Solution file for additional exercise 10.11 ------------------------------------------- Data on dietary regimes for 12 hospital patients, for which plasma ascorbic acid was measured at weeks 1,2 (pre-treatment), 6,10,14 (treatment), and 15,16 (post-treatment). - notation: y_ij = measurement for patient i at week j, i = 1,...,12 (patient), j = 1,2,3,4,5,6,7 (week 1,2,6,10,14,15,16), - repeated measures data with 7 measurements on each patient, - may also be viewed as a block design with * weeks = "treatments" * patient = blocks, however that would assume equal correlation between all pairs of measurements for each patient, which probably is quite unrealistic with a series of this length, - model: y_ij = mu + alpha_i + beta_j + eps_ij, where eps_ij's are assumed i.i.d. N(0,sigma^2), we could take person effects as random if there was interest in a variation between persons. Note that this will not change the inference for weeks, except for standard errors and CIs for week means/margins (which in a random effects model would include the between-person variability). Effects of interest: comparison of pre- and post-treatments levels with treatment levels. MTB > WOpen "H:\VHM\VHM802\Data_csv\hs10_11.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_11.csv’ Worksheet was saved on 26/03/2016 MTB > Plot 'placid'*'week'; SUBC> Symbol 'patient'; SUBC> Connect 'patient'. Scatterplot of placid vs week Comments: --------- The profile plot shows somewhat similar patterns for the 12 patients, with in almost cases highest values in the treatment period and lower values before and after. The time points are quite differently spaced, presumably a deliberate decision by the investigators. The variation (between patients) seems reasonably constant over time. MTB > GLM; SUBC> Response 'placid'; SUBC> Nodefault; SUBC> Categorical 'patient' 'week'; SUBC> Terms patient week; SUBC> Means week; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: placid versus patient, week Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values patient Fixed 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 week Fixed 7 1, 2, 6, 10, 14, 15, 16 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value patient 11 3.682 25.76% 3.682 0.33473 4.97 0.000 week 6 6.165 43.12% 6.165 1.02742 15.24 0.000 Error 66 4.449 31.12% 4.449 0.06741 Total 83 14.296 100.00% Model Summary S R-sq R-sq(adj) PRESS R-sq(pred) 0.259636 68.88% 60.86% 7.20682 49.59% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 0.8488 0.0283 ( 0.7922, 0.9054) 29.96 0.000 patient 1 -0.2902 0.0940 (-0.4778, -0.1027) -3.09 0.003 1.83 2 -0.1617 0.0940 (-0.3493, 0.0259) -1.72 0.090 1.83 3 0.0098 0.0940 (-0.1778, 0.1973) 0.10 0.918 1.83 4 -0.1888 0.0940 (-0.3764, -0.0012) -2.01 0.049 1.83 5 0.0383 0.0940 (-0.1493, 0.2259) 0.41 0.685 1.83 6 -0.1045 0.0940 (-0.2921, 0.0831) -1.11 0.270 1.83 7 0.0455 0.0940 (-0.1421, 0.2331) 0.48 0.630 1.83 8 0.4140 0.0940 ( 0.2265, 0.6016) 4.41 0.000 1.83 9 -0.2631 0.0940 (-0.4507, -0.0755) -2.80 0.007 1.83 10 0.3326 0.0940 ( 0.1450, 0.5202) 3.54 0.001 1.83 11 0.0855 0.0940 (-0.1021, 0.2731) 0.91 0.366 1.83 week 1 -0.3346 0.0694 (-0.4732, -0.1961) -4.82 0.000 1.71 2 -0.2680 0.0694 (-0.4065, -0.1294) -3.86 0.000 1.71 6 0.3562 0.0694 ( 0.2176, 0.4947) 5.13 0.000 1.71 