Stat 5303 (Oehlert): Split Plots 1

Transcription

1 Stat 50 (Oehlert): Split Plots Cmd> print(y,format:"f7.",labels:f) y: More emulsion data. Eight samples of unprocessed gum from acacia senegal trees are obtained. Four samples come from one batch of gum, and the other four come from a second batch of gum. Within each batch, the four samples are randomly assigned to the factor/level combinations of two factors. The first factor is whether or not the gum is demineralized; the second factor is whether or not the gum is pasteurized. An emulsion is made using each sample of gum. Each emulsion is then split into three smaller parts, with the parts randomly assigned to be ph adjusted to.5, 4.5, or 5.5 using citric acid. The measured response is the time (in hours) till the emulsion fails, called the breakage time Cmd> print(past,format:"f4.0",labels:f) past: Cmd> print(demin,format:"f4.0",labels:f) demin: Cmd> print(ph,format:"f4.0",labels:f) ph: Cmd> print(block,format:"f4.0",labels:f) block:

2 Stat 50 (Oehlert): Split Plots Cmd> anova("y=block+past*demin+e(block.past.demin)+ph*past*demin",fstats:t) This is a split plot, with batch as block, sample as whole plot, and part of the emulsion as split plot. Blocks and whole plot treatments together enumerate all whole plots, so any variation among the block by whole plot treatment combinations that remains after removing block and whole plot treatments must be whole plot error. The E() notation just relabels the term as an ERROR term. Model used is y=block+past*demin+e(block.past.demin)+ph*past*demin CONSTANT 8.558e e block past demin past.demin ERROR ph demin.ph past.ph past.demin.ph ERROR Cmd> resvsyhat() Decreasing variance? Box-Cox suggests squaring, which does improve normality somwhat, but does little to help with nonconstant variance. We ll use the original data. Standardized Residuals vs Fitted values (Yhat) Standardized Residuals Fitted values (Yhat)

3 Stat 50 (Oehlert): Split Plots Cmd> tabs(y,past,demin,mean:t) (,) (,) Here are the means for the whole plot treatments. The breakage time is much greater when both demineralization and pasteurization are used; the other three treatments are roughly equal. Cmd> tabs(y,past,demin,ph,mean:t) Here is the full table of treatment means. ph has very little effect except when both pasteurization and demineralization are used. So what appears to be happening is that nothing much is going on except when pasteurization and demineralization are used, in which case ph also has an effect. (,,) (,,) (,,) (,,) Cmd> interactplot(y,ph,past,demin,xaxis:f,yaxis:f) Interaction plot of y vs ph by past.demin. Means of y Levels of ph

4 Stat 50 (Oehlert): Split Plots 4 Cmd> pd<-rep(0,4);pd[past== && demin==]<- pd is a dummy variable indicating when both pasteurization and demineralization are used. Cmd> phpd<-makefactor(ph*pd) phpd is factor with four levels, one each for the three ph levels when pd is, and a level for all other combinations. Cmd> anova("y=block+pd+past.demin+e(block.past.demin)+ phpd+ph+ph.past+ph.demin+ph.past.demin",fstats:t) We can see that pd accounts for about 99% of the whole plot SS, and pdph accounts for about 9% of the split plot variation. Model used is y=block+pd+past.demin+e(block.past.demin)+phpd+ph+ph.past+ ph.demin+ph.past.demin WARNING: summaries are sequential CONSTANT 8.558e e block pd past.demin ERROR phpd ph past.ph demin.ph past.demin.ph 0 0 undefined undefined undefined ERROR Cmd> -cumf(458//8.,,) Because the pd and phpd terms were suggested by the data, we cannot trust the p-values given in the table above; we must adjust for mutiple comparisons. Here we use Scheffé (very conservative) to assess the pd term. () Cmd> -cumf(9756.9/8/4.4,8,8) Scheffé for the splitplot effects suggested by the data. () Cmd> # We could have used HSD on the pasteurization/demineralization means, treating each as an average of six readings, and using ERROR (with its df) as the estimate of error. This simple HSD approach does not work for the ph/demin/past means, because some comparisons are split plot comparisons and some comparisons are whole plot comparisons. That is, we cannot just use ERROR or ERROR. What you can do is use methods of section.7 to get the standard error for each pairwise difference, and then use Bonferroni to adjust for multiple comparisons.

