Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 1

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1 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 1 Cmd> a<-factor(1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2) We have a 2 4 factorial; 4 factors, each at two levels. We will be entering the data in standard order, with factor A varying fastest, B next fastest, etc. This order is not needed for anova() commands, but is needed for the yates() command. Cmd> b<-factor(1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2) Cmd> c<-factor(1,1,1,1,2,2,2,2,1,1,1,1,2,2,2,2) Cmd> d<-factor(1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2) Cmd> y<-vector(1.68,1.98,3.28,3.44,4.98,5.7,9.97,9.07,\ 2.07,2.44,4.09,4.53,7.77,9.43,11.75,16.3) These data are from Daniel (1976). The experiment was on a stone drill. Factor A is the load on the drill; B is the flow rate through the drill; C is the rotational speed; D is the type of mud used. The response is the rate of drill advance. There was just one replication. Cmd> anova("y=(a+b+c+d)ˆ3",fstats:t) Because we only have one replication, we can try to use the four-factor interaction as error. This doesn t work well, since it only leaves 1 df for error. Note that c has an F ratio of about 150, but is only marginally significant! Model used is y=(a+b+c+d)ˆ3 CONSTANT a b c d a.b a.c a.d b.c b.d c.d a.b.c a.b.d a.c.d b.c.d ERROR

2 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 2 Cmd> anova("y=(a+b+c+d)ˆ2",fstats:t) Let s try something more reasonable, where we only fit main effects and two factor interactions, pooling the three and four way interactions for error. The p-values we get are at best a guideline. It looks like b, c, d, and some two factor interactions might be significant. Model used is y=(a+b+c+d)ˆ2 CONSTANT e-06 a b c e-05 d a.b a.c a.d b.c b.d c.d ERROR Cmd> resvsyhat() Looking at the residuals, there may be some nonconstant variance; we ll check with Box- Cox. Standardized Resids Standardized Residuals vs Fitted Values (Yhat) Fitted Values (Yhat) Cmd> boxcoxvec("y=(a+b+c+d)ˆ2") Box-Cox suggests the power Both -1 and 0 are well within the confidence interval, so let s look at both. Note that the reciprocal is also easy to understand as a time to reach a drilling depth (rather than a drilling rate). component: power (1) (6) (11) component: SS (1)

3 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 3 (6) (11) Cmd> ry<-1/y Note that treatments with a big ry (time) have a low rate y. Cmd> anova("ry=(a+b+c+d)ˆ2",fstats:t) Model used is ry=(a+b+c+d)ˆ2 CONSTANT < 1e-08 a b e-06 c e-07 d a.b a.c a.d e e b.c e-05 b.d c.d ERROR Cmd> resvsyhat() Better. Standardized Residuals vs Fitted Values (Yhat) Standardized Resids Fitted Values (Yhat)

4 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 4 Cmd> anova("ly=(a+b+c+d)ˆ2",fstats:t) Log doesn t do quite as well as reciprocal for constancy of variance, but most of the interaction goes away. We are fairly certain that b, c, and d main effects are present. There is some evidence for a and c.d effects, but we temper our enthusiasm by recalling 1) the p-values are questionable to begin with, since we don t have an estimate of pure error, 2) these are just the smallest 2 of the other 7 p-values, so we may wish to use Bonferroni or some other multiple comparisons procedure. Model used is ly=(a+b+c+d)ˆ2 CONSTANT < 1e-08 a b e-05 c e-06 d a.b a.c a.d b.c b.d c.d ERROR Cmd> resvsyhat() Residuals not quite as good on log scale as on reciprocal, but not too bad here. Standardized Residuals vs Fitted Values (Yhat) Standardized Resids Fitted Values (Yhat)

5 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 5 Cmd> resvsrankits() Not bad here. Standardized Residuals vs Normal Scores Standardized Resids Normal Scores Cmd> anova("y=a+b+c+d",silent:t);tukey<-(y-residuals)ˆ2/2/coefs("constant") Cmd> anova("y=a+b+c+d+tukey") DF SS MS CONSTANT a b c d tukey ERROR Cmd> coefs("tukey") (1) The log transformation is also suggested by the Tukey one degree of freedom procedure using a additive model as the simple model. Together with the pretty good residuals, I think I ll work on the log scale.

