SUGI 28: Multilevel Designs and Their Analyses

Multilevel designs are used by researchers in many areas (Raudenbush and Bryk (2002) and Milliken and Johnson (1992)) and those researchers have used different terminologies to describe the designs. Multilevel designs are designs that involve more than one size of experimental unit, thus the name, multilevel. Split-plot designs are multilevel designs and have been used by physical and biological science researchers since the 1930s. Hierarchical designs and now multilevel designs have been used by social science researchers the past few years. The structures are identical and the models needed to describe resulting data are identical. This presentation provides a general frame work within which to identify and then analyze multilevel designs whether called split-plot, hierarchical, or multilevel. Repeated measures designs are multilevel designs. The stripplot is also a multilevel design that is not a hierarchical design and is a structure not generally considered by those using multilevel designs in the social sciences. A collection of examples is used to demonstrate the similarities and differences of various designs and the needed analyses. All of the models required to provide appropriate analyses of these designs are members of the class of mixed models. Proc Mixed of the SAS system is used to fit all of the models. INTRODUCTION Different terminologies have evolved for describing designs that involve more than one size of experimental unit or sampling unit. The goal of this presentation is to provide a unified structure that can be used to bring the varying descriptions together. The three main terminologies correspond to those used to describe (1) splitplot or repeated measures types of designs, (2) hierarchical types of designs and (3) multi-level types of designs. A unified structure is first described and then each of these designs are fit into that structure, BASICS OF DESIGNED EXPERIMENTS USING BLOCKS The discussion centers abound the design of experiments or observational studies. The unified structure involves classifying the factors in an experiment or study as belonging to the treatment structure or design structure (Milliken and Johnson (1992)). The treatment structure consists of those factors in the experiment that were selected to be studied and should have an influence on the response of interest. The design structure of a study consists of the factors used to form groups or blocks of the experimental units. Examples of treatment structures are oneway, two-way factorial arrangement, three-way factorial arrangment, 2 factorial arrangement, 2 fractional factorial, a set of combinations obtained for an optimal design, a two-way factorial plus a control, etc. Examples of design structures are completely randomized (CR) design, randomized complete block (RCB) design, incomplete block design (ICB), etc. Two important assumptions about the relationship between elements of the treatment structure and elements of the design structure are (1) there are no interactions between the factors in the treatment structure and the factors in the design structure and (2) the levels of the factors in the design structure are random effects. The first assumption is very important and is often overlooked by most researchers when factors are selected for forming blocks. Most text books define blocking factors as those that are not of interest in the current study. But one has to be more careful and make sure the blocking factors will not interact with the factors in the treatment structure. The second assumption implies there is a population of experimental units to which inferences are to made. As a consequence of the second assumption, all necessary models are mixed models unless the design structure is CR. DESIGNS WITH ONE SIZE OF EXPERIMENTAL UNIT Designs with one size of experimental unit can be formed with any of the design structures, i.e., CR, RCB, or ICB, and are called one level designs. Suppose the treatment structure consists of four treatments and the design structure consists of twelve experimental units. Each of the four treatments can be assigned to three experimental units, thus there can be three replications of each treatment. The CR design is constructed by randomly assigning each of the treatments to three of the experimental units, as demonstrated in Figure 1. A model one could use to describe data from a CR design is (1) y i j where N ij i ij ij = + + = = μ τ ε ε σε , , , , , , , , ~ ( , ), 1 2 3 4 1 2 3 0 2 : denotes the mean of the response, Ji denotes the effect of the ith treatment and ,ij is the experimental unit error. For the RCB design structure, construct three blocks of four experimental units and then randomly assign the treatments to an experimental unit within each of the blocks. A graphic demonstrating the construction of a RCB design structure in displayed in Figure 2. A model one could use to describe data from a RCB design is (2) y b i j where b N N ij i j ij j blk ij = + + + = = μ τ ε σ ε σε , , , , , , , , ~ ( , ), ~ ( , ), 1 2 3 4 12 3 0 0 2 2 and in addition to the terms described for model 1, bj represents the random block effect. The estimate of F, is computed from the treatment by block interaction. This computation implies it is very important that the elements of the treatment structure must not interact with the elements of the design structure. Incomplete block design structures occur when the block size is smaller than the number of treatments in the treatment structure, i.e., not all treatments can occur in each block or an incomplete set of treatments occur in each block, thus, the name. Figure 3 contains a graphic demonstrating the construction of an incomplete block design structure with blocks of size 3 and Figure 4 contains the graphic for an incomplete block design structure with blocks of size 2. An incomplete block design is constructed by selecting a block size and determine the number of blocks needed in the study. Treatment patterns, one for each possible block, are determined to provide the appropriate relationship among the treatments. One such relationship is to select a pattern such that each pair of treatments occur together within a block an equal number of times and such an arrangement is called a balanced incomplete block design SUGI 28 Statistics and Data Analysis