Randomized Complete Block Design (RCBD) Block--a nuisance factor included in an experiment to account for variation among eu’s Block--a nuisance factor included in an experiment to account for variation among eu’s Presumably, eu’s are homogenous within a block Presumably, eu’s are homogenous within a block Treatments are randomly assigned to eu’s within each block Treatments are randomly assigned to eu’s within each block

RCBD The model and hypotheses The model and hypotheses

RCBD Blocks can be modeled as both fixed and random effects (Soil example) Blocks can be modeled as both fixed and random effects (Soil example) –Block: Soil type (fixed or random?) –Treatment: Nitrogen x Watering Regimen –Response: IR/R reflection

RCBD There is some controversy as to whether fixed block effects should be tested There is some controversy as to whether fixed block effects should be tested –F test is considered at best approximate Additivity of the block and factor effects Additivity of the block and factor effects –Error includes lack-of-fit –Practical considerations Both block and factor could have a factorial structure Both block and factor could have a factorial structure

Missing values in RCBD’s Missing values result in a loss of orthogonality (generally) Missing values result in a loss of orthogonality (generally) A single missing value can be imputed A single missing value can be imputed –The missing cell (y i*j* =x) can be estimated by profile least squares

Imputation The error df should be reduced by one, since x was estimated The error df should be reduced by one, since x was estimated SAS can compute the F statistic, but the p- value will have to be computed separately SAS can compute the F statistic, but the p- value will have to be computed separately The method is efficient only when a couple cells are missing The method is efficient only when a couple cells are missing

Imputation The usual Type III analysis is available, but be careful of interpretation The usual Type III analysis is available, but be careful of interpretation Little and Rubin use MLE and simulation- based approaches Little and Rubin use MLE and simulation- based approaches PROC MI in SAS v9 implements Little and Rubin approaches PROC MI in SAS v9 implements Little and Rubin approaches

Power analysis Power calculations change little Power calculations change little –b replaces n in formulas –The error df is (a-1)(b-1)