10 0.2304 0.0694 ( 0.0918, 0.3689) 3.32 0.001 1.71 14 0.3145 0.0694 ( 0.1760, 0.4531) 4.53 0.000 1.71 15 -0.0921 0.0694 (-0.2307, 0.0464) -1.33 0.189 1.71 Regression Equation placid = 0.8488 - 0.2902 patient_1 - 0.1617 patient_2 + 0.0098 patient_3 - 0.1888 patient_4 + 0.0383 patient_5 - 0.1045 patient_6 + 0.0455 patient_7 + 0.4140 patient_8 - 0.2631 patient_9 + 0.3326 patient_10 + 0.0855 patient_11 + 0.0826 patient_12 - 0.3346 week_1 - 0.2680 week_2 + 0.3562 week_6 + 0.2304 week_10 + 0.3145 week_14 - 0.0921 week_15 - 0.2063 week_16 Fits and Diagnostics for Unusual Observations Obs placid Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 26 1.480 0.975 0.120 (0.735, 1.214) 0.505 2.20 2.26 0.214286 0.07 44 1.090 0.626 0.120 (0.386, 0.866) 0.464 2.01 2.06 0.214286 0.06 56 0.410 1.057 0.120 (0.817, 1.297) -0.647 -2.81 -2.97 0.214286 0.12 64 1.400 0.847 0.120 (0.607, 1.087) 0.553 2.40 2.50 0.214286 0.09 65 1.400 0.913 0.120 (0.673, 1.153) 0.487 2.11 2.17 0.214286 0.07 Obs DFITS 26 1.18232 R 44 1.07787 R 56 -1.55170 R 64 1.30419 R 65 1.13476 R R Large residual Means Fitted Term Mean SE Mean week 1 0.5142 0.0750 2 0.5808 0.0750 6 1.2050 0.0750 10 1.0792 0.0750 14 1.1633 0.0750 15 0.7567 0.0750 16 0.6425 0.0750 Residual Plots for placid Comments: --------- The residual plots look good, and no residuals are extreme. The model shows strong effects of both patients and weeks, and the week means show an increased level during treatment, whereas pre- and post-treatment look quite similar. We could do pairwise comparisons between weeks, but the method is perhaps less attractive because there are many pairs (21), and any adjustment for multiple comparisons will decrease power substantially. A more logical approach would be to construct two orthogonal contrasts to estimate the difference between (i) treated versus non-treated, and (ii) pre versus post. For this solution, we will leave the details for the reader, and instead assess the significance of the model reduction where week effects are replaced by stage effects (stages being pre, treated, post). student. The listing for this model is shown below, and from it we compute the F-test for the model reduction: F = (4.653-4.449)/(70-66) / 0.06741 = 0.76 which corresponds to P=0.56 in F(4,66). We conclude there is no loss in fit by replacing weeks by stages, and we could also work out that the stage effect includes 5.961/6.165=97% of the variation between weeks. In this instance it would seem reasonable to base all further inference on the model with stage effects instead of week effects. This does however ignore the fact that in a sense the experimental unit for stage is week. If we wanted to build this into the model, we should include random week effects and nest stages in weeks. Minitab can fit such a model, but the variance component for week(stage) is estimated at 0 (actually a negative value); see Addendum below. Also in a linear mixed model in Stata (see Stata do-file), this variance component is estimated at zero. Therefore, we are effectively unable to include the variation between weeks within stages in the inference. The parametrization below with stage 0 (pre) as the reference category allows us to immediately read off a very strong significance (P<0.0005) for the treated stage