Step-down Procedure Research Articles

A Step‐down Procedure of Eliminating Variables in Multivariate Analysis of Variance

Biometrical JournalVolume 19, Issue 6 p. 449-453 Article A Step-down Procedure of Eliminating Variables in Multivariate Analysis of Variance T. Caliński, T. Caliński Academy of Agriculture and the Institute of Plant Genetics of the Polish Academy of Sciences, Poznan, PolandSearch for more papers by this authorZ. Kaczmarek, Z. KaczmarekSearch for more papers by this author T. Caliński, T. Caliński Academy of Agriculture and the Institute of Plant Genetics of the Polish Academy of Sciences, Poznan, PolandSearch for more papers by this authorZ. Kaczmarek, Z. KaczmarekSearch for more papers by this author First published: 1977 https://doi.org/10.1002/bimj.4710190608AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume19, Issue61977Pages 449-453 RelatedInformation

Selecting Concomitant Variables Using a Likelihood Ratio Step-Down Procedure and a Method of Testing Goodness of Fit in an Exponential Survival Model

In 1960, the Veterans Administration Co-operative Urological Research Group (VACURG) undertook a study designed to compare several standard modes of treatment for cancer of the prostate. In 1967 the group analyzed, by treatment, survival experience through August 15, 1966 and published these data along with a detailed description of the study design (Mellinger [1967]). Meanwhile, Feigl and Zelen [1965] published a paper dealing with follow-up studies using an exponential model with concomitant information to estimate survival probabilities. This model was extended in 1966 to censored data by Zippen and Armitage [1966]. A more recent paper by the VACURG group (Huse et al. [1974]) modified this model to determine the relative importance of presenting signs and symptoms in relation to survival and to suggest optimum treatment depending upon symptoms presented at diagnosis. Using the same basic model (see Appendix 1), we have reanalyzed the VACURG data to obtain estimates of the parameters of eight variables (Table 1) felt to be important in predicting survival of patients with prostatic cancer and have evaluated the fit of the model to the data. We also introduce a log likelihood ratio step-down procedure for elimination of variables.

Nested step-down multiple decision procedures for testing homogeneity in Bernoulli trials

Abstract A multistage multiple decision procedure is developed to test if a number of sets of independent variables are binomially distributed and if so is the case, to test if they are homogeneous with respect to the binomial parameter p. The homogeneity hypothesis is decomposed in a hierarchy of homogeneity hypotheses of the nested type. The binomial distributions are either untruncated or truncated in different ways. In the multiple procedure the goodness of fit test is included as an integrated part. For those test situations which cannot be considered as two-sample tests, chi-square tests are applied throughout. Small-sample and large-sample versions of all tests are given, also of the goodness of fit tests. Special merits of the procedure are its power to exclude deviating and heterogeneous sets of variables and to separate different sets of variables where the variables within each set are homogeneous with respect to p, but the different sets have different values of p. The significance level of th...

Maximizing the validity of a unit-weight composite as a function of relative component lengths with a fixed total testing time

An explicit solution is given to the problem of assigning relative lengths to the subtests of a test so as to maximize the correlation of the unit weight composite with a specified criterion when the total testing time is fixed. This solution is valid and unique whenever it specifies nonnegative times for all variables. A step-down procedure is suggested for cases in which some of the testing times are zero. This procedure does not necessarily provide an optimal allocation. However in examples studied it is found to provide near optimum results. Algorithms are also developed for the determination of the least total testing time required to attain specified multiple and composite correlations. A numerical example is given illustrating the use of the unit weight procedure in combination with the regression weight algorithm.

