Supplementary Exercises 12.54 and 12.55 of IPS7e ------------------------------------------------ Data: measurements of vitamin contents in loaves immediately after baking and after storage for 1, 3, 5 and 7 days. Two loaves were measured for each storage condition. In this exercise we consider two outcomes: concentrations of vitamin A and vitamin C. Model: if we denote the vitamin concentration in loaf j subjected to storage condition i (where i=1,2,3,4,5 and j=1,2), the statistical model is that the X_ij's are normally distributed N(mu_i,sigma), and all obs. are independent. With only two replicates per group, it is almost impossible to check the assumptions of the model. The standard deviations within groups are highly variable when based only on two observations, and the IPS/PSLS guideline for the homogeneity of standard deviations can be dispensed with. The best descriptive statistic is a dotplot to get a visual impression of whether the groups looks reasonable similar. Minitab commands, with comments below. MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 12\ex12_054.mtw". Retrieving worksheet from file: 'H:\VHM\VHM801\Datasets\Minitab\Chapter 12\ex12_054.mtw' Worksheet was saved on 02/11/2014 MTB > Describe 'vita' 'vite'; SUBC> By 'condition'. Descriptive Statistics: vita, vite Variable condition N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum vita 0 2 0 3.3500 0.0100 0.0141 3.3400 * 3.3500 * 3.3600 1 2 0 3.2400 0.0400 0.0566 3.2000 * 3.2400 * 3.2800 3 2 0 3.2100 0.0500 0.0707 3.1600 * 3.2100 * 3.2600 5 2 0 3.3050 0.0550 0.0778 3.2500 * 3.3050 * 3.3600 7 2 0 2.9650 0.0450 0.0636 2.9200 * 2.9650 * 3.0100 vite 0 2 0 95.300 0.700 0.990 94.600 * 95.300 * 96.000 1 2 0 94.45 1.25 1.77 93.20 * 94.45 * 95.70 3 2 0 95.85 1.55 2.19 94.30 * 95.85 * 97.40 5 2 0 96.35 1.35 1.91 95.00 * 96.35 * 97.70 7 2 0 93.70 1.40 1.98 92.30 * 93.70 * 95.10 MTB > Dotplot ( 'vita' 'vite' ) * 'condition'. Dotplot of vita vs condition Dotplot of vite vs condition Comments for descriptive analysis: ---------------------------------- The variation is reasonably constant across the groups, and except for the 7 day storage group and the vitamin A outcome the groups also seem to have approximately the same location. The null hypothesis tested is: H0: mu1=mu2=mu3=mu4=mu5 (same mean vitamin level in all storage groups) Ha: not H0, that is, some storage groups differ. MTB > OneWay; SUBC> Response 'vita'; SUBC> Categorical 'condition'; SUBC> IType 0; SUBC> GMCI; SUBC> GIntPlot; SUBC> TMethod; SUBC> TFactor; SUBC> TANOVA; SUBC> TSummary; SUBC> TMeans; SUBC> Nodefault. One-way ANOVA: vita versus condition Method Null hypothesis All means are equal Alternative hypothesis At least one mean is different Significance level a = 0.05 Equal variances were assumed for the analysis. Factor Information Factor Levels Values condition 5 0, 1, 3, 5, 7 Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value condition 4 0.17894 0.044735 12.09 0.009 Error 5 0.01850 0.003700 Total 9 0.19744 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.0608276 90.63% 83.13% 62.52% Means condition N Mean StDev 95% CI 0 2 3.3500 0.0141 (3.2394, 3.4606) 1 2 3.2400 0.0566 (3.1294, 3.3506) 3 2 3.2100 0.0707 (3.0994, 3.3206) 5 2 3.3050 0.0778 (3.1944, 3.4156) 7 2 2.9650 0.0636 (2.8544, 3.0756) Pooled StDev = 0.0608276 Interval Plot of vita vs condition MTB > OneWay; SUBC> Response 'vite'; SUBC> Categorical 'condition'; SUBC> IType 0; SUBC> GMCI; SUBC> GIntPlot; SUBC> TMethod; SUBC> TFactor; SUBC> TANOVA; SUBC> TSummary; SUBC> TMeans; SUBC> Nodefault. One-way ANOVA: vite versus condition Method Null hypothesis All means are equal Alternative hypothesis At least one mean is different Significance level a = 0.05 Equal variances were assumed for the analysis. Factor Information Factor Levels Values condition 5 0, 1, 3, 5, 7 Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value condition 4 9.086 2.272 0.69 0.630 Error 5 16.475 3.295 Total 9 25.561 Model Summary S R-sq R-sq(adj) R-sq(pred) 1.81521 35.55% 0.00% 0.00% Means condition N Mean StDev 95% CI 0 2 95.300 0.990 (92.001, 98.599) 1 2 94.45 1.77 ( 91.15, 97.75) 3 2 95.85 2.19 ( 92.55, 99.15) 5 2 96.35 1.91 ( 93.05, 99.65) 7 2 93.70 1.98 ( 90.40, 97.00) Pooled StDev = 1.81521 Interval Plot of vite vs condition Comments: --------- The ANOVA tables