compared to pre and also a weak significance (P=0.045) for the post stage compared to pre. MTB > GLM; SUBC> Response 'placid'; SUBC> Nodefault; SUBC> Categorical 'patient' 'stage'; SUBC> Binary; SUBC> Terms patient stage; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: placid versus patient, stage Method Factor coding (1, 0) Factor Information Factor Type Levels Values patient Fixed 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 stage Fixed 3 0, 1, 2 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value patient 11 3.682 25.76% 3.682 0.33473 5.04 0.000 stage 2 5.961 41.70% 5.961 2.98053 44.84 0.000 Error 70 4.653 32.55% 4.653 0.06647 Lack-of-Fit 22 2.397 16.77% 2.397 0.10895 2.32 0.008 Pure Error 48 2.256 15.78% 2.256 0.04699 Total 83 14.296 100.00% S R-sq R-sq(adj) PRESS R-sq(pred) 0.257809 67.45% 61.41% 6.72333 52.97% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 0.257 0.107 ( 0.044, 0.471) 2.40 0.019 patient 2 0.129 0.138 (-0.146, 0.403) 0.93 0.354 1.83 3 0.300 0.138 ( 0.025, 0.575) 2.18 0.033 1.83 4 0.101 0.138 (-0.173, 0.376) 0.74 0.464 1.83 5 0.329 0.138 ( 0.054, 0.603) 2.38 0.020 1.83 6 0.186 0.138 (-0.089, 0.461) 1.35 0.182 1.83 7 0.336 0.138 ( 0.061, 0.611) 2.44 0.017 1.83 8 0.704 0.138 ( 0.429, 0.979) 5.11 0.000 1.83 9 0.027 0.138 (-0.248, 0.302) 0.20 0.844 1.83 10 0.623 0.138 ( 0.348, 0.898) 4.52 0.000 1.83 11 0.376 0.138 ( 0.101, 0.651) 2.73 0.008 1.83 12 0.373 0.138 ( 0.098, 0.648) 2.71 0.009 1.83 stage 1 0.6017 0.0679 (0.4662, 0.7372) 8.86 0.000 1.43 2 0.1521 0.0744 (0.0037, 0.3005) 2.04 0.045 1.43 Regression Equation placid = 0.257 + 0.0 patient_1 + 0.129 patient_2 + 0.300 patient_3 + 0.101 patient_4 + 0.329 patient_5 + 0.186 patient_6 + 0.336 patient_7 + 0.704 patient_8 + 0.027 patient_9 + 0.623 patient_10 + 0.376 patient_11 + 0.373 patient_12 + 0.0 stage_0 + 0.6017 stage_1 + 0.1521 stage_2 Fits and Diagnostics for Unusual Observations Obs placid Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 13 1.030 0.538 0.107 (0.324, 0.752) 0.492 2.10 2.15 0.172619 0.07 26 1.480 0.960 0.103 (0.756, 1.165) 0.520 2.20 2.26 0.158730 0.07 44 1.090 0.593 0.107 (0.379, 0.807) 0.497 2.12 2.18 0.172619 0.07 56 0.410 1.114 0.107 (0.900, 1.327) -0.704 -3.00 -3.19 0.172619 0.13 64 1.400 0.880 0.107 (0.666, 1.094) 0.520 2.22 2.28 0.172619 0.07 65 1.400 0.880 0.107 (0.666, 1.094) 0.520 2.22 2.28 0.172619 0.07 Obs DFITS 13 0.98302 R 26 0.98220 R 44 0.99356 R 56 -1.45766 R 64 1.04262 R 65 1.04262 R R Large residual Residual Plots for placid Comments: --------- The potential problem with this model and analysis is that it does not reflect the time ordering of the repeated measures on each patient. If patients had been modelled by random effects (corresponding to a variance component of 0.038 or 36% of the total unexplained variation), then the correlations within patients had been the same between all weeks (and equal to 0.36). This is probably quite unrealistic. The repeated measures ANOVA methods tell us that the inference within patients (across weeks) would be affected; here this is exactly the inference we're interested in. In order to explore this, we have a choice between repeated measures ANOVA or linear mixed models with correlation structure. It is relatively easy to run a repeated measures ANOVA