When it is necessary to apply several different treatments in succession to a given subject, the residual effect of one treatment on another must be taken into consideration. A number of various designs have been developed for this purpose. A number of them are presented in this paper and can be summarized as follows:
Type I: Balanced for first-order residual effects. For n, the number of treatments, even, any number of Latin squares can be used; for n odd, an even number of squares is necessary.
Type II: Formed by repeating the final period of Type I designs. Direct and residual effects are orthogonal.
Type III: Formed from p<n corresponding rows of n-1 orthogonal nxn Latin squares.
Type IV: Complete orthogonality except for subjects and residuals. Very efficient but large numbers of observations are necessary.
Type V: Designs balanced for first and second order effects. Also formed from orthogonal Latin squares.
Type VI: Designs orthogonal for direct, first and second order residuals. Designs presented for n=2, 3 and 5.
Type VII: Orthogonal for linear, quadratic, ...components of direct and linear component of residual effects. Analysis includes linear direct x linear residual interaction. Designs given for n = 4, 5.
Type VIII: Type II designs analyzed under model for Type VII designs. Less efficiency, but designs available for all n.
Type IX: Designs useful for testing more than one treatment and direct x residual interactions.
Analysis for most designs includes normal equations, analysis of variance, variances of estimates, expected mean squares, efficiencies and missing value formulas.
A list of designs is presented in an appendix.