Statistics Review - Mathematics and Statistics for the Environment - page 158

= +−
−−

bB tT G

X

( bt 11 )( )

where
b = Number of blocks.
B = Total of all other units in the block with a missing plot.
t = Number of treatments.
T = Total of all other units that received the same treatment as the missing plot.
G = Total of all observed units.

If more than one plot is missing, the customary procedure is to insert guessed values for all but
one of the missing units, which is then estimated by the above formula. This estimate is used in
obtaining an estimated value for one of the guessed plots, and so on through each missing unit.
Then the process is repeated, with the first estimates replacing the guessed values. The cycle should
be repeated until the new approximations differ little from the previous estimates.
The estimated values are now applied in the usual analysis-of-variance calculations. For each
missing unit one degree of freedom is deducted from the total and from the error term.
A similar procedure is used with the Latin square design, but the formula for a missing plot is

rR CT ( ++−

−−

2
12

) G

X =

( r )( r )

where
r = Number of rows.
R = Total of all observed units in the row with the missing plot.
C = Total of all observed units in the column with the missing plot.
T = Total of all observed units in the missing plot treatment.
G = Grand total of all observed units.

With the split-plot design, missing plots can cause trouble. A single missing subplot value can be
estimated by the following equation:

+ () − ()
−−

rP mT T

ij i

X =

( r 1 )( m 1 )

where
r = Number of replications of major plot treatments.
P = Total of all observed subplots in the major plot having a missing subplot.
m = Number of subplot treatments.
T ij = Total of all subplots having the same treatment combination as the missing unit.
T i = Total of all subplots having the same major plot treatment as the missing unit.

For more than one missing subplot the iterative process described for randomized blocks must
be used. In the analysis, one degree of freedom will be deducted from the total and subplot error
terms for each missing subplot.
When data for missing plots are estimated, the treatment mean square for all designs is biased
upwards. If the proportion of missing plots is small, the bias can usually be ignored. Where the pro-
portion is large, adjustments can be made as described in the standard references on experimental
designs.