to adjust the significance of the F-test for week (not too surprisingly, it remains highly significant), but this does not help very much. We need adjustments for the post-ANOVA inference. It turns out that the model with stages instead of weeks cannot be handled easily by the repeated measures ANOVA method (see Stata do-file). Our main focus in the following is therefore on linear mixed models. The mixed command in Stata does not deal well with non-hierarchical structures, so we will only include random effects and correlation structure for patients, and not further explore random effects for weeks. One first question is whether we should base correlation structures on the measured (non-equidistant) weeks, or whether we should convert these to equidistant times (= within-patient observation numbers). Some understanding of the biological mechanisms of the treatment would be helpful; sometimes one may argue that time points are "biologically equidistant". That does not seem obvious here, but we can compare the fits with both types of time points. It turns out, perhaps surprisingly, that a model with equidistant time points fits much better with the ar(1) correlation structure. Inference in this model shows strongly significant differences between treated and both non-treated stages, whereas the difference between pre- and post-stages is not quite significant (by unadjusted pairwise comparisons). Addendum: additional random effects models in Minitab -------- We show results for a model with random effects of both patients and weeks (nested in stages), and another expanded model with an additional patient*stage effect, accouting for the variation in the patient's responses to the treatment. It would be a random term as well (because patient is random). Such an analysis cannot easily be run in Stata, but SAS proc mixed can be used, and can at the same time include the autocorrelation within patients discussed above. MTB > GLM; SUBC> Response 'placid'; SUBC> Nodefault; SUBC> Categorical 'patient' 'week' 'stage'; SUBC> Binary; SUBC> Nest week(stage); SUBC> Random patient week; SUBC> Terms patient stage week; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TEMS; SUBC> TVariance; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: placid versus patient, week, stage Method Factor coding (1, 0) Factor Information Factor Type Levels Values patient Random 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 week(stage) Random 7 1(0), 2(0), 6(1), 10(1), 14(1), 15(2), 16(2) stage Fixed 3 0, 1, 2 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value patient 11 3.6820 25.76% 3.6820 0.33473 4.97 0.000 stage 2 5.9611 41.70% 2.9482 1.47412 28.98 0.004 week(stage) 4 0.2035 1.42% 0.2035 0.05087 0.75 0.559 Error 66 4.4491 31.12% 4.4491 0.06741 Total 83 14.2957 100.00% S R-sq R-sq(adj) PRESS R-sq(pred) 0.259636 68.88% 60.86% 7.20682 49.59% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 0.224 0.120 (-0.016, 0.464) 1.86 0.067 patient 2 0.129 0.139 (-0.149, 0.406) 0.93 0.358 * 3 0.300 0.139 ( 0.023, 0.577) 2.16 0.034 * 4 0.101 0.139 (-0.176, 0.379) 0.73 0.467 * 5 0.329 0.139 ( 0.051, 0.606) 2.37 0.021 * 6 0.186 0.139 (-0.091, 0.463) 1.34 0.185 * 7 0.336 0.139 ( 0.059, 0.613) 2.42 0.018 * 8 0.704 0.139 ( 0.427, 0.981) 5.07 0.000 * 9 0.027 0.139 (-0.250, 0.304) 0.20 0.846 * 10 0.623 0.139 ( 0.346, 0.900) 4.49 0.000 * 11 0.376 0.139 ( 0.099, 0.653) 2.71 0.009 * 12 0.373 0.139 ( 0.096, 0.650) 2.69 0.009 * stage 1 0.691 0.106 ( 0.479, 0.902) 6.52 0.000 3.43 2 0.243 0.106 ( 0.031, 0.454) 2.29 0.025 2.86 week(stage) 2(0) 0.067 0.106 (-0.145, 0.278) 0.63 0.532 * 10(1) -0.126 0.106 (-0.337, 0.086) -1.19 0.239 * 14(1) -0.042 0.106 (-0.253, 0.170) -0.39 0.696 * 16(2) -0.114 0.106 (-0.326, 0.097) -1.08 0.285 * Regression Equation placid = 0.224 + 0.0 patient_1 + 0.129 patient_2 + 0.300 patient_3 + 0.101 patient_4 + 0.329 patient_5 + 0.186 patient_6 + 0.336 patient_7 + 0.704 patient_8 + 0.027 patient_9 + 0.623 patient_10 + 0.376 patient_11 + 0.373 patient_12 + 0.0 stage_0 + 0.691 stage_1 + 0.243 stage_2 + 0.0 week(stage)_1(0) + 0.067 week(stage)_2(0) + 0.0 week(stage)_6(1) - 0.126 week(stage)_10(1) - 0.042 week(stage)_14(1) + 0.0 week(stage)_15(2) - 0.114 week(stage)_16(2) Equation treats random terms as though they are fixed. Fits and Diagnostics for Unusual Observations Obs placid Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 26 1.480 0.975 0.120 (0.735, 1.214) 0.505 2.20 2.26 0.214286 0.07 44 1.090 0.626 0.120 (0.386, 0.866) 0.464 2.01 2.06 0.214286 0.06 56 0.410 1.057 0.120 (0.817, 1.297) -0.647 -2.81 -2.97 0.214286 0.12 64 1.400 0.847 0.120 (0.607, 1.087) 0.553 2.40 2.50 0.214286 0.09 65 1.400 0.913 0.120 (0.673, 1.153) 0.487 2.11 2.17 0.214286 0.07 Obs DFITS 26 1.18232 R 44 1.07787 R 56 -1.55170 R 64 1.30419 R 65 1.13476 R R Large residual Expected Mean Squares, using Adjusted SS Expected Mean Square for Source Each Term 1 patient (4) + 7.0000 (1) 2 stage (4) + 12.0000 (3) + Q[2] 3 week(stage) (4) + 12.0000 (3) 4 Error (4) Variance Components, using Adjusted SS Source Variance % of Total StDev % of Total patient 0.0381884 36.16% 0.195418 60.14% week(stage) -0.00137824* 0.00% 0.000000 0.00% Error 0.0674108 63.84% 0.259636 79.90% Total 0.105599 0.324960 * Value is negative, and is estimated by zero. Residual Plots for placid MTB > GLM; SUBC> Response 'placid'; SUBC> Nodefault; SUBC> Categorical 'patient' 'week' 'stage'; SUBC> Binary; SUBC> Nest week(stage); SUBC> Random patient week; SUBC> Terms patient stage patient*stage week; SUBC> TExpand; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TEMS; SUBC> TVariance; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: placid versus patient, week, stage Method Factor coding (1, 0) Factor Information Factor Type Levels Values patient Random 12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 week(stage) Random 7 1(0), 2(0), 6(1), 10(1), 14(1), 15(2), 16(2) stage Fixed 3 0, 1, 2 Analysis of Variance Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value patient 11 3.6820 25.76% 3.1528 0.28662 2.83 0.015 x stage 2 5.9611 41.70% 0.7051 0.35256 3.41 0.050 x patient*stage 22 2.3969 16.77% 2.3969 0.10895 2.34 0.008 week(stage) 4 0.2035 1.42% 0.2035 0.05087 1.09 0.373 Error 44 2.0522 14.36% 2.0522 0.04664 Total 83 14.2957 100.00% x Not an exact F-test. S R-sq R-sq(adj) PRESS R-sq(pred) 0.215963 85.64% 72.92% 7.87838 44.89% Coefficients Term Coef SE Coef 95% CI T-Value P-Value VIF Constant 0.077 0.159 ( -0.244, 0.397) 0.48 0.632 patient 2 -0.020 0.216 ( -0.455, 0.415) -0.09 0.927 * 3 0.440 0.216 ( 