13.10 Graphical Methods and Model CheckingIn ‘several c;hapterswe make reference to graphical procedures displaying data ailalYJiI:alresults. In early chapters we used tem- nd-leaf and box-and-whis plots as viSuals that aid in summarizing samples, We use similar diagnostics • better &ndl!rStandthe data n two sample problems in Chapters 9 and 10, Chap~r 9 We introduce the notion of residual plots (ordinary and studentizec residuals):to detect. viOlations of standard assumptions. In recent years rn !lttention hi data. arialyais’bas centered on raphical methods. Like regressi anaI)-sisQf.railance lends itself to gl’l\phics that aid in summari:dng data as ~ . ¥ det~ting ‘i~latioftS. For example, a simple plotting of the raw ob6ervati -around each treatment mean can’ give th~ analyst & feel for ariability berw sample means and within samples. Figure 13.6 depicts such a plot for the Ilggre data of Table 13.1. By the ppearance of the plot one migh~even gain a grap . ·insight about w~ich aggregates (if any) stand out from the others. It: is cI -,tpat ggreg~te 4 stands out from the others. Aggregates 3 and 5 certainly form . homogeneous group, as do aggregates 1 and 2. , . ‘As i’n he case of regression, residuals can. be helpful in anal)’Sis of vari in .providing a diagnOflticthat may detect violations of ssumptions. To form ~duals, we merely need to consider the model of the one-factor problem. Figure 13.6: Plot. of data around the mean for the aggregate data of Table 13.1Figure 3.7: Plots of residuals for five aggregates, using data in Table 13.1. It is straightforward to determine that the estimate of J.’i is iii.. Hence the ijth . residual is Yi. – Y… This is easily extendable to the andomized complete block model. It may be instructive to have the residuals plotted for each aggregate in order to gain some nsight regarding the homogeneous variance assumption. This plot is shown in Figure 13.7. T:ends in plots such as these may eveal difficulties in some situations, particularly when the violation of a particular assumption is graphic. In the case of Ftgure 3.7, the residuals seem to indicate that the withirt”treatment variances are reasonably homogeneousapart from aggregate 1. here s some graphical evidence th,.t the variance ·foraggregate 1 is larger than the rest. What Is a Residual for a RCB Design? The andomized Completeblock is another experimental situation in which graphic: aIdisplays can make the analyst feel comfortable ith·an ‘ideal picture,” or perhaps highlight difficulties. Recall that the model for the randomised completeblock is i = 1, … ,k, j = 1, … ,b, with the imposed constrai,nts To determine what indeed constitutes a residual, consider that Cti Jli. Jl, (3; = Jl.j – Jl and that Jl is estimated by Y.. ,Jli. is estimated by fk, and ‘Jl.j is estimated by ij.;. As a result, the predicted or fitted alue Yij is given byS=’-rl:+O’+• (3-‘ J =Y-•’. +1/-.J’-Y-”and thus the residual at the (i; j). observation is given by Note that ij, the fitted-value, is an estimate of the mean /lij. This is consiste with the partitioning of variability given in Theorem 13.3, where he error sum squares isSSE = L L(Yij – Yi. – Y.; +y..)2. The visual displays in the randomized complete block involve plotting the esinuals separately for each treatment and for each block. The analyst should expec: rougbly equal variability if the omogeneous ariance assumption holds. The read should recall that in Chapter 12 we discuss plots where the residuals are plott for the purpose f detecting model misspeci6cation. In the case of the randomizec complete block, the serious model misspeci6cation may be elated to our assumption of additivity (i.e., no interaction). If no interaction is present, a rando pattern should appear. Consider he data of Example 13.6, in which treatments are four machines blocks are six operators. Figures 13.8 and 13.9 give the residual lots for sep treatments and separate blocks. Figure 13.10 shows a plot ofthe residuals ag . the 6tted values, Figure 13.8 reveals hat he error variance may not be the s for all machines. The same may be true for error variance at each of the operators, However, o unusually large residuals appear to produce the ap difficulty. Figure 13.10 reveals a plot of residuals that shows reasonable evid of a random behavior. However, the two large residuals displayed earlier still out. Figure 13.10: Residuals plotted against fitted values for the data of Example 13.6.,