Supplementary Exercise 13.31 of IPS7e ------------------------------------- (a) Minitab commands: MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 13\ex13_031.mtw". Retrieving worksheet from file: 'H:\VHM\VHM801\Datasets\Minitab\Chapter 13\ex13_031.mtw' Worksheet was saved on 21/11/2014 MTB > Table 'promo' 'percent'; SUBC> Layout 1 1; SUBC> DMissing 'promo' 'percent'; SUBC> Means 'eprice'; SUBC> StDev 'eprice'; SUBC> Counts. Tabulated statistics: promo, percent Rows: promo Columns: percent 10 20 30 40 All 1 4.920 4.689 4.225 4.423 4.564 0.1520 0.2331 0.3856 0.1848 0.3621 10 10 10 10 40 3 4.756 4.524 4.097 4.284 4.415 0.2429 0.2707 0.2346 0.2040 0.3406 10 10 10 10 40 5 4.393 4.251 3.890 4.058 4.148 0.2685 0.2648 0.1629 0.1760 0.2887 10 10 10 10 40 7 4.269 4.094 3.760 3.780 3.976 0.2699 0.2407 0.2618 0.2144 0.3225 10 10 10 10 40 All 4.585 4.390 3.993 4.136 4.276 0.3518 0.3376 0.3196 0.3099 0.3990 40 40 40 40 160 Cell Contents: eprice : Mean eprice : Standard deviation Count MTB > Interact 'promo' 'percent'; SUBC> Response 'eprice'; SUBC> Full. Interaction Plot for eprice Comments for (a): ----------------- The means for promotion groups show that the expected price decreases with the number of promotions, almost linearly. Also, the expected price decreases with the discount, except for 30% and 40% where the values are interchanged. The lines in the plot are almost parallel, indicating that the interaction is small relative to the main effects. (b) The two-way ANOVA model is Xijk = mu + alpha_i + beta_j + gamma_ij + epsilon_ijk where mu= overall mean alpha_i= effect of promotion group i beta_j= effect of discount group j gamma_ij= interaction effect of (promotion,discount) group (i,j) epsilon_ijk= error, assumed to be i.i.d. and N(0,sigma) Written as a model formula, Price = Promotion + Discount + Promotion*Discount + Error Minitab commands for analysis: MTB > Name c5 "RESI1" MTB > ANOVA 'eprice' = promo percent promo* percent; SUBC> Means promo percent; SUBC> Residuals 'RESI1'; SUBC> GFourpack. ANOVA: eprice versus promo, percent Factor Type Levels Values promo fixed 4 1, 3, 5, 7 percent fixed 4 10, 20, 30, 40 Analysis of Variance for eprice Source DF SS MS F P promo 3 8.3605 2.7868 47.73 0.000 percent 3 8.3069 2.7690 47.42 0.000 promo*percent 9 0.2306 0.0256 0.44 0.912 Error 144 8.4087 0.0584 Total 159 25.3067 S = 0.241648 R-Sq = 66.77% R-Sq(adj) = 63.31% Means promo N eprice 1 40 4.5642 3 40 4.4153 5 40 4.1480 7 40 3.9758 percent N eprice 10 40 4.5845 20 40 4.3895 30 40 3.9930 40 40 4.1362 Residual Plots for eprice MTB > GSummary 'RESI1'. Summary Report for RESI1 MTB > VarTest 'eprice' 'promo' 'percent'; SUBC> Confidence 95.0; SUBC> GInterval; SUBC> NoDefault; SUBC> TMethod; SUBC> TBonferroni; SUBC> TTest. Test for Equal Variances: eprice versus promo, percent Method Null hypothesis All variances are equal Alternative hypothesis At least one variance is different Significance level a = 0.05 95% Bonferroni Confidence Intervals for Standard Deviations promo percent N StDev CI 1 10 10 0.152023 (0.071699, 0.45755) 1 20 10 0.233069 (0.049296, 1.56416) 1 30 10 0.385609 (0.202106, 1.04435) 1 40 10 0.184755 (0.083423, 0.58081) 3 10 10 0.242908 (0.126194, 0.66370) 3 20 10 0.270727 (0.105097, 0.98993) 3 30 10 0.234618 (0.118820, 0.65760) 3 40 10 0.204026 (0.107508, 0.54962) 5 10 10 0.268537 (0.070749, 1.44683) 5 20 10 0.264846 (0.125048, 0.79623) 5 30 10 0.162891 (0.097383, 0.38675) 5 40 10 0.175992 (0.076193, 0.57704) 