Estimation Of A Missing Observation In A Rbd 14

1. Core Concept Overview & Intuition

A randomized block design (RBD) is an experimental layout in which the experimental units are first grouped into homogeneous sets called blocks, and within each block the treatments are assigned at random. The block structure removes a known source of variability from the error term, so that treatment comparisons are made more precisely than in a completely randomized design. The standard additive model for an RBD with $v$ treatments and $b$ blocks, with a single observation per treatment–block combination, is

$$
y_{ij}=\mu+\beta_i+t_j+e_{ij},
$$

where $y_{ij}$ is the observation recorded in block $i$ on treatment $j$, $\mu$ is the general mean, $\beta_i$ is the effect of block $i$, $t_j$ is the effect of treatment $j$, and $e_{ij}$ is the random error component. The indices run over $i=1,2,\dots,b$ and $j=1,2,\dots,v$.

In practice an observation can be lost: a plot may be damaged, a measurement may fail, or a sample may be spoiled. Deleting the whole block or the whole treatment is wasteful, because the remaining data still carry information about the lost cell. The problem addressed here is to replace the single missing value by a number computed from the surviving observations, so that the usual analysis of variance can be carried out on a complete table.

The intuition rests on the additive structure. If the cell $(i,j)$ were present, its value would be approximately the sum of the block contribution and the treatment contribution, corrected by the general mean. The least-squares principle therefore chooses the replacement value that makes the residual sum of squares of the completed table as small as possible. Because the fitted value for a cell in the additive model is the block mean plus the treatment mean minus the general mean, the least-squares replacement is exactly that fitted value, computed from the incomplete table. This replacement is the best linear unbiased estimate of the lost observation under the assumed model.

Two consequences follow immediately and must be kept in mind throughout. First, the completed table is not genuine data: one degree of freedom has been spent in estimating the missing value, so the error degrees of freedom of the analysis must be reduced by one. Second, the treatment sum of squares computed from the completed table is slightly inflated, and the standard analysis is therefore approximate; the bias is small when only one observation is missing and the design is reasonably large.

2. Mathematical Formalization & Core Definitions

> Definition (Randomized Block Design). An RBD consists of $b$ blocks, each containing $v$ homogeneous experimental units, with the $v$ treatments assigned at random to the units within every block. The observation in block $i$ receiving treatment $j$ is modelled as $y_{ij}=\mu+\beta_i+t_j+e_{ij}$, with $e_{ij}$ independent errors having mean zero and common variance $\sigma^2$.

> Definition (Missing Observation). A cell $(i,j)$ of the RBD table is missing when the value $y_{ij}$ is unavailable, while all other $bv-1$ observations are recorded. The design is then called an incomplete RBD with one missing value.

> Definition (Least-Squares Estimate of a Missing Value). The estimate $\hat{y}_{ij}$ of the missing observation is the value which, when substituted into the cell $(i,j)$, minimises the residual sum of squares of the completed table under the additive model.

The derivation proceeds by writing the residual sum of squares of the completed table as a function of the unknown cell value. Let $x$ denote the value to be inserted in the missing cell $(i,j)$. The block totals and treatment totals of the completed table are linear functions of $x$:

$$
B_i'=B_i+x,
$$

$$
T_j'=T_j+x,
$$

where $B_i$ is the total of the observed values in block $i$ and $T_j$ is the total of the observed values in treatment $j$. The grand total of the completed table is $G'=G+x$, where $G$ is the grand total of the observed values. The residual sum of squares of the completed table, expressed through the computational identity for an RBD, is

$$
SSE(x)=\sum_{i}\sum_{j}y_{ij}^2-\frac{1}{v}\sum_{i}(B_i')^2-\frac{1}{b}\sum_{j}(T_j')^2+\frac{(G')^2}{bv},
$$

where the sum of squares of all entries is unaffected by $x$ except through the single term $x^2$. Differentiating with respect to $x$ and equating to zero gives the estimating equation

$$
2x-\frac{2(B_i+x)}{v}-\frac{2(T_j+x)}{b}+\frac{2(G+x)}{bv}=0.
$$

Solving,

$$
x\left(1-\frac{1}{v}-\frac{1}{b}+\frac{1}{bv}\right)=\frac{B_i}{v}+\frac{T_j}{b}-\frac{G}{bv}.
$$

The multiplier on the left is

$$
1-\frac{1}{v}-\frac{1}{b}+\frac{1}{bv}=\frac{(v-1)(b-1)}{bv},
$$

so that

$$
\hat{y}_{ij}=\frac{vB_i+bT_j-G}{(v-1)(b-1)}.
$$

This is the standard estimating formula for a single missing observation in an RBD. It can be read as the fitted value of the additive model: the block mean $B_i/v$ plus the treatment mean $T_j/b$ minus the general mean $G/(bv)$, rescaled by the factor $bv/[(v-1)(b-1)]$.

> Definition (Adjusted Analysis). The analysis of variance performed on the completed table, with the error degrees of freedom reduced by one to account for the estimated cell, is called the adjusted analysis of the incomplete RBD.

The analysis of variance table for the completed table takes the usual RBD form. With $b$ blocks and $v$ treatments, the sums of squares are

$$
TSS=\sum_{i}\sum_{j}y_{ij}^2-\frac{(G')^2}{bv},
$$

$$
SSBL=\frac{1}{v}\sum_{i}(B_i')^2-\frac{(G')^2}{bv},
$$

$$
SST=\frac{1}{b}\sum_{j}(T_j')^2-\frac{(G')^2}{bv},
$$

$$
SSE=TSS-SSBL-SST.
$$

The degrees of freedom are $b-1$ for blocks, $v-1$ for treatments, and $(b-1)(v-1)-1$ for error, the subtraction of one reflecting the estimated cell. The treatment mean square is $MST=SST/(v-1)$ and the error mean square is $MSE=SSE/[(b-1)(v-1)-1]$, and the test statistic is $F_c=MST/MSE$.