Block Design Analysis

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 r_1, r_2, r_3, ..., r_v times respectively, and b blocks with k_1,k_2,...k_b units respectively. We indicate such a block design as d(v,b,<span class=r,k)" align="absmiddle" />, where <span class=r=(r_1,r_2,...,r_v)" align="absmiddle" /> and<span class=k=(k_1,k_2,...,k_b)" align="absmiddle" />. Thus \sum_{i=1}^vr_i=\sum_{j=1}^bk_j=n, where n is the total number of observations. Let n_{ij} be the number of times i^{th} treatment appears in j^{th} block.Assuming an additive model for the block design, an appropriate model for observations is:

y_{ijl}=\mu +\tau_{i}+\beta_j+\epsilon_{ijl},                                  i=1,2,..v; j=1,2,…b; l=1,2,…n_{ij}.

where \tau_i is the effect of i^{th} treatment, \beta_j is the effect of j^{th} block, and \epsilon_{ijl} are error associated with observation y_{ijl}. We can treat \epsilon_{ijl} as random observations with mean 0, and variance \sigma^2.

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 v \times b matrix N, called Treatment-Block Incidence Matrix.

N=\left \{ n_{ij} \right \},  n_{ij}=number of times i^{th} treatment is applied to j^{th} block.

Note: n_{ij} 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.

\Omega:     <span class=Y=X\theta+\epsilon" align="absmiddle" />,    where E(\epsilon)=0, and D(\epsilon)=\sigma^2 I_n.

\theta_{p \times 1}=\begin{pmatrix} \mu_{1 \times 1} \\ \tau_{v \times 1} \\ \beta_{b \times 1} \end{pmatrix}p=1+v+b,  and X=\begin{bmatrix} 1_n & X_{\tau} & X_{\beta} \end{bmatrix}

Normal Equations:  X'X \theta= X'Y

Observe That:

  • X_{\tau}'X_{\tau}=R
  • X_{\beta}'X_{\beta}=K
  • X_{\tau}'X_{\beta}=N_{v \times b}
  • 1_n'X_{\tau}=r
  • 1_n'X_{\beta}=k
  • T=X_{\tau}'Y=\begin{pmatrix} T_1 \\ \vdots \\ T_v \end{pmatrix} , where T_ii^{th} treatment total
  • B=X_{\beta}'Y=\begin{pmatrix} B_1 \\ \vdots \\ B_v \end{pmatrix}, where B_jj^{th} treatment total

Thus giving the normal equation to be:                          \begin{pmatrix} n & r' & k' \\ r & R & N \\ k & N' & K \end{pmatrix} \begin{pmatrix} \hat{\mu} \\ \hat{\tau} \\ \hat{\beta} \end{pmatrix}=\begin{pmatrix} G \\ T \\ B \end{pmatrix}

This equation can be reduced for \tau to be : C \hat{\tau}= Q, where C= R-NK^{-1}N', and Q=T-NK^{-1}B

On further inspection, it can be seen that,

c_{ii'}=\delta_{ii'}r_i - \sum_{j=1}^b\frac{n_{ij}n_{i'j}}{k_j}, and Q_i=T_i-\sum_{j=1}^bn_{ij}\frac{B_j}{K_j}.

The matrix C is also called the C-matrix, or the Information Matrix. Q_i 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. T_i is the unadjusted treatment total.

We can see that: C_{v \times v}= X_{\tau}' [I_n - P_{X_{\beta}}]X_{\tau}, where I_n-P_{X_{\beta}} is the projection matrix onto the space orthogonal to the column space of X_{\beta}. So the C-matrix is symmetric and non-negative definite.

Also, C1_v=0, implying that rank(C) \leq v-1, 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 \beta_j‘s, and it is easy to see that :

  • D \hat{\beta}=P
  • D= K -N'R^{-1}N
  • P=B-N'R^{-1}T, the vector of adjusted block totals.

The C-matrix and D-matrix have several interesting properties which I will discuss on my next post.



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