Explore data:
- 548 observations in 7 variables.
- Sales in Thousands is the dependent variable, additional to which there are six variables: MarketID, MarketSize, LocationID, AgeOfStore, Promotion and Week.
- MarketSize is the only categorical variable with categories: small, medium and large.
- MarketID: There are 10 ID’s.
- LocationID: Numeric representation of the location of each store.
- AgeOfStore: The age of stores range from 1year to 28years.
- Promotion: The stores have been promoted atleast once upto three times.
- Week: Represents time span within which sales were recorded.
- SalesInThousands: Represents sales in thousands over weeks
## MarketID MarketSize LocationID AgeOfStore
## Min. : 1.000 Length:548 Min. : 1.0 Min. : 1.000
## 1st Qu.: 3.000 Class :character 1st Qu.:216.0 1st Qu.: 4.000
## Median : 6.000 Mode :character Median :504.0 Median : 7.000
## Mean : 5.715 Mean :479.7 Mean : 8.504
## 3rd Qu.: 8.000 3rd Qu.:708.0 3rd Qu.:12.000
## Max. :10.000 Max. :920.0 Max. :28.000
## Promotion Week SalesInThousands
## Min. :1.000 Min. :1.00 Min. :17.34
## 1st Qu.:1.000 1st Qu.:1.75 1st Qu.:42.55
## Median :2.000 Median :2.50 Median :50.20
## Mean :2.029 Mean :2.50 Mean :53.47
## 3rd Qu.:3.000 3rd Qu.:3.25 3rd Qu.:60.48
## Max. :3.000 Max. :4.00 Max. :99.65
## var frequency percentage cumulative_perc
## 1 Medium 320 58.39 58.39
## 2 Large 168 30.66 89.05
## 3 Small 60 10.95 100.00
Creating Initial Model and Selecting Appropriate Variables for the Model:
- The linear model equation tests SalesInThousands as a factor of all the variables from the training data set: trainlm.
- Variables MarketID and Promotion are significantly related to Sales.
- R Squared for initial model (fit1) is 0.90. Thus, the model explains 90 percent of the variance in the data set.
- This models reveals that variables:
- AgeofStore and
- Week are not significantly related with Sales.
- Ergo, it is safe to eliminate them from our final model.
##
## Call:
## lm(formula = SalesInThousands ~ ., data = trainlm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.6813 -3.3325 -0.1337 3.1093 14.1222
##
## Coefficients: (2 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 40.376472 1.053230 38.336 < 2e-16 ***
## MarketID2 25.763553 1.441758 17.870 < 2e-16 ***
## MarketID3 50.099687 1.027675 48.751 < 2e-16 ***
## MarketID4 19.347784 1.275623 15.167 < 2e-16 ***
## MarketID5 16.205559 1.098480 14.753 < 2e-16 ***
## MarketID6 2.037251 1.084460 1.879 0.061 .
## MarketID7 10.606854 1.083625 9.788 < 2e-16 ***
## MarketID8 13.200319 1.154628 11.433 < 2e-16 ***
## MarketID9 17.673032 1.230141 14.367 < 2e-16 ***
## MarketID10 19.525676 1.009458 19.343 < 2e-16 ***
## MarketSizeMedium NA NA NA NA
## MarketSizeSmall NA NA NA NA
## AgeOfStore -0.003655 0.037670 -0.097 0.923
## Promotion2 -10.041430 0.620137 -16.192 < 2e-16 ***
## Promotion3 -5.039170 0.636692 -7.915 2.27e-14 ***
## Week -0.166360 0.220784 -0.753 0.452
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.104 on 416 degrees of freedom
## Multiple R-squared: 0.9088, Adjusted R-squared: 0.906
## F-statistic: 319.1 on 13 and 416 DF, p-value: < 2.2e-16
Final Model with Significant Variables and Interpreting the Coefficients of the model.
- The linear equation tests variable ‘SalesInThousands’ as a factor of variables MarketID and Promotion from the training data set: trainlm.
- The R Squared for the final model (fit2) is 0.90.
- Thus, the model explains 90 percent of the variance in the data set using significant variables.
- MarketID 3 performs the best and shows the highest estimate at 50.09 followed by MarketID 2 at 25.71
- Promotion 1 shows the best results as its coefficient is 0.
- Although Promotion 2 and 3 are significantly related to sales, they show negative estimates to the sales. Hence, they should not be considered.
##
## Call:
## lm(formula = SalesInThousands ~ MarketID + Promotion, data = trainlm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.5894 -3.3428 -0.1003 3.2096 14.2140
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.9386 0.8507 46.950 < 2e-16 ***
## MarketID2 25.7183 1.4281 18.008 < 2e-16 ***
## MarketID3 50.0952 1.0253 48.857 < 2e-16 ***
## MarketID4 19.3497 1.2734 15.195 < 2e-16 ***
## MarketID5 16.1848 1.0949 14.782 < 2e-16 ***
## MarketID6 2.0295 1.0825 1.875 0.0615 .
## MarketID7 10.5980 1.0816 9.798 < 2e-16 ***
## MarketID8 13.1561 1.1427 11.513 < 2e-16 ***
## MarketID9 17.6263 1.2262 14.374 < 2e-16 ***
## MarketID10 19.5234 1.0068 19.392 < 2e-16 ***
## Promotion2 -10.0387 0.6190 -16.217 < 2e-16 ***
## Promotion3 -5.0267 0.6350 -7.916 2.23e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.096 on 418 degrees of freedom
## Multiple R-squared: 0.9087, Adjusted R-squared: 0.9063
## F-statistic: 378.3 on 11 and 418 DF, p-value: < 2.2e-16
Testing Interactions Between Variables:
- This tests interactions between the variables for gaining deeper insights.
- R Squared for model is 0.91. Thus, the model explains 91 percent of the variance in the data set.
- Promotion 3 in interaction with MarketID3 is significant and shows positive addition to the estimated sales.
- MarketID9 and AgeOfStore are significant and add positively to estimated sales.
- MarkedID8 and Promotion 3 in interaction are singificant as well as add positively to estimated sales.
- Small and Large markets perform better.
- The following variables and their interactions were removed during the backward elimination method: MarketID:MarketSize, MarketID:Promotion, MarketID:AgeOfStore, MarketID:Week, MarketSize:Promotion, MarketSize, Promotion:Week, AgeOfStore:Week, MarketSize:Week and Week.
##
## Call:
## lm(formula = SalesInThousands ~ (.)^2, data = trainlm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.7173 -3.2565 -0.1168 3.2114 15.7946
