Solution file for additional exercise 7.4 ----------------------------------------- Data: survival times of goldfish exposed different concentrations of cyanide and oxygen, at different temperatures. Notation: y_i = survival times (averaged over 10 fish) for solution i, i=1,...,45, or y_ijk = survival times (averaged over 10 fish) for solution with oxygen level j and cyanide level k, at temperature i, i=5,15,25; j=0.5,3,9; k=0.16,0.8,4.20,100. The design is a complete 3-factorial without replication (because the replications of goldfish are summarized into a single, averaged response). In absence of replication and with no factor seemingly less relevant for interactions, the most natural model takes the 3-factor interaction as the error term. This gives the statistical model y_i = mu + alpha_temp(i) + beta_oxygen(i) + gamma_cyanide(i) + (alpha beta)_temp*oxygen(i) + (alpha gamma)_temp*cyanide(i) + (beta gamma)_oxygen*cyanide(i) + eps_i, or y_ijk = mu + alpha_i + beta_j + gamma_k + (alpha beta)_ij + (alpha gamma)_ik + (beta gamma)_jk + eps_ijk, depending on the chosen notation. MTB > WOpen "H:\VHM\VHM802\Data_csv\hs07_4.csv"; SUBC> FType; SUBC> CSV; SUBC> DecSep; SUBC> Period; SUBC> Field; SUBC> Comma; SUBC> TDelimiter; SUBC> DoubleQuote. Retrieving worksheet from file: ‘H:\VHM\VHM802\Data_csv\hs07_4.csv’ Worksheet was saved on 17/02/2011 MTB > Name C5 "SRES". MTB > GLM; SUBC> Response 'surv'; SUBC> Nodefault; SUBC> Categorical 'temp' 'oxygen' 'cyanide'; SUBC> Terms temp oxygen cyanide temp*oxygen temp*cyanide oxygen*cyanide; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TFactor; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK; SUBC> SResiduals 'SRES'. General Linear Model: surv versus temp, oxygen, cyanide Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values temp Fixed 3 5, 15, 25 oxygen Fixed 3 1.5, 3.0, 9.0 cyanide Fixed 5 0.16, 0.80, 4.00, 20.00, 100.00 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value temp 2 571.16 45.15% 571.161 285.581 441.51 0.000 oxygen 2 37.59 2.97% 37.588 18.794 29.06 0.000 cyanide 4 555.45 43.91% 555.455 138.864 214.68 0.000 temp*oxygen 4 0.97 0.08% 0.971 0.243 0.38 0.823 temp*cyanide 8 36.86 2.91% 36.859 4.607 7.12 0.000 oxygen*cyanide 8 52.65 4.16% 52.645 6.581 10.17 0.000 Error 16 10.35 0.82% 10.349 0.647 Total 44 1265.03 100.00% S R-sq R-sq(adj) PRESS R-sq(pred) 0.804259 99.18% 97.75% 81.8648 93.53% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 10.840 0.120 (10.586, 11.094) 90.41 0.000 temp 5 4.567 0.170 ( 4.207, 4.926) 26.93 0.000 1.33 15 -0.440 0.170 (-0.799, -0.081) -2.60 0.020 1.33 oxygen 1.5 -1.193 0.170 (-1.553, -0.834) -7.04 0.000 1.33 3.0 0.167 0.170 (-0.193, 0.526) 0.98 0.340 1.33 cyanide 0.16 6.460 0.240 ( 5.952, 6.968) 26.94 0.000 1.60 0.80 0.560 0.240 ( 0.052, 1.068) 2.34 0.033 1.60 4.00 -1.429 0.240 (-1.937, -0.921) -5.96 0.000 1.60 20.00 -1.796 0.240 (-2.304, -1.287) -7.49 0.000 1.60 temp*oxygen 5 1.5 -0.033 0.240 (-0.542, 0.475) -0.14 0.891 1.78 5 3.0 0.147 0.240 (-0.362, 0.655) 0.61 0.549 1.78 15 1.5 0.053 0.240 (-0.455, 0.562) 0.22 0.827 1.78 15 3.0 0.093 0.240 (-0.415, 0.602) 0.39 0.702 1.78 temp*cyanide 5 0.16 2.067 0.339 ( 1.348, 2.786) 6.09 0.000 2.13 5 0.80 0.167 0.339 (-0.552, 0.886) 0.49 0.630 2.13 5 4.00 -0.044 0.339 (-0.763, 0.674) -0.13 0.897 2.13 5 20.00 -0.511 0.339 (-1.230, 0.208) -1.51 0.151 2.13 15 0.16 -0.560 0.339 (-1.279, 0.159) -1.65 0.118 2.13 15 0.80 0.107 0.339 (-0.612, 0.826) 0.31 0.757 2.13 15 4.00 -0.104 0.339 (-0.823, 0.614) -0.31 0.762 2.13 15 20.00 0.296 0.339 (-0.423, 1.014) 0.87 0.396 2.13 oxygen*cyanide 1.5 0.16 -2.640 0.339 (-3.359, -1.921) -7.79 0.000 2.13 1.5 0.80 0.360 0.339 (-0.359, 1.079) 1.06 0.304 2.13 1.5 4.00 0.682 0.339 (-0.037, 1.401) 2.01 0.061 2.13 1.5 20.00 1.149 0.339 ( 0.430, 1.868) 3.39 0.004 2.13 3.0 0.16 0.300 0.339 (-0.419, 1.019) 0.88 0.389 2.13 3.0 0.80 -0.600 0.339 (-1.319, 0.119) -1.77 0.096 2.13 3.0 4.00 0.022 0.339 (-0.697, 0.741) 0.07 0.949 2.13 3.0 20.00 0.089 0.339 (-0.630, 0.808) 0.26 0.797 2.13 Fits and Diagnostics for Unusual Observations Obs surv Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 21 15.800 16.860 0.646 (15.491, 18.229) -1.060 -2.21 -2.57 0.644444 0.31 26 20.700 19.520 0.646 (18.151, 20.889) 1.180 2.46 3.02 0.644444 0.38 36 12.900 11.893 0.646 (10.525, 13.262) 1.007 2.10 2.39 0.644444 0.28 41 14.200 15.293 0.646 (13.925, 16.662) -1.093 -2.28 -2.69 0.644444 0.32 Obs DFITS 21 -3.45697 R 26 4.06817 R 36 3.21447 R 41 -3.61683 R R Large residual Residual Plots for surv MTB > PPlot 'SRES'; SUBC> Normal; SUBC> Symbol; SUBC> FitD; SUBC> Grid 2; SUBC> Grid 1; SUBC> MGrid 1. Probability Plot of SRES The P-value for the Anderson-Darling test of normality is 0.014 Comments: --------- The residual plot does not look good: there is a clear, narrower shape for residuals corresponding to low fitted values: a fan shape to the right. Nor does the normality of the residuals look good, and the normality test gives a P-value below 0.05. Despite the fact that the observations are already transformed, a further transformation is suggested. We try below a Box-Cox analysis to determine a suitable transformation. It can also be noted that the ANOVA table shows two strongly significant interactions; the only non-significant term is temp*oxygen. No single residual is alarmingly large. We perform a Box-Cox analysis to get guidance on a useful transformation: MTB > GLM; SUBC> Response 'surv'; SUBC> Nodefault; SUBC> Categorical 'temp' 'oxygen' 'cyanide'; SUBC> Terms temp oxygen cyanide temp*oxygen temp*cyanide oxygen*cyanide; SUBC> Boxcox; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TFactor; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: surv versus temp, oxygen, cyanide Box-Cox transformation Rounded lambda 1 Estimated lambda 0.762848 95% CI for lambda (0.344348, 1.12135) ... Comments: --------- The surprising result is that the Box-Cox analysis does not strongly suggest a transformation. The optimal value of lambda seems to be around 