Lattice squares

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    Can someone please help me with lattice square design.  For example, If I am running an experiment with 8 trials & taking 5 samples from each trial, what would be the treatment combination to compare the samples be.  Or what would happen if I wanted to take 7 samples per trial. 
    I used this once and was given the appropriate lattice square by the MBB consultant who was training us, but need info on how to do this myself.  I have a boock by cochran & Cox which goes into it, but I’m no statistician or mathematician.
    Can someone explain to a lay man?
    Thanks, David


    Robert Butler

    I may be wrong but I think you will have to provide more detail before anyone can offer any constructive thoughts.  Lattice squares are pretty restrictive with respect to treatments, replicates, and such.  Based on what I know of them 8 treatments are not permitted.  The book you reference, Cochran & Cox ,indicates that “the number of treatments must be an exact square”.  In the edition I use they further list the useful plans that they knew of at the time of printing.  These consisted of designs for 9, 16, 25, 49, 64, and 81 treatments.  Furthermore, this type of design, like the Latin Square, is used primarily in those cases where the treatment cannot be viewed as a continuous variable – for example 4 different tire types and 4 different driving styles.  If you could provide some more details concerning the experimental effort you are attempting I, for one, would be willing to try to offer some suggestions.


    Robert Butler

    Thanks David, that gives me a better understanding of what you are trying to do.  Over lunch I sat down and re-read Chapter 10 of Cochran and Cox and based on your description of the Multiple Subjective Evaluation Technique it appears that it is wedded to the Lattice Square protocol.  This would explain why you had to have 4 dummie samples in order to get up to 9 treatments with 3 samples each.  Going by the book, this would also mean that an increase to 5 samples would violate that protocol and, at least using Lattice Squares, I can’t see a way around this.  However, there are other possibilities-more on this in a minute. 
      To continue with this thought, a full factorial with 16 treatments would require 4 samples per treatment and, again based on the book, it would appear that your choices of designs and levels are dictated by the Lattice Square requirements.
      The main problem I’m having is trying to understand why one would use a Lattice Square to set up a rating plan.  I can understand using such a plan to guarantee randomization with respect to raters and samples but if one is going to use a DOE this usually means that one is interested in expressing a given rating as a function of process variables.  I can’t offer any more on this line but I am curious and I’ll have to look into this some more.
      If you are interested in expressing a rating as a function of process variables you can run a  regression on the discrete Y’s.  Many Six Sigma courses take the very conservative aproach to regression and state that this is incorrect. This is just to make sure that you don’t make too many mistakes when you first use regression methods. If your attribute data is in rank form, for example 1-5, best to worst, rating on a scale from 1-100, a bunch of defects, not so many defects, a few defects, etc. You can use the numbers or assign meaningful rank numbers to the verbal scores and run your regression on these responses.  If you do this, you can take advantage of more sample ratings (which is what I’m assuming you want to do when you asked about increasing the samples from 3 to 5) without having to worry about the restrictions of  Lattice Squares. 
      There are a number of things that you should keep in mind if you try this:
    1. As with any attribute protocol, all of your raters of the attribute in question must be trained by the same person, with the same materials, so that their ratings will be consistent.
    2. The discrete nature of the Y’s will probably mask interaction effects so if you are trying to fine tune a process, as opposed to looking for major factors impacting the process, there is a chance that your success will be limited.
    3. Your residual analysis, particularly your residual plots of residuals vs. predicted are going to look odd.  They will consist of a series of parallel lines each exhibiting a slope of -1.
    4. The final model may indeed be able to discrminate between one rating and the next but a much more probable outcome will be the ability to only discriminate between good, neutral, and bad or even just good and bad.
      An example of running a regression on the Y response of “How you rate your supervisor” can be found in Chapter 3 of Regression Analysis by Example by Chatterjee and Price and if you want to check up on the residual patterns you can read Parallel Lines in Residual Plots by Searle in the American Statistician for August 1988.

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