##
## Coefficients: (29 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 44.60635 3.70737 12.032 < 2e-16 ***
## MarketID2 18.03050 6.90821 2.610 0.009416 **
## MarketID3 46.93987 3.99285 11.756 < 2e-16 ***
## MarketID4 8.64526 4.53749 1.905 0.057504 .
## MarketID5 15.13694 4.62834 3.270 0.001173 **
## MarketID6 -3.01540 4.17233 -0.723 0.470305
## MarketID7 8.90180 4.65345 1.913 0.056514 .
## MarketID8 8.26477 4.39165 1.882 0.060618 .
## MarketID9 6.77028 4.68005 1.447 0.148836
## MarketID10 13.68350 4.00424 3.417 0.000702 ***
## MarketSizeMedium NA NA NA NA
## MarketSizeSmall NA NA NA NA
## AgeOfStore -0.24500 0.26559 -0.922 0.356872
## Promotion2 -9.38912 2.76420 -3.397 0.000755 ***
## Promotion3 -9.50224 2.98034 -3.188 0.001551 **
## Week -0.57790 0.80031 -0.722 0.470684
## MarketID2:MarketSizeMedium NA NA NA NA
## MarketID3:MarketSizeMedium NA NA NA NA
## MarketID4:MarketSizeMedium NA NA NA NA
## MarketID5:MarketSizeMedium NA NA NA NA
## MarketID6:MarketSizeMedium NA NA NA NA
## MarketID7:MarketSizeMedium NA NA NA NA
## MarketID8:MarketSizeMedium NA NA NA NA
## MarketID9:MarketSizeMedium NA NA NA NA
## MarketID10:MarketSizeMedium NA NA NA NA
## MarketID2:MarketSizeSmall NA NA NA NA
## MarketID3:MarketSizeSmall NA NA NA NA
## MarketID4:MarketSizeSmall NA NA NA NA
## MarketID5:MarketSizeSmall NA NA NA NA
## MarketID6:MarketSizeSmall NA NA NA NA
## MarketID7:MarketSizeSmall NA NA NA NA
## MarketID8:MarketSizeSmall NA NA NA NA
## MarketID9:MarketSizeSmall NA NA NA NA
## MarketID10:MarketSizeSmall NA NA NA NA
## MarketID2:AgeOfStore 0.36669 0.32134 1.141 0.254553
## MarketID3:AgeOfStore 0.08197 0.26494 0.309 0.757203
## MarketID4:AgeOfStore 0.43623 0.27863 1.566 0.118272
## MarketID5:AgeOfStore 0.22881 0.26792 0.854 0.393647
## MarketID6:AgeOfStore 0.42755 0.27572 1.551 0.121827
## MarketID7:AgeOfStore -0.01964 0.29817 -0.066 0.947516
## MarketID8:AgeOfStore 0.14006 0.26291 0.533 0.594534
## MarketID9:AgeOfStore 0.51215 0.30601 1.674 0.095037 .
## MarketID10:AgeOfStore 0.13564 0.26597 0.510 0.610351
## MarketID2:Promotion2 NA NA NA NA
## MarketID3:Promotion2 1.20362 2.89574 0.416 0.677901
## MarketID4:Promotion2 4.19112 3.00383 1.395 0.163761
## MarketID5:Promotion2 -0.49693 3.29827 -0.151 0.880321
## MarketID6:Promotion2 1.58319 2.77704 0.570 0.568951
## MarketID7:Promotion2 0.41935 3.27729 0.128 0.898252
## MarketID8:Promotion2 2.39968 3.12967 0.767 0.443711
## MarketID9:Promotion2 4.67490 3.33406 1.402 0.161691
## MarketID10:Promotion2 0.27141 2.56838 0.106 0.915896
## MarketID2:Promotion3 6.29297 4.58590 1.372 0.170806
## MarketID3:Promotion3 4.99323 2.87310 1.738 0.083043 .
## MarketID4:Promotion3 7.00450 4.66881 1.500 0.134382
## MarketID5:Promotion3 2.83457 3.68315 0.770 0.442019
## MarketID6:Promotion3 3.78951 3.43239 1.104 0.270280
## MarketID7:Promotion3 3.66743 3.12666 1.173 0.241557
## MarketID8:Promotion3 7.29697 3.19481 2.284 0.022928 *
## MarketID9:Promotion3 2.97228 3.35959 0.885 0.376874
## MarketID10:Promotion3 3.82679 3.14627 1.216 0.224635
## MarketID2:Week 0.37156 1.25334 0.296 0.767046
## MarketID3:Week 0.40685 0.91043 0.447 0.655220
## MarketID4:Week 1.71023 1.14705 1.491 0.136805
## MarketID5:Week -0.12308 0.97234 -0.127 0.899339
## MarketID6:Week 0.03055 0.98913 0.031 0.975375
## MarketID7:Week 0.41412 0.97465 0.425 0.671158
## MarketID8:Week 0.37061 1.02380 0.362 0.717561
## MarketID9:Week 1.95631 1.08603 1.801 0.072449 .
## MarketID10:Week 1.59633 0.90246 1.769 0.077728 .
## MarketSizeMedium:AgeOfStore NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA NA NA
## MarketSizeMedium:Promotion2 NA NA NA NA
## MarketSizeSmall:Promotion2 NA NA NA NA
## MarketSizeMedium:Promotion3 NA NA NA NA
## MarketSizeSmall:Promotion3 NA NA NA NA
## MarketSizeMedium:Week NA NA NA NA
## MarketSizeSmall:Week NA NA NA NA
## AgeOfStore:Promotion2 -0.11635 0.12461 -0.934 0.351050
## AgeOfStore:Promotion3 0.10674 0.11699 0.912 0.362160
## AgeOfStore:Week 0.01458 0.03375 0.432 0.666006
## Promotion2:Week -0.55491 0.55942 -0.992 0.321864
## Promotion3:Week -0.48595 0.57671 -0.843 0.399979
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.093 on 376 degrees of freedom
## Multiple R-squared: 0.918, Adjusted R-squared: 0.9064
## F-statistic: 79.41 on 53 and 376 DF, p-value: < 2.2e-16
## Backward Elimination Method
## ---------------------------
##
## Candidate Terms:
##
## 1 . MarketID
## 2 . MarketSize
## 3 . AgeOfStore
## 4 . Promotion
## 5 . Week
## 6 . MarketID:MarketSize
## 7 . MarketID:AgeOfStore
## 8 . MarketID:Promotion
## 9 . MarketID:Week
## 10 . MarketSize:AgeOfStore
## 11 . MarketSize:Promotion
## 12 . MarketSize:Week
## 13 . AgeOfStore:Promotion
## 14 . AgeOfStore:Week
## 15 . Promotion:Week
##
## We are eliminating variables based on p value...
##
## - AgeOfStore
##
## Backward Elimination: Step 1
##
## Variable AgeOfStore Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.958 RMSE 5.093
## R-Squared 0.918 Coef. Var 9.574
## Adj. R-Squared 0.906 MSE 25.935
## Pred R-Squared 0.894 MAE 3.777
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## -----------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## -----------------------------------------------------------------------
## Regression 109154.818 53 2059.525 79.41 0.0000
## Residual 9751.669 376 25.935
## Total 118906.487 429
## -----------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 44.606 3.707 12.032 0.000 37.317 51.896
## MarketID2 18.031 6.908 0.228 2.610 0.009 4.447 31.614
## MarketID3 46.940 3.993 1.005 11.756 0.000 39.089 54.791
## MarketID4 8.645 4.537 0.124 1.905 0.058 -0.277 17.567
## MarketID5 15.137 4.628 0.284 3.270 0.001 6.036 24.238
## MarketID6 -3.015 4.172 -0.057 -0.723 0.470 -11.219 5.189
## MarketID7 8.902 4.653 0.170 1.913 0.057 -0.248 18.052
## MarketID8 8.265 4.392 0.143 1.882 0.061 -0.370 16.900
## MarketID9 6.770 4.680 0.104 1.447 0.149 -2.432 15.973
## MarketID10 13.683 4.004 0.297 3.417 0.001 5.810 21.557
## MarketSizeMedium NA 2.764 -0.278 -3.397 0.001 NA NA
## MarketSizeSmall NA 2.980 -0.177 -3.188 0.002 NA NA
## Promotion2 -9.389 0.800 -0.017 -0.722 0.471 -14.824 -3.954
## Promotion3 -9.502 0.266 -0.007 -0.922 0.357 -15.362 -3.642
## Week -0.578 0.242 0.008 0.503 0.615 -2.152 0.996
## MarketID2:MarketSizeMedium NA 0.152 0.000 -1.076 0.283 NA NA
## MarketID3:MarketSizeMedium NA 0.194 0.000 0.986 0.325 NA NA
## MarketID4:MarketSizeMedium NA 0.180 0.000 -0.090 0.928 NA NA
## MarketID5:MarketSizeMedium NA 0.184 0.003 0.990 0.323 NA NA
## MarketID6:MarketSizeMedium NA 0.193 -0.005 -1.369 0.172 NA NA
## MarketID7:MarketSizeMedium NA 0.146 -0.002 -0.720 0.472 NA NA
## MarketID8:MarketSizeMedium NA 0.219 0.005 1.221 0.223 NA NA
## MarketID9:MarketSizeMedium NA 0.138 -0.002 -0.793 0.428 NA NA
## MarketID10:MarketSizeMedium NA 2.896 0.000 0.416 0.678 NA NA
## MarketID2:MarketSizeSmall NA 3.004 0.053 1.395 0.164 NA NA
## MarketID3:MarketSizeSmall NA 3.298 0.000 -0.151 0.880 NA NA
## MarketID4:MarketSizeSmall NA 2.777 0.023 0.570 0.569 NA NA
## MarketID5:MarketSizeSmall NA 3.277 0.000 0.128 0.898 NA NA
## MarketID6:MarketSizeSmall NA 3.130 0.000 0.767 0.444 NA NA
## MarketID7:MarketSizeSmall NA 3.334 0.000 1.402 0.162 NA NA
## MarketID8:MarketSizeSmall NA 2.568 0.000 0.106 0.916 NA NA
## MarketID9:MarketSizeSmall NA 4.586 0.000 1.372 0.171 NA NA
## MarketID10:MarketSizeSmall NA 2.873 0.000 1.738 0.083 NA NA
## MarketID1:AgeOfStore -0.245 4.669 1.170 1.500 0.134 -0.767 0.277
## MarketID2:AgeOfStore 0.122 3.683 0.551 0.770 0.442 -0.354 0.597
## MarketID3:AgeOfStore -0.163 3.432 0.838 1.104 0.270 -0.461 0.135
## MarketID4:AgeOfStore 0.191 3.127 0.609 1.173 0.242 -0.190 0.573
## MarketID5:AgeOfStore -0.016 3.195 1.700 2.284 0.023 -0.370 0.337
## MarketID6:AgeOfStore 0.183 3.360 0.613 0.885 0.377 -0.180 0.545
## MarketID7:AgeOfStore -0.265 3.146 0.725 1.216 0.225 -0.645 0.115
## MarketID8:AgeOfStore -0.105 1.253 0.101 0.296 0.767 -0.392 0.182
## MarketID9:AgeOfStore 0.267 0.910 0.062 0.447 0.655 -0.163 0.697
## MarketID10:AgeOfStore -0.109 1.147 0.359 1.491 0.137 -0.381 0.162
## MarketID2:Promotion2 NA 0.972 0.000 -0.127 