Solution file for additional exercise 10.9 ------------------------------------------ Data on visual acuity (response time to see a flash light) which was measured for the left and right eye of 7 persons, and with 4 different lenses of different powers placed before the eyes. - notation: y_ijk = response time for person i and eye j with lens of power k, i = 1,2,3,4,5,6,7 (persons), j = 1,2 (side: left, right), k = 1,2,3,4 (powers: 6/6, 6/18, 6/36, 6/60), - repeated measures data with 4 measurements on each eye (order unknown), - may be viewed as a split-plot design with * eyes = whole plots, * side = whole plot factor, * measurements = subplots, * power = subplot factor, * persons = blocks, - model: y_ijk = mu + A_i + beta_j + (AB)_ij + gamma_k + (alpha gamma)_jk + eps_ijk, where A_I's are assumed i.i.d. N(0,sigma^2), where AB_ij's are assumed i.i.d. N(0,sigma_AB^2), and where eps_ijk's are assumed i.i.d. N(0,sigma^2), we take here person effects as random because there could be some interest in a variation between persons. MTB > WOpen "H:\VHM\VHM802\Data_csv\hs10_9.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\hs10_9.csv’ Worksheet was saved on 03/03/2011 MTB > Code ("6|18") 2 ("6|36") 3 ("6|6") 1 ("6|60") 4 'power'; SUBC> TSummary; SUBC> After. Code Summary Original Recoded Number Value Value of Rows 6|18 2 14 6|36 3 14 6|6 1 14 6|60 4 14 Source data column power Recoded data column Coded power MTB > Plot 'acuity'*'Coded power'; SUBC> Symbol 'person'; SUBC> Connect 'person'; SUBC> Panel 'eye'. Scatterplot of acuity vs Coded power Comments: --------- In order to construct a profile plot, we had to assign numerical values (1-4) to the powers. The plot shows fairly similar and noisy curves for all persons, except 6, with no obvious differences between powers or eyes. Person 6 has a single very low value on the left eye, as well as low values at all powers on the right eye. MTB > GLM; SUBC> Response 'acuity'; SUBC> Nodefault; SUBC> Categorical 'person' 'eye' 'power'; SUBC> Random person; SUBC> Terms person eye power person*eye eye*power; SUBC> Means eye power eye*power; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TEMS; SUBC> TVariance; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: acuity versus person, eye, power Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values person Random 7 1, 2, 3, 4, 5, 6, 7 eye Fixed 2 left, right power Fixed 4 6|18, 6|36, 6|6, 6|60 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value person 6 1379.43 53.64% 1379.43 229.90 3.86 0.062 eye 1 46.45 1.81% 46.45 46.45 0.78 0.411 power 3 140.77 5.47% 140.77 46.92 2.78 0.055 person*eye 6 357.43 13.90% 357.43 59.57 3.53 0.007 eye*power 3 40.63 1.58% 40.63 13.54 0.80 0.500 Error 36 606.86 23.60% 606.86 16.86 Total 55 2571.55 100.00% Model Summary S R-sq R-sq(adj) PRESS R-sq(pred) 4.10575 76.40% 63.95% 1468.44 42.90% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 113.411 0.549 (112.298, 114.523) 206.71 0.000 person 1 5.09 1.34 ( 2.36, 7.81) 3.79 0.001 * 2 -2.54 1.34 ( -5.26, 0.19) -1.89 0.067 * 3 6.46 1.34 ( 3.74, 9.19) 4.81 0.000 * 4 1.84 1.34 ( -0.89, 4.56) 1.37 0.180 * 5 1.34 1.34 ( -1.39, 4.06) 1.00 0.326 * 6 -9.16 1.34 ( -11.89, -6.44) -6.82 0.000 * eye left 0.911 0.549 ( -0.202, 2.023) 1.66 0.106 