A randomized complete block experimental design was used to determine the joint effects of temperature and concentration of herbicide on absorption of 2 herbicides on a commercial charcoal material. There were 2 blocks and a total of 20 treatment combinations—2 temperatures by 5 concentrations by 2 herbicides. (The data are used with permission of Dr. J. B. Weber, North Carolina State University.)
The usual linear model for a randomized complete block experiment, Yij
= µ + γi
+ τj
+ ϵij, where γi
is the effect of the ith block and τj
is the effect of the jth treatment, can be expanded to include the main and interaction effects of the three factors:
where Tj
, Hk, and Cl
refer to the effects of temperature, herbicide, and concentration, respectively. The combinations of letters refer to the corresponding interaction effects.
(a) Show the form of X and β for the usual RCB model, the model containing γi
and τj. Assume the data in Y are listed in the order that would be obtained if successive rows of data in the table were appended into one vector. What is the order of X and how many singularities does it have? Use γ2
= 0 and τ20
= 0 to reparameterize the model and compute the sums of squares for blocks and treatments.
(b) Define K for the singular model in Part (a) for the composite null hypothesis that there is no temperature effect at any of the combinations of herbicide and concentration. (Note: τ1
is the effect for the treatment having temperature 10°, herbicide A, and concentration 20 × 10−5. τ11
is the effect for the similar treatment except with 55° temperature. The null hypothesis states that these two effects must be equal, or their difference must be zero, and similarly for all other combinations of herbicide and concentration.) How many degrees of freedom does this sum of squares have? Relate these degrees of freedom to degrees of freedom in the conventional factorial analysis of variance. Define K for the null hypothesis that the average effect of temperature is zero. How many degrees of freedom does this sum of squares have and how does it relate to the analysis of variance?
(c) Show the form of X and β if the factorial model with only the main effects Tj, Hk, and Cl
is used. How many singularities does this X matrix contain? Show the form of X∗
if the “sum” constraints are used. Use this reparameterized form to compute the sums of squares due to temperature, due to herbicides, and due to concentration.
(d) Demonstrate how X in Part (c) is augmented to include the (T H)jk
effects. How many columns are added to X? How many additional singularities does this introduce? How many columns would be added to X to accommodate the (T C)jl
effects? The (HC)kl
effects? The (THC)jkl
effects? How many singularities does each introduce?
(e) Use PROC ANOVA in SAS, or a similar computer package, to compute the full factorial analysis of variance. Regard blocks, temperature, herbicide, and concentration as class variables.