Solution file for additional exercise 10.3 ------------------------------------------ (see additional exercise 10.1 for discussion of model, design and notation) MTB > WOpen "h:\VHM\VHM802\Data_csv\hs10_3.csv"; SUBC> FType; SUBC> Text; SUBC> VNames; SUBC> None; SUBC> DecSep; SUBC> Period; SUBC> TDelimiter; SUBC> DoubleQuote. Retrieving worksheet from file: 'h:\VHM\VHM802\Data_csv\hs10_3.csv' Worksheet was saved on 03/03/2011 MTB > Name c4 "SRES1" c5 "TRES1" MTB > GLM 'conc' = material|lab; SUBC> Random 'lab'; SUBC> Brief 2 ; SUBC> EMS; SUBC> Means material; SUBC> SResiduals 'SRES1'; SUBC> TResiduals 'TRES1'; SUBC> GFourpack; SUBC> RType 2 . General Linear Model: conc versus material, lab Factor Type Levels Values material fixed 3 1, 2, 3 lab random 11 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Analysis of Variance for conc, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P material 2 751.888 751.888 375.944 2938.10 0.000 lab 10 2.961 2.961 0.296 2.31 0.053 material*lab 20 2.559 2.559 0.128 2.44 0.011 Error 33 1.730 1.730 0.052 Total 65 759.138 S = 0.228963 R-Sq = 99.77% R-Sq(adj) = 99.55% Unusual Observations for conc Obs conc Fit SE Fit Residual St Resid 8 22.0000 21.5500 0.1619 0.4500 2.78 R 19 21.1000 21.5500 0.1619 -0.4500 -2.78 R 32 12.1000 12.4500 0.1619 -0.3500 -2.16 R 43 12.8000 12.4500 0.1619 0.3500 2.16 R R denotes an observation with a large standardized residual. Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 material (4) + 2.0000 (3) + Q[1] 2 lab (4) + 2.0000 (3) + 6.0000 (2) 3 material*lab (4) + 2.0000 (3) 4 Error (4) ... Variance Components, using Adjusted SS Estimated Source Value lab 0.02802 material*lab 0.03777 Error 0.05242 Least Squares Means for conc material Mean 1 21.12 2 12.88 3 16.42 Residual Plots for conc MTB > GLM 'conc' = material|lab; SUBC> Random 'lab'; SUBC> SMeans C4000; SUBC> Brief 0; SUBC> Interact 'material' 'lab'. MTB > GFInt 'material' 'lab'; SUBC> Responses 'conc'; SUBC> FMeans C4000; SUBC> Full. Interaction Plot (fitted means) for conc MTB > Erase C4000. MTB > NormTest 'SRES1'. Probability Plot of SRES1 The P-value for the Anderson-Darling test for normality is 0.038. Comments and answers to questions: ---------------------------------- With two replications, the (error) residuals come in pairs (positive and negative), and furthermore the fitted values are clearly divided by the materials. Taking this into account, the (error) residual plots look fine. The weakly significant test for normality may reflect that the largest residual (2.78) is mirrored by another equally large negative residual (-2.78). There does not seem to be any obvious model deviations. We give an additional residual analysis for the two other random effects below. The ANOVA shows a significant material*lab interaction, but as this is modelled by a random effect, we are still allowed to look at the main effects. The interaction plot seems in fact very regular (almost parallel lines). There is, as expected, a huge difference between materials, and there are also some differences between laboratories. Note that it is of no interest here to eliminate non-significant random terms, because the the random effects parameters are of major interest. The estimated variance components are: sigma^2 (error): 0.052 sigma^2_AB (L*M): 0.038 sigma^2_A (Labs): 0.028 We can use these values to compute the repeatability and reproducibility: r = 2.83 sqrt(0.052) = 0.65 R = 2.83 sqrt(0.052+0.038+0.028) = 0.97 Note that the Minitab least squares means are correct, but the standard errors are missing. In this case, we have no particular interest in intervals for the the materials anyway. The comparisons in the Comparison menu would be ok, if we wanted to use them. Minitab commands for residual analysis for Lab*Mat and Lab: ----------------------------------------------------------- MTB > Name c6 "ByVar1" c7 "ByVar2" c8 "Mean1" MTB > Statistics 'conc'; SUBC> By 'lab' 'material'; SUBC> GValues 'ByVar1'-'ByVar2'; SUBC> Mean 'Mean1'. MTB > Name c9 "SRES2" MTB > GLM 'Mean1' = ByVar1 ByVar2; SUBC> Brief 2 ; SUBC> SResiduals 'SRES2'; SUBC> GFourpack; SUBC> RType 2 . General Linear Model: Mean1 versus ByVar1, ByVar2 Factor Type Levels Values ByVar1 fixed 11 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ByVar2 fixed 3 1, 2, 3 Analysis of Variance for Mean1, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P ByVar1 10 1.480 1.480 0.148 2.31 0.053 ByVar2 2 375.944 375.944 187.972 2938.10 0.000 Error 20 1.280 1.280 0.064 Total 32 378.704 S = 0.252937 R-Sq = 99.66% R-Sq(adj) = 99.46% Unusual Observations for Mean1 Obs Mean1 Fit SE Fit Residual St Resid 6 15.9500 16.4667 0.1588 -0.5167 -2.62 R R denotes an observation with a large standardized residual. Residual Plots for Mean1 MTB > NormTest 'SRES2'. Probability Plot of SRES2 The P-value for the Anderson-Darling test for normality is 0.544 Comments: --------- This analysis for the lab*mat means gives the same F-statistics as above. The residuals in the analysis can be used to check the assumptions for the Lab*Mat random effects. In this case the residual plots look similar to the previous residual plots, and are no cause of concern. MTB > Name c10 "ByVar3" c11 "Mean3" MTB > Statistics 'conc'; SUBC> By 'lab'; SUBC> GValues 'ByVar3'; SUBC> Mean 'Mean3'. MTB > NormTest 'Mean3'. Probability Plot of Mean3 The P-value for the Anderson-Darling test for normality is 0.032. Comments: --------- The laboratory means themselves should, if the model is correct, correspond to a sample from a normal distribution as well. However, the normal plot does not look too good - there are two labs with values clearly above the rest, and in particular lab no. 4 looks suspicious. Caution should be exercised to not include outlying labs from the calculations of reproducibility, because they will lead to overestimation of the variability between laboratories. The conclusion here is not obvious, but certainly one may express doubt about the procedures at lab no. 4.