# Combined intra- and inter-block analysis for factorials in incomplete block designs

 dc.contributor.author Walpole, Ronald E. en dc.contributor.department Statistics en dc.contributor.department Statistics en dc.date.accessioned 2019-01-31T19:03:49Z en dc.date.available 2019-01-31T19:03:49Z en dc.date.issued 1958 en dc.description.abstract The work of Kramer and Bradley on the use of factorials in incomplete block designs bas been extended to permit both the intra-block and combined intra- and inter-block analyses of factorials in balanced and partially balanced incomplete block designs. In particular, we have obtained a combined intra- and inter-block analysis for factorials in balanced incomplete block designs, group divisible designs, and Latin Square types of partially balanced, incomplete block designs. For the class of Latin Square sub-type L₃ designs both the intra-block and combined intra- and inter-block analyses have been developed. In general, factorial treatment combinations were assigned to the association schemes by permitting the rows of the association schemes to represent the levels of one factor and the columns to represent the levels of a second factor. The extension to multifactor factorials was then carried out by sub-dividing the levels of the basic two-factor factorial, the levels in the sub-divisions representing the levels of the multi-factor factorials. Estimators for the factorial effects have been obtained along with their variances and covariances. Sums of squares in terms of the factorial estimators have been derived and can be used to carry out tests of significance. These sums of squares were shown, for the combined intra- and inter-block analyses, to be independently distributed as χ²-variates with the appropriate numbers of degrees of freedom. Suitable sums of squares for tests of significance are not possible in general for Latin Square sub-type L₃ designs. In situations such as these, we can only consider contrasts among the estimators and use their variances to perform tests of significance. However, for the special cases of factorials in the 4 x 4 Latin Square sub-type L₃ design, a complete analysis yielding the adjusted sums of squares for the factorial effects is possible if the factorial treatments are applied to the association scheme in a different manner. Single-degree-of-freedom contrasts are obtained in much the usual way as for factorials in complete block designs. The method of incorporating a fractional replicate of a factorial into incomplete block designs is also considered. Numerical examples have been worked in detail for a group divisible design and a Latin Square sub-type L₂ design. The procedure for analyzing a Latin Square sub-type L₃ design is also discussed. en dc.description.degree Ph. D. en dc.format.extent 129, [2] leaves en dc.format.mimetype application/pdf en dc.identifier.uri http://hdl.handle.net/10919/87309 en dc.language.iso en_US en dc.publisher Virginia Polytechnic Institute en dc.relation.isformatof OCLC# 20405660 en dc.rights In Copyright en dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ en dc.subject.lcc LD5655.V856 1958.W346 en dc.subject.lcsh Incomplete block designs en dc.subject.lcsh Block designs en dc.title Combined intra- and inter-block analysis for factorials in incomplete block designs en dc.type Dissertation en dc.type.dcmitype Text en thesis.degree.discipline Statistics en thesis.degree.grantor Virginia Polytechnic Institute en thesis.degree.level doctoral en thesis.degree.name Ph. D. en

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