0.899 NA NA
## MarketID3:Promotion2 1.204 0.989 0.000 0.031 0.975 -4.490 6.897
## MarketID4:Promotion2 4.191 0.975 0.004 0.425 0.671 -1.715 10.098
## MarketID5:Promotion2 -0.497 1.024 0.006 0.362 0.718 -6.982 5.988
## MarketID6:Promotion2 1.583 1.086 0.025 1.801 0.072 -3.877 7.044
## MarketID7:Promotion2 0.419 0.902 0.016 1.769 0.078 -6.025 6.863
## MarketID8:Promotion2 2.400 0.125 -0.001 -0.934 0.351 -3.754 8.554
## MarketID9:Promotion2 4.675 0.117 0.001 0.912 0.362 -1.881 11.231
## MarketID10:Promotion2 0.271 0.034 0.000 0.432 0.666 -4.779 5.322
## MarketID2:Promotion3 6.293 0.559 -0.006 -0.992 0.322 -2.724 15.310
## MarketID3:Promotion3 4.993 0.577 -0.007 -0.843 0.400 -0.656 10.643
## MarketID4:Promotion3 7.004 NA 0.074 NA NA -2.176 16.185
## MarketID5:Promotion3 2.835 NA 0.465 NA NA -4.408 10.077
## MarketID6:Promotion3 3.790 NA 0.078 NA NA -2.960 10.539
## MarketID7:Promotion3 3.667 NA 0.201 NA NA -2.481 9.815
## MarketID8:Promotion3 7.297 NA -0.035 NA NA 1.015 13.579
## MarketID9:Promotion3 2.972 NA 0.092 NA NA -3.634 9.578
## MarketID10:Promotion3 3.827 NA 0.075 NA NA -2.360 10.013
## MarketID2:Week 0.372 NA 0.241 NA NA -2.093 2.836
## MarketID3:Week 0.407 NA 0.801 NA NA -1.383 2.197
## MarketID4:Week 1.710 NA -0.361 NA NA -0.545 3.966
## MarketID5:Week -0.123 NA -0.497 NA NA -2.035 1.789
## MarketID6:Week 0.031 NA -0.029 NA NA -1.914 1.975
## MarketID7:Week 0.414 NA -0.013 NA NA -1.502 2.331
## MarketID8:Week 0.371 NA 0.006 NA NA -1.642 2.384
## MarketID9:Week 1.956 NA -0.007 NA NA -0.179 4.092
## MarketID10:Week 1.596 NA 0.011 NA NA -0.178 3.371
## MarketSizeMedium:AgeOfStore NA NA -0.007 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA 0.046 NA NA NA NA
## MarketSizeMedium:Promotion2 NA NA -0.007 NA NA NA NA
## MarketSizeSmall:Promotion2 NA NA -0.001 NA NA NA NA
## MarketSizeMedium:Promotion3 NA NA 0.006 NA NA NA NA
## MarketSizeSmall:Promotion3 NA NA -0.001 NA NA NA NA
## MarketSizeMedium:Week NA NA 0.109 NA NA NA NA
## MarketSizeSmall:Week NA NA 0.215 NA NA NA NA
## Promotion2:AgeOfStore -0.116 NA -0.168 NA NA -0.361 0.129
## Promotion3:AgeOfStore 0.107 NA 0.564 NA NA -0.123 0.337
## Week:AgeOfStore 0.015 NA 0.536 NA NA -0.052 0.081
## Promotion2:Week -0.555 NA 0.197 NA NA -1.655 0.545
## Promotion3:Week -0.486 NA 0.372 NA NA -1.620 0.648
## ---------------------------------------------------------------------------------------------------------
##
##
## - MarketID:Promotion
##
## Backward Elimination: Step 2
##
## Variable MarketID:Promotion Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.957 RMSE 5.070
## R-Squared 0.916 Coef. Var 9.532
## Adj. R-Squared 0.907 MSE 25.708
## Pred R-Squared 0.898 MAE 3.873
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108906.036 40 2722.651 105.906 0.0000
## Residual 10000.451 389 25.708
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 41.861 2.913 14.372 0.000 36.135 47.588
## MarketID2 21.022 5.581 0.266 3.767 0.000 10.049 31.995
## MarketID3 50.103 3.236 1.072 15.484 0.000 43.741 56.465
## MarketID4 11.459 3.932 0.164 2.914 0.004 3.728 19.190
## MarketID5 16.478 3.213 0.309 5.129 0.000 10.161 22.795
## MarketID6 -0.083 3.304 -0.002 -0.025 0.980 -6.580 6.414
## MarketID7 11.072 3.253 0.212 3.403 0.001 4.676 17.469
## MarketID8 12.340 3.465 0.213 3.561 0.000 5.527 19.153
## MarketID9 11.102 3.787 0.170 2.932 0.004 3.656 18.548
## MarketID10 16.189 3.229 0.351 5.013 0.000 9.840 22.538
## MarketSizeMedium NA 1.986 -0.260 -4.424 0.000 NA NA
## MarketSizeSmall NA 1.899 -0.100 -2.843 0.005 NA NA
## Promotion2 -8.787 0.795 -0.015 -0.645 0.519 -12.692 -4.882
## Promotion3 -5.400 0.224 -0.003 -0.399 0.690 -9.134 -1.665
## Week -0.513 0.214 0.007 0.490 0.625 -2.075 1.050
## MarketID2:MarketSizeMedium NA 0.147 0.000 -1.133 0.258 NA NA
## MarketID3:MarketSizeMedium NA 0.190 0.000 1.042 0.298 NA NA
## MarketID4:MarketSizeMedium NA 0.175 0.000 -0.184 0.854 NA NA
## MarketID5:MarketSizeMedium NA 0.169 0.003 0.978 0.328 NA NA
## MarketID6:MarketSizeMedium NA 0.168 -0.004 -1.356 0.176 NA NA
## MarketID7:MarketSizeMedium NA 0.142 -0.001 -0.460 0.646 NA NA
## MarketID8:MarketSizeMedium NA 0.205 0.003 0.779 0.437 NA NA
## MarketID9:MarketSizeMedium NA 0.136 -0.002 -0.832 0.406 NA NA
## MarketID10:MarketSizeMedium NA 1.245 0.000 0.222 0.824 NA NA
## MarketID2:MarketSizeSmall NA 0.904 0.004 0.356 0.722 NA NA
## MarketID3:MarketSizeSmall NA 1.138 0.000 1.455 0.147 NA NA
## MarketID4:MarketSizeSmall NA 0.963 -0.002 -0.163 0.871 NA NA
## MarketID5:MarketSizeSmall NA 0.977 0.000 -0.043 0.965 NA NA
## MarketID6:MarketSizeSmall NA 0.967 0.000 0.345 0.730 NA NA
## MarketID7:MarketSizeSmall NA 1.016 0.000 0.212 0.832 NA NA
## MarketID8:MarketSizeSmall NA 1.078 0.000 1.620 0.106 NA NA
## MarketID9:MarketSizeSmall NA 0.895 0.000 1.706 0.089 NA NA
## MarketID10:MarketSizeSmall NA 1.424 0.000 0.563 0.574 NA NA
## MarketID1:AgeOfStore -0.089 2.335 0.569 1.460 0.145 -0.530 0.351
## MarketID2:AgeOfStore 0.105 1.460 -0.134 -0.473 0.637 -0.316 0.525
## MarketID3:AgeOfStore -0.166 3.006 0.382 0.575 0.566 -0.455 0.122
## MarketID4:AgeOfStore 0.198 0.117 -0.020 -1.017 0.310 -0.175 0.571
## MarketID5:AgeOfStore -0.032 0.106 0.034 1.379 0.169 -0.376 0.312
## MarketID6:AgeOfStore 0.165 0.034 0.003 0.381 0.703 -0.167 0.497
## MarketID7:AgeOfStore -0.228 0.556 -0.097 -0.925 0.355 -0.558 0.103
## MarketID8:AgeOfStore -0.065 0.570 -0.115 -0.744 0.457 -0.343 0.213
## MarketID9:AgeOfStore 0.160 NA 3.183 NA NA -0.244 0.563
## MarketID10:AgeOfStore -0.113 NA 10.518 NA NA -0.381 0.155
## MarketID2:Week 0.277 NA 0.408 NA NA -2.171 2.726
## MarketID3:Week 0.322 NA 0.965 NA NA -1.455 2.099
## MarketID4:Week 1.655 NA -0.003 NA NA -0.582 3.892
## MarketID5:Week -0.157 NA 0.579 NA NA -2.051 1.737
## MarketID6:Week -0.042 NA 0.629 NA NA -1.964 1.879
## MarketID7:Week 0.334 NA 0.579 NA NA -1.568 2.236
## MarketID8:Week 0.215 NA 0.796 NA NA -1.782 2.213
## MarketID9:Week 1.747 NA -0.396 NA NA -0.373 3.867
## MarketID10:Week 1.528 NA -0.321 NA NA -0.232 3.288
## MarketSizeMedium:AgeOfStore NA NA -0.211 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA -0.022 NA NA NA NA
## MarketSizeMedium:Promotion2 0.802 NA 0.003 NA NA -1.998 3.601
## MarketSizeSmall:Promotion2 3.408 NA -0.002 NA NA -1.182 7.999
## MarketSizeMedium:Promotion3 -0.690 NA 0.005 NA NA -3.560 2.180
## MarketSizeSmall:Promotion3 1.728 NA 0.000 NA NA -4.182 7.639
## MarketSizeMedium:Week NA NA 0.015 NA NA NA NA
## MarketSizeSmall:Week NA NA -0.012 NA NA NA NA
## Promotion2:AgeOfStore -0.119 NA -0.022 NA NA -0.350 0.111
## Promotion3:AgeOfStore 0.147 NA 0.057 NA NA -0.063 0.356
## Week:AgeOfStore 0.013 NA -0.145 NA NA -0.053 0.079
## Promotion2:Week -0.514 NA 0.023 NA NA -1.607 0.579
## Promotion3:Week -0.424 NA 0.026 NA NA -1.544 0.696
## ---------------------------------------------------------------------------------------------------------
##
##
## - AgeOfStore:Week
##
## Backward Elimination: Step 3
##
## Variable AgeOfStore:Week Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.957 RMSE 5.065
## R-Squared 0.916 Coef. Var 9.521
## Adj. R-Squared 0.907 MSE 25.652
## Pred R-Squared 0.899 MAE 3.875
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108902.301 39 2792.367 108.857 0.0000
## Residual 10004.185 390 25.652
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 41.597 2.826 14.720 0.000 36.041 47.153