0.75 (more precisely 0.76), but CIs are wide and include 1 in both Minitab and Stata (and the likelihood-ratio test for lambda=1 in Stata gives P=0.206). One possible explanation of the apparent discrepancy to our residual analysis, is that the residuals here are quite strongly correlated because they correspond to the 3-factor interaction effects. There are several possible routes to continue the analysis. One is to stick with the untransformed survival values. Another is to use the optimal value of lambda (0.75), and the third one which we will follow here is to square-root transform the survivals. This is hardly worse than using the untransformed data (0.5 is about as far from the optimal lambda as 1); however, as will be shown below, the residuals look markedly better and the ANOVA table allows for further model simplification. MTB > Name C6 'rootsurv' MTB > Let 'rootsurv' = sqrt('surv') MTB > Name C7 "SRES_1". MTB > GLM; SUBC> Response 'rootsurv'; SUBC> Nodefault; SUBC> Categorical 'temp' 'oxygen' 'cyanide'; SUBC> Terms temp oxygen cyanide temp*oxygen temp*cyanide oxygen*cyanide; SUBC> Means temp oxygen cyanide temp*oxygen temp*cyanide oxygen*cyanide; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TFactor; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK; SUBC> SResiduals 'SRES_1'. General Linear Model: rootsurv versus temp, oxygen, cyanide Factor Information Factor Type Levels Values temp Fixed 3 5, 15, 25 oxygen Fixed 3 1.5, 3.0, 9.0 cyanide Fixed 5 0.16, 0.80, 4.00, 20.00, 100.00 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value temp 2 13.5422 50.16% 13.5422 6.77108 404.53 0.000 oxygen 2 0.7783 2.88% 0.7783 0.38915 23.25 0.000 cyanide 4 11.3155 41.92% 11.3155 2.82887 169.01 0.000 temp*oxygen 4 0.0855 0.32% 0.0855 0.02138 1.28 0.320 temp*cyanide 8 0.1784 0.66% 0.1784 0.02230 1.33 0.297 oxygen*cyanide 8 0.8277 3.07% 0.8277 0.10346 6.18 0.001 Error 16 0.2678 0.99% 0.2678 0.01674 Total 44 26.9954 100.00% S R-sq R-sq(adj) PRESS R-sq(pred) 0.129377 99.01% 97.27% 2.11844 92.15% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 3.2000 0.0193 ( 3.1591, 3.2409) 165.92 0.000 temp 5 0.6800 0.0273 ( 0.6222, 0.7379) 24.93 0.000 1.33 15 -0.0166 0.0273 (-0.0745, 0.0412) -0.61 0.550 1.33 oxygen 1.5 -0.1726 0.0273 (-0.2304, -0.1148) -6.33 0.000 1.33 3.0 0.0263 0.0273 (-0.0315, 0.0841) 0.96 0.349 1.33 cyanide 0.16 0.8962 0.0386 ( 0.8144, 0.9779) 23.23 0.000 1.60 0.80 0.1229 0.0386 ( 0.0411, 0.2047) 3.19 0.006 1.60 4.00 -0.1885 0.0386 (-0.2703, -0.1067) -4.89 0.000 1.60 20.00 -0.2441 0.0386 (-0.3258, -0.1623) -6.33 0.000 1.60 temp*oxygen 5 1.5 0.0317 0.0386 (-0.0501, 0.1135) 0.82 0.423 1.78 5 3.0 0.0148 0.0386 (-0.0669, 0.0966) 0.38 0.706 1.78 15 1.5 0.0115 0.0386 (-0.0702, 0.0933) 0.30 0.769 1.78 15 3.0 0.0283 0.0386 (-0.0535, 0.1101) 0.73 0.474 1.78 temp*cyanide 5 0.16 0.1068 0.0545 (-0.0089, 0.2224) 1.96 0.068 2.13 5 0.80 0.0123 0.0545 (-0.1033, 0.1280) 0.23 0.824 2.13 5 4.00 0.0399 0.0545 (-0.0758, 0.1555) 0.73 0.475 2.13 5 20.00 -0.0170 0.0545 (-0.1326, 0.0987) -0.31 0.760 2.13 15 0.16 -0.0642 0.0545 (-0.1798, 0.0514) -1.18 0.256 2.13 15 0.80 0.0194 0.0545 (-0.0962, 0.1351) 0.36 0.727 2.13 15 4.00 -0.0199 0.0545 (-0.1355, 0.0958) -0.36 0.720 2.13 15 20.00 0.0438 0.0545 (-0.0718, 0.1595) 0.80 0.433 2.13 oxygen*cyanide 1.5 0.16 -0.3185 0.0545 (-0.4341, -0.2028) -5.84 0.000 2.13 1.5 0.80 0.0523 0.0545 (-0.0633, 0.1679) 0.96 0.352 2.13 1.5 4.00 0.0904 0.0545 (-0.0252, 0.2061) 1.66 0.117 2.13 1.5 20.00 0.1657 0.0545 ( 0.0501, 0.2813) 3.04 0.008 2.13 3.0 0.16 0.0530 0.0545 (-0.0626, 0.1686) 0.97 0.346 2.13 3.0 0.80 -0.1142 0.0545 (-0.2298, 0.0015) -2.09 0.053 2.13 3.0 4.00 0.0020 0.0545 (-0.1136, 0.1177) 0.04 0.971 2.13 3.0 20.00 0.0127 0.0545 (-0.1030, 0.1283) 0.23 0.819 2.13 Fits and Diagnostics for Unusual Observations Obs rootsurv Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 26 4.550 4.387 0.104 (4.167, 4.607) 0.162 2.11 2.40 0.644444 0.28 36 3.592 3.426 0.104 (3.206, 3.647) 0.165 2.14 2.46 0.644444 0.29 37 2.324 2.497 0.104 (2.277, 2.717) -0.173 -2.24 -2.62 0.644444 0.31 Obs DFITS 26 3.22948 R 36 3.30743 R 37 -3.52974 R R Large residual Means Fitted Term Mean SE Mean temp 5 3.8800 0.0334 15 3.1834 0.0334 25 2.5366 0.0334 oxygen 1.5 3.0274 0.0334 3.0 3.2263 0.0334 9.0 3.3463 0.0334 cyanide 0.16 4.0962 0.0431 0.80 3.3229 0.0431 4.00 3.0115 0.0431 20.00 2.9559 0.0431 100.00 2.6135 0.0431 temp*oxygen 5 1.5 3.7391 0.0579 5 3.0 3.9212 0.0579 5 9.0 3.9798 0.0579 15 1.5 3.0223 0.0579 15 3.0 3.2380 0.0579 15 9.0 3.2898 0.0579 25 1.5 2.3208 0.0579 25 3.0 2.5198 0.0579 25 9.0 2.7693 0.0579 temp*cyanide 5 0.16 4.8830 0.0747 5 0.80 4.0153 0.0747 5 4.00 3.7314 0.0747 5 20.00 3.6190 0.0747 5 100.00 3.1516 0.0747 15 0.16 4.0153 0.0747 15 0.80 3.3257 0.0747 15 4.00 2.9750 0.0747 15 20.00 2.9832 0.0747 15 100.00 2.6177 0.0747 25 0.16 3.3902 0.0747 25 0.80 2.6278 0.0747 25 4.00 2.3281 0.0747 25 20.00 2.2657 0.0747 25 100.00 2.0713 0.0747 oxygen*cyanide 1.5 0.16 3.6051 0.0747 1.5 0.80 3.2026 0.0747 1.5 4.00 2.9293 0.0747 1.5 20.00 2.9491 0.0747 1.5 100.00 2.4509 0.0747 3.0 0.16 4.1755 0.0747 3.0 0.80 3.2350 0.0747 3.0 4.00 3.0399 0.0747 3.0 20.00 2.9949 0.0747 3.0 100.00 2.6863 0.0747 9.0 0.16 4.5079 0.0747 9.0 0.80 3.5311 0.0747 9.0 4.00 3.0654 0.0747 9.0 20.00 2.9239 0.0747 9.0 100.00 2.7034 0.0747 Residual Plots for rootsurv MTB > PPlot 'SRES_1'; SUBC> Normal; SUBC> Symbol; SUBC> FitD; SUBC> Grid 2; SUBC> Grid 1; SUBC> MGrid 1. Probability Plot of SRES_1 The P-value of the Anderson-Darling test of normality is 0.477. MTB > FacPlot 'rootsurv'; SUBC> Factors temp oxygen cyanide; SUBC> GMain; SUBC> GInt; SUBC> Full. Main Effects Plot for rootsurv Interaction Plot for rootsurv Comments: --------- The residuals for the square-root transformed data look better: less/no indication of fan shape, and the points in the normal plot much closer to a straight line (and a P-value>>0.10). For this exercise we will therefore analyse the square-root transformed values. As the original values