7 10 10 0.269916 (0.139903, 0.73919) 7 20 10 0.240749 (0.135867, 0.60554) 7 30 10 0.261789 (0.150101, 0.64811) 7 40 10 0.214372 (0.117067, 0.55723) Individual confidence level = 99.6875% Tests Test Method Statistic P-Value Multiple comparisons - 0.347 Levene 1.04 0.416 Test for Equal Variances: eprice vs promo, percent Comments: --------- The ANOVA analysis shows indeed that the interaction is small and insignificant, whereas the two main effects are strongly significant. The residual plots show no reasons for concern, and the normality test for the residuals is clearly non-significant. The standard deviations are similar across the 16 groups and only mildly violate the IPS rule (for 1-Way ANOVA). The tests for equal variance are both clearly non-significant so this is no reason for concern. In order to continue the analysis with multiple comparisons, we have to do a LSD calculation by hand, or reanalyze using the General Linear Model menu. Both promo and discount have 4 levels, so for both we need to take into account a total of 4*3/2=6 multiple comparisons. We need a tstar value from a t-distribution with DF=144. MTB > InvCDF .00417; SUBC> T 144. Inverse Cumulative Distribution Function Student's t distribution with 144 DF P( X <= x ) x 0.00417 -2.67491 The relevant percentile was computed as 0.025/6=0.00417. Now LSD = 2.67491*0.2416*sqrt(2/40) = 0.1445. The LSD comparisons show that all levels of the main effects are significantly different at an overall 5% level, except the 30% and 40% discount groups where the difference is 0.143 and just short of the LSD threshold for significance. We can also use the more comprehensive General Linear Model menu (note that you have to include the interaction in the Model submenu because it will not be included by default), and request the Bonferroni-adjusted comparisons directly. MTB > GLM; SUBC> Response 'eprice'; SUBC> Nodefault; SUBC> Categorical 'promo' 'percent'; SUBC> Terms promo percent promo*percent; SUBC> Means promo percent; 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: eprice versus promo, percent Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values promo Fixed 4 1, 3, 5, 7 percent Fixed 4 10, 20, 30, 40 Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value promo 3 8.3605 2.78683 47.73 0.000 percent 3 8.3069 2.76898 47.42 0.000 promo*percent 9 0.2306 0.02562 0.44 0.912 Error 144 8.4087 0.05839 Total 159 25.3067 Model Summary S R-sq R-sq(adj) R-sq(pred) 0.241648 66.77% 63.31% 58.98% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 4.2758 0.0191 223.82 0.000 promo 1 0.2884 0.0331 8.72 0.000 1.50 3 0.1394 0.0331 4.21 0.000 1.50 5 -0.1278 0.0331 -3.86 0.000 1.50 percent 10 0.3087 0.0331 9.33 0.000 1.50 20 0.1137 0.0331 3.44 0.001 1.50 30 -0.2828 0.0331 -8.55 0.000 1.50 promo*percent 1 10 0.0471 0.0573 0.82 0.413 2.25 1 20 0.0111 0.0573 0.19 0.847 2.25 1 30 -0.0564 0.0573 -0.98 0.326 2.25 3 10 0.0321 0.0573 0.56 0.577 2.25 3 20 -0.0049 0.0573 -0.09 0.931 2.25 3 30 -0.0354 0.0573 -0.62 0.537 2.25 5 10 -0.0637 0.0573 -1.11 0.268 2.25 5 20 -0.0107 0.0573 -0.19 0.852 2.25 5 30 0.0248 0.0573 0.43 0.666 2.25 Regression Equation eprice = 4.2758 + 0.2884 promo_1 + 0.1394 promo_3 - 0.1278 promo_5 - 0.3001 promo_7 + 0.3087 percent_10 + 0.1137 percent_20 - 0.2828 percent_30 - 0.1396 percent_40 + 0.0471 promo*percent_1 10 + 0.0111 promo*percent_1 20 - 0.0564 promo*percent_1 30 - 0.0017 promo*percent_1 40 + 0.0321 promo*percent_3 10 - 0.0049 promo*percent_3 