1.00 power 6|18 0.018 0.950 ( -1.909, 1.945) 0.02 0.985 1.50 6|36 -1.768 0.950 ( -3.695, 0.159) -1.86 0.071 1.50 6|6 -0.768 0.950 ( -2.695, 1.159) -0.81 0.424 1.50 person*eye 1 left -0.66 1.34 ( -3.39, 2.06) -0.49 0.626 * 2 left 0.46 1.34 ( -2.26, 3.19) 0.35 0.732 * 3 left -2.04 1.34 ( -4.76, 0.69) -1.51 0.139 * 4 left -2.16 1.34 ( -4.89, 0.56) -1.61 0.117 * 5 left -0.91 1.34 ( -3.64, 1.81) -0.68 0.502 * 6 left 5.84 1.34 ( 3.11, 8.56) 4.34 0.000 * eye*power left 6|18 0.232 0.950 ( -1.695, 2.159) 0.24 0.808 1.50 left 6|36 -1.411 0.950 ( -3.338, 0.517) -1.48 0.146 1.50 left 6|6 0.304 0.950 ( -1.624, 2.231) 0.32 0.751 1.50 Regression Equation acuity = 113.411 + 5.09 person_1 - 2.54 person_2 + 6.46 person_3 + 1.84 person_4 + 1.34 person_5 - 9.16 person_6 - 3.04 person_7 + 0.911 eye_left - 0.911 eye_right + 0.018 power_6|18 - 1.768 power_6|36 - 0.768 power_6|6 + 2.518 power_6|60 - 0.66 person*eye_1 left + 0.66 person*eye_1 right + 0.46 person*eye_2 left - 0.46 person*eye_2 right - 2.04 person*eye_3 left + 2.04 person*eye_3 right - 2.16 person*eye_4 left + 2.16 person*eye_4 right - 0.91 person*eye_5 left + 0.91 person*eye_5 right + 5.84 person*eye_6 left - 5.84 person*eye_6 right - 0.54 person*eye_7 left + 0.54 person*eye_7 right + 0.232 eye*power_left 6|18 - 1.411 eye*power_left 6|36 + 0.304 eye*power_left 6|6 + 0.875 eye*power_left 6|60 - 0.232 eye*power_right 6|18 + 1.411 eye*power_right 6|36 - 0.304 eye*power_right 6|6 - 0.875 eye*power_right 6|60 Equation treats random terms as though they are fixed. Fits and Diagnostics for Unusual Observations Obs acuity Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 41 119.00 110.54 2.45 (105.56, 115.51) 8.46 2.57 2.81 0.357143 0.18 43 94.00 107.82 2.45 (102.85, 112.80) -13.82 -4.20 -5.80 0.357143 0.49 Obs DFITS 41 2.09145 R 43 -4.31942 R R Large residual Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 person (6) + 4.0000 (4) + 8.0000 (1) 2 eye (6) + 4.0000 (4) + Q[2, 5] 3 power (6) + Q[3, 5] 4 person*eye (6) + 4.0000 (4) 5 eye*power (6) + Q[5] 6 Error (6) Means Term Fitted Mean eye left 114.321 right 112.500 power 6|18 113.429 6|36 111.643 6|6 112.643 6|60 115.929 eye*power left 6|18 114.571 left 6|36 111.143 left 6|6 113.857 left 6|60 117.714 right 6|18 112.286 right 6|36 112.143 right 6|6 111.429 right 6|60 114.143 Variance Components, using Adjusted SS Source Variance % of Total StDev % of Total person 21.2917 43.61% 4.61429 66.03% person*eye 10.6786 21.87% 3.26781 46.77% Error 16.8571 34.52% 4.10575 58.76% Total 48.8274 6.98766 Residual Plots for acuity MTB > FacPlot 'acuity'; SUBC> Factors eye power; SUBC> GInt; SUBC> Full. Interaction Plot for acuity Comments: --------- The analysis shows a reasonably good looking residual plot, except for one extreme outlier: patient 6, left eye, power 6/36. This value clearly sticks out as different by just looking at the data, and the standardized and deletion residuals are -4.20 and -5.80, respectively. The P-value for the usual outlier test is therefore: P=2*P(t(35)<-5.80)*56 = 0.0001. We decide to remove this value. MTB > Copy 'acuity' c5; SUBC> Varnames. MTB > let c5(43)='*' MTB > GLM; SUBC> Response 'acuity_1'; SUBC> Nodefault; SUBC> Categorical 'person' 'eye' 'power'; SUBC> Random person; SUBC> Terms person