were already log-transformed (perhaps before averaging, this is not clear from the text), it might however be that another transformation should have been applied to those original values. The ANOVA table for the square-root transformed values allows for more simplification: the interaction temp*cyanide is clearly nonsignificant, only the interaction oxygen*cyanide remains significant at the 5% level. Therefore, in addition to improved compliance with model assumptions, we also get a simpler description of the data. The factor and interactions plots are visually very informative. The temperature plot seems almost perfectly linear, so it is naturally suggested to examine this further by fitting temp as a covariate (continuous predictor). Before doing that, we drop the non-significant interactions (in order to avoid nuisance effects from assuming linearity in these interactions). MTB > GLM; SUBC> Response 'rootsurv'; SUBC> Nodefault; SUBC> Categorical 'temp' 'oxygen' 'cyanide'; SUBC> Terms temp oxygen cyanide oxygen*cyanide; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TFactor; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: rootsurv versus temp, oxygen, cyanide Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value temp 2 13.5422 50.16% 13.5422 6.77108 356.53 0.000 oxygen 2 0.7783 2.88% 0.7783 0.38915 20.49 0.000 cyanide 4 11.3155 41.92% 11.3155 2.82887 148.96 0.000 oxygen*cyanide 8 0.8277 3.07% 0.8277 0.10346 5.45 0.000 Error 28 0.5318 1.97% 0.5318 0.01899 Total 44 26.9954 100.00% S R-sq R-sq(adj) PRESS R-sq(pred) 0.137809 98.03% 96.90% 1.37348 94.91% MTB > GLM; SUBC> Response 'rootsurv'; SUBC> Nodefault; SUBC> Continuous 'temp'; SUBC> Categorical 'oxygen' 'cyanide'; SUBC> Unstandardized; SUBC> Terms temp oxygen cyanide oxygen*cyanide; SUBC> Means oxygen cyanide oxygen*cyanide; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TFactor; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: rootsurv versus temp, oxygen, cyanide Factor Information Factor Type Levels Values oxygen Fixed 3 1.5, 3.0, 9.0 cyanide Fixed 5 0.16, 0.80, 4.00, 20.00, 100.00 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value temp 1 13.5359 50.14% 13.5359 13.5359 729.65 0.000 oxygen 2 0.7783 2.88% 0.7783 0.3891 20.98 0.000 cyanide 4 11.3155 41.92% 11.3155 2.8289 152.49 0.000 oxygen*cyanide 8 0.8277 3.07% 0.8277 0.1035 5.58 0.000 Error 29 0.5380 1.99% 0.5380 0.0186 Total 44 26.9954 100.00% S R-sq R-sq(adj) PRESS R-sq(pred) 0.136203 98.01% 96.98% 1.30858 95.15% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 4.2076 0.0425 ( 4.1207, 4.2944) 99.08 0.000 temp -0.06717 0.00249 (-0.07226, -0.06209) -27.01 0.000 1.00 oxygen 1.5 -0.1726 0.0287 ( -0.2313, -0.1139) -6.01 0.000 1.33 3.0 0.0263 0.0287 ( -0.0324, 0.0850) 0.92 0.367 1.33 cyanide 0.16 0.8962 0.0406 ( 0.8131, 0.9792) 22.07 0.000 1.60 0.80 0.1229 0.0406 ( 0.0398, 0.2060) 3.03 0.005 1.60 4.00 -0.1885 0.0406 ( -0.2715, -0.1054) -4.64 0.000 1.60 20.00 -0.2441 0.0406 ( -0.3271, -0.1610) -6.01 0.000 