6 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 6 Cmd> anova("ly=a+b+c*d",pvals:t) Refitting with just the significant terms, we get a different error estimate and thus different p-values. This comes of not having an estimate of pure error and having to pool various terms into error (and different terms in different analyses). Model used is ly=a+b+c*d CONSTANT < 1e-08 a b e-08 c < 1e-08 d e-05 c.d ERROR Cmd> yts<-yates(ly);yts Now we look at the data an alternate way. Since the data were in standard order, the yates() function will produce the total factorial effects in standard order: A, B, AB, C, AC, BC, ABC, D, AD, BD, ABD, CD, ACD, BCD, ABCD. (1) (6) (11) Cmd> yatesplot(ly,fullnorm:t) Factors or interactions that are zero should have estimated effects that look like normal noise, while factors or interactions that are not zero should have estimated effects that look like outliers. Here, C, B, and D look like outliers. A looks pretty much like the rest of the pack. Normal scores plot of effects C Effects B D A CD AD ABCDABDACD AC ABC BC BCD AB BD Normal scores

7 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 7 Cmd> yatesplot(ly) A half normal plot is another way of looking at these effects; it folds the plot over and looks at the absolute values of the effects. Again, C, B, and D look like outliers, while A and CD, which were marginally significant, are back in the pack. Half Normal plot of absolute effects C Absolute effects B D A CD AB BC AD ACABCBD BCDABCD ABDACD Half normal scores Cmd> s1<-describe(abs(yts),median:t)*3.75;s1 Now we compute Lenth s pseudo standard error. First we get 3.75 time the median of the absolute effects. (1) Cmd> yts2<-yts[abs(yts)<s1] Now we select those absolute effects that are less than s1. Cmd> pse<-describe(abs(yts2),median:t)*1.5;pse The pseudo standard error is 1.5 time the median of the absolute effects that are less than s1. Use it as a standard error with 15/3 = 5 degrees of freedom. (1) Cmd> invstu(.975,5) Ratio of effect to pse should be 2.6 or greater to be significant at the.05 level. (1) Cmd> yts/pse Using the PSE, only B, C, and D are significant. A has a p-value of about.1 (1) (6) (11)

8 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 8 Cmd> yatesplot(ry) For completeness, let s look at the reciprocal scale. On this scale, C, B, D, BC, and perhaps A look like outliers. Absolute effects Half Normal plot of absolute effects C B BC D A AC AB ABC CD BD ABD BCDABCDAD ACD Half normal scores Cmd> y<-vector(8,4,53,43,31,9,12,36,79,68,73,8,77,38,49,23)/100 Data from Davies (1954) via Box and Meyer (1986). Response is the yield of isatin under different production conditions. Factors are: A acid strength; B time; C amount of acid; D temperature. Cmd> anova("y=(a+b+c+d)ˆ2",pvals:t) Doesn t look like much going on here. Alternatively, everything is significant, but we can t see it because we are using a large interaction for error, making other significant effects look insignificant. Without an estimate of pure error, we cannot tell which situation we are in. Model used is y=(a+b+c+d)ˆ2 CONSTANT a b c d a.b e e a.c a.d b.c b.d c.d ERROR

9 Stat 5303 (Oehlert): Unreplicated 2-Series Factorials 9 Cmd> yts<-yates(y);yts Data were entered in standard order, so we can use yates() to get effects in standard order. Not shown here, but the Lenth PSE is.114, and the t-statistic for D is 2.4, with a p-value of about.06. (1) (6) (11) Cmd> yatesplot(y) Looks like a whole lot of nothing. Half Normal plot of absolute effects D Absolute effects BD A AD ABC BCD ABD C BC AC B CD ABCD AB ACD Half normal scores