In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) that are similar to one another. Compared to a completely randomized design, this design reduces variability within treatment conditions and potential confounding, producing a better estimate of treatment effects.
Suppose we have v treatments replicated times respectively, and b blocks with units respectively. We indicate such a block design as r,k)" align="absmiddle" />, where r=(r_1,r_2,...,r_v)" align="absmiddle" /> andk=(k_1,k_2,...,k_b)" align="absmiddle" />. Thus , where n is the total number of observations. Let be the number of times treatment appears in block.Assuming an additive model for the block design, an appropriate model for observations is:
, i=1,2,..v; j=1,2,…b; l=1,2,….
where is the effect of treatment, is the effect of block, and are error associated with observation . We can treat as random observations with mean 0, and variance .
Let us consider all effects except the error components to be fixed, i.e. non-random. The analysis under such a fixed effects model is called intra-block analysis. The name is derived from the fact fact that contrasts in the treatment effects are estimated as linear combinations of comparisons of observations in the same block. In this way, the block effects are eliminated and estimates are functions of treatment effects and error only. The errors are termed as intra-block errors. The treatment effects are so computed because the comparison among treatment effects and comparison among block effects are not orthogonal to each other.
Let us define a matrix N, called Treatment-Block Incidence Matrix.
N=, =number of times treatment is applied to block.
Note: may be 0 if all treatments do not occur in each block. Such a design is called an incomplete block design.
Model: Let us construct the model for the block design described earlier.
: Y=X\theta+\epsilon" align="absmiddle" />, where , and .
, , and
- , where : treatment total
- , where : treatment total
Thus giving the normal equation to be:
This equation can be reduced for to be : , where , and
On further inspection, it can be seen that,
, and .
The matrix C is also called the C-matrix, or the Information Matrix. is called the adjusted treatments total, the adjustment being due to the fact that the treatment do not occur the same number of times in the blocks. is the unadjusted treatment total.
We can see that: , where is the projection matrix onto the space orthogonal to the column space of . So the C-matrix is symmetric and non-negative definite.
Also, , implying that , i.e. C is positive semi definite with at least one eigen value 0.
In the same manner, we can obtain the reduced normal equations for ‘s, and it is easy to see that :
- , the vector of adjusted block totals.
The C-matrix and D-matrix have several interesting properties which I will discuss on my next post.