5 Stat 50 (Oehlert): Split Plots 5 Cmd> # Next is Example 6.7 of the text. Tables are blocks, nitrogen is the whole plot treatment factor; wetlands are split plots, and weed seeding is the split plot treatment factor; sections of wetlands are split split plots, and clipping is the split split plot treatment factor. Cmd> anova("y=table+n+e(table.n)+weed*n+e(table.weed.n)+clip*weed*n") Here is the basic anova. We label two terms as ERROR terms for whole and split plot error; split split plot error is computed automatically. Residuals look fine. N is tested against ERROR (whole plot error); weed and n.weed are tested against ER- ROR (split plot error); all others are tested against ERROR (split split plot error). We can see that whole plot error is substantially larger than split plot error which is substantially larger than split split plot error. Nitrogen and weeds interact, and there is a main effect of clipping. Model used is y=table+n+e(table.n)+weed*n+e(table.weed.n)+clip*weed*n CONSTANT.9804e e e-05 table n ERROR weed < e-08 n.weed e-05 ERROR clip e-07 weed.clip n.clip n.weed.clip ERROR Cmd> interactplot(y,n,weed,xaxis:f,yaxis:f) Nonweed biomass decreases as nitrogen increases, but the decrease is much larger for the weed-seeded treatments. Means of y Interaction plot of y vs n by weed 4 Levels of n

6 Stat 50 (Oehlert): Split Plots 6 Cmd> coefs("n") Main effects of nitrogen. () Cmd> anova("y=table+n+e(tray)+weed*n+e(wet)+clip*weed*n",fstats:t) Here is an alternate formulation that gives the same ANOVA. Instead of denoting each tray as the table by nitrogen combinations, we can have a factor enumerating the trays. Similarly, instead of denoting each wetland as the table by nitrogen by weed combinations, we can have a factor that enumerates the wetlands. This gives the same ANOVA as before. Model used is y=table+n+e(tray)+weed*n+e(wet)+clip*weed*n WARNING: summaries are sequential CONSTANT.9804e e e-05 table n ERROR weed e-09 n.weed e-05 ERROR clip e-07 weed.clip n.clip n.weed.clip ERROR Cmd> coefs("n") Note that tray (ERROR) has eight levels, but only three degrees of freedom; also, anova() reports that the sums of squares are sequential. One consequence of this is that the treatment coefficients are not the same as before (and you want the other ones). In fact, the coefficients you get for n depend on how you number the trays. Stick with the other method if you want to look at coefficients. () Cmd> print(y,format:"f7.",labels:f) More emulsion data. Three emulsifiers: milk protein concentrate (MPC), modified milk protein concentrate (mmpc), and sodium caseinate (NC). An emulsion is made from each of the emulsifiers and the turbidity is measured at 0,,,, 4, 5, 6,, 4, 6, and 48 hours. A good emulsifier should keep the turbidity high. The experiment is then repeated two more times. Here are the turbidities. y:

7 Stat 50 (Oehlert): Split Plots 7 Cmd> print(protein,format:"f4.0",labels:f) protein: Cmd> print(time,format:"f4.0",labels:f) time: Cmd> print(block,format:"f4.0",labels:f) block: Cmd> hrs<-vector(run(0,6),,4,6,48);h <- hrs[time] Numerical values for the times. Cmd> anova("y=block+protein+e(block.protein)+protein*time") This is the univariate analysis of a repeated measure. For a single repeated measure (time in this case), the analysis is just like a split plot. Model used is y=block+protein+e(block.protein)+protein*time CONSTANT < e-08 block protein ERROR time < e-08 protein.time < e-08 ERROR

8 Stat 50 (Oehlert): Split Plots 8 Cmd> plotresids("standardized","yhat",symbol:"\0",xaxis:f,yaxis:f ) Maybe a slight increase in variance? Maybe an outlier. In fact, the outlier is significant at the.04 level, but I m going to leave it in for now. Also, Box-Cox does not suggest a transformation. Standardized Residuals vs Fitted values (Yhat) Standardized Residuals Fitted values (Yhat) Cmd> pairwise("protein",.0,hsd:t,error:"error") We can do pairwise comparisons for whole plot treatment means provided we choose the correct error term Cmd> interactplot(y,time,protein) Turbidity decreases with time, but MPC decreases much more quickly than the other two, which are very similar. Means of y Interaction plot of y vs time by protein Levels of time

9 Stat 50 (Oehlert): Split Plots 9 Cmd> plot(hrs,tabs(y,time,protein,mean:t)) We can also plot against actual time hrs Cmd> anova("y=block+protein+e(block.protein)+protein*time",fstats:t, wts:*(protein>)) If we just look at proteins and, there are no protein effects or interactions. Model used is y=block+protein+e(block.protein)+protein*time WARNING: cases with zero weight completely removed Model fit by weighted least squares WARNING: summaries are sequential CONSTANT e-06 block protein ERROR time protein.time ERROR