DOE to study yield of an experiment

  Since you haven’t been very specific concerning your definition of yield I’ll assume you are talking about trying to identify an optimum combination of processing conditions and raw material levels that will maximize a final product output and minimize waste. 
  Sample size calculations are based on the size of the difference you wish to detect and the expected variability of the process.  If either one of these are missing you are wasting your time trying to compute a sample size.
 If the following assumptions are met
  1. You have adequate control over your process so that you can actually vary your raw material and process variables as planned.
  2.  You are interested in big changes in percent conversion.
  3. You know how to set up a design and can enforce the randomization of the experimental runs. 
  then the best way to both get an estimate of existing process variation and gain an understanding of changes in percent yield caused by changing your process and raw materials is to do the following:
  1. Set up the design
  2. Pick a couple of points (if you have genuine center points choose those) for replication.
  3. Run the randomized design and the replicate points once.
  4. Analyze the results – use standard regression methods to gain an understanding of those variables impacting mean shift and use the Box-Meyers method to gain an understanding of those variables impacting yield variance.
  Run this way you will quickly learn several things:
  1. If the variables of interest do matter and if the measured variance of the system isn’t too large you will detect significant trends. The information gleaned from your efforts can be quickly tested to see if the trends are real and if changes in the direction indicated do, in fact improve yield.
  2. If there are no significant effects you may want to confirm a run or two and if they do confirm you will have to ask additional questions:
   a. How big is the residual variation? 
   b. Use this estimate of variance to compute the minimum change in yield you could see if the variables of interest really had an impact on your process.
      1. If the variance in b does allow for the detection of “reasonable” changes in your percent yield and if the residual analysis indicates that nothing is amiss with the final models then there is a high probability that the variables and/or the ranges of the variables you chose really don’t impact yield.
     2. If the variance in b doesn’t allow for the detection of “reasonable” changes then you need to stop and think very hard about identifying and eliminating the variation in your process that is masking your ability to detect the changes you wish to see. 
   A final thought – you need to remember that samples as thought of  in sample size calculations are independent samples.  This means that if you want to take two samples from a given experimental condition you will need to set up and run that condition twice. Two samples, separated in time, taken from the same run are not independent.  If you attempt to treat them as such you will go wrong with great assurance.  There are methods to handle measurements made this way – the techniques are called repeat measure analysis – but they require a lot of skill and statistical understanding for proper use.