Addendum to ‘The Incremental Progression From Fixed to Random Factors in the Analysis of Variance: A New Synthesis’

Summary Classically, the distinction between a fixed versus a random factor in analysis of variance has been considered a binary choice. Here we consider that any given factor can also occur along an incremental series of steps between these two extremes, depending on the sampling fraction of its levels from the wider population. Fixed factors occur where all possible levels are drawn, and random factors occur in the limit as the population of possible levels approaches infinity. When some identifiable fraction of a finite population of possible levels is drawn, the factor can be thought of as something in between fixed and random, and can be analysed explicitly as finite directly within the analysis of variance (ANOVA) framework. Requiring explicit specification of the population size from which observed levels are drawn for each factor, we provide a unified approach to derive expectations of mean squares (EMS) in ANOVA for any types of factors along the entire graded progression from fixed to random, inclusive, that may be nested within or crossed with one another, from balanced, asymmetrical or unbalanced designs, including multi‐level hierarchical sampling designs, mixed models and interactions. Implications for estimation of variance components, tailored bootstrap methods and tests of hypotheses under minimal assumptions of exchangeability are described and further extended to multivariate dissimilarity‐based settings.

Analysis of Variance Through Examples

The contraction of floating collagen gels is suggested to mimic the reorganization of collagenous matrix during development and tissue healing. Here, we have studied two osteogenic cell lines, namely MG-63 and HOS, and a chemically transformed subclone of HOS cells, HOS-MNNG. Transforming growth factor-beta (TGF-beta), a putative regulator of bone fracture healing, increased collagen gel contraction by MG-63 and HOS-MNNG, but not by HOS cells. Our data show that TGF-beta-induced fibronectin synthesis is not sufficient for the process. Instead, anti-beta 1 integrin antibodies could prevent the contraction. There are three different integrin heterodimers that are known to mediate the cell-collagen interaction, namely alpha 1 beta 1, alpha 2 beta 1, and alpha 3 beta 1. In MG-63 cells TGF-beta increased the expression of alpha 2 beta 1 integrin and decreased the expression of alpha 3 beta 1 integrin, whereas alpha 1 beta 1 integrin is not expressed. HOS cells had no alpha 2 beta 1 integrin, neither did TGF-beta induce its expression. However, HOS-MNNG cells expressed more alpha 2 beta 1 integrin when treated with TGF-beta. Thus, we suggest that the mechanism of the enhanced collagen gel contraction by TGF-beta is the increased expression of alpha 2 beta 1 integrin heterodimer. To further test this hypothesis, we expressed a full-length alpha 2 integrin cDNA in HOS cells and in MG-63 cells. We obtained HOS cell clones that expressed alpha 2 beta 1 heterodimer, and the ability of these cells to contract collagen gels was greatly enhanced. Furthermore, the contraction by MG-63 cells transfected with alpha 2 integrin cDNA was enhanced, and the contraction by cells transfected with antisense oriented alpha 2 integrin cDNA was decreased. Thus, both in MG-63 and HOS cells the increased alpha 2 integrin expression alone was sufficient for the enhanced contraction of collagen gels. Furthermore, the amount of alpha 2 integrin is critical for the process, and its decrease leads to diminished ability to contract gels.

Further difficulties with multifactorial analysis of variance: Random and nested factors and independence of data

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

Analysis of variance is used to test the significance of the difference between the means of two or more samples, but due to the influence of uncontrollable random factors and controllable factors imposed on the results in the study, the data obtained in the study presents a fluctuating state. In this paper, the single factor analysis of variance in the least significant difference method is used to analyze the aroma components of the collected Hubei Qingzhuan tea and Yunnan Pu'er tea, and the T test is used to complete the paired comparison between each group, which improves the sensitivity of the test and makes each level It is also possible to detect the slight difference in the mean value among them, which provides a new way of thinking for the scientific and reasonable evaluation index system.

