A Probability Plotting Procedure for General Analysis of Variance

Given a collection of analysis of variance mean squares, not all of which necessarily have the same degrees of freedom, the present paper describes a method of "mapping" them so as to facilitate the statistical structuring of the mean squares. Even under a null model of no real effects, the mean squares do not have the same distribution because their degrees of freedom may differ, and the ordered mean squares cannot be regarded as the usual order statistics of a sample from a single common distribution. If the ordered mean squares in a general orthogonal analysis of variance are $0 < S_1 \leqq S_2 \leqq \cdots \leqq S_K$ with corresponding degrees of freedom, $\nu_1,\nu_2, \cdots, \nu_K$, then the inferential reference set in the present approach is one obtained by so-called complete conditioning, i.e., repeated sampling from a set of $K$ populations such that the $i$th ordered mean square will be considered to have come from the population associated with $v_i$ degrees of freedom, for $i = 1,2, \cdots, K$. The approach consists of obtaining from each of the ordered mean squares, in turn, a maximum likelihood estimate of a presumed common error variance based on an order statistics formulation which employs complete conditioning of the mean squares. Methods of obtaining the sequence of maximum likelihood estimates as well as two graphical modes of displaying them are described. Illustrative examples are included.

Analysis of Variance Through Examples

This study focuses on optimizing the process of biofuel production from citrus peel using the Design of Experiments (DOE) technique. This study aims to determine the optimal values for the variables that have a significant impact on the production of biofuel. The variance within and between data groups was determined using the analysis of variance (ANOVA) table. The ANOVA table shows how much of the response variable's variation (biofuel production) can be explained by the independent variables (A, B, C, D, E, AB, AC, AD, AE, and BJ) and how much is caused by random error. The ANOVA table comprises of three primary parts: the F-statistic, the p-value, the df, the mean square (MS), the source of variation, and the sum of squares (SS). The wellspring of variety alludes to the beginning of the information variety, which can be either the lingering or the model. The amount of squares estimates the information's changeability, with the absolute amount of squares addressing the amount of the squared deviations of the genuine qualities from the mean worth. The residual is the sum of the squared deviations from the predicted values of the actual values, while the model's sum of squares is the sum of the squared deviations from the mean of the predicted values. The model has 10 degrees of freedom (the number of independent variables) and the residual has 4 degrees of freedom (the number of observations minus the number of independent variables). These degrees of freedom represent the number of independent pieces of information used to estimate a parameter. The mean square, which indicates the typical amount of variation for each variation source, is calculated by dividing the sum of squares by the degrees of freedom. The degree to which the model explains the variation in the data is indicated by the F-statistic, which is the ratio of the model's mean square to the residual's mean square. The probability of obtaining an F-statistic that is as large as the one observed if the null hypothesis is true is represented by the p-value. The independent variables' insignificant impact on biofuel production is the null hypothesis in this instance. The model's p-esteem in this study is under 0.05, demonstrating that the free factors essentially affect biofuel creation and that the model is genuinely huge. In addition, the model is significant because the F-statistic is relatively large in comparison to the F-distribution for the 10 and 4 degrees of freedom, respectively. The estimated coefficients for the linear regression model used to investigate the production of biofuel from citrus peel can be found in the ANOVA coefficients table. The table provides a list of the intercept and independent variables' coefficients, standard errors, t-values, and p-values. When all of the independent variables are zero, the intercept has a coefficient of 0.0672, indicating the estimated value of the response variable. The fact that the intercept does not differ significantly from zero is supported by the fact that its p-value is not significant. The fact that the coefficients of the independent variables A, E, AC, AD, AE, and BJ are not statistically significant indicates that these variables have little impact on the response variable. On the other hand, the positive coefficients and significant p-values of the independent variables B and C suggest that an increase in their values could result in an increase in the production of biofuel from citrus peel. In conclusion, the key variables that influence the production of biofuel from citrus peel have been identified thanks to the use of the Design of Experiments (DOE) method. According to the findings of this study, an increase in the production of biofuel from citrus peel may result from an increase in the values of the independent variables B and C. The development of environmentally friendly energy sources and the optimization of biofuel production processes will benefit greatly from these findings