## MarketID2 20.863 5.559 0.264 3.753 0.000 9.933 31.793
## MarketID3 50.140 3.231 1.073 15.519 0.000 43.788 56.492
## MarketID4 11.460 3.928 0.164 2.917 0.004 3.737 19.182
## MarketID5 16.433 3.207 0.308 5.124 0.000 10.128 22.739
## MarketID6 -0.103 3.300 -0.002 -0.031 0.975 -6.592 6.385
## MarketID7 11.092 3.249 0.212 3.414 0.001 4.704 17.481
## MarketID8 12.197 3.441 0.211 3.544 0.000 5.432 18.963
## MarketID9 11.084 3.783 0.170 2.930 0.004 3.647 18.521
## MarketID10 16.235 3.223 0.352 5.037 0.000 9.898 22.573
## MarketSizeMedium NA 1.984 -0.259 -4.421 0.000 NA NA
## MarketSizeSmall NA 1.897 -0.100 -2.849 0.005 NA NA
## Promotion2 -8.769 0.746 -0.012 -0.549 0.584 -12.669 -4.869
## Promotion3 -5.405 0.207 -0.002 -0.276 0.783 -9.135 -1.675
## Week -0.409 0.196 0.009 0.702 0.483 -1.876 1.057
## MarketID2:MarketSizeMedium NA 0.122 0.000 -1.108 0.268 NA NA
## MarketID3:MarketSizeMedium NA 0.170 0.000 1.350 0.178 NA NA
## MarketID4:MarketSizeMedium NA 0.151 0.000 0.010 0.992 NA NA
## MarketID5:MarketSizeMedium NA 0.144 0.004 1.371 0.171 NA NA
## MarketID6:MarketSizeMedium NA 0.145 -0.004 -1.347 0.179 NA NA
## MarketID7:MarketSizeMedium NA 0.108 -0.001 -0.280 0.780 NA NA
## MarketID8:MarketSizeMedium NA 0.180 0.003 1.097 0.273 NA NA
## MarketID9:MarketSizeMedium NA 0.109 -0.001 -0.755 0.451 NA NA
## MarketID10:MarketSizeMedium NA 1.233 0.000 0.276 0.783 NA NA
## MarketID2:MarketSizeSmall NA 0.902 0.004 0.345 0.731 NA NA
## MarketID3:MarketSizeSmall NA 1.137 0.000 1.460 0.145 NA NA
## MarketID4:MarketSizeSmall NA 0.961 -0.002 -0.142 0.887 NA NA
## MarketID5:MarketSizeSmall NA 0.976 0.000 -0.035 0.972 NA NA
## MarketID6:MarketSizeSmall NA 0.966 0.000 0.340 0.734 NA NA
## MarketID7:MarketSizeSmall NA 1.007 0.000 0.260 0.795 NA NA
## MarketID8:MarketSizeSmall NA 1.077 0.000 1.614 0.107 NA NA
## MarketID9:MarketSizeSmall NA 0.893 0.000 1.693 0.091 NA NA
## MarketID10:MarketSizeSmall NA 1.422 0.000 0.556 0.579 NA NA
## MarketID1:AgeOfStore -0.057 2.332 0.567 1.455 0.146 -0.465 0.351
## MarketID2:AgeOfStore 0.137 1.458 -0.134 -0.474 0.636 -0.247 0.522
## MarketID3:AgeOfStore -0.135 3.003 0.381 0.574 0.566 -0.375 0.105
## MarketID4:AgeOfStore 0.230 0.117 -0.020 -1.036 0.301 -0.105 0.564
## MarketID5:AgeOfStore 0.002 0.106 0.034 1.370 0.172 -0.295 0.298
## MarketID6:AgeOfStore 0.198 0.555 -0.106 -0.922 0.357 -0.086 0.482
## MarketID7:AgeOfStore -0.196 0.569 -0.079 -0.733 0.464 -0.481 0.090
## MarketID8:AgeOfStore -0.030 NA 5.652 NA NA -0.242 0.182
## MarketID9:AgeOfStore 0.197 NA 7.592 NA NA -0.156 0.551
## MarketID10:AgeOfStore -0.082 NA 2.406 NA NA -0.296 0.132
## MarketID2:Week 0.340 NA 0.586 NA NA -2.084 2.764
## MarketID3:Week 0.311 NA -0.006 NA NA -1.463 2.085
## MarketID4:Week 1.660 NA 0.426 NA NA -0.575 3.894
## MarketID5:Week -0.136 NA 0.638 NA NA -2.025 1.753
## MarketID6:Week -0.034 NA 0.565 NA NA -1.953 1.885
## MarketID7:Week 0.329 NA 0.846 NA NA -1.571 2.228
## MarketID8:Week 0.262 NA -0.431 NA NA -1.718 2.243
## MarketID9:Week 1.737 NA -0.244 NA NA -0.379 3.854
## MarketID10:Week 1.512 NA -0.024 NA NA -0.244 3.269
## MarketSizeMedium:AgeOfStore NA NA -0.023 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA 0.034 NA NA NA NA
## MarketSizeMedium:Promotion2 0.790 NA -0.003 NA NA -2.005 3.586
## MarketSizeSmall:Promotion2 3.394 NA 0.002 NA NA -1.191 7.978
## MarketSizeMedium:Promotion3 -0.691 NA 0.000 NA NA -3.559 2.176
## MarketSizeSmall:Promotion3 1.723 NA 0.002 NA NA -4.181 7.628
## MarketSizeMedium:Week NA NA -0.018 NA NA NA NA
## MarketSizeSmall:Week NA NA -0.002 NA NA NA NA
## Promotion2:AgeOfStore -0.121 NA 0.067 NA NA -0.352 0.109
## Promotion3:AgeOfStore 0.146 NA -0.029 NA NA -0.063 0.355
## Promotion2:Week -0.512 NA 0.028 NA NA -1.604 0.580
## Promotion3:Week -0.417 NA 0.025 NA NA -1.535 0.701
## ---------------------------------------------------------------------------------------------------------
##
##
## - Promotion:Week
##
## Backward Elimination: Step 4
##
## Variable Promotion:Week Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.957 RMSE 5.058
## R-Squared 0.916 Coef. Var 9.509
## Adj. R-Squared 0.908 MSE 25.585
## Pred R-Squared 0.899 MAE 3.873
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108877.331 37 2942.631 115.016 0.0000
## Residual 10029.155 392 25.585
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------------------------
## (Intercept) 42.341 2.718 15.579 0.000 36.998 47.684
## MarketID2 20.979 5.497 0.266 3.816 0.000 10.172 31.787
## MarketID3 50.109 3.218 1.073 15.572 0.000 43.782 56.435
## MarketID4 11.426 3.913 0.164 2.920 0.004 3.733 19.119
## MarketID5 16.797 3.180 0.315 5.282 0.000 10.545 23.049
## MarketID6 -0.071 3.295 -0.001 -0.022 0.983 -6.549 6.407
## MarketID7 11.190 3.236 0.214 3.459 0.001 4.829 17.551
## MarketID8 12.254 3.426 0.212 3.576 0.000 5.517 18.990
## MarketID9 11.179 3.770 0.171 2.965 0.003 3.767 18.591
## MarketID10 16.209 3.212 0.351 5.046 0.000 9.893 22.524
## MarketSizeMedium NA 1.438 -0.297 -6.979 0.000 NA NA
## MarketSizeSmall NA 1.341 -0.119 -4.775 0.000 NA NA
## Promotion2 -10.036 0.688 -0.020 -1.006 0.315 -12.863 -7.209
## Promotion3 -6.404 0.207 -0.002 -0.289 0.773 -9.040 -3.767
## Week -0.692 0.195 0.009 0.703 0.483 -2.044 0.661
## MarketID2:MarketSizeMedium NA 0.122 0.000 -1.121 0.263 NA NA
## MarketID3:MarketSizeMedium NA 0.170 0.000 1.370 0.171 NA NA
## MarketID4:MarketSizeMedium NA 0.150 0.000 0.022 0.983 NA NA
## MarketID5:MarketSizeMedium NA 0.144 0.004 1.380 0.168 NA NA
## MarketID6:MarketSizeMedium NA 0.145 -0.004 -1.368 0.172 NA NA
## MarketID7:MarketSizeMedium NA 0.108 -0.001 -0.265 0.791 NA NA
## MarketID8:MarketSizeMedium NA 0.179 0.003 1.076 0.283 NA NA
## MarketID9:MarketSizeMedium NA 0.109 -0.001 -0.746 0.456 NA NA
## MarketID10:MarketSizeMedium NA 1.193 0.000 0.238 0.812 NA NA
## MarketID2:MarketSizeSmall NA 0.891 0.004 0.339 0.735 NA NA
## MarketID3:MarketSizeSmall NA 1.132 0.000 1.451 0.148 NA NA
## MarketID4:MarketSizeSmall NA 0.948 -0.004 -0.297 0.767 NA NA
## MarketID5:MarketSizeSmall NA 0.974 0.000 -0.056 0.955 NA NA
## MarketID6:MarketSizeSmall NA 0.956 0.000 0.314 0.754 NA NA
## MarketID7:MarketSizeSmall NA 0.996 0.000 0.237 0.813 NA NA
## MarketID8:MarketSizeSmall NA 1.069 0.000 1.599 0.111 NA NA
## MarketID9:MarketSizeSmall NA 0.891 0.000 1.684 0.093 NA NA
## MarketID10:MarketSizeSmall NA 1.419 0.000 0.524 0.600 NA NA
## MarketID1:AgeOfStore -0.060 2.328 0.576 1.480 0.140 -0.467 0.347
## MarketID2:AgeOfStore 0.137 1.454 -0.148 -0.524 0.601 -0.247 0.521
## MarketID3:AgeOfStore -0.137 2.995 0.369 0.557 0.578 -0.376 0.103
## MarketID4:AgeOfStore 0.233 0.117 -0.020 -1.049 0.295 -0.101 0.566
## MarketID5:AgeOfStore 0.003 0.106 0.034 1.366 0.173 -0.292 0.299
## MarketID6:AgeOfStore 0.199 NA 4.324 NA NA -0.084 0.483
## MarketID7:AgeOfStore -0.198 NA 9.497 NA NA -0.484 0.087
## MarketID8:AgeOfStore -0.029 NA 3.095 NA NA -0.240 0.183
## MarketID9:AgeOfStore 0.193 NA 2.543 NA NA -0.160 0.546
## MarketID10:AgeOfStore -0.081 NA -0.015 NA NA -0.294 0.132
## MarketID2:Week 0.284 NA 0.399 NA NA -2.061 2.629
## MarketID3:Week 0.302 NA 0.717 NA NA -1.450 2.054
## MarketID4:Week 1.642 NA 0.430 NA NA -0.583 3.867
## MarketID5:Week -0.282 NA 0.848 NA NA -2.145 1.582
## MarketID6:Week -0.055 NA -0.511 NA NA -1.970 1.861
## MarketID7:Week 0.300 NA -0.334 NA NA -1.579 2.180
## MarketID8:Week 0.236 NA -0.034 NA NA -1.723 2.194
## MarketID9:Week 1.710 NA -0.003 NA NA -0.393 3.813
## MarketID10:Week 1.500 NA 0.008 NA NA -0.251 3.251
## MarketSizeMedium:AgeOfStore NA NA -0.056 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA 0.058 NA NA NA NA
## MarketSizeMedium:Promotion2 0.744 NA 0.000 NA NA -2.046 3.534
## MarketSizeSmall:Promotion2 3.446 NA 0.002 NA NA -1.131 8.024
## MarketSizeMedium:Promotion3 -0.762 NA -0.005 NA NA -3.620 2.097
## MarketSizeSmall:Promotion3 1.669 NA 0.000 NA NA -4.219 7.556
## MarketSizeMedium:Week NA NA 0.017 NA NA NA NA
## MarketSizeSmall:Week NA NA -0.004 NA NA NA NA
## Promotion2:AgeOfStore -0.123 NA 0.096 NA NA -0.352 0.107
## Promotion3:AgeOfStore 0.145 NA 0.108 NA NA -0.064 0.354
## ----------------------------------------------------------------------------------------------------------
##
##
## - MarketSize:Promotion
##
## Backward Elimination: Step 5
##
## Variable MarketSize:Promotion Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.957 RMSE 5.053
## R-Squared 0.915 Coef. Var 9.499
## Adj. R-Squared 0.908 MSE 25.533
## Pred R-Squared 0.900 MAE 3.891
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108795.553 33 3296.835 129.122 0.0000
## Residual 10110.934 396 25.533
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 42.102 2.630 16.008 0.000 36.931 47.273