Why Stepdown Procedures in Variable Selection

Recent reviews have dealt with the subject of which variables to select and which to discard in multiple regression problems. Lindley (1968) emphasized that the method to be employed in any analysis should be related to the use intended for the finally fitted regression. In the report by Beale et al. (1967), the emphasis is on selecting the best subset for any specified number of retained independent variables. Here we will be concerned with pointing out the advantages of the variable selection scheme in which independent variables are successively discarded one at a time from the original full set. While these advantages are not unknown to workers in this field, they are however not appreciated by the statistical community in general. For the purposes of this demonstration it is assumed that we are in the nonsingular case so that the number of observations exceeds the number of regressor variables. Let us begin by considering economy of effort. Suppose that we were using a step-up regression procedure, ignoring for the while its theoretical deficiencies (to be discussed later). We should then first fit k simple regressions, one for each of the k regressor variables considered, selecting the single most significant individual regressor variable. Having made this selection we would proceed with k - 1 additional fits to determine which of the remaining variables in conjunction with the first selected yielded the greatest reduction in residual variation. This process is continued on so as to provide a successive selection and ordering of variables. We may even require the ordering of all k variables, leaving for later decision what critical juncture is to be employed in determining which of the k variables to retain, which to reject-if we do so we shall have made a total of k(k + 1)/2 fits, albeit they may have differed greatly in their degree of complexity. A complete stepdown regression procedure however requires but k fits, as will now be indicated. Suppose we have done a multiple regression on all k variables and wish to consider the k possible multiple regressions on all sets of k - 1 variables, that is where 1 variable has been deleted. The results for these k possible multiple regressions are implicit in the initial k-variable regression, provided we have secured the inverse matrix, or at least its diagonal, necessary for testing the significance of the fitted partial regression coefficients. The case

Why Stepdown Procedures in Variable Selection

Why Stepdown Procedures in Variable Selection

Development of competence of Haemophilus influenzae.

Spencer, Hugh T. (The Johns Hopkins University School of Hygiene and Public Health, Baltimore, Md.), and Roger M. Herriott. Development of competence of Haemophilus influenzae. J. Bacteriol. 90:911-920. 1965.-A chemically defined nongrowth medium was developed for the induction of competence of Haemophilus influenzae by a stepdown procedure. Cells grown logarithmically in Heart Infusion Broth became competent after being transferred to a medium which consisted of amino acids, sodium fumarate, and inorganic salts. Chloramphenicol (2 mug/ml) or l-valine (1 mug/ml) in the nongrowth medium inhibited development of competence. The inhibitory action of l-valine was reversed by comparable concentrations of l-isoleucine. Kinetic studies of the development of competence showed a variable capacity of competent cells to take up deoxyribonucleic acid and reaffirmed earlier findings that competence was not transmissible in H. influenzae. Addition of nicotinamide adenine dinucleotide, thiamine, calcium pantothenate, uracil, and hypoxanthine to the medium for competence resulted in a minimal growth medium in which reduced levels of competence were developed.

Some Nonparametric Median Procedures

Under the nonparametric approach, various methods have been suggested to avoid the assumption of normality and homoscedasticity implicit in the analysis of variance. For the one-way classification, i.e., to decide whether $c$ samples come from the same population, Kruskal and Wallis [9] have proposed the $H$-test based on ranks; Mood and Brown [10] have proposed the $M$-test, utilizing the numbers of observations above the median of the combined sample; while the present author [2] has offered the $V$-test based on the number of $c$-plets that can be formed by choosing one observation from each sample such that the observation from the $k$th sample is the least, $k = 1, 2, \cdots, c$. For the two-way classification, Friedman [8] has made use of ranks. His statistic, to test the hypothesis that the rankings by $m$ "observers" of $n$ "objects" are independent, essentially offers a test for the two-way classification with one observation per cell. Durbin [7] has given a generalization for the balanced incomplete block design. Benard and Van Elteren [1] have generalized it still further. Mood and Brown [6, 10] also have considered the two-way classification with one observation per cell or the same number of observations per cell. In the first part of the present paper, their test has been extended to cover incomplete block situations. Mood and Brown [6, 10] have also considered some simple regression problems. In the present paper their methods are extended to discuss some additional regression problems. Next some bivariate analysis of variance problems are considered. The "step-down procedure" [11, 12] is used to reduce the problem to the univariate case with the other variate as a concomitant variate. The regression methods developed earlier are used here in these bivariate problems. The method seems to be perfectly general and could be extended to three or more variates, that is, to the multivariate situation. Most of the tests are offered on heuristic considerations. They are expected to be good for large samples.