eye power person*eye eye*power; SUBC> Means eye power eye*power; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TEMS; SUBC> TVariance; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: acuity_1 versus person, eye, power Method Factor coding (-1, 0, +1) Rows unused 1 Factor Information Factor Type Levels Values person Random 7 1, 2, 3, 4, 5, 6, 7 eye Fixed 2 left, right power Fixed 4 6|18, 6|36, 6|6, 6|60 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value person 6 1115.87 51.00% 972.351 162.058 1.71 0.266 eye 1 71.70 3.28% 91.325 91.325 0.96 0.364 x power 3 105.24 4.81% 88.968 29.656 3.35 0.030 person*eye 6 580.79 26.55% 569.636 94.939 10.73 0.000 eye*power 3 4.63 0.21% 4.631 1.544 0.17 0.913 Error 35 309.70 14.15% 309.696 8.848 Total 54 2187.93 100.00% x Not an exact F-test. S R-sq R-sq(adj) PRESS R-sq(pred) 2.97464 85.85% 78.16% 772.007 64.72% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 113.795 0.403 (112.977, 114.613) 282.38 0.000 person 1 4.705 0.976 ( 2.724, 6.687) 4.82 0.000 * 2 -2.920 0.976 ( -4.901, -0.938) -2.99 0.005 * 3 6.080 0.976 ( 4.099, 8.062) 6.23 0.000 * 4 1.455 0.976 ( -0.526, 3.437) 1.49 0.145 * 5 0.955 0.976 ( -1.026, 2.937) 0.98 0.334 * 6 -6.86 1.05 ( -8.99, -4.72) -6.52 0.000 * eye left 1.295 0.403 ( 0.477, 2.113) 3.21 0.003 1.01 power 6|18 -0.366 0.692 ( -1.770, 1.038) -0.53 0.600 1.51 6|36 -0.616 0.717 ( -2.071, 0.839) -0.86 0.396 1.57 6|6 -1.152 0.692 ( -2.556, 0.252) -1.67 0.105 1.51 person*eye 1 left -1.045 0.976 ( -3.026, 0.937) -1.07 0.292 * 2 left 0.080 0.976 ( -1.901, 2.062) 0.08 0.935 * 3 left -2.420 0.976 ( -4.401, -0.438) -2.48 0.018 * 4 left -2.545 0.976 ( -4.526, -0.563) -2.61 0.013 * 5 left -1.295 0.976 ( -3.276, 0.687) -1.33 0.193 * 6 left 8.14 1.05 ( 6.01, 10.28) 7.74 0.000 * eye*power left 6|18 -0.152 0.692 ( -1.556, 1.252) -0.22 0.828 1.51 left 6|36 -0.259 0.717 ( -1.714, 1.196) -0.36 0.720 1.57 left 6|6 -0.080 0.692 ( -1.485, 1.324) -0.12 0.908 1.51 Regression Equation acuity_1 = 113.795 + 4.705 person_1 - 2.920 person_2 + 6.080 person_3 + 1.455 person_4 + 0.955 person_5 - 6.86 person_6 - 3.420 person_7 + 1.295 eye_left - 1.295 eye_right - 0.366 power_6|18 - 0.616 power_6|36 - 1.152 power_6|6 + 2.134 power_6|60 - 1.045 person*eye_1 left + 1.045 person*eye_1 right + 0.080 person*eye_2 left - 0.080 person*eye_2 right - 2.420 person*eye_3 left + 2.420 person*eye_3 right - 2.545 person*eye_4 left + 2.545 person*eye_4 right - 1.295 person*eye_5 left + 1.295 person*eye_5 right + 8.14 person*eye_6 left - 8.14 person*eye_6 right - 0.920 person*eye_7 left + 0.920 person*eye_7 right - 0.152 eye*power_left 6|18 - 0.259 eye*power_left 6|36 - 0.080 eye*power_left 6|6 + 0.491 eye*power_left 6|60 + 0.152 eye*power_right 6|18 + 0.259 eye*power_right 6|36 + 0.080 eye*power_right 6|6 - 0.491 eye*power_right 6|60 Equation treats random terms as though they are fixed. Fits and Diagnostics for Unusual Observations Obs acuity_1 Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 51 105.00 109.88 1.82 (106.18, 113.57) -4.88 -2.07 -2.18 0.375000 0.13 54 105.00 109.79 1.78 (106.18, 113.39) -4.79 -2.01 -2.10 0.357143 0.11 55 115.00 109.64 1.78 (106.03, 113.25) 5.36 2.25 2.39 0.357143 0.14 Obs DFITS 51 -1.68977 R 54 -1.56698 R 55 1.78366 R R Large