THE analysis of variance may be thought of as a method of ranking a set of independent mean sums of squares (mean squares), or more specifically, of separating the significantly larger mean squares from a null set of homogeneous ones. When only a poor estimate of a2, the expected mean square under the null hypothesis, is available, or when there is no estimate of U2 at all, there is no standard way of doing this analysis. In such cases, the method to be presented below may be useful as a supplementary tool in the analysis of variance. As in other ranking problems, the first step will be to bring the mean squares onto a common scale which is well understood statistically. This is not quite straightforward in the general case when U2 is unknown and the individual mean squares have different degrees of freedom. It is instructive first to consider the problem in the simpler situation when the degrees of freedom are all equal to vo, say. Then one can plot the ordered sums of squares against the corresponding quantities of the chi-squared distribution with vo degrees of freedom. The plot may then be analysed in the same way as the half-normal plot of Daniel (1959) which represents the case for vo = 1. Another simple case is when a2 is known. Let the independent sums of squares be SS ,..., SSp

Linear regression.

The analysis of clinical studies: Comparison of means, part I

-Repeatability is a useful tool for the population geneticist or genetical ecologist, but several papers have carried errors in its calculation. We outline the correct calculation of repeatability, point out the common mistake, show how the incorrectly calculated value relates to repeatability, and provide a method for checking published values and calculating approximate repeatability values from the F ratio (mean squares among groups/ mean squares within groups). Received 6 February 1986, accepted 25 August 1986. REPEATABILITY is a measure used in quantitative genetics to describe the proportion of variance in a character that occurs among rather than within individuals. Repeatability, r, is given by: r = (VG + VEg)/ VP, (1) where VG is the genotypic variance, VEg the general environmental variance, and Vp the phenotypic variance (Falconer 1960, 1981). In addition to its use in assessing the reliability of multiple measurements on the same individual, repeatability may be used to set an upper limit to the value of heritability (Falconer 1960, 1981) and to separate, for instance, the effects of self and mate on a character such as clutch size (van Noordwijk et al. 1980). Repeatability is therefore a useful statistic for population geneticists and genetical ecologists. Recently, we have noticed an increasing number of published papers and unpublished manuscripts in which repeatability was miscalculated. Our purpose is fivefold: (1) to outline the correct method of calculating repeatability; (2) to point out a common mistake in calculating repeatability; (3) to show how much this mistake affects values of repeatability; (4) to provide a quick way of checking published estimates, and to calculate an approximate value of repeatability from published F ratios and degrees of freedom; and (5) to make recommendations for authors, referees, editors, and readers to prevent the promulgation and propagation of incorrect repeatability values in the literature. CALCULATION OF REPEATABILITY Repeatability is the intraclass correlation coefficient (Sokal and Rohlf 1981), which is based on variance components derived from a one-way analysis of variance (ANOVA). The intraclass correlation coefficient is given by some statistical packages; otherwise it can be calculated from an ANOVA. ANOVA is described in most statistics textbooks (e.g. Sokal and Rohlf 1981; Kirk 1968 gives a detailed treatment of more complex designs of ANOVA), so we will not repeat it here, but give the general form of the results from such an analysis in Table 1. Repeatability, r, is given by r = sA / (S + SA)' (2) where S2A is the among-groups variance component and s2 is the within-group variance component. These variance components are calculated from the mean squares in the analysis of variance as:

Analysis of Cultivar X Environment Interactions for Kansas Growing Wheat Using Regression, Variance Component, and Clustering Methods

