Randomization analysis of experimental designs under non standard conditions
Often the basic assumptions of the ANOVA for an experimental design are not met or the statistical model is incorrectly specified. Randomization of treatments to experimental units is expected to protect against such shortcomings. This paper uses randomization theory to examine the impact on the expectations of mean squares, treatment means, and treatment differences for two model mis·specifications: Systematic response shifts and correlated experimental units.
Systematic response shifts are presented in the context of the randomized complete block design (RCBD). In particular fixed shifts are added to the responses of experimental units in the initial and final positions of each block. The fixed shifts are called border shifts. It is shown that the RCBD is an unbiased design under randomization theory when border shifts are present. Treatment means are biased but treatment differences are unbiased. However the estimate of error is biased upwards and the power of the F test is reduced.
Alternative designs to the RCBD under border shifts are the Latin square, semi-Latin square, and two-column designs. Randomization analysis demonstrates that the Latin square is an unbiased design with an unbiased estimate of error and of treatment differences. The semi-Latin square has each of the t treatments occurring only once per row and column, but t is a multiple of the number of rows or columns. Thus each row-column combination contains more than one experimental unit. The semi-Latin square is a biased design with a biased estimate of error even when no border shifts are present. Row-column interaction is responsible for the bias. Border shifts do not contaminate the expected mean squares or treatment differences, and thus the semi-Latin square is a viable alternative when the border shift overwhelms the row-column interaction. The two columns of the two-column design correspond to the border and interior experimental units respectively. Results similar to that for the semi-Latin square are obtained. Simulation studies for the RCBD and its alternatives indicate that the power of the F test is reduced for the RCBD when border shifts are present. When no row-column interaction is present, the semi-Latin square and two-column designs provide good alternatives to the RCBD.
Similar results are found for the split plot design when border shifts occur in the sub plots. A main effects plan is presented for situations when the number of whole plot units equals the number of sub plot units per whole plot.
The analysis of designs in which the experimental units occur in a sequence and exhibit correlation is considered next. The Williams Type Il(a) design is examined in conjunction with the usual ANOVA and with the method of first differencing. Expected mean squares, treatment means, and treatment differences are obtained under randomization theory for each analysis. When only adjacent experimental units have non negligible correlation, the Type Il(a) design provides an unbiased error estimate for the usual ANOVA. However the expectation of the treatment mean square is biased downwards for a positive correlation. First differencing results in a biased test and a biased error estimate. The test is approximately unbiased if the correlation between units is close to a half.