The purpose of this article is to investigate the decision qualities of the Bayes factor (BF) method compared with the p value-based null hypothesis significance testing (NHST). The performance of the 2 methods is assessed in terms of the false- and true-positive rates, as well as the false-discovery rates and the posterior probabilities of the null hypothesis for 2 different models: an independent-samples t test and an analysis of variance (ANOVA) model with 2 random factors. Our simulation study results showed the following: (a) The common BF > 3 criterion is more conservative than the NHST α = .05 criterion, and it corresponds better with the α = .01 criterion. (b) An increasing sample size has a different effect on the false-positive rate and the false-discovery rate, depending on whether the BF or NHST approach is used. (c) When effect sizes are randomly sampled from the prior, power curves tend to be flat compared with when effect sizes are prespecified. (d) The larger the scale factor (or the wider the prior), the more conservative the inferential decision is. (e) The false-positive and true-positive rates of the BF method are very sensitive to the scale factor when the effect size is small. (f) While the posterior probabilities of the null hypothesis ideally follow from the BF value, they can be surprisingly high using NHST. In general, these findings were consistent independent of which of the 2 different models was used. (PsycINFO Database Record

114. Effect of grazing and feeding management on milk mineral concentrations

Ignoring a nested factor can influence the validity of statistical decisions about treatment effectiveness. Previous discussions have centered on consequences of ignoring nested factors versus treating them as random factors on Type I errors and measures of effect size (B. E. Wampold & R. C. Serlin). The authors (a) discuss circumstances under which the treatment of nested provider effects as fixed as opposed to random is appropriate; (b) present 2 formulas for the correct estimation of effect sizes when nested factors are fixed; (c) present the results of Monte Carlo simulations of the consequences of treating providers as fixed versus random on effect size estimates, Type I error rates, and power; and (d) discuss implications of mistaken considerations of provider effects for the study of differential treatment effects in psychotherapy research.

Organic (OA) and low-input (LI) farming rely on genotypes with high adaptability that maintain good performance over a broad range of agronomic and environmental conditions. Two synthetic varieties of Brassica oleracea var. italica Plenck (broccoli) were developed from a landrace. Their performance and stability under LI and OA farming conditions were then assessed and compared to a F1 hybrid variety. Identical experiments were carried out over a period of 2 years in three locations in Italy having different management and pedo-climatic conditions. Initially, an analysis of variance, carried out using a linear mixed model (LMM), with “Genotype” (“G”) and “Location” (“L”) as fixed factors and “Year” (“Y”) as a random factor, showed that the “Genotype” effect was significant for days to heading (DH), head number (HN), plant diameter (PD), plant vigour (PV) and plant height (PH). The “L” effect was significant for PD and PV. “G × L” interaction was significant for DH, PV and for yield. To obtain a better understanding of entry performances across years and locations, each location—year combination was considered as “Environment” and the additive main effects and multiplicative interaction analysis was used to dissect the “G × E” interaction. Synthetic varieties had good performances and always had a higher stability than the F1 hybrid. The data discussed in this study suggest that heterogeneous varieties developed from adapted materials are suitable for OA and LI because of their stability.

Factors affecting the reproductive performance of Finnish trotter stallions

A statistical analysis tool for variation simulation modeling

Allometric coefficients for body measures and morphometric indexes in a meat-type sheep breed

In his text on the analysis of variance, Sheffe (1959) presents a detailed development of the mixed model for two factors. The model, as initially described, appears to be quite general and is capable of accomodating a wide variety of experimental situations in which one factor is assumed fixed and the other random. Other authors have proposed different models which appear to differ from Scheff6's model in various ways. These models have been the subject of considerable discussion, with much of the controversy focusing on the expressions for the expected value of the mean square associated with the random factor. The purpose of this paper is to relate some of the existing models to the Scheffe model in an attempt to provide the user with some guidance in the choice of an appropriate model. The question of the correct expressions for the expected mean squares is also resolved. Although primary consideration is given to the relations between contending, infinite population models, one relation to the limiting form of the finite model is also noted. For ease of reference, the models are formally stated in Section 2. These statements also contain the basic relations between the models. These relations are then discussed in Section 3, and in Section 4, the correct expressions for the expected means squares are given.

Impact du traitement par interféron α sur le métabolisme du tryptophane chez des patients porteurs d’hépatite C chronique: Résultats de la phase d’étude pilote sur dix patients