Notice how closely these estimates of block and treatment effects correspond to those from the complete
block data set, which contains the same data points as the incomplete block data plus four extra
observations.


Complete


Incomplete


Block 1 +98 +99
Block 2 -52 -49
Block 3 +63 +65
Block 4 -112 -112

Treatment A -148 -150
Treatment B -33 -35
Treatment C +79 +81
Treatment D +104 +103


We would expect such agreement because we are trying to estimate the same quantities, the only change
being that in the incomplete design we have less information on which to base our estimation. The reduced
information is clearly sufficient to obtain estimates close to those based on fuller information. We can also
compare the final residuals for the complete and incomplete cases. Again the agreement is good and not
surprising.


 Complete Incomplete

+6 +6 -9 -4 +5 +3 -9
-9 +1 -4 +11 -10 -2 +11
+9 -11 -7 +9 +4 -10 +5
-6 +4 +18 -16 0 +18 -17


Finally we would wish to compare the treatment mean yields. The calculation of treatment means is
already almost completed. We merely have to add the overall mean to the estimates of the treatment
effects. The calculation of standard errors is more difficult and is one aspect of the sweeping technique
where an exact solution is not possible with manual calculation. However, we can use an approximation
which generally gives excellent results.

We calculate two variances for each treatment pair, one based on the total number of observations for each
treatment (MIN), the other based on the number of blocks in which both treatments appear (MAX). If the
number of treatments in the experiment is t, then the variance for a difference between two treatment
means is

 MIN + (MAX MIN)/t.

For our incomplete block design

 MIN = 2(219)/3 = 146

 MAX = 2(219)/2 = 219

 Var = 146 +(219 146)/4 = 164.25.