B-6 APPENDIX B 



Discussion of Method of Analysis 



Omission of some valid data: Only the balanced data ignoring test laboratories was 

 included in the analyses variance. A more ambitious program might have tried to use 

 standard least-squares methods (multiple regression, including all two-factor interactions) 

 on the complete and unbalanced set of data. However the considerable number of bad values 

 turned up in the balanced analyses warns that mere insistence on including all data does not 

 in itself guarantee greater validity. Since the spotting of bad values is much more difficult 

 in unbalanced data, it is not considered likely that much information was lost by omitting 

 the panels that were added later to the balanced plan. (These were the four panels of Ml 

 added to form a complete sequence of thicknesses, and the sixteen panels added for 

 "duplicates"). These omitted data were not entirely wasted, since the computed Lower 

 Tolerance Limits were compared with the lowest values in all the relevant data. In most 

 cases no values lay below the computed L.T.L. In some cases several values lay below, 

 usually coming from related panels. These values were judged bad, and therefore rejected. 



Rejection of some duplicate data: Certain pieces of data were rejected on the grounds 

 that they did not fit the pattern clearly formed by the remaining pieces. This practice, while 

 it gave neater conclusions, is especially questionable since judgments on future variability 

 are to be made. Unless enough data is taken in future qualifying tests by fabricators, so 

 that similar rejection of out-of-pattern pieces can be made, then even wider limits of 

 tolerance will have to be used. Less than ten panels of each type and thickness of material 

 from a new and untried fabricator should not be considered. Schedules for producing such 

 test panels should be prepared with statistical caution. 



Omission of test laboratories as a factor: Except for a mistake in one test laboratory's 

 measurements of specific gravity, no serious differences among laboratories were noted. 

 However a full analysis including test laboratories was not made. If there were any serious 

 laboratory differences, they would show in the analyses of variance made as MFT inter- 

 actions. Whenever MFT appeared larger than MF, MT or FT, a search was made to see if 

 the larger discrepancies were attributable to systematic laboratory differences. No such 

 attributions were made. There is of course always the chance that MFT was in fact inflated 

 by its confounding with the laboratories and that at the same time MF, MT and FT were 

 large for other reasons. If this unfortunate combination of circumstances was obtained in any 

 case, we would miss all four, and would report a gross over-estimate of the panel error. 



Inappropriateness of Lieberman's Tolerance Limit Table: This table is exact when the 

 error distribution is normal, and when a simple mean value and its associated standard 

 deviation are to be used. This case is different since we usually have a mean value estab- 

 lished by a regression-equation, but with variance a known multiple of the error variance. 

 Professor Henry Scheffe has worked out the exact distribution required. It turns out to be 

 a case of the "non-central t-distribution. " Because of the spotty tabulation of this distribu- 

 tion in the literature, two checks of the exact distribution were made and compared with the 

 values corresponding from Liebermans' Table. The differences were negligible and so the 

 easier tabulation was used in all other cases. 



Conclusions 



One of the commonest conclusions from a large statistical study is that a great deal of 

 care must be used in planning future extensions. This conclusion holds here. Much has been 

 learned about the systematic differences among fabricators and among materials. Much 

 more needs to be learned about the cause and cure of the random variability of the materials, 

 especially among duplicate panels produced by the same fabricator. 