SUMMARY Tests for fixed-effect factors in unbalanced mixed models have previously used t-tests on a contrastby-contrast basis or Wald statistics without a universally accepted method of calculating the denominator degrees of freedom. This situation has arisen because the variances of different contrasts are differently weighted sums of the variance components with associated degrees of freedom that are not necessarily equal. A simultaneous F-test for differences between all levels of a fixed-effect factor can be derived by forming new contrasts, by rotation of the original contrasts, with variances that are close to being the same weighted sum of variance components. The associated degrees of freedom for these new contrasts are nearly equal. A small simulation study shows the appropriateness of a X2 approximation to the distribution of the weighted sums of variance components. Three simple examples are used to demonstrate the effects of rotation. The last of these examples is also used to compare the proposed simultaneous F-test with the distribution of the Wald statistic obtained by numerical simulation. The method of rotations is then applied to data on the range size of mountain hares (Lepus timidus) to assess the evidence for a two-way interaction between season and habitat. Analysis of data from balanced experiments is traditionally performed by an analysis of variance in which the total sum of squared deviations of observations about their mean is partitioned into sums of squares attributable to either the different treatment effects or the different random effects. Under standard distributional assumptions, each random effect sum of squares follows a multiple of a X2-distribution, this multiple being the product of the degrees of freedom (the number of independent error contrasts whose square has the required expectation) and a linear combination of the different variance components, the coefficients in this linear combination being fixed by the design. Under any null hypothesis concerning the absence of treatment effects, the corresponding treatment sum of squares also follows such a multiple of a X2-distribution. Furthermore, the linear combination of variance components in the multiple for the treatment sum of squares is identical to the linear combination for one of the random-effect sum of squares; hence, hypotheses about the treatment effects can be tested by dividing treatment mean squares by the appropriate error mean square to form a variance ratio with an F-distribution. The benefits of experimental design accrue, first, through the variance of all contrasts for any given treatment factor or interaction having the same expectation under the null hypothesis; second, through this variance being as small as possible; and, third, through there being as many error contrasts as possible whose variance is the same as that for the treatment factor or interaction. Analysis of data that lacks these properties of balance is much less straightforward, yet there are many areas in which, due to the nature of experimental material, the treatment effects cannot be applied in a balanced fashion, and so such unbalanced data are the norm rather than the

This note describes a function written in APL/360 that computes an analysis of variance table for many kinds of experimental designs (excluding those with unequal n or missing observations). The only argument required by the function is the name of the array containing the data to be analyzed. No declaration of type of design whatsoever is needed. Instead, the Subjects factor is included as a dimension of the data array, an approach described by Lindman (1974, p. 189) as treating Error (subjects as a source of variability) "as a random factor nested in all factors in the design." Beyond this, the function also treats all data arrays as if they were fully crossed designs, with n = 1. In these respects, the function is similar to a FORTRAN routine written by Ogilvie (Note 1), and is also a generalization of computational methods described by Clifford (1968). As an example of the use of this function, consider a classification with factors A (two levels), B (three levels), and Subjects (four). The data are assigned to X, a 2.by 3 by 4 array (see Figure 1). A call to the function (APLAOV X) produces a table of sums of squares, degrees of freedom, and mean squares for the following "effects:" A, B, AB, S, AS, BS, ABS. The function automatically assigns the letters A, If, c, ... , to the first, second, third, ... , dimension of the data array, with the letter "S" assigned to the last dimension. (The data array must be structured so that Subjects is the last dimension.) Denominators of F ratios would be formed from the last four terms by the user. If, for instance, the design was completely randomized (four subjects in each of the six cells), addition of the last four sums of squares, and division by the sum of their degreesof freedom, would produce the usual mean square within cells (assuming fixed effects for A and B). If factor A was between subjects and factor B within subjects, combining the S and AS terms would give the error for testing A, while the BS term alone would be the error for B, etc. In this latter case, the four levels of the Subjects factor would refer to the number of subjects in each level of A, or to the total number of subjects if the designwas entirely repeated measures,with N =4. The function would also produce the necessary sums of squares if the design was hierarchical, if, for instance, factor B was groups nested under A, with four subjects in each nested group. In this case, the sums,of squares for the nested factor would be gotten from A and AB. The procedure generalizes to designs with any number of bases of classification, effectively limited only by workspace size. (Arrays up to rank 63 can be defined in APL.) At the cost of requiring the user to form error terms and perform the final F tests, the function achieves considerable generality and ease of use, due largely to the array definition and manipulation capabilities of APL. It has also proven quite useful didactically. Large ANOVA programs that produce not only F ratios, but p values as well, may not be entirely desirable in a first course analysis of variance, where the student might profit more from having to examine sources of variability in performing F tests. v IPUOV I I I J V-( (pS I.'C£ I pS-O-( 1."-12. ('C£-pH-pl ))-1100 01 rp-'CU'AlCU"O' 13J rpl.c£J-'S' 10J .-"1 Bl ":~I I. J C,-(tv -0I" ptv-tar-(leU2) TI-I.I 17] OIJ J-"I·rIJ-I.1: J-IV IIJ c-,o.o IIJ .«pc,.rplc,lI'OIISQ llOl rC:O-./I (VI,c,ICJI-«C·C.1 )-1) 1 D (11 J ·(cc pC, IIfC 1I2 J SQ,sIJl.(.1 ( ('1 (pOI )pD).2)' ('I£vIC,JI 113J ,O.'D.SIIJ'"I.oIJl IloJ +<1'011,. (UJ ••1 I a J n: +1I-(p£o.(ol I. JcolJ J 11,.01 IIJ) ) ISV 117J U.-l.(oIJj.Of£OI•••• JJll 111 J ,/I·'D'S.·SI£of. J l-DI £0 Ib J J. (. /l vI £01.]; J'I)A (Y(J I J'1I1 (1IJ +1, 120' sr :IIS.16S-Slll'ID) 'D'·-' l£V( ,,-(YIJ; J-IIIIPY(J; j J.II (21l "'(IIh' I;IS;I 'ID1;' ';'" 122J .(""11,. v

