates of the missing values, which are determined so that the residual sum of squares for the completed data shall be a minimum. The correct estimates of treatment effects and other parameters in the linear model are then given by the standard formulae for the experimental design, and the correct residual sum of squares for the incomplete data is given by the standard analysis of variance (with the degrees of freedom appropriately reduced). However, the other components of this analysis are only approximations (though usually reasonably good) to the corresponding components of a correct least squares analysis of variance, since the latter involves in principle the fitting of one or more auxiliary models, for which appropriate estimates of the missing values would be necessary. It should also be noted that the usual formulae for standard errors of treatment comparisons need to be modified to take into account the loss in precision due to the missing values. The non-standard aspects of this missing value procedure will be discussed by the present writer in two papers: the present paper, dealing with the estimation of missing values (which enables completion of the first phase of analysis), and a second paper [11] which deals with correction of the analysis of variance and the derivation of standard errors. In connection with incomplete block designs, lattice squares,

Estimation of Variance and Covariance Components

The theory of variance component analysis has been discussed recently by Crump (1946, 1951) and by Eisenhart (1947). These papers and, indeed, most of the published works on estimating variance components deal with the one-way classification, with nested classifications, and with factorial classifications having equal subclass numbers. Also most papers on this subject are concerned with what Eisenhart (1947) has called Model II; that is, all elements of the linear model save gi are regarded as random variables. In the above cases, estimation of variance components is usually accomplished by computing the mean squares in the standard analysis of variance, equating these mean squares to their expectations, and solving for the unknown variances. These techniques are described in many statistical textbooks. Unfortunately, research workers in some of those fields in which much use is made of variance component estimates are unable to obtain data which have the above described characteristics. This is particularly true in those fields in which survey data must be used or where, even in a well-planned experiment, the subclasses are of quite unequal size due, for example, to differences in litter numbers. Also,