residual Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 person (6) + 3.9048 (4) + 7.8095 (1) 2 eye (6) + 3.8919 (4) + Q[2, 5] 3 power (6) + Q[3, 5] 4 person*eye (6) + 3.9048 (4) 5 eye*power (6) + Q[5] 6 Error (6) Means Term Fitted Mean eye left 115.089 right 112.500 power 6|18 113.429 6|36 113.179 6|6 112.643 6|60 115.929 eye*power left 6|18 114.571 left 6|36 114.214 left 6|6 113.857 left 6|60 117.714 right 6|18 112.286 right 6|36 112.143 right 6|6 111.429 right 6|60 114.143 Variance Components, using Adjusted SS Source Variance % of Total StDev % of Total person 8.59451 21.76% 2.93164 46.65% person*eye 22.0477 55.83% 4.69550 74.72% Error 8.84847 22.41% 2.97464 47.34% Total 39.4907 6.28416 Residual Plots for acuity_1 MTB > FacPlot 'acuity_1'; SUBC> Factors eye power; SUBC> GInt; SUBC> Full. Interaction Plot for acuity_1 Comments: --------- The changes in the ANOVA table and the estimated variance components are substantial. The distribution of variance on the three random effects is shifted completely, and not surprisingly has the error variance gone down to about half of its previous value. The R^2 has increased from 76% to 86%. The interaction of the reduced dataset shows almost completely parallel curves, but in the original analysis there was a clear deviation from parallel curves for eye=left and power=6|36. The design without this one value becomes unbalanced, and the ANOVA-based methods only gives approximate inference for testing the eye effect. In practice, this small deviation from a balanced design is not critical. The ANOVA table shows only a significant effect of power, and it is rather weak with P=0.03. The estimated variance components show the variation between eyes to be of major importance. The residuals look nice. The least squares means for the 4 powers indicate 6/60 to give higher response times than the others. It might be tempting to create a contrast for this particular comparison. Because of the unbalancedness, the explicit formulae in the GO textbook for calculations involving contrasts will no longer work. Therefore, it is natural to use statistical software that can manage mixed models (Stata, SAS, R) to get inference for such a contrast, see Stata and SAS programs. Unless the Scheffe method is available in the software to adjust for contrasts suggested by the data, the inference will however either be too liberal or need an assumption that the contrast was of interest prior to the analysis. Note that in the balanced design without removing the extreme observation, the methods from GO would still have worked here because the contrast will be assessed entirely against the within-subject (or error) variation (by the guidelines for inference in a split-plot design). Additional note: ---------------- Crowder & Hand (1990): Analysis of Repeated Measures, give a different analysis of these data. First, it is not analyzed as a split-plot design but by a more complex random effects model with an additional person*power random effect. This leads to a different ANOVA table and changes the residuals of the model, so that obs. 43 has no longer an extreme residual. Therefore, a different analysis results, and in this analysis there is NO effect of power (or of anything else). One may ask which analysis