1.60 oxygen*cyanide 1.5 0.16 -0.3185 0.0574 ( -0.4359, -0.2010) -5.55 0.000 2.13 1.5 0.80 0.0523 0.0574 ( -0.0651, 0.1698) 0.91 0.370 2.13 1.5 4.00 0.0904 0.0574 ( -0.0270, 0.2079) 1.57 0.126 2.13 1.5 20.00 0.1657 0.0574 ( 0.0482, 0.2832) 2.89 0.007 2.13 3.0 0.16 0.0530 0.0574 ( -0.0645, 0.1705) 0.92 0.364 2.13 3.0 0.80 -0.1142 0.0574 ( -0.2316, 0.0033) -1.99 0.056 2.13 3.0 4.00 0.0020 0.0574 ( -0.1154, 0.1195) 0.04 0.972 2.13 3.0 20.00 0.0127 0.0574 ( -0.1048, 0.1301) 0.22 0.827 2.13 Fits and Diagnostics for Unusual Observations Obs rootsurv Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 15 3.1464 3.3751 0.0825 (3.2064, 3.5437) -0.2286 -2.11 -2.25 0.366667 0.16 37 2.3238 2.5633 0.0825 (2.3947, 2.7320) -0.2395 -2.21 -2.38 0.366667 0.18 45 2.2804 2.0316 0.0825 (1.8630, 2.2003) 0.2487 2.29 2.49 0.366667 0.19 Obs DFITS 15 -1.71402 R 37 -1.81190 R 45 1.89624 R R Large residual Means Fitted Term Mean SE Mean oxygen 1.5 3.0274 0.0352 3.0 3.2263 0.0352 9.0 3.3463 0.0352 cyanide 0.16 4.0962 0.0454 0.80 3.3229 0.0454 4.00 3.0115 0.0454 20.00 2.9559 0.0454 100.00 2.6135 0.0454 oxygen*cyanide 1.5 0.16 3.6051 0.0786 1.5 0.80 3.2026 0.0786 1.5 4.00 2.9293 0.0786 1.5 20.00 2.9491 0.0786 1.5 100.00 2.4509 0.0786 3.0 0.16 4.1755 0.0786 3.0 0.80 3.2350 0.0786 3.0 4.00 3.0399 0.0786 3.0 20.00 2.9949 0.0786 3.0 100.00 2.6863 0.0786 9.0 0.16 4.5079 0.0786 9.0 0.80 3.5311 0.0786 9.0 4.00 3.0654 0.0786 9.0 20.00 2.9239 0.0786 9.0 100.00 2.7034 0.0786 Covariate Data Mean StDev temp 15.00 8.26 Residual Plots for rootsurv MTB > FacPlot 'rootsurv'; SUBC> Factors oxygen cyanide; SUBC> GInt; SUBC> Full. Interaction Plot for rootsurv Comments: --------- The F-test for linearity of the effect of temperature is F = [0.5380-0.5318]/[29-28] / 0.0190 = 0.33, which is far from significant in F(1,28). The estimated regression coefficient shows the rootsurv values to decrease 0.067 units per temperature increase. As the next step, we examine whether the effect of oxygen could be modelled as linear as well, as might seem plausible from the interaction plot. However, as the following analysis shows, the effect of oxygen does not seem linear when the actual concentrations are taken into account. MTB > GLM; SUBC> Response 'rootsurv'; SUBC> Nodefault; SUBC> Continuous 'temp' 'oxygen'; SUBC> Categorical 'cyanide'; SUBC> Unstandardized; SUBC> Terms temp oxygen cyanide oxygen*cyanide; SUBC> Means cyanide; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TFactor; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: rootsurv versus temp, oxygen, cyanide Factor Information Factor Type Levels Values cyanide Fixed 5 0.16, 0.80, 4.00, 20.00, 100.00 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value temp 1 13.5359 50.14% 13.5359 13.5359 517.50 0.000 oxygen 1 0.6153 2.28% 0.6153 0.6153 23.52 0.000 cyanide 4 11.3155 41.92% 2.0450 0.5113 19.55 0.000 oxygen*cyanide 4 0.6394 2.37% 0.6394 0.1598 6.11 0.001 Error 34 0.8893 3.29% 0.8893 0.0262 Total 44 26.9954 100.00% S R-sq R-sq(adj) PRESS