## MarketID2 23.673 4.296 0.300 5.511 0.000 15.228 32.119
## MarketID3 50.546 3.056 1.082 16.541 0.000 44.538 56.554
## MarketID4 13.187 3.622 0.189 3.641 0.000 6.066 20.308
## MarketID5 16.892 3.172 0.317 5.325 0.000 10.655 23.129
## MarketID6 0.125 3.270 0.002 0.038 0.970 -6.304 6.553
## MarketID7 11.139 3.231 0.213 3.448 0.001 4.788 17.490
## MarketID8 12.368 3.409 0.214 3.629 0.000 5.667 19.069
## MarketID9 11.391 3.750 0.175 3.038 0.003 4.019 18.764
## MarketID10 16.252 3.037 0.352 5.352 0.000 10.282 22.223
## MarketSizeMedium NA 1.094 -0.273 -8.431 0.000 NA NA
## MarketSizeSmall NA 1.111 -0.126 -6.122 0.000 NA NA
## Promotion2 -9.223 0.687 -0.020 -0.991 0.322 -11.374 -7.073
## Promotion3 -6.798 0.202 -0.001 -0.238 0.812 -8.982 -4.615
## Week -0.681 0.167 0.005 0.404 0.686 -2.031 0.670
## MarketID2:MarketSizeMedium NA 0.119 0.000 -1.324 0.186 NA NA
## MarketID3:MarketSizeMedium NA 0.167 0.000 1.436 0.152 NA NA
## MarketID4:MarketSizeMedium NA 0.147 0.000 0.019 0.985 NA NA
## MarketID5:MarketSizeMedium NA 0.142 0.003 1.279 0.202 NA NA
## MarketID6:MarketSizeMedium NA 0.142 -0.004 -1.359 0.175 NA NA
## MarketID7:MarketSizeMedium NA 0.106 -0.001 -0.403 0.687 NA NA
## MarketID8:MarketSizeMedium NA 0.178 0.003 0.968 0.333 NA NA
## MarketID9:MarketSizeMedium NA 0.108 -0.001 -0.833 0.406 NA NA
## MarketID10:MarketSizeMedium NA 1.191 0.000 0.238 0.812 NA NA
## MarketID2:MarketSizeSmall NA 0.890 0.003 0.304 0.761 NA NA
## MarketID3:MarketSizeSmall NA 1.124 0.000 1.405 0.161 NA NA
## MarketID4:MarketSizeSmall NA 0.946 -0.004 -0.320 0.749 NA NA
## MarketID5:MarketSizeSmall NA 0.973 0.000 -0.047 0.962 NA NA
## MarketID6:MarketSizeSmall NA 0.955 0.000 0.313 0.754 NA NA
## MarketID7:MarketSizeSmall NA 0.995 0.000 0.241 0.809 NA NA
## MarketID8:MarketSizeSmall NA 1.068 0.000 1.577 0.116 NA NA
## MarketID9:MarketSizeSmall NA 0.888 0.000 1.631 0.104 NA NA
## MarketID10:MarketSizeSmall NA 0.114 0.000 -1.061 0.289 NA NA
## MarketID1:AgeOfStore -0.048 0.100 0.026 1.562 0.119 -0.446 0.350
## MarketID2:AgeOfStore 0.067 NA 4.601 NA NA -0.260 0.395
## MarketID3:AgeOfStore -0.158 NA 11.175 NA NA -0.392 0.076
## MarketID4:AgeOfStore 0.240 NA 2.188 NA NA -0.089 0.569
## MarketID5:AgeOfStore 0.003 NA 3.935 NA NA -0.287 0.292
## MarketID6:AgeOfStore 0.181 NA 0.026 NA NA -0.097 0.460
## MarketID7:AgeOfStore -0.193 NA 2.111 NA NA -0.473 0.086
## MarketID8:AgeOfStore -0.043 NA 3.351 NA NA -0.251 0.166
## MarketID9:AgeOfStore 0.172 NA 1.725 NA NA -0.178 0.522
## MarketID10:AgeOfStore -0.090 NA 3.412 NA NA -0.301 0.122
## MarketID2:Week 0.283 NA -0.329 NA NA -2.059 2.625
## MarketID3:Week 0.271 NA -0.398 NA NA -1.479 2.020
## MarketID4:Week 1.580 NA -0.026 NA NA -0.630 3.789
## MarketID5:Week -0.302 NA -0.003 NA NA -2.162 1.558
## MarketID6:Week -0.046 NA 0.003 NA NA -1.960 1.867
## MarketID7:Week 0.299 NA -0.008 NA NA -1.578 2.176
## MarketID8:Week 0.240 NA 0.012 NA NA -1.716 2.196
## MarketID9:Week 1.685 NA 0.000 NA NA -0.415 3.785
## MarketID10:Week 1.449 NA 0.011 NA NA -0.297 3.196
## MarketSizeMedium:AgeOfStore NA NA -0.079 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA -0.011 NA NA NA NA
## MarketSizeMedium:Week NA NA 0.016 NA NA NA NA
## MarketSizeSmall:Week NA NA -0.005 NA NA NA NA
## Promotion2:AgeOfStore -0.121 NA 0.096 NA NA -0.345 0.103
## Promotion3:AgeOfStore 0.157 NA 0.096 NA NA -0.041 0.354
## ---------------------------------------------------------------------------------------------------------
##
##
## - Week
##
## Backward Elimination: Step 6
##
## Variable Week Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.957 RMSE 5.053
## R-Squared 0.915 Coef. Var 9.499
## Adj. R-Squared 0.908 MSE 25.533
## Pred R-Squared 0.900 MAE 3.891
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108795.553 33 3296.835 129.122 0.0000
## Residual 10110.934 396 25.533
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 42.102 2.630 16.008 0.000 36.931 47.273
## MarketID2 23.673 4.296 0.300 5.511 0.000 15.228 32.119
## MarketID3 50.546 3.056 1.082 16.541 0.000 44.538 56.554
## MarketID4 13.187 3.622 0.189 3.641 0.000 6.066 20.308
## MarketID5 16.892 3.172 0.317 5.325 0.000 10.655 23.129
## MarketID6 0.125 3.270 0.002 0.038 0.970 -6.304 6.553
## MarketID7 11.139 3.231 0.213 3.448 0.001 4.788 17.490
## MarketID8 12.368 3.409 0.214 3.629 0.000 5.667 19.069
## MarketID9 11.391 3.750 0.175 3.038 0.003 4.019 18.764
## MarketID10 16.252 3.037 0.352 5.352 0.000 10.282 22.223
## MarketSizeMedium NA 1.094 -0.273 -8.431 0.000 NA NA
## MarketSizeSmall NA 1.111 -0.126 -6.122 0.000 NA NA
## Promotion2 -9.223 0.202 -0.001 -0.238 0.812 -11.374 -7.073
## Promotion3 -6.798 0.167 0.002 0.404 0.686 -8.982 -4.615
## MarketID2:MarketSizeMedium NA 0.119 0.000 -1.324 0.186 NA NA
## MarketID3:MarketSizeMedium NA 0.167 0.000 1.436 0.152 NA NA
## MarketID4:MarketSizeMedium NA 0.147 0.000 0.019 0.985 NA NA
## MarketID5:MarketSizeMedium NA 0.142 0.003 1.279 0.202 NA NA
## MarketID6:MarketSizeMedium NA 0.142 -0.004 -1.359 0.175 NA NA
## MarketID7:MarketSizeMedium NA 0.106 -0.001 -0.403 0.687 NA NA
## MarketID8:MarketSizeMedium NA 0.178 0.003 0.968 0.333 NA NA
## MarketID9:MarketSizeMedium NA 0.108 -0.001 -0.833 0.406 NA NA
## MarketID10:MarketSizeMedium NA 0.687 0.000 -0.991 0.322 NA NA
## MarketID2:MarketSizeSmall NA 0.973 -0.005 -0.409 0.683 NA NA
## MarketID3:MarketSizeSmall NA 0.566 0.000 -0.724 0.469 NA NA
## MarketID4:MarketSizeSmall NA 0.889 0.013 1.011 0.313 NA NA
## MarketID5:MarketSizeSmall NA 0.651 0.000 -1.510 0.132 NA NA
## MarketID6:MarketSizeSmall NA 0.690 0.000 -1.053 0.293 NA NA
## MarketID7:MarketSizeSmall NA 0.663 0.000 -0.576 0.565 NA NA
## MarketID8:MarketSizeSmall NA 0.720 0.000 -0.612 0.541 NA NA
## MarketID9:MarketSizeSmall NA 0.817 0.000 1.228 0.220 NA NA
## MarketID10:MarketSizeSmall NA 0.564 0.000 1.363 0.174 NA NA
## MarketID1:AgeOfStore -0.048 0.114 -0.020 -1.061 0.289 -0.446 0.350
## MarketID2:AgeOfStore 0.067 0.100 0.030 1.562 0.119 -0.260 0.395
## MarketID3:AgeOfStore -0.158 NA 5.234 NA NA -0.392 0.076
## MarketID4:AgeOfStore 0.240 NA 8.387 NA NA -0.089 0.569
## MarketID5:AgeOfStore 0.003 NA 3.072 NA NA -0.287 0.292
## MarketID6:AgeOfStore 0.181 NA 3.482 NA NA -0.097 0.460
## MarketID7:AgeOfStore -0.193 NA 0.024 NA NA -0.473 0.086
## MarketID8:AgeOfStore -0.043 NA 3.018 NA NA -0.251 0.166
## MarketID9:AgeOfStore 0.172 NA 1.873 NA NA -0.178 0.522
## MarketID10:AgeOfStore -0.090 NA 2.391 NA NA -0.301 0.122
## MarketID1:Week -0.681 NA 0.786 NA NA -2.031 0.670
## MarketID2:Week -0.398 NA -0.329 NA NA -2.311 1.516
## MarketID3:Week -0.410 NA -0.398 NA NA -1.522 0.703
## MarketID4:Week 0.899 NA -0.002 NA NA -0.850 2.647
## MarketID5:Week -0.983 NA 0.004 NA NA -2.263 0.297
## MarketID6:Week -0.727 NA -0.008 NA NA -2.084 0.630
## MarketID7:Week -0.382 NA 0.013 NA NA -1.686 0.922
## MarketID8:Week -0.441 NA 0.000 NA NA -1.856 0.974
## MarketID9:Week 1.004 NA 0.008 NA NA -0.603 2.611
## MarketID10:Week 0.768 NA -0.011 NA NA -0.340 1.877
## MarketSizeMedium:AgeOfStore NA NA -0.018 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA 0.043 NA NA NA NA
## MarketSizeMedium:Week NA NA -0.008 NA NA NA NA
## MarketSizeSmall:Week NA NA -0.035 NA NA NA NA
## Promotion2:AgeOfStore -0.121 NA -0.134 NA NA -0.345 0.103
## Promotion3:AgeOfStore 0.157 NA -0.146 NA NA -0.041 0.354
## ---------------------------------------------------------------------------------------------------------
##
##
## - MarketID:Week
##
## Backward Elimination: Step 7
##
## Variable MarketID:Week Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.956 RMSE 5.055
## R-Squared 0.913 Coef. Var 9.502
## Adj. R-Squared 0.908 MSE 25.550
## Pred R-Squared 0.902 MAE 3.902
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108609.919 26 4177.305 163.497 0.0000
## Residual 10296.568 403 25.550
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 41.521 2.105 19.728 0.000 37.383 45.658