Step-Down Procedure in Multivariate Analysis

Test criteria for (i) multivariate analysis of variance, (ii) comparison of variance-covariance matrices, and (iii) multiple independence of groups of variates when the parent population is multivariate normal are usually derived either from the likelihood-ratio principle [6] or from the "union-intersection" principle [2]. An alternative procedure, called the "step-down" procedure, has been recently used by Roy and Bargmann [5] in devising a test for problem (iii). In this paper the step-down procedure is applied to problems (i) and (ii) in deriving new tests of significance and simultaneous confidence-bounds on a number of "deviation-parameters." The essential point of the step-down procedure in multivariate analysis is that the variates are supposed to be arranged in descending order of importance. The hypothesis concerning the multivariate distribution is then decomposed into a number of hypotheses--the first hypothesis concerning the marginal univariate distribution of the first variate, the second hypothesis concerning the conditional univariate distribution of the second variate given the first variate, the third hypothesis concerning the conditional univariate distribution of the third variate given the first two variates, and so on. For each of these component hypotheses concerning univariate distributions, well known test procedures with good properties are usually available, and these are made use of in testing the compound hypothesis on the multivariate distribution. The compound hypothesis is accepted if and only if each of the univariate hypotheses are accepted. It so turns out that the component univariate tests are independent, if the compound hypothesis is true. It is therefore possible to determine the level of significance of the compound test in terms of the levels of significance of the component univariate tests and to derive simultaneous confidence-bounds on certain meaningful parametric functions on the lines of [3] and [4]. The step-down procedure obviously is not invariant under a permutation of the variates and should be used only when the variates can be arranged on a priori grounds. Some advantages of the step-down procedure are (i) the procedure uses widely known statistics like the variance-ratio, (ii) the test is carried out in successive stages and if significance is established at a certain stage, one can stop at that stage and no further computations are needed, and (iii) it leads to simultaneous confidence-bounds on certain meaningful parametric functions. 1.1 Notations. The operator $\varepsilon$ applied to a matrix of random variables is used to generate the matrix of expected values of the corresponding random variables. The form of a matrix is denoted by a subscript; thus $A_{n \times m}$ indicates that the matrix $A$ has $n$ rows and $m$ columns. The maximum latent root of a square matrix $B$ is denoted by $\lambda_{\max}(B)$. Given a vector $a = (a_1, a_2, \cdots, a_t)'$ and a subset $T$ of the natural numbers $1, 2, \cdots, t$, say $T = (j_1, j_2, \cdots, j_u)$ where $j_1 < j_2 < \cdots j_u$, the notation $T\lbrack a\rbrack$ will be used to denote the positive quantity: $$T\lbrack a\rbrack = + \{a^2_{j_1} + a^2_{j_2} + \cdots + a^2_{j_t}\}^{1/2}.$$ $T\lbrack a\rbrack$ will be called the $T$-norm of $a$. Similarly, given a matrix $B_{t \times t}$, we shall write $B_{(T)}$ for the $u \times u$ submatrix formed by taking the $j_1$th, $j_2$th, $\cdots, j_u$th rows and columns of $B$. We shall call $B_{(T)}$ the $T$-submatrix of $B$.

On permutation procedures for strong control in multiple testing with gene expression data

We consider two popular, permutation-based, step-down procedures of p-values adjustment in multiple testing problems known as min P and max T and intended for strong control of the family-wise error rate, under the so-called subset pivotality property (SPP). We examine key but subtle issues involved in ascertaining validity of these methods, and also introduce a new, slightly narrower notion of strong control which ensures proper bounds on the family-wise error rate in min P and max T without SPP.