# Multiple comparison procedures in factorial designs using the aligned rank transformation

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A factorial design is used for an experiment that involves the study of two or more factors simultaneously, with each factor having two or more levels. The importance of factorial designs is that they permit simultaneous examination of the effects of individual factors and their interactions. All possible combinations of the levels of the factors are investigated in each replication of an experiment. A main effect is defined as the change in response produced by a change in the level of one factor while keeping the remaining factors at a fixed level. Interaction exists between two factors if the difference in response between the levels of one factor is not the same at all levels of the other factors. Thus, the preliminary focus of analysis is testing hypotheses about interaction, equality of row treatment effects, and column treatment effects. If interaction exists, main effects exists as well. If interaction does not exist. then testing main effects should proceed. In Chapter II, we present the analysis of variance for a two-factor factorial fixed effects factorial design.

The hypotheses do not provide detailed information about the difference in interactions and main eflfects. Multiple comparison procedures are capable of responding to specific questions about more meaningful comparisons on any of the above effects. These procedures allow the comparisons between groups or pairs of treatment means. The comparisons are made in terms of treatment totals or treatment averages. It must be noted that multiple comparison techniques are not dependent on the rejection of null hypothesis. The testing of interaction between two factors and main eflfects can also be performed by multiple comparison methods. Common multiple comparison procedures that will be implemented are Tukey's Studentized Range Test and Scheflfe's Method. These multiple comparison procedures are discussed in Chapter II. Additionally, we will employ a macro called %SimPower in order to perform multiple comparisons. This macro was developed by Tobias (see Westfall et al.. 1999).

%SimPower simulates power for multiple comparisons. It uses complete, minimal, and proportional power definitions. Complete power is defined as the probability of rejecting all false null hypotheses. Minimal power is the probability that at least one false null hypothesis is rejected implying a significant result. Proportional power is the proportion of false null hypotheses detected to all false null hypotheses, that is false nulls expected to be detected. Further discussion is provided in Chapter V.

One of the main purposes of this investigation is to carry out multiple comparisons for tw^o-factor factorial designs based on the aligned rank transformation. The aligned rank transform procedure provides a robust and powerful alternative method of data analysis to the classical least squares method. In this investigation, we will study the validity and power of the aligned rank transform technique for multiple comparisons. In Chapter III, we define the classical least squares F-statistic and the aligned rank transform technique.

Analysis of two applications based on the least squares and aligned rank transform methods will be examined in Chapter IV. The final conclusio will be provided in Chapter VI. We will utilize the SAS programming language to perform the classical least squares method and the aligned rank transform technique. PROC GLM, PROG REG, and PROC RANK will execute the two techniques. These ideas will be explained in more detail later on.