Resource equation

The Resource equation method (Mead 1988) of determining sample size is appropriate for experiments which can be analysed  using the analysis of variance such as:

  1. Exploratory experiments
  2. Complex biological experiments with several factors and treatments
  3. Any experiment where the power analysis method is not possible or  practicable

However, anyone doing a clinical trial or other relatively simple but  expensive experiment should try to use power analysis.

The method depends on the law of diminishing returns. Adding one experimental unit to a small experiment gives good returns, while adding it to a large experiment does not do so. It has been used by statisticians for  decades, but has been explicitly justified by Mead (1988) who states that "The total information in an experiment involving N experimental units may  be represented by the total variation based on (N-1) degrees of freedom (df). In the general experimental situation this total variation is divided into  three components, each serving a different function."  These three  components consist of:

  1. The treatment component, T, corresponding to the questions being asked
  2. The blocking component B, representing environmental effects allowed for in the design
  3. The error component E, being used to estimate the variance, Swhich is used for calculating the standard errors for treatment  comparisons.

Mead goes on to say that "..to obtain a good estimate of error it is necessary to have at least 10 df, and many statisticians would take 12 or 15  df as their preferred lower limit....." .... "But equally, if E is allowed to be large, say greater than 20, then the experimenter is wasting  resources..."





The figure shows the 5% critical value of Student's t  plotted against the degrees of freedom. This decreases dramatically as the number of df increases from two to 10, but by the time it has reached 20 it has flattened out. Thus Mead's suggestion is that experiments should be designed to give a good estimate of error, but should not be so big that they waste resources, i.e. the error degrees of freedom should be somewhere between  10 and 20.

Note that with n subjects, treatments, blocks etc. the df (given in upper  case letters) are always N=n-1. So if there are 20 rats in an experiment the total df will be 19. If there are 6 treatments, then the treatments df will  be 5.

The method is extremely easy to use as it boils down to the very simple  equation:

where E is the error df and should be between 10 and 20, N is the total df, B is the blocks df, and T is the treatments df.

In a non-blocked design the equation reduces to E=N-T should be  10-20. which is simply:

The total number of animals minus the number of treatments should be between ten and twenty.

Example: suppose an experiment is planned with four treatments, with  eight animals per group (32 rats total). In this case N=31, B=0, T=3, so E=28. Conclusion: this experiment is  a bit too large, and six animals per group  might be more appropriate.

A problem with blocking

There is one problem with this simple equation. It appears as though blocking is "bad" because it reduces the error df.  If the above example the experiment was going to be done in eight blocks, then N=31, B=7, T=3 and E= 31-7-3 =21 instead of 28. However,  blocking nearly always reduces variation which more than compensates for the decrease in the error df unless the experiment is very small. Provided the error df is not less than about 6 in a blocked design, then the experiment is probably of an adequate size.

 Mead R. 1988. The design of experiments. Cambridge, New York: Cambridge University Press. 620 p.