0.005, 0.875) 2.04 0.048 * 4 0.165 0.216 ( -0.270, 0.600) 0.76 0.449 * 5 0.370 0.216 ( -0.065, 0.805) 1.71 0.094 * 6 0.120 0.216 ( -0.315, 0.555) 0.56 0.581 * 7 0.585 0.216 ( 0.150, 1.020) 2.71 0.010 * 8 0.890 0.216 ( 0.455, 1.325) 4.12 0.000 * 9 0.315 0.216 ( -0.120, 0.750) 1.46 0.152 * 10 1.290 0.216 ( 0.855, 1.725) 5.97 0.000 * 11 0.590 0.216 ( 0.155, 1.025) 2.73 0.009 * 12 0.505 0.216 ( 0.070, 0.940) 2.34 0.024 * stage 1 0.796 0.208 ( 0.376, 1.216) 3.82 0.000 19.14 2 0.600 0.225 ( 0.147, 1.053) 2.67 0.011 18.57 patient*stage 2 1 0.170 0.279 ( -0.392, 0.732) 0.61 0.545 * 2 2 0.265 0.305 ( -0.351, 0.881) 0.87 0.390 * 3 1 -0.253 0.279 ( -0.815, 0.309) -0.91 0.368 * 3 2 -0.110 0.305 ( -0.726, 0.506) -0.36 0.720 * 4 1 0.125 0.279 ( -0.437, 0.687) 0.45 0.656 * 4 2 -0.410 0.305 ( -1.026, 0.206) -1.34 0.186 * 5 1 0.130 0.279 ( -0.432, 0.692) 0.47 0.643 * 5 2 -0.340 0.305 ( -0.956, 0.276) -1.11 0.272 * 6 1 0.303 0.279 ( -0.259, 0.865) 1.09 0.283 * 6 2 -0.225 0.305 ( -0.841, 0.391) -0.74 0.465 * 7 1 -0.322 0.279 ( -0.884, 0.240) -1.15 0.255 * 7 2 -0.390 0.305 ( -1.006, 0.226) -1.28 0.208 * 8 1 0.027 0.279 ( -0.535, 0.589) 0.10 0.924 * 8 2 -0.690 0.305 ( -1.306, -0.074) -2.26 0.029 * 9 1 -0.275 0.279 ( -0.837, 0.287) -0.99 0.329 * 9 2 -0.595 0.305 ( -1.211, 0.021) -1.95 0.058 * 10 1 -0.760 0.279 ( -1.322, -0.198) -2.73 0.009 * 10 2 -1.195 0.305 ( -1.811, -0.579) -3.91 0.000 * 11 1 -0.253 0.279 ( -0.815, 0.309) -0.91 0.368 * 11 2 -0.370 0.305 ( -0.986, 0.246) -1.21 0.232 * 12 1 -0.152 0.279 ( -0.714, 0.410) -0.54 0.589 * 12 2 -0.235 0.305 ( -0.851, 0.381) -0.77 0.446 * week(stage) 2(0) 0.0667 0.0882 (-0.1110, 0.2444) 0.76 0.454 * 10(1) -0.1258 0.0882 (-0.3035, 0.0519) -1.43 0.161 * 14(1) -0.0417 0.0882 (-0.2194, 0.1360) -0.47 0.639 * 16(2) -0.1142 0.0882 (-0.2919, 0.0635) -1.29 0.202 * Regression Equation ... Equation treats random terms as though they are fixed. Fits and Diagnostics for Unusual Observations Obs placid Fit SE Fit 95% CI Resid Std Resid Del Resid HI Cook’s D 24 0.740 1.163 0.135 (0.891, 1.434) -0.423 -2.50 -2.67 0.388889 0.10 26 1.480 1.121 0.135 (0.849, 1.392) 0.359 2.13 2.22 0.388889 0.07 43 0.300 0.662 0.159 (0.341, 0.982) -0.362 -2.47 -2.64 0.541667 0.18 44 1.090 0.728 0.159 (0.408, 1.049) 0.362 2.47 2.64 0.541667 0.18 55 1.230 0.877 0.159 (0.557, 1.197) 0.353 2.41 2.56 0.541667 0.17 56 0.410 0.763 0.159 (0.443, 1.083) -0.353 -2.41 -2.56 0.541667 0.17 Obs DFITS 24 -2.13103 R 26 1.77128 R 43 -2.86508 R 44 2.86508 R 55 2.78503 R 56 -2.78503 R R Large residual Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 patient (5) + 2.0000 (3) + 2.0000 (1) 2 stage (5) + 2.0599 (4) + 2.0599 (3) + Q[2] 3 patient*stage (5) + 2.2857 (3) 4 week(stage) (5) + 12.0000 (4) 5 Error (5) Error Terms for Tests, using Adjusted SS Source Error DF Error MS Synthesis of Error MS 1 patient 24.73 0.1012 0.8750 (3) + 0.1250 (5) 2 stage 23.42 0.1035 0.9012 (3) + 0.1717 (4) - 0.0729 (5) 3 patient*stage 44.00 0.0466 (5) 4 week(stage) 44.00 0.0466 (5) Variance Components, using Adjusted SS Source Variance % of Total StDev % of Total patient 0.0927298 55.53% 0.304516 74.52% patient*stage 0.0272615 16.33% 0.165111 40.41% week(stage) 0.0003527 0.21% 0.018779 4.60% Error 0.0466401 27.93% 0.215963 52.85% Total 0.166984 0.408637 Residual Plots for placid