is then the correct one. A residual analysis of the estimated variance components using proc Mixed in SAS shows that there are now 3 sets of moderately extreme residuals for obs. 43: for error residual, for person*eye and for person*power. To keep this observation in the model still seems questionable, but it has become more difficult to detect. If this observation is removed, the F-test for power become F=3.40 with a P-value of 0.039. Therefore, it seems that the conclusion of the split-plot model holds also for the more elaborate model in Crowder & Hand, but that the conclusion (both our and theirs) hinge on a single observation. Below the Minitab analyses with ANOVA tables for the model with an additional random effect of person*power, both when excluding and including obs. 43. Note also the considerable differences in the estimates for the variance components. MTB > GLM; SUBC> Response 'acuity'; SUBC> Nodefault; SUBC> Categorical 'person' 'eye' 'power'; SUBC> Random person; SUBC> Terms person eye power person*eye person*power eye*power; SUBC> Means eye power eye*power; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TEMS; SUBC> TVariance; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: acuity versus person, eye, power Factor Information Factor Type Levels Values person Random 7 1, 2, 3, 4, 5, 6, 7 eye Fixed 2 left, right power Fixed 4 6|18, 6|36, 6|6, 6|60 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value person 6 1379.43 53.64% 1379.43 229.90 3.40 0.064 x eye 1 46.45 1.81% 46.45 46.45 0.78 0.411 power 3 140.77 5.47% 140.77 46.92 2.25 0.118 person*eye 6 357.43 13.90% 357.43 59.57 4.64 0.005 person*power 18 375.86 14.62% 375.86 20.88 1.63 0.155 eye*power 3 40.63 1.58% 40.63 13.54 1.06 0.393 Error 18 231.00 8.98% 231.00 12.83 Total 55 2571.55 100.00% x Not an exact F-test. S R-sq R-sq(adj) PRESS R-sq(pred) 3.58236 91.02% 72.55% 2235.85 13.05% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 113.411 0.479 (112.405, 114.416) 236.91 0.000 person 1 5.09 1.17 ( 2.63, 7.55) 4.34 0.000 * 2 -2.54 1.17 ( -5.00, -0.07) -2.16 0.044 * 3 6.46 1.17 ( 4.00, 8.93) 5.51 0.000 * 4 1.84 1.17 ( -0.62, 4.30) 1.57 0.134 * 5 1.34 1.17 ( -1.12, 3.80) 1.14 0.268 * 6 -9.16 1.17 ( -11.62, -6.70) -7.81 0.000 * eye left 0.911 0.479 ( -0.095, 1.916) 1.90 0.073 1.00 power 6|18 0.018 0.829 ( -1.724, 1.760) 0.02 0.983 1.50 6|36 -1.768 0.829 ( -3.510, -0.026) -2.13 0.047 1.50 6|6 -0.768 0.829 ( -2.510, 0.974) -0.93 0.367 1.50 person*eye 1 left -0.66 1.17 ( -3.12, 1.80) -0.56 0.580 * 2 left 0.46 1.17 ( -2.00, 2.93) 0.40 0.697 * 3 left -2.04 1.17 ( -4.50, 0.43) -1.74 0.100 * 4 left -2.16 1.17 ( -4.62, 0.30) -1.84 0.082 * 5 left -0.91 1.17 ( -3.37, 1.55) -0.78 0.447 * 6 left 5.84 1.17 ( 3.38, 8.30) 4.98 0.000 * person*power 1 6|18 -0.52 2.03 ( -4.78, 3.75) -0.25 0.802 * 1 6|36 -1.73 2.03 ( -6.00, 2.53) -0.85 0.405 * 1 6|6 0.27 2.03 ( -4.00, 4.53) 0.13 0.897 * 2 6|18 0.11 2.03 ( -4.16, 4.37) 0.05 0.959 * 2 6|36 2.89 2.03 ( -1.37, 7.16) 1.42 0.171 * 2 6|6 -2.11 2.03 ( -6.37, 2.16) -1.04 0.313 * 3 6|18 -0.89 2.03 ( -5.16, 3.37) -0.44 0.665 * 3 6|36 1.89 2.03 ( -2.37, 6.16) 0.93 0.364 * 3 6|6 -0.61 2.03 ( -4.87, 3.66) -0.30 0.768 * 4 6|18 0.73 2.03 ( -3.53, 5.00) 0.36 0.723 * 4 6|36 2.02 2.03 ( -2.25, 6.28) 0.99 0.334 * 4 