Finally if we want to think about precision for interaction effects or other effects than the simple difference
for two treatment means, then our formula for the standard error will change. For an interaction of two 2-
level factors, which is the difference between two differences,

 SE = 4(2s2/n + 2s2/n) = (4s2/n).

 2.2 Replication, Hidden and Total

 The actual number of observations relevant to a comparison may involve hidden replication as well as the
 apparent, explicit replication. Further, we may have to think which replication is relevant to our question.
 Suppose we have a set of 16 treatments comprising all combinations of 24 factorial structure with factors
 density, weed control, nitrogen and phosphorus. The full set of combinations are represented in Table 1 for
 a single site with 2 replications of each combination. Suppose further that we have 5 sites.

 If we are interested in detecting an average nitrogen effect at one specific site, we have 16-fold replication
 for each N level (n=16) being made up of 2 explicit replicates x 8 hidden replicates since each N level
 occurs for the same eight combinations of density, weed control and phosphorus.

 If we are interested in the average nitrogen effect averaging over all five sites than we have 80-fold
 replication.

 If we are interested in the density x nitrogen combinations at one site we have 8-fold replication.

 If we are interested in comparing the nitrogen effect at high density, no weed control and no P average
 across sites, we have 10-fold replication.

 Thus the amount of replication, and equivalently, the information, for any comparison must be assessed for
 each specific comparison of interest.

 The use of replications in assessing precision in the previous section has a further potential complication
 because the variability between plots within a site may be different (smaller) from the variability between
 sites. Thus, if we are considering the 80-fold replication for the N effect average over all sites our CV will
 tend to be higher than the 20% within site CV. The increase will depend on the CV of sites and if this were,
 say, 50% the overall CV would be approximately.

 4(20%2 + 1 (50%2 20%2)) = 23%
 16

Note that the increase in the SD or CV depends on the proportion of site replication in the total replication
which is 1/16 in the example but would be 1/2 for the fourth example above with a much larger consequent
increase in the CV. The amount by which the between-site CV is greater than the within-site CV may vary
widely, but a factor of 2 or 3 might be a reasonable guess.

2.3 Efficient Use of Resources

In section 2.1 we considered the choice of n, the treatment replication, which must be considered in the
context of different forms or replication (2.2). We now consider the efficiency of use of total resources in
an experiment.