User manual MATLAB STATISTICS TOOLBOX 7

[. . . ] Statistics ToolboxTM 7 User's Guide How to Contact The MathWorks Web Newsgroup www. mathworks. com/contact_TS. html Technical Support www. mathworks. com comp. soft-sys. matlab suggest@mathworks. com bugs@mathworks. com doc@mathworks. com service@mathworks. com info@mathworks. com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. Statistics ToolboxTM User's Guide © COPYRIGHT 1993­2010 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. [. . . ] terms is a string consisting of words formed from the letters a-z, separated by spaces. Single-character words indicate main effects to be estimated; multiple-character words indicate interactions. Alternatively, terms is an m-by-n matrix of 0s and 1s where m is the number of model terms to be estimated and n is the number of factors. For example, if terms contains rows [0 1 0 0] and [1 0 0 1], then the factor b and the interaction between factors a and d are included in the model. Pass generators to fracfact to produce the fractional-factorial design and corresponding confounding pattern. generators = fracfactgen(terms, k) returns generators for a two-level fractional-factorial design with 2k-runs, if possible. generators = fracfactgen(terms, k, R) finds a design with resolution R, if possible. The default resolution is 3. A design of resolution R is one in which no n-factor interaction is confounded with any other effect containing less than R ­ n factors. Thus a resolution III design does not confound main effects with one another but may confound them with two-way interactions, while a resolution IV design does not confound either main effects or two-way interactions but may confound two-way interactions with each other. If fracfactgen is unable to find a design at the requested resolution, it tries to find a lower-resolution design sufficient to calibrate the model. 18-442 fracfactgen If it is successful, it returns the generators for the lower-resolution design along with a warning. If it fails, it returns an error. generators = fracfactgen(terms, k, R, basic) also accepts a vector basic specifying the indices of factors that are to be treated as basic. These factors receive full-factorial treatments in the design. The default includes factors that are part of the highest-order interaction in terms. Examples Suppose you wish to determine the effects of four two-level factors, for which there may be two-way interactions. The fracfactgen function finds generators for a resolution IV (separating main effects) fractional-factorial design that requires only 23 = 8 runs: generators = fracfactgen('a b c d', 3, 4) generators = 'a' 'b' 'c' 'abc' The more economical design and the corresponding confounding pattern are returned by fracfact: [dfF, confounding] = fracfact(generators) dfF = -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 confounding = 'Term' 'Generator' 'Confounding' 'X1' 'a' 'X1' 'X2' 'b' 'X2' 18-443 fracfactgen 'X3' 'X4' 'X1*X2' 'X1*X3' 'X1*X4' 'X2*X3' 'X2*X4' 'X3*X4' 'c' 'abc' 'ab' 'ac' 'bc' 'bc' 'ac' 'ab' 'X3' 'X4' 'X1*X2 'X1*X3 'X1*X4 'X1*X4 'X1*X3 'X1*X2 + + + + + + X3*X4' X2*X4' X2*X3' X2*X3' X2*X4' X3*X4' The confounding pattern shows, for example, that the two-way interaction between X1 and X2 is confounded by the two-way interaction between X3 and X4. References See Also [1] Box, G. Hoboken, NJ: Wiley-Interscience, 1978. fracfact, hadamard 18-444 friedman Purpose Syntax Friedman's test p = friedman(X, reps) p = friedman(X, reps, displayopt) [p, table] = friedman(. . . ) [p, table, stats] = friedman(. . . ) p = friedman(X, reps) performs the nonparametric Friedman's test Description to compare column effects in a two-way layout. Friedman's test is similar to classical balanced two-way ANOVA, but it tests only for column effects after adjusting for possible row effects. Friedman's test is appropriate when columns represent treatments that are under study, and rows represent nuisance effects (blocks) that need to be taken into account but are not of any interest. If there is more than one observation for each combination of factors, input reps indicates the number of replicates in each "cell, " which must be constant. The matrix below illustrates the format for a set-up where column factor A has three levels, row factor B has two levels, and there are two replicates (reps=2). The subscripts indicate row, column, and replicate, respectively. x111 x 112 x211 x212 x121 x122 x221 x222 x131 x132 x231 x232 Friedman's test assumes a model of the form xijk = + i + j + ijk 18-445 friedman where is an overall location parameter, i represents the column effect, j represents the row effect, and ijk represents the error. This test ranks the data within each level of B, and tests for a difference across levels of A. The p that friedman returns is the p value for the null hypothesis that i = 0 . A sufficiently small p value suggests that at least one column-sample median is significantly different than the others; i. e. , there is a main effect due to factor A. The choice of a critical p value to determine whether a result is "statistically significant" is left to the researcher. [. . . ] "Overview and Recent Advances in Partial Least Squares. " Subspace, Latent Structure and Feature Selection: Statistical and Optimization Perspectives Workshop (SLSFS 2005), Revised Selected Papers (Lecture Notes in Computer Science 3940). "Population marginal means in the linear model: an alternative to least-squares means. " American Statistician. "ECM Algorithms that Converge at the Rate of EM. " Biometrika. "Large sample properties of simulations using latin hypercube sampling. " Technometrics. [. . . ]