show that the F-tests have (4,5) degrees of freedom, and have values of 12.09 and 0.69 for vitamin A and E, respectively. The former is clearly significant and the latter is clearly nonsignificant. For vitamin A we conclude that some differences among groups exist, and looking at the means it does in fact seem as if the bread has lost some of its vitamin A after a 7-day storage (but not after 5 days). For vitamin E we conclude that there is no evidence of a difference among the storage groups; the bread does not seem to loose its vitamin E by storage (up to 7 days). Supplementary Exercise 12.55: ----------------------------- (a) It is inappropriate to perform multiple comparisons when the overall F-test is nonsignificant. When there is no evidence of a difference between the groups, it makes no sense to go hunting after specific differences between groups. (b) Assuming that no particular group differences are of special interest, we may use the Bonferroni method to account for a total of 5*4/2=10 pairwise comparisons. Therefore, the individual error rate is 0.05/10=0.005. An alternative to using Minitab is to compute an adjusted LSD directly. From the t-distribution table with df=5 (DFE) and confidence level C=99.5% (corresponding to an error level of 0.5%) we get t*=4.773. Therefore, LSD(.995) = 4.773*0.06083*sqrt(2/2) = 0.29. Among the group means, only those of group 7 and 0 days, and of group 7 and 5 days, differ more than 0.29. Therefore these are the only significant comparisons. Further comments below the Minitab print, showing only the extra listing resulting from the Comparisons submenu, where the Fisher error rate has been set to 0.005 (not 0.5, as one would intuitively think). MTB > OneWay; SUBC> Response 'vita'; SUBC> Categorical 'condition'; SUBC> IType 0; SUBC> Fisher 0.005; SUBC> GMCI; SUBC> TGrouping; SUBC> TMTest; SUBC> GIntPlot; SUBC> TMethod; SUBC> TFactor; SUBC> TANOVA; SUBC> TSummary; SUBC> TMeans; SUBC> Nodefault. One-way ANOVA: vita versus condition ... Fisher Pairwise Comparisons Grouping Information Using the Fisher LSD Method and 99.5% Confidence condition N Mean Grouping 0 2 3.3500 A 5 2 3.3050 A 1 2 3.2400 A B 3 2 3.2100 A B 7 2 2.9650 B Means that do not share a letter are significantly different. Fisher Individual Tests for Differences of Means Difference Difference SE of Adjusted of Levels of Means Difference 99.5% CI T-Value P-Value 1 - 0 -0.1100 0.0608 (-0.4004, 0.1804) -1.81 0.130 3 - 0 -0.1400 0.0608 (-0.4304, 0.1504) -2.30 0.070 5 - 0 -0.0450 0.0608 (-0.3354, 0.2454) -0.74 0.493 7 - 0 -0.3850 0.0608 (-0.6754, -0.0946) -6.33 0.001 3 - 1 -0.0300 0.0608 (-0.3204, 0.2604) -0.49 0.643 5 - 1 0.0650 0.0608 (-0.2254, 0.3554) 1.07 0.334 7 - 1 -0.2750 0.0608 (-0.5654, 0.0154) -4.52 0.006 5 - 3 0.0950 0.0608 (-0.1954, 0.3854) 1.56 0.179 7 - 3 -0.2450 0.0608 (-0.5354, 0.0454) -4.03 0.010 7 - 5 -0.3400 0.0608 (-0.6304, -0.0496) -5.59 0.003 Simultaneous confidence level = 97.46% Fisher Individual 99.5% CIs Comments: --------- The 99.5% confidence intervals lead to the same conclusion as described above, namely that the only significant comparisons are 7-0 and 7-5 (these are the only CIs not including zero). The grouping information represents that conclusion in a diagram. The column labelled "Adjusted P-value" is a bit misleading here, because the Fisher method involves no adjustment. Therefore the P-values are unadjusted for multiple testing, and our choice of a 0.5% error level has no impact on these P-values. The correct way to interpret these is therefore at a 0.5% significance level, showing us that only the two comparisons 7-0 and 7-5 are significant. Our results are that at the overall 5% level there is a significant difference between 7-days storage and storage of 0 and 5 days, but not 1 and 3 days. Biologically this does not make much sense, so a more reasonable conclusion is that the data show a lower vitamin A content for day 7 than all other days, and this effect is significant at the overall 5% level for 0 and 5 days, and close to significant for days 1 and 3. The Bonferroni method is conservative, so other methods might give significance for more days (for example, the Tukey method shows significance for all comparisons with 7 days), but in practice this should be of little importance for the conclusions from the study.