R-sq(pred) 0.161729 96.71% 95.74% 1.56502 94.20% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 4.0452 0.0605 ( 3.9222, 4.1682) 66.83 0.000 temp -0.06717 0.00295 (-0.07317, -0.06117) -22.75 0.000 1.00 oxygen 0.03609 0.00744 ( 0.02097, 0.05121) 4.85 0.000 1.00 cyanide 0.16 0.6004 0.0825 ( 0.4327, 0.7681) 7.28 0.000 4.69 0.80 0.0811 0.0825 ( -0.0866, 0.2488) 0.98 0.333 4.69 4.00 -0.0899 0.0825 ( -0.2576, 0.0778) -1.09 0.284 4.69 20.00 -0.0557 0.0825 ( -0.2234, 0.1120) -0.67 0.505 4.69 oxygen*cyanide 0.16 0.0657 0.0149 ( 0.0355, 0.0960) 4.42 0.000 4.69 0.80 0.0093 0.0149 ( -0.0209, 0.0395) 0.62 0.537 4.69 4.00 -0.0219 0.0149 ( -0.0522, 0.0083) -1.47 0.150 4.69 20.00 -0.0419 0.0149 ( -0.0721, -0.0116) -2.81 0.008 4.69 Fits and Diagnostics for Unusual Observations Obs rootsurv Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 6 4.9598 4.6152 0.0663 (4.4803, 4.7500) 0.3447 2.34 2.51 0.168254 0.10 31 2.8107 3.1190 0.0792 (2.9581, 3.2799) -0.3083 -2.19 -2.32 0.239683 0.14 36 3.5917 3.2717 0.0663 (3.1369, 3.4066) 0.3199 2.17 2.30 0.168254 0.09 Obs DFITS 6 1.13018 R 31 -1.30457 R 36 1.03537 R R Large residual Means Fitted Term Mean SE Mean cyanide 0.16 4.0962 0.0539 0.80 3.3229 0.0539 4.00 3.0115 0.0539 20.00 2.9559 0.0539 100.00 2.6135 0.0539 Covariate Data Mean StDev temp 15.00 8.26 oxygen 4.50 3.28 Comments: --------- The F-test for linearity of the effect of oxygen is F = [0.8893-0.5380]/[34-29] / 0.0186 = 3.78 which corresponds to P=0.06 in F(1,29). There is some evidence of non-linearity in the effect of oxygen, and we take this as a reason to not explore linear effects of oxygen further. Thus, the best way to represent the interaction between oxygen and cyanide, as well as their main effects, is by the interaction plot, where each estimate has a standard error of 0.079 (as shown above). These estimates can be backtransformed (squaring them). Overall, higher oxygen is associated with increased survival, while - not surprisingly - survival decreases with increasing levels of cyanide. If cyanide as kept as categorical, the interaction cyanide*oxygen is best represented by the interaction plot, a table of means with standard errors and suitable pairwise comparisons among these means. Minitab offers Bonferroni-adjusted comparisons for all comparisons among the 3*5=15 means - very large listing and a very strong correction (because there are 15*14/2=105 comparisons). It is not clear that all these comparisons are of (same) interest. Sometimes one would want to limit the comparisons within an interaction to comparisons with one factor kept constant. In this case, this corresponds to: - oxygen constant: 3*(5*4/2) = 30 comparisons - cyanide constant: 5*(3*2/2) = 15 comparisons These 45 comparisons comparisons could be considered among all the comparisons in the (full) listing, but the Bonferroni adjustment must then be redone with 45 instead of 105 comparisons. That is, the P-values based on the full adjustment must be multiplied by 45/105.