Stat 502 Design and Analysis of Experiments Large Factorial Designs

Transcription

1 1 Stat 502 Design and Analysis of Experiments Large Factorial Designs Fritz Scholz Department of Statistics, University of Washington December 3, 2013

2 2 Exploration of Many Factors The typical setting is in industrial applications. Investigate manufacturing process improvements. Often want to screen a multitude of factors. No idea about important drivers for the response. Or one does not want to miss out on anything important. For k factors with respective numbers of levels L 1,..., L k, one would need to run M = L 1 L 2... L k units, just to get one observation for each factor combination. No degree of freedom for error estimation. What to do? We focus here of L i = 2, i = 1,..., k. Levels correspond to high/low (+/ ) settings of each factor.

3 3 Speedometer Data From Box, Bisgaard, and Fung (1988), An explanation and critique of Taguchi's contributions to quality engineering. Quality and Reliability Engineering International 4, Speedometer cables can be noisy because of shrinkage in plastic casing material. Engineers identied 15 factors as possible shrinkage causes. For the 15 factors see the above reference. Response = shrinkage percentage per specimen = runs to try all factor combinations once. Here we run just 16 combinations and assume no interactions. This allows estimation of overall mean and 15 main eects. Nothing left for error estimation. Data set speedo is part of the faraway package.

4 4 Speedometer Data > speedo h d l b j f n a i e m c k g o y

5 Comments on Design Matrix X 5 Think of the + and in previous slide as +1 and 1. Add to that matrix a rst column of all +1 and call it X. All columns x i of X are orthogonal to each other. The main eects model design matrix X is (here with k = 16) Y = X µ + e = k µ i x i + e, e 1,..., e k i.i.d. N (0, σ 2 ) i=1 The least squares estimates of µ i are ˆµ i = x i Y / x i 2, since ˆµ i x i is the projection of Y onto x i. Y = αx i + (Y αx i ) x i Y = α x i 2 αx i = ˆµ i x i. Note x i 2 = k = var(ˆµ i ) = σ 2 /k, i = 1,..., k. The vector of tted values is k Ŷ = ˆµ i x i = Y ê = Y Ŷ = 0 since Y R k i=1 i.e., no residuals for error estimation.

6 6 Independence of ˆµ i = x i Y / x i 2 The independence of the Y j, the orthogonality of the x i and the (approximate, via CLT) normality of the x i Y = independence of the ˆµ i. Proof: For i j we have cov(x i Y, x j Y ) = { E x i[y E(Y )]x j[y E(Y )] } { } = E x im [Y m E(Y m )]x jn [Y n E(Y n )] = n = n n m x im x jn E {[Y m E(Y m )][Y n E(Y n )]} m x in x jn σ 2 = 0

7 7 Main Eects Model g <- lm(y ~.,speedo) > summary(g) Residuals: ALL 16 residuals are 0: no residual degrees of freedom! Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) NA NA NA h NA NA NA d NA NA NA l NA NA NA b NA NA NA j NA NA NA f NA NA NA n NA NA NA a NA NA NA i NA NA NA e NA NA NA m NA NA NA c NA NA NA k NA NA NA g NA NA NA o NA NA NA

8 8 Main Eects Model continued Residual standard error: NaN on 0 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: NaN F-statistic: NaN on 15 and 0 DF, p-value: NA We have the 16 estimates of mean and main eects. No p-values for the least squares estimates. Note that we stipulated a main eects model. Can we do more? All ˆµ i have variance σ 2 /k and they are independent. If the main eects are all zero then these estimates should behave like an i.i.d. N (0, σ 2 /k). Since we are dealing with linear combinations there should be a strong CLT eect, even if the Y i are not normal.

9 9 How to Judge Signicance of Main Eects This can be examined with a normal probability plot. Any points far away from the straight line pattern would suggest that the corresponding main eect are not zero. Normal Q Q Plot g Sample Quantiles c f n o m l i k a h d j b e Theoretical Quantiles

10 10 Commands for the Plot > library(faraway) > data(speedo) > g <- lm(y ~., speedo) > qqnorm(g$coeff[2:16],pch=names(g$coeff[2:16])) > qqline(g$coeff[2:16]) The dot in ~. stands for all other variables/factors in the data frame speedo besides the response y.

11 11 How to Generate Fractional Factorial Designs There are many experimental design tools discussed in The package FrF2 contains many functions, one of which is the function FrF2. It generates fractional factorial designs, given inputs for nfactors, number of factors nruns, number of runs = 2 p [4, 4096], if given, resolution, the design resolution. resolution=5 means that all 2-factor interaction are unconfounded with each other and with main eects. See documentation for FrF2.

12 12 Further Sources on Fractional Factorial Designs Box, G.E.; Hunter, J.S., Hunter,W.G. (2005) Statistics for Experimenters: Design, Innovation, and Discovery, 2nd Edition. Wiley. National Institute of Standards and Technology Vijay Nair An excellent introduction and reference for using linear models in S (a precursor to R and applicable to it) is Statistical Models in S (1991) edited by Chambers and Hastie, Chapman and Hall/CRC