## MarketID2 22.405 3.542 0.284 6.325 0.000 15.442 29.369
## MarketID3 49.663 2.427 1.063 20.462 0.000 44.891 54.434
## MarketID4 15.217 2.922 0.218 5.207 0.000 9.472 20.962
## MarketID5 16.255 2.198 0.305 7.395 0.000 11.934 20.577
## MarketID6 0.030 2.188 0.001 0.014 0.989 -4.271 4.331
## MarketID7 11.877 2.214 0.227 5.364 0.000 7.524 16.229
## MarketID8 12.966 2.277 0.224 5.695 0.000 8.491 17.442
## MarketID9 15.977 2.439 0.245 6.550 0.000 11.182 20.772
## MarketID10 18.270 2.402 0.396 7.605 0.000 13.547 22.993
## MarketSizeMedium NA 1.093 -0.274 -8.467 0.000 NA NA
## MarketSizeSmall NA 1.110 -0.126 -6.093 0.000 NA NA
## Promotion2 -9.257 0.202 -0.001 -0.229 0.819 -11.406 -7.108
## Promotion3 -6.762 0.167 0.002 0.420 0.675 -8.944 -4.581
## MarketID2:MarketSizeMedium NA 0.119 0.000 -1.323 0.187 NA NA
## MarketID3:MarketSizeMedium NA 0.167 0.000 1.421 0.156 NA NA
## MarketID4:MarketSizeMedium NA 0.146 0.000 -0.074 0.941 NA NA
## MarketID5:MarketSizeMedium NA 0.142 0.003 1.269 0.205 NA NA
## MarketID6:MarketSizeMedium NA 0.142 -0.004 -1.356 0.176 NA NA
## MarketID7:MarketSizeMedium NA 0.106 -0.001 -0.401 0.689 NA NA
## MarketID8:MarketSizeMedium NA 0.178 0.003 0.909 0.364 NA NA
## MarketID9:MarketSizeMedium NA 0.108 -0.001 -0.817 0.414 NA NA
## MarketID10:MarketSizeMedium NA 0.400 0.000 0.455 0.650 NA NA
## MarketID2:MarketSizeSmall NA 0.286 -0.006 -1.570 0.117 NA NA
## MarketID3:MarketSizeSmall NA 0.657 0.000 0.471 0.638 NA NA
## MarketID4:MarketSizeSmall NA 0.114 -0.002 -1.030 0.304 NA NA
## MarketID5:MarketSizeSmall NA 0.100 0.000 1.544 0.123 NA NA
## MarketID6:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID7:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID8:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID9:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID10:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID1:AgeOfStore -0.046 NA 1.984 NA NA -0.444 0.352
## MarketID2:AgeOfStore 0.070 NA 2.520 NA NA -0.257 0.397
## MarketID3:AgeOfStore -0.157 NA 3.532 NA NA -0.391 0.077
## MarketID4:AgeOfStore 0.238 NA 3.031 NA NA -0.091 0.566
## MarketID5:AgeOfStore -0.011 NA -2.156 NA NA -0.298 0.277
## MarketID6:AgeOfStore 0.180 NA -1.394 NA NA -0.099 0.458
## MarketID7:AgeOfStore -0.193 NA -0.009 NA NA -0.472 0.087
## MarketID8:AgeOfStore -0.042 NA 0.019 NA NA -0.251 0.166
## MarketID9:AgeOfStore 0.162 NA -0.024 NA NA -0.188 0.511
## MarketID10:AgeOfStore -0.088 NA 0.050 NA NA -0.299 0.123
## MarketSizeMedium:AgeOfStore NA NA -0.004 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA 0.045 NA NA NA NA
## MarketSizeLarge:Week 0.182 NA -0.015 NA NA -0.604 0.968
## MarketSizeMedium:Week -0.448 NA -0.004 NA NA -1.009 0.113
## MarketSizeSmall:Week 0.310 NA 0.008 NA NA -0.982 1.601
## Promotion2:AgeOfStore -0.117 NA -0.030 NA NA -0.341 0.106
## Promotion3:AgeOfStore 0.155 NA 0.065 NA NA -0.042 0.352
## ---------------------------------------------------------------------------------------------------------
##
##
## - MarketSize:Week
##
## Backward Elimination: Step 8
##
## Variable MarketSize:Week Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.955 RMSE 5.054
## R-Squared 0.913 Coef. Var 9.501
## Adj. R-Squared 0.908 MSE 25.543
## Pred R-Squared 0.902 MAE 3.940
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108535.982 23 4718.956 184.745 0.0000
## Residual 10370.505 406 25.543
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 40.361 1.971 20.476 0.000 36.486 44.236
## MarketID2 24.307 3.040 0.308 7.996 0.000 18.331 30.283
## MarketID3 51.254 2.089 1.097 24.535 0.000 47.147 55.361
## MarketID4 17.142 2.344 0.246 7.315 0.000 12.535 21.749
## MarketID5 16.382 2.196 0.307 7.458 0.000 12.064 20.700
## MarketID6 0.089 2.187 0.002 0.041 0.967 -4.210 4.389
## MarketID7 11.937 2.213 0.228 5.393 0.000 7.586 16.288
## MarketID8 12.957 2.276 0.224 5.692 0.000 8.482 17.432
## MarketID9 15.889 2.438 0.243 6.517 0.000 11.096 20.682
## MarketID10 19.863 2.070 0.431 9.597 0.000 15.794 23.931
## MarketSizeMedium NA 1.093 -0.274 -8.477 0.000 NA NA
## MarketSizeSmall NA 1.109 -0.125 -6.056 0.000 NA NA
## Promotion2 -9.266 0.202 -0.001 -0.196 0.845 -11.414 -7.117
## Promotion3 -6.715 0.167 0.002 0.428 0.669 -8.895 -4.536
## MarketID2:MarketSizeMedium NA 0.119 0.000 -1.338 0.182 NA NA
## MarketID3:MarketSizeMedium NA 0.167 0.000 1.396 0.164 NA NA
## MarketID4:MarketSizeMedium NA 0.146 0.000 -0.165 0.869 NA NA
## MarketID5:MarketSizeMedium NA 0.142 0.003 1.247 0.213 NA NA
## MarketID6:MarketSizeMedium NA 0.142 -0.004 -1.381 0.168 NA NA
## MarketID7:MarketSizeMedium NA 0.106 -0.001 -0.410 0.682 NA NA
## MarketID8:MarketSizeMedium NA 0.178 0.003 0.921 0.358 NA NA
## MarketID9:MarketSizeMedium NA 0.107 -0.001 -0.824 0.411 NA NA
## MarketID10:MarketSizeMedium NA 0.114 0.000 -0.987 0.324 NA NA
## MarketID2:MarketSizeSmall NA 0.100 0.002 1.535 0.126 NA NA
## MarketID3:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeSmall NA NA 0.735 NA NA NA NA
## MarketID5:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID6:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID7:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID8:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID9:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID10:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID1:AgeOfStore -0.040 NA 3.318 NA NA -0.437 0.358
## MarketID2:AgeOfStore 0.071 NA -1.801 NA NA -0.256 0.399
## MarketID3:AgeOfStore -0.159 NA -1.485 NA NA -0.393 0.075
## MarketID4:AgeOfStore 0.233 NA -0.007 NA NA -0.095 0.562
## MarketID5:AgeOfStore -0.024 NA 0.017 NA NA -0.311 0.263
## MarketID6:AgeOfStore 0.176 NA -0.033 NA NA -0.102 0.455
## MarketID7:AgeOfStore -0.196 NA 0.044 NA NA -0.476 0.083
## MarketID8:AgeOfStore -0.043 NA -0.007 NA NA -0.251 0.165
## MarketID9:AgeOfStore 0.164 NA 0.027 NA NA -0.186 0.513
## MarketID10:AgeOfStore -0.089 NA -0.041 NA NA -0.300 0.123
## MarketSizeMedium:AgeOfStore NA NA -0.018 NA NA NA NA
## MarketSizeSmall:AgeOfStore NA NA 0.041 NA NA NA NA
## Promotion2:AgeOfStore -0.112 NA -0.030 NA NA -0.336 0.111
## Promotion3:AgeOfStore 0.154 NA -0.040 NA NA -0.043 0.351
## ---------------------------------------------------------------------------------------------------------
##
##
## - MarketID:AgeOfStore
##
## Backward Elimination: Step 9
##
## Variable MarketID:AgeOfStore Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.955 RMSE 5.049
## R-Squared 0.911 Coef. Var 9.492
## Adj. R-Squared 0.908 MSE 25.492
## Pred R-Squared 0.904 MAE 3.975
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108378.252 16 6773.641 265.715 0.0000
## Residual 10528.234 413 25.492
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 40.142 1.126 35.666 0.000 37.930 42.355
## MarketID2 23.286 2.027 0.295 11.488 0.000 19.302 27.271
## MarketID3 51.166 1.267 1.095 40.393 0.000 48.676 53.656
## MarketID4 17.982 1.561 0.258 11.517 0.000 14.913 21.052
## MarketID5 16.311 1.101 0.306 14.815 0.000 14.146 18.475
## MarketID6 1.833 1.087 0.034 1.686 0.092 -0.304 3.969
## MarketID7 10.631 1.072 0.203 9.914 0.000 8.523 12.739
## MarketID8 12.690 1.159 0.219 10.947 0.000 10.411 14.969
## MarketID9 17.443 1.222 0.267 14.269 0.000 15.040 19.846
## MarketID10 20.343 1.220 0.441 16.669 0.000 17.944 22.742
## MarketSizeMedium NA 0.999 -0.278 -9.411 0.000 NA NA
## MarketSizeSmall NA 1.073 -0.123 -6.153 0.000 NA NA
## Promotion2 -9.405 0.083 -0.004 -1.781 0.076 -11.369 -7.441
## Promotion3 -6.603 0.077 -0.001 -0.353 0.724 -8.712 -4.493
## MarketID2:MarketSizeMedium NA 0.119 0.000 1.164 0.245 NA NA
## MarketID3:MarketSizeMedium NA 0.094 0.000 -0.693 0.489 NA NA
## MarketID4:MarketSizeMedium NA 0.094 0.000 1.921 0.055 NA NA
## MarketID5:MarketSizeMedium NA NA 0.437 NA NA NA NA
## MarketID6:MarketSizeMedium NA NA 0.960 NA NA NA NA
## MarketID7:MarketSizeMedium NA NA 0.344 NA NA NA NA
## MarketID8:MarketSizeMedium NA NA 0.282 NA NA NA NA
## MarketID9:MarketSizeMedium NA NA 0.028 NA NA NA NA
## MarketID10:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID2:MarketSizeSmall NA NA 0.161 NA NA NA NA
## MarketID3:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeSmall NA NA 0.292 NA NA NA NA
## MarketID5:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID6:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID7:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID8:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID9:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID10:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketSizeLarge:AgeOfStore -0.148 NA 0.052 NA NA -0.311 0.015
## MarketSizeMedium:AgeOfStore -0.027 NA 9.566 NA NA -0.179 0.125
## MarketSizeSmall:AgeOfStore 0.139 NA 12.845 NA NA -0.096 0.374
## Promotion2:AgeOfStore -0.065 NA 6.079 NA NA -0.250 0.119
## Promotion3:AgeOfStore 0.181 NA 5.808 NA NA -0.004 0.365
## ---------------------------------------------------------------------------------------------------------
##
##
## - MarketSize:AgeOfStore
##
## Backward Elimination: Step 10
##
## Variable MarketSize:AgeOfStore Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.954 RMSE 5.069
## R-Squared 0.910 Coef. Var 9.530
## Adj. R-Squared 0.907 MSE 25.696
## Pred R-Squared 0.904 MAE 4.005
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108242.481 14 7731.606 300.883 0.0000
