Modelling ties in the sign test

If ties occur in the sign test, the procedure recommended by Coakley and Heise (1996, Biometrics 52, 1242-1251) is the asymptotic uniformly most powerful nonrandomised test due to Putter (1955, Annals of Mathematical Statistics 26, 368-386). It may be shown that this is a consequence of how the probability of a tie is modelled. Other models with different optimal procedures can be constructed.     

Signed-Rank Tests in the Presence of Ties (with extended tables)

We begin with a review of the areas of application of the signed-rank tests (SRTs) and we conclude that the results are exact only if no ties of non-null differences exist. In order to apply the SRTs according to WILCOXON and according to PRATT also in the presence of ties, by assigning midranks, we derive their null distributions. As special cases the null distributions for the problem without ties are obtained. In order to save the practising statistician the time-consuming calculations of the distribution functions, we compute tables of critical values (for reasons of volume they will be published as part of the reprints only). For N₀ = 0 (1) 5 null differences and M = = 1(1) 10 non-null differences the critical values of all distributions with all possible tie vectors are calculated. Instructions are provided and an example serves to illustrate the use of the table. The extension of the tables are obtained by means of counting formulas given in the text. Approximations are provided in order to make the application of tests possible for larger samples as well. It is shown that the approximation of the null distribution in the presence of ties by the null distributions under the assumption of no ties in some cases overstates and sometimes understates the exact rejection probability. For N₀ = 0 (1) 10 and M = 1 (1) 10 all distributions with all possible tie vectors for the SRTs with WILCOXON and PRATT ranking are examined with respect to the lattice type of the test statistic. The result is given in table 6. It is evident that the portion of PRATT -distributions with lattice character decreases as the number of null differences increases. Continuity corrections are obtained for the asymptotic normal distribution which take into account the lattice character of the distribution of the test statistic.     

Using matrix of thresholding partial correlation coefficients to infer regulatory network

DNA arrays measure the expression levels for thousands of genes simultaneously under different conditions. These measurements reflect many aspects of the underlying biological processes. A method based on the matrix of thresholding partial correlation coefficients (MTPCC) is proposed for network inference from expression profiles. It includes three main parts: (1) hierarchical cluster analysis, (2) cluster boundaries establishment, and (3) regulatory network inference. The method was applied to the expression data of 2467 genes in Saccharomyces cerevisiae measured under 79 different conditions [Eisen, M.B., Spellman, P.T., Brown, P.O., Botstein, D., 1998. Cluster analysis and display of genome-wide expression patterns. Proc. Natl. Acad. Sci. 95, 14863-14868]. Using hierarchical clustering and cluster boundaries establishment, the 2467 genes were grouped into 12 clusters. The expression profiles of each cluster were expressed as a set of expression levels average over the cluster that constituted genes of each condition. Then the expression data of these clusters were subjected to the analysis of partial correlation, and the significance of each element in the obtained partial correlation coefficient matrix (PCCM) was examined by a permutation test. The corresponding undirected dependency graph (UDG) was obtained as a model of the regulatory network of S. cerevisiae. The veracity of the network was evidenced by the consistency of our results with the collected results from experimental studies.     

A general class of correlation coefficients for the 2 x 2 crossover design

The Pearson correlation coefficient and the Kendall correlation coefficient are two popular statistics for assessing the correlation between two variables in a bivariate sample. We indicate how both of these statistics are special cases of a general class of correlation statistics that is parameterized by gamma element of [0, 1]. The Pearson correlation coefficient is characterized by gamma = 1 and the Kendall correlation coefficient by gamma = 0, so they yield the upper and lower extremes of the class, respectively. The correlation coefficient characterized by gamma = 0.5 is of special interest because it only requires that first-order moments exist for the underlying bivariate distribution, whereas the Pearson correlation coefficient requires that second-order moments exist. We derive the asymptotic theory for the general class of sample correlation coefficients and then describe the use of this class of correlation statistics within the 2 x 2 crossover design. We illustrate the methodology using data from the CLIC trial of the Childhood Asthma Research and Education (CARE) Network.     

The Influence of Noise and Non-Normality on Biserial Correlation Coefficients

The estimation of any biserial correlation coefficient is biased if noise-like processes are overlaid. We've carried out a computer simulation to study this effect. The effect can be described in a universal way. Additionally, we've varied the type of distribution within the simulation and compared this effect with the effect of noise. Based on this comparison, we've been able to draw some conclusions on the use of biserial correlation coefficients.     