6|6 -0.98 2.03 ( -5.25, 3.28) -0.48 0.635 * 5 6|18 0.73 2.03 ( -3.53, 5.00) 0.36 0.723 * 5 6|36 2.02 2.03 ( -2.25, 6.28) 0.99 0.334 * 5 6|6 -0.48 2.03 ( -4.75, 3.78) -0.24 0.815 * 6 6|18 2.73 2.03 ( -1.53, 7.00) 1.35 0.195 * 6 6|36 -8.48 2.03 ( -12.75, -4.22) -4.18 0.001 * 6 6|6 6.02 2.03 ( 1.75, 10.28) 2.96 0.008 * eye*power left 6|18 0.232 0.829 ( -1.510, 1.974) 0.28 0.783 1.50 left 6|36 -1.411 0.829 ( -3.153, 0.331) -1.70 0.106 1.50 left 6|6 0.304 0.829 ( -1.438, 2.046) 0.37 0.719 1.50 Regression Equation ... Equation treats random terms as though they are fixed. Fits and Diagnostics for Unusual Observations Obs acuity Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 43 94.00 99.34 2.95 (93.14, 105.54) -5.34 -2.63 -3.25 0.678571 0.38 47 94.00 88.66 2.95 (82.46, 94.86) 5.34 2.63 3.25 0.678571 0.38 Obs DFITS 43 -4.72939 R 47 4.72939 R R Large residual Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 person (7) + 2.0000 (5) + 4.0000 (4) + 8.0000 (1) 2 eye (7) + 4.0000 (4) + Q[2, 6] 3 power (7) + 2.0000 (5) + Q[3, 6] 4 person*eye (7) + 4.0000 (4) 5 person*power (7) + 2.0000 (5) 6 eye*power (7) + Q[6] 7 Error (7) Error Terms for Tests, using Adjusted SS Synthesis of Source Error DF Error MS Error MS 1 person 7.32 67.6190 (4) + (5) - (7) 2 eye 6.00 59.5714 (4) 3 power 18.00 20.8810 (5) 4 person*eye 18.00 12.8333 (7) 5 person*power 18.00 12.8333 (7) 6 eye*power 18.00 12.8333 (7) Means Term Fitted Mean eye left 114.321 right 112.500 power 6|18 113.429 6|36 111.643 6|6 112.643 6|60 115.929 eye*power left 6|18 114.571 left 6|36 111.143 left 6|6 113.857 left 6|60 117.714 right 6|18 112.286 right 6|36 112.143 right 6|6 111.429 right 6|60 114.143 Variance Components, using Adjusted SS Source Variance % of Total StDev % of Total person 20.2857 41.55% 4.50397 64.46% person*eye 11.6845 23.93% 3.41826 48.92% person*power 4.02381 8.24% 2.00594 28.71% Error 12.8333 26.28% 3.58236 51.27% Total 48.8274 6.98766 Residual Plots for acuity MTB > GLM; SUBC> Response 'acuity_1'; SUBC> Nodefault; SUBC> Categorical 'person' 'eye' 'power'; SUBC> Random person; SUBC> Terms person eye power person*eye person*power eye*power; SUBC> Means eye power eye*power; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TEMS; SUBC> TVariance; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: acuity_1 versus person, eye, power Method Factor coding (-1, 0, +1) Rows unused 1 Factor Information Factor Type Levels Values person Random 7 1, 2, 3, 4, 5, 6, 7 eye Fixed 2 left, right power Fixed 4 6|18, 6|36, 6|6, 6|60 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value person 6 1115.87 51.00% 977.537 162.923 2.17 0.183 x eye 1 71.70 3.28% 77.333 77.333 1.05 0.344 x power 3 105.24 4.81% 94.762 31.587 3.40 0.039 x person*eye 6 580.79 26.55% 445.537 74.256 8.87 0.000 person*power 18 164.20 7.50% 167.388 9.299 1.11 0.416 eye*power 3 7.82 0.36% 7.816 2.605 0.31 0.817 Error 17 142.31 6.50% 142.309 8.371 Total 54 2187.93 100.00% x Not an exact F-test. S R-sq R-sq(adj) PRESS R-sq(pred) 2.89328 93.50% 79.34% * * Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 113.707 0.397 (112.869, 114.545) 286.25 0.000 person 1 4.793 0.951 ( 2.785, 6.800) 5.04 0.000 * 2 -2.832 0.951 ( -4.840, -0.825) -2.98 0.008 * 3 6.168 0.951 ( 4.160, 8.175) 6.48 0.000 * 4 1.543 0.951 ( -0.465, 3.550) 1.62 0.123 * 5 1.043 0.951 ( -0.965, 3.050) 1.10 0.288 * 6 -7.38 1.09 ( -9.69, -5.07) -6.75 0.000 * eye left 1.207 0.397 ( 0.369, 2.045) 3.04 0.007 1.04 power 6|18 -0.279 0.676 ( -1.705, 1.147) -0.41 0.685 1.53 6|36 -0.878 0.723 ( -2.404, 0.648) -1.21 0.241 1.69 6|6 -1.064 0.676 ( -2.490, 0.361) -1.58 0.134 1.53 person*eye 1 left -0.957 0.951 ( -2.965, 1.050) -1.01 0.328 * 2 left 0.168 0.951 ( -1.840, 2.175) 0.18 0.862 * 3 left -2.332 0.951 ( -4.340, -0.325) -2.45 0.025 * 4 left -2.457 0.951 ( -4.465, -0.450) -2.58 0.019 * 5 left -1.207 0.951 ( -3.215, 0.800) -1.27 0.222 * 6 left 7.62 1.09 ( 5.31, 9.93) 6.97 0.000 * person*power 1 6|18 -0.22 1.64 ( -3.69, 3.24) -0.13 0.894 * 1 6|36 -2.62 1.66 ( -6.13, 0.89) -1.58 0.133 * 1 6|6 0.56 1.64 ( -2.90, 4.03) 0.34 0.735 * 2 6|18 0.40 1.64 ( -3.06, 3.87) 0.25 0.809 * 2 6|36 2.00 1.66 ( -1.51, 5.51) 1.20 0.245 * 2 6|6 -1.81 1.64 ( -5.28, 1.66) -1.10 0.286 * 3 6|18 -0.60 1.64 ( -4.06, 2.87) -0.36 0.721 * 3 6|36 1.00 1.66 ( -2.51, 4.51) 0.60 0.554 * 3 6|6 -0.31 1.64 ( -3.78, 3.16) -0.19 0.852 * 4 6|18 1.03 1.64 ( -2.44, 4.49) 0.63 0.539 * 4 6|36 1.13 1.66 ( -2.38, 4.64) 0.68 0.507 * 4 6|6 -0.69 1.64 ( -4.15, 2.78) -0.42 0.682 * 5 6|18 1.03 1.64 ( -2.44, 4.49) 0.63 0.539 * 5 6|36 1.13 1.66 ( -2.38, 4.64) 0.68 0.507 * 5 6|6 -0.19 1.64 ( -3.65, 3.28) -0.11 0.911 * 6 6|18 0.95 1.73 ( -2.70, 4.60) 0.55 0.589 * 6 6|36 -3.14 2.32 ( -8.04, 1.75) -1.35 0.193 * 6 6|6 4.24 1.73 ( 0.59, 7.89) 2.45 0.025 * eye*power left 6|18 -0.064 0.676 ( -1.490, 1.361) -0.10 0.925 1.53 left 6|36 -0.521 0.723 ( -2.047, 1.005) -0.72 0.481 1.69 left 6|6 0.007 0.676 ( -1.419, 1.433) 0.01 0.992 1.53 Regression Equation ... Equation treats random terms as though they are fixed. Fits and Diagnostics for Unusual Observations Obs acuity_1 Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 47 94.00 94.00 2.89 ( 87.90, 100.10) -0.00 * * 1.0000 * 51 105.00 109.85 2.40 (104.79, 114.92) -4.85 -3.00 -4.25 0.6875 0.52 55 115.00 110.15 2.40 (105.08, 115.21) 4.85 3.00 4.25 0.6875 0.52 Obs DFITS 47 * X 51 -6.29832 R 55 6.29832 R R Large residual X Unusual X Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 person (7) + 1.9167 (5) + 3.8333 (4) + 7.6667 (1) 2 eye (7) + 3.7895 (4) + Q[2, 6] 3 power (7) + 1.9048 (5) + Q[3, 6] 4 person*eye (7) + 3.8333 (4) 5 person*power (7) + 1.9444 (5) 6 eye*power (7) + Q[6] 7 Error (7) Error Terms for Tests, using Adjusted SS Source Error DF Error MS Synthesis of Error MS 1 person 6.09 75.1711 (4) + 0.9857 (5) - 0.9857 (7) 2 eye 6.02 73.5023 0.9886 (4) + 0.0114 (7) 3 power 18.67 9.2804 0.9796 (5) + 0.0204 (7) 4 person*eye 17.00 8.3711 (7) 5 person*power 17.00 8.3711 (7) 6 eye*power 17.00 8.3711 (7) Means Term Fitted Mean eye left 114.915 right 112.500 power 6|18 113.429 6|36 112.829 6|6 112.643 6|60 115.929 eye*power left 6|18 114.571 left 6|36 113.516 left 6|6 113.857 left 6|60 117.714 right 6|18 112.286 right 6|36 112.143 right 6|6 111.429 right 6|60 114.143 Variance Components, using Adjusted SS Source Variance % of Total StDev % of Total person 11.4459 30.54% 3.38317 55.26% person*eye 17.1874 45.86% 4.14577 67.72% person*power 0.477379 1.27% 0.69093 11.29% Error 8.37109 22.33% 2.89328 47.26% Total 37.4817 6.12223 Residual Plots for acuity_1