## Residual 10664.006 415 25.696
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------------------------
## (Intercept) 40.573 1.061 38.255 0.000 38.488 42.658
## MarketID2 25.611 1.433 0.324 17.870 0.000 22.794 28.428
## MarketID3 50.165 1.032 1.074 48.615 0.000 48.137 52.193
## MarketID4 19.305 1.273 0.277 15.166 0.000 16.803 21.807
## MarketID5 16.278 1.104 0.305 14.748 0.000 14.108 18.448
## MarketID6 1.768 1.091 0.033 1.621 0.106 -0.376 3.912
## MarketID7 10.650 1.076 0.203 9.894 0.000 8.534 12.766
## MarketID8 12.701 1.161 0.219 10.943 0.000 10.420 14.982
## MarketID9 17.397 1.226 0.267 14.185 0.000 14.987 19.808
## MarketID10 19.410 1.012 0.421 19.179 0.000 17.421 21.400
## MarketSizeMedium NA 0.970 -0.294 -10.259 0.000 NA NA
## MarketSizeSmall NA 1.063 -0.130 -6.559 0.000 NA NA
## Promotion2 -9.949 0.066 -0.002 -0.952 0.342 -11.855 -8.042
## Promotion3 -6.971 0.062 -0.002 -1.268 0.205 -9.060 -4.882
## MarketID2:MarketSizeMedium NA 0.067 0.000 2.179 0.030 NA NA
## MarketID3:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID5:MarketSizeMedium NA NA 0.362 NA NA NA NA
## MarketID6:MarketSizeMedium NA NA 0.305 NA NA NA NA
## MarketID7:MarketSizeMedium NA NA 0.034 NA NA NA NA
## MarketID8:MarketSizeMedium NA NA 0.184 NA NA NA NA
## MarketID9:MarketSizeMedium NA NA 0.195 NA NA NA NA
## MarketID10:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID2:MarketSizeSmall NA NA 0.246 NA NA NA NA
## MarketID3:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeSmall NA NA -0.100 NA NA NA NA
## MarketID5:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID6:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID7:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID8:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID9:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID10:MarketSizeSmall NA NA 0.000 NA NA NA NA
## Promotion1:AgeOfStore -0.063 NA 5.412 NA NA -0.193 0.067
## Promotion2:AgeOfStore -0.079 NA 0.598 NA NA -0.202 0.043
## Promotion3:AgeOfStore 0.145 NA 3.792 NA NA 0.014 0.276
## ----------------------------------------------------------------------------------------------------------
##
##
## - AgeOfStore:Promotion
##
## Backward Elimination: Step 11
##
## Variable AgeOfStore:Promotion Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.953 RMSE 5.096
## R-Squared 0.909 Coef. Var 9.579
## Adj. R-Squared 0.906 MSE 25.966
## Pred R-Squared 0.903 MAE 4.044
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108052.847 11 9822.986 378.307 0.0000
## Residual 10853.640 418 25.966
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## -----------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## -----------------------------------------------------------------------------------------------------------
## (Intercept) 39.939 0.851 46.950 0.000 38.267 41.611
## MarketID2 25.718 1.428 0.326 18.008 0.000 22.911 28.525
## MarketID3 50.095 1.025 1.072 48.857 0.000 48.080 52.111
## MarketID4 19.350 1.273 0.277 15.195 0.000 16.847 21.853
## MarketID5 16.185 1.095 0.304 14.782 0.000 14.033 18.337
## MarketID6 2.029 1.083 0.038 1.875 0.062 -0.098 4.157
## MarketID7 10.598 1.082 0.203 9.798 0.000 8.472 12.724
## MarketID8 13.156 1.143 0.227 11.513 0.000 10.910 15.402
## MarketID9 17.626 1.226 0.270 14.374 0.000 15.216 20.037
## MarketID10 19.523 1.007 0.423 19.392 0.000 17.544 21.502
## MarketSizeMedium NA 0.619 -0.297 -16.217 0.000 NA NA
## MarketSizeSmall NA 0.635 -0.093 -7.916 0.000 NA NA
## Promotion2 -10.039 NA 0.739 NA NA -11.255 -8.822
## Promotion3 -5.027 NA 1.412 NA NA -6.275 -3.779
## MarketID2:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID3:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID5:MarketSizeMedium NA NA 0.199 NA NA NA NA
## MarketID6:MarketSizeMedium NA NA 0.247 NA NA NA NA
## MarketID7:MarketSizeMedium NA NA 0.337 NA NA NA NA
## MarketID8:MarketSizeMedium NA NA 0.337 NA NA NA NA
## MarketID9:MarketSizeMedium NA NA -0.154 NA NA NA NA
## MarketID10:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID2:MarketSizeSmall NA NA 0.326 NA NA NA NA
## MarketID3:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeSmall NA NA 0.277 NA NA NA NA
## MarketID5:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID6:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID7:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID8:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID9:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID10:MarketSizeSmall NA NA 0.000 NA NA NA NA
## -----------------------------------------------------------------------------------------------------------
##
##
##
## No more variables satisfy the condition of p value = 0.05
##
##
## Variables Removed:
##
## - AgeOfStore
## - MarketID:Promotion
## - AgeOfStore:Week
## - Promotion:Week
## - MarketSize:Promotion
## - Week
## - MarketID:Week
## - MarketSize:Week
## - MarketID:AgeOfStore
## - MarketSize:AgeOfStore
## - AgeOfStore:Promotion
##
##
## Final Model Output
## ------------------
##
## Model Summary
## --------------------------------------------------------------
## R 0.953 RMSE 5.096
## R-Squared 0.909 Coef. Var 9.579
## Adj. R-Squared 0.906 MSE 25.966
## Pred R-Squared 0.903 MAE 4.044
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108052.847 11 9822.986 378.307 0.0000
## Residual 10853.640 418 25.966
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## -----------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## -----------------------------------------------------------------------------------------------------------
## (Intercept) 39.939 0.851 46.950 0.000 38.267 41.611
## MarketID2 25.718 1.428 0.326 18.008 0.000 22.911 28.525
## MarketID3 50.095 1.025 1.072 48.857 0.000 48.080 52.111
## MarketID4 19.350 1.273 0.277 15.195 0.000 16.847 21.853
## MarketID5 16.185 1.095 0.304 14.782 0.000 14.033 18.337
## MarketID6 2.029 1.083 0.038 1.875 0.062 -0.098 4.157
## MarketID7 10.598 1.082 0.203 9.798 0.000 8.472 12.724
## MarketID8 13.156 1.143 0.227 11.513 0.000 10.910 15.402
## MarketID9 17.626 1.226 0.270 14.374 0.000 15.216 20.037
## MarketID10 19.523 1.007 0.423 19.392 0.000 17.544 21.502
## MarketSizeMedium NA 0.619 -0.297 -16.217 0.000 NA NA
## MarketSizeSmall NA 0.635 -0.093 -7.916 0.000 NA NA
## Promotion2 -10.039 NA 0.739 NA NA -11.255 -8.822
## Promotion3 -5.027 NA 1.412 NA NA -6.275 -3.779
## MarketID2:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID3:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID5:MarketSizeMedium NA NA 0.199 NA NA NA NA
## MarketID6:MarketSizeMedium NA NA 0.247 NA NA NA NA
## MarketID7:MarketSizeMedium NA NA 0.337 NA NA NA NA
## MarketID8:MarketSizeMedium NA NA 0.337 NA NA NA NA
## MarketID9:MarketSizeMedium NA NA -0.154 NA NA NA NA
## MarketID10:MarketSizeMedium NA NA 0.000 NA NA NA NA
## MarketID2:MarketSizeSmall NA NA 0.326 NA NA NA NA
## MarketID3:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID4:MarketSizeSmall NA NA 0.277 NA NA NA NA
## MarketID5:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID6:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID7:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID8:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID9:MarketSizeSmall NA NA 0.000 NA NA NA NA