Data transformation for rank reduction in multi-trait MACE model for international bull comparison

Since many countries use multiple lactation random regression test day models in national evaluations for milk production traits, a random regression multiple across-country evaluation (MACE) model permitting a variable number of correlated traits per country should be used in international dairy evaluations. In order to reduce the number of within country traits for international comparison, three different MACE models were implemented based on German daughter yield deviation data and compared to the random regression MACE. The multiple lactation MACE model analysed daughter yield deviations on a lactation basis reducing the rank from nine random regression coefficients to three lactations. The lactation breeding values were very accurate for old bulls, but not for the youngest bulls with daughters with short lactations. The other two models applied principal component analysis as the dimension reduction technique: one based on eigenvalues of a genetic correlation matrix and the other on eigenvalues of a combined lactation matrix. The first one showed that German data can be transformed from nine traits to five eigenfunctions without losing much accuracy in any of the estimated random regression coefficients. The second one allowed performing rank reductions to three eigenfunctions without having the problem of young bulls with daughters with short lactations.     

Kendall's and Spearman's correlation coefficients in the presence of a blocking variable

The use of a weighted sum of Kendall's taus or a weighted sum of Spearman's rhos for testing association in the presence of a blocking variable is discussed. In a Monte Carlo study the two are shown to have essentially the same power with the optimal choice of weights. In the presence of ties, the weighted sum of Spearman's rhos is preferred because its variance has a much simpler form. An example is given from the field of radiation therapy.     

Random marginal agreement coefficients: rethinking the adjustment for chance when measuring agreement

Agreement coefficients quantify how well a set of instruments agree in measuring some response on a population of interest. Many standard agreement coefficients (e.g. kappa for nominal, weighted kappa for ordinal, and the concordance correlation coefficient (CCC) for continuous responses) may indicate increasing agreement as the marginal distributions of the two instruments become more different even as the true cost of disagreement stays the same or increases. This problem has been described for the kappa coefficients; here we describe it for the CCC. We propose a solution for all types of responses in the form of random marginal agreement coefficients (RMACs), which use a different adjustment for chance than the standard agreement coefficients. Standard agreement coefficients model chance agreement using expected agreement between two independent random variables each distributed according to the marginal distribution of one of the instruments. RMACs adjust for chance by modeling two independent readings both from the mixture distribution that averages the two marginal distributions. In other words, both independent readings represent first a random choice of instrument, then a random draw from the marginal distribution of the chosen instrument. The advantage of the resulting RMAC is that differences between the two marginal distributions will not induce greater apparent agreement. As with the standard agreement coefficients, the RMACs do not require any assumptions about the bivariate distribution of the random variables associated with the two instruments. We describe the RMAC for nominal, ordinal and continuous data, and show through the delta method how to approximate the variances of some important special cases.     

Estimating the generalized concordance correlation coefficient through variance components

The intraclass correlation coefficient (ICC) and the concordance correlation coefficient (CCC) are two of the most popular measures of agreement for variables measured on a continuous scale. Here, we demonstrate that ICC and CCC are the same measure of agreement estimated in two ways: by the variance components procedure and by the moment method. We propose estimating the CCC using variance components of a mixed effects model, instead of the common method of moments. With the variance components approach, the CCC can easily be extended to more than two observers, and adjusted using confounding covariates, by incorporating them in the mixed model. A simulation study is carried out to compare the variance components approach with the moment method. The importance of adjusting by confounding covariates is illustrated with a case example.     

Impact of biparental mating on correlation coefficients in bread wheat

Summary Phenotypic and genotypic correlation coefficients and path-coefficients were studied in the biparental (BIPs) and F₃ self progenies of the two wheat crosses. A comparison of correlation coefficients in the BIPs and the F₃'s revealed that as many as twelve new significant correlations were noticed in case of the BIPs in cross I although some of them occurred in the undesirable direction. On the other hand, only three new correlations were observed in the BIPs of cross II, although as many as fifteen correlations were not significant. Results suggested that intermating in the F₂ was effective in breaking the linkages. Path-coefficient analysis further revealed that the direct effect of tillers/ plant on grain yield was important and remained unchanged in both populations of cross I. In cross II, the direct effect of tillers/plants on grain yield was also high and it increased in the BIPs. Intermating seemed to have influenced considerably both the direct and indirect effects.