ON THE z-TEST IN RANDOMIZED BLOCKS AND LATIN SQUARES

Welch, B. L.

doi:10.1093/biomet/29.1-2.21

Cited by 251 publications

(51 citation statements)

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“…Alternatively, one may rely on the fundamental notion of randomization tests, and use as approximations to randomization distributions certain procedures suggested by normal theory. (See Welch [18], Pitman [12J, Kempthorne [7,8], Wilk [19, 20J. ) • We use the word "additivity" in the sense of "negligible interaction effects.• We prefer the latter viewpoint of mathematical approximation to the former of statistical assumptions because (i) it is less likely to obscurethe tenuous relation between the physical situation and mathematical or statistical abstractions; and (ii) it faces honestly the fact that more than desire for mathematical elegance is required to validate statistical assumptions.…”

Section: Discussionmentioning

confidence: 99%

Fixed, Mixed, and Random Models*

Wilk

Kempthorne

1955

Journal of the American Statistical Association

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Some explicit questions are raised regarding the adequacy of assumed linear models as a basis for the interpretation of the analysis of variance of randomized experiments. A generally applicable method for the derivation of a linear statistical model, based on the experimental situation and the design of the experiment, is exemplified. The central features of the method are the notion of "experimental unit," the concept of "true response," and the use of randomization in the design. A model is derived for the case where two factors having A and B levels respectively, are to be examined with respect to a population of P experimental units, where selection of levels of the factors to be tested, selection of experimental units to be used, and the allocation of selected treatment combinations to units is at random. First a linear population model is given, based on the structure of the experimental situation, whose components are (unknown) parameters of the population of (conceptual) "true" responses. Then the conditions of the design are imposed to obtain a linear statistical model whose components involve the parameters of the population model and some defined random variables reflecting the experimental procedure and design. The derived statistical model is then used to obtain expected mean squares in the analyses of variance. A second illustration of the general methodology is given for a more complex example originated by Vaurio and Daniel and discussed statistically by Scheffe. Some differences from Bcheffe's results are noted.

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Section: Discussionmentioning

confidence: 99%

Fixed, Mixed, and Random Models*

Wilk

Kempthorne

1955

Journal of the American Statistical Association

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“…Other relevant investigations are those of Scheffe [15] and Smith [16]. Work indirectly related to the present context is that of Fisher [7], Welch [18] and Pitman [14] on randomization tests.…”

Section: Iowa State Collegementioning

confidence: 99%

Non-Additivities in a Latin Square Design

Wilk

Kempthorne

1957

Journal of the American Statistical Association

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A brief history of the Latin square design is given. A generalization of the design to the case in which the rows, columns and treatments represented in the experiment are samples from populations of rows, columns and treatments respectively is studied. A possible frame of reference for the interpretation of the experimental results is described.The leads to what is termed a "population model," various population parameters, means and components of variation which are of interest to the experimenter. No assumptions are made about additivity of experimental units and treatments. Results on expectations of mean squares in the analysis of variance are given in terms of quantities (denoted by 2:'s) which result in a concise description and in terms of the components of variation. Biases in the estimation of components of variation by means of the analysis of variance are discussed and assessed. Comparisons of randomized block designs and Latin square designs are given for the general case of non-additive treatments for random, mixed, or fixed population of rows and columns. A generalization of the design is discussed. The mathematical machinery which is used to derive the results is presented briefly. Finally linear estimates and errors of estimates are discussed. One main conclusion of the study is that the Latin square analysis of variance may overestimate the error of treatment comparisons and underestimate the component of variation associated with treatment main effects.

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“…The use of tests of significance based on randomization of the observations was given by in the first edition of The Design of Experiments [21]. Here the problem was the comparison of two paired samples of 15 observations.…”

Section: Iowa State Collegementioning

confidence: 99%

“…It was shown by Welch [21] and Pitman [17] that the randomization test may be approximated by the normal theory F test with (s-1) and (r-1) (s-l) degrees of freedom. Hence under some circumstances the randomization distribution is probably represented fairly accurately by the corresponding normal theory distribution, so that the standard F test may be used.…”

Section: Complete Randomized Blocksmentioning

confidence: 99%

“…The quantity U treatment sum of squares error sum of squares plus treatment sum of squares is easily seen to have an expectation of l/et-l), by the argument given given by Fisher [10]. Welch [21] found that the variance of U, depended on the population of Latin Squares from which the random one was chosen, in fact, on the transformation set, and on functions of the eu/s. Welch the F table could be in error appreciably.…”

Section: U = ------------------mentioning

confidence: 99%

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