20 - 0.0354 promo*percent_3 30 + 0.0083 promo*percent_3 40 - 0.0637 promo*percent_5 10 - 0.0107 promo*percent_5 20 + 0.0248 promo*percent_5 30 + 0.0496 promo*percent_5 40 - 0.0154 promo*percent_7 10 + 0.0046 promo*percent_7 20 + 0.0671 promo*percent_7 30 - 0.0562 promo*percent_7 40 Fits and Diagnostics for Unusual Observations Obs eprice Fit Resid Std Resid 11 3.5700 4.2250 -0.6550 -2.86 R 30 5.2400 4.6890 0.5510 2.40 R 65 3.9600 4.5240 -0.5640 -2.46 R 108 3.7600 4.2510 -0.4910 -2.14 R 115 3.7400 4.3930 -0.6530 -2.85 R R Large residual Means Fitted Term Mean SE Mean promo 1 4.5643 0.0382 3 4.4153 0.0382 5 4.1480 0.0382 7 3.9757 0.0382 percent 10 4.5845 0.0382 20 4.3895 0.0382 30 3.9930 0.0382 40 4.1363 0.0382 Residual Plots for eprice MTB > Compare 'eprice'; SUBC> Pairwise promo percent; SUBC> Bonferroni; SUBC> GIntPlot; SUBC> NoDefault; SUBC> TGrouping; SUBC> TMTest. Comparisons for eprice Bonferroni Pairwise Comparisons: Response = eprice, Term = promo Grouping Information Using the Bonferroni Method and 95% Confidence promo N Mean Grouping 1 40 4.56425 A 3 40 4.41525 B 5 40 4.14800 C 7 40 3.97575 D Means that do not share a letter are significantly different. Bonferroni Simultaneous Tests for Differences of Means Difference of promo Difference SE of Simultaneous 95% Adjusted Levels of Means Difference CI T-Value P-Value 3 - 1 -0.1490 0.0540 (-0.2936, -0.0044) -2.76 0.039 5 - 1 -0.4163 0.0540 (-0.5608, -0.2717) -7.70 0.000 7 - 1 -0.5885 0.0540 (-0.7331, -0.4439) -10.89 0.000 5 - 3 -0.2673 0.0540 (-0.4118, -0.1227) -4.95 0.000 7 - 3 -0.4395 0.0540 (-0.5841, -0.2949) -8.13 0.000 7 - 5 -0.1722 0.0540 (-0.3168, -0.0277) -3.19 0.011 Individual confidence level = 99.17% Bonferroni Simultaneous 95% CIs Bonferroni Pairwise Comparisons: Response = eprice, Term = percent Grouping Information Using the Bonferroni Method and 95% Confidence percent N Mean Grouping 10 40 4.58450 A 20 40 4.38950 B 40 40 4.13625 C 30 40 3.99300 C Means that do not share a letter are significantly different. Bonferroni Simultaneous Tests for Differences of Means Difference of percent Difference SE of Simultaneous 95% Adjusted Levels of Means Difference CI T-Value P-Value 20 - 10 -0.1950 0.0540 (-0.3396, -0.0504) -3.61 0.003 30 - 10 -0.5915 0.0540 (-0.7361, -0.4469) -10.95 0.000 40 - 10 -0.4482 0.0540 (-0.5928, -0.3037) -8.30 0.000 30 - 20 -0.3965 0.0540 (-0.5411, -0.2519) -7.34 0.000 40 - 20 -0.2532 0.0540 (-0.3978, -0.1087) -4.69 0.000 40 - 30 0.1432 0.0540 (-0.0013, 0.2878) 2.65 0.054 Individual confidence level = 99.17% Bonferroni Simultaneous 95% CIs Comments: -------- The Bonferroni comparisons show that all levels of the main effects are significantly different at an overall 5% level, except the 30% and 40% discount groups with a P-value of 0.0535. We also note that there were 5 standardized residuals outside of (-2,2), but this is actually slightly less than expected for a data set of this size. (c) The analysis showed no interaction effect of Discount and Promotion on the expected price. Both Discount and Promotion had clearly significant effects. The expected price decreases with the number of promotions and is significantly different between all groups. The expected price decreases when the discount increases from 10% to 20% and 30%, also significantly so. However, the expected price takes a jump upwards from 30% to 40% even if this jump is only close to significant. The value at 40% is still significantly less than that at 20%.