In the analysis of variance it is not unusual to form a denominator of an approximate t-statistic from a linear combination of mean squares. Two examples include the Behrens-Fisher problem and the repeated measures analysis of variance. One solution to the problem of finding the appropriate degrees of freedom is to use Satterthwaite's approximation while another solution, due to Cochran, is to form a weighted t-statistic. Based upon computer simulations I have found that the magnitude of the bias of the Satterthwaite approximation was less than that of the Cochran approximation in 68/75 cases considered. When the bias of the Cochran approximation was smaller than the bias of the Satterthwaite approximation, I found that the estimated bias of the Satterthwaite approximation no more than 0.5 per cent in the cases considered. I recommend performing the additional calculations required for the Satterthwaite approximations when combining two mean squares, especially when one mean square is based upon 12 or fewer degrees of freedom.

The design of the experiment is a test or series of tests, using both descriptive statistics and inferential statistics that aims to transform the input variables into an output which is the response of the experiment. The Graeco Latin Square Design was built to control the diversity of component units of local control experiment of three is a row, column, and Greek letters. Terms the Graeco Latin Square Design is if the rows, columns, Latin letters, and Greek letters have the same level and each Greek letter appears only once in each row, column, and Latin letter. The steps in the analysis of the test Graeco Latin Square Design to test the normality of the error, homogeneity of variance test, determine the degrees of freedom, calculating Sum of Squares and Mean Square every factor. Next calculate the value of F for test row, column, treatments Latin letter, and treatment of Greek letters, draw up a table of variance analysis, and conclude whether there is any effect on the response variance of each source. If there is impact, it is necessary to further test using the Duncan test

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.

This paper was written to find a table of Analysis of Variance (ANOVA) for four factors experiments for fixed models in completely randomized design. The things that must be determined are Source of Variance (SV), Degree of Freedom (df), Some of Square (SS), Mean Square (MS), Expected Values of Mean Square (EMS), F_0, and F tables. This four-factor experiment can be applied directly to experimental units with the experimental unit requirements used in the research in uniform relatively. The result of this research can found an ANOVA Table for Completely Randomized Factorial (CRF)-2222 Design for Fixed Model independently where consists of 16 of SV, 16 of df, 16 of SS, 16 of MS, 16 of EMS, 15 of F_0, and 15 of table F.

Psychometric Analyses of the Eating and Food Literacy Behaviors Questionnaire with University Students