## MarketID10:MarketSizeSmall NA NA 0.000 NA NA NA NA
## -----------------------------------------------------------------------------------------------------------
##
## Call:
## lm(formula = paste(response, "~", paste(preds, collapse = " + ")),
## data = l)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.5894 -3.3428 -0.1003 3.2096 14.2140
##
## Coefficients: (20 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 39.9386 0.8507 46.950 < 2e-16 ***
## MarketID2 25.7183 1.4281 18.008 < 2e-16 ***
## MarketID3 50.0952 1.0253 48.857 < 2e-16 ***
## MarketID4 19.3497 1.2734 15.195 < 2e-16 ***
## MarketID5 16.1848 1.0949 14.782 < 2e-16 ***
## MarketID6 2.0295 1.0825 1.875 0.0615 .
## MarketID7 10.5980 1.0816 9.798 < 2e-16 ***
## MarketID8 13.1561 1.1427 11.513 < 2e-16 ***
## MarketID9 17.6263 1.2262 14.374 < 2e-16 ***
## MarketID10 19.5234 1.0068 19.392 < 2e-16 ***
## MarketSizeMedium NA NA NA NA
## MarketSizeSmall NA NA NA NA
## Promotion2 -10.0387 0.6190 -16.217 < 2e-16 ***
## Promotion3 -5.0267 0.6350 -7.916 2.23e-14 ***
## MarketID2:MarketSizeMedium NA NA NA NA
## MarketID3:MarketSizeMedium NA NA NA NA
## MarketID4:MarketSizeMedium NA NA NA NA
## MarketID5:MarketSizeMedium NA NA NA NA
## MarketID6:MarketSizeMedium NA NA NA NA
## MarketID7:MarketSizeMedium NA NA NA NA
## MarketID8:MarketSizeMedium NA NA NA NA
## MarketID9:MarketSizeMedium NA NA NA NA
## MarketID10:MarketSizeMedium NA NA NA NA
## MarketID2:MarketSizeSmall NA NA NA NA
## MarketID3:MarketSizeSmall NA NA NA NA
## MarketID4:MarketSizeSmall NA NA NA NA
## MarketID5:MarketSizeSmall NA NA NA NA
## MarketID6:MarketSizeSmall NA NA NA NA
## MarketID7:MarketSizeSmall NA NA NA NA
## MarketID8:MarketSizeSmall NA NA NA NA
## MarketID9:MarketSizeSmall NA NA NA NA
## MarketID10:MarketSizeSmall NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.096 on 418 degrees of freedom
## Multiple R-squared: 0.9087, Adjusted R-squared: 0.9063
## F-statistic: 378.3 on 11 and 418 DF, p-value: < 2.2e-16
## Backward Elimination Method
## ---------------------------
##
## Candidate Terms:
##
## 1 . MarketID
## 2 . MarketSize
## 3 . AgeOfStore
## 4 . Promotion
## 5 . Week
## 6 . MarketID:MarketSize
## 7 . MarketID:AgeOfStore
## 8 . MarketID:Promotion
## 9 . MarketID:Week
## 10 . MarketSize:AgeOfStore
## 11 . MarketSize:Promotion
## 12 . MarketSize:Week
## 13 . AgeOfStore:Promotion
## 14 . AgeOfStore:Week
## 15 . Promotion:Week
##
## Step 0: AIC = 2730.493
## SalesInThousands ~ MarketID + MarketSize + AgeOfStore + Promotion + Week + MarketID:MarketSize + MarketID:AgeOfStore + MarketID:Promotion + MarketID:Week + MarketSize:AgeOfStore + MarketSize:Promotion + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week + Promotion:Week
##
## -------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## -------------------------------------------------------------------------------------
## MarketID 1 2730.493 0.000 9751.669 0.918 0.906
## Week 1 2730.493 0.000 9751.669 0.918 0.906
## MarketSize 1 2730.493 0.000 9751.669 0.918 0.906
## MarketID:MarketSize 1 2694.493 0.000 9751.669 0.918 0.906
## MarketSize:AgeOfStore 1 2726.493 0.000 9751.669 0.918 0.906
## MarketSize:Promotion 1 2722.493 0.000 9751.669 0.918 0.906
## MarketSize:Week 1 2726.493 0.000 9751.669 0.918 0.906
## Promotion 1 2730.493 0.000 9751.669 0.918 0.906
## AgeOfStore 1 2730.493 0.000 9751.669 0.918 0.906
## AgeOfStore:Week 1 2728.706 4.840 9756.508 0.918 0.907
## Promotion:Week 1 2727.837 30.541 9782.209 0.918 0.907
## AgeOfStore:Promotion 1 2730.615 93.940 9845.608 0.917 0.906
## MarketID:Week 1 2721.283 201.406 9953.075 0.916 0.906
## MarketID:AgeOfStore 1 2722.235 223.468 9975.137 0.916 0.906
## MarketID:Promotion 1 2705.325 248.782 10000.451 0.916 0.907
## -------------------------------------------------------------------------------------
##
##
## Variables Removed:
##
## - MarketID:MarketSize
##
##
## Step 1 : AIC = 2694.493
## SalesInThousands ~ MarketID + MarketSize + AgeOfStore + Promotion + Week + MarketID:AgeOfStore + MarketID:Promotion + MarketID:Week + MarketSize:AgeOfStore + MarketSize:Promotion + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week + Promotion:Week
##
## --------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## --------------------------------------------------------------------------------------
## Week 1 2694.493 0.000 9751.669 0.918 0.906
## MarketSize 1 2690.493 0.000 9751.669 0.918 0.906
## MarketSize:AgeOfStore 1 2690.493 0.000 9751.669 0.918 0.906
## MarketSize:Promotion 1 2686.493 0.000 9751.669 0.918 0.906
## MarketSize:Week 1 2690.493 0.000 9751.669 0.918 0.906
## Promotion 1 2694.493 0.000 9751.669 0.918 0.906
## AgeOfStore 1 2694.493 0.000 9751.669 0.918 0.906
## AgeOfStore:Week 1 2692.706 4.840 9756.508 0.918 0.907
## Promotion:Week 1 2691.837 30.541 9782.209 0.918 0.907
## AgeOfStore:Promotion 1 2694.615 93.940 9845.608 0.917 0.906
## MarketID:Week 1 2685.283 201.406 9953.075 0.916 0.906
## MarketID:AgeOfStore 1 2686.235 223.468 9975.137 0.916 0.906
## MarketID:Promotion 1 2669.325 248.782 10000.451 0.916 0.907
## MarketID 1 2845.154 4683.594 14435.263 0.879 0.864
## --------------------------------------------------------------------------------------
##
## - MarketID:Promotion
##
##
## Step 2 : AIC = 2669.325
## SalesInThousands ~ MarketID + MarketSize + AgeOfStore + Promotion + Week + MarketID:AgeOfStore + MarketID:Week + MarketSize:AgeOfStore + MarketSize:Promotion + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week + Promotion:Week
##
## --------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## --------------------------------------------------------------------------------------
## MarketSize 1 2665.325 0.000 10000.451 0.916 0.907
## Week 1 2669.325 0.000 10000.451 0.916 0.907
## MarketSize:AgeOfStore 1 2665.325 0.000 10000.451 0.916 0.907
## MarketSize:Week 1 2665.325 0.000 10000.451 0.916 0.907
## AgeOfStore 1 2669.325 0.000 10000.451 0.916 0.907
## Promotion 1 2669.325 0.000 10000.451 0.916 0.907
## AgeOfStore:Week 1 2667.486 3.734 10004.185 0.916 0.907
## Promotion:Week 1 2666.415 25.376 10025.827 0.916 0.907
## MarketSize:Promotion 1 2664.719 79.238 10079.689 0.915 0.907
## AgeOfStore:Promotion 1 2672.014 156.784 10157.235 0.915 0.906
## MarketID:AgeOfStore 1 2658.066 158.005 10158.456 0.915 0.907
## MarketID:Week 1 2659.195 184.712 10185.163 0.914 0.907
## MarketID 1 2853.838 6015.594 16016.045 0.865 0.854
## --------------------------------------------------------------------------------------
##
## - MarketID:AgeOfStore
##
##
## Step 3 : AIC = 2658.066
## SalesInThousands ~ MarketID + MarketSize + AgeOfStore + Promotion + Week + MarketID:Week + MarketSize:AgeOfStore + MarketSize:Promotion + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week + Promotion:Week
##
## --------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## --------------------------------------------------------------------------------------
## MarketSize 1 2654.066 0.000 10158.456 0.915 0.907
## Week 1 2658.066 0.000 10158.456 0.915 0.907
## MarketSize:Week 1 2654.066 0.000 10158.456 0.915 0.907
## Promotion 1 2658.066 0.000 10158.456 0.915 0.907
## AgeOfStore 1 2658.066 0.000 10158.456 0.915 0.907
## AgeOfStore:Week 1 2656.287 5.228 10163.684 0.915 0.908
## Promotion:Week 1 2655.177 26.280 10184.736 0.914 0.908
## MarketSize:Promotion 1 2653.382 78.638 10237.094 0.914 0.908
## MarketSize:AgeOfStore 1 2659.287 124.094 10282.551 0.914 0.907
## MarketID:Week 1 2647.731 182.704 10341.161 0.913 0.907
## AgeOfStore:Promotion 1 2662.306 196.543 10354.999 0.913 0.906
## MarketID 1 2874.597 7368.286 17526.742 0.853 0.843
## --------------------------------------------------------------------------------------
##
## - MarketID:Week
##
##
## Step 4 : AIC = 2647.731
## SalesInThousands ~ MarketID + MarketSize + AgeOfStore + Promotion + Week + MarketSize:AgeOfStore + MarketSize:Promotion + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week + Promotion:Week
##
## ---------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------------
## MarketSize 1 2643.731 0.000 10341.161 0.913 0.907
## Week 1 2647.731 0.000 10341.161 0.913 0.907
## AgeOfStore 1 2647.731 0.000 10341.161 0.913 0.907
## Promotion 1 2647.731 0.000 10341.161 0.913 0.907
## AgeOfStore:Week 1 2645.774 1.025 10342.186 0.913 0.908
## Promotion:Week 1 2645.208 35.579 10376.739 0.913 0.908
## MarketSize:Week 1 2646.198 59.507 10400.667 0.913 0.907
## MarketSize:Promotion 1 2642.776 73.481 10414.641 0.912 0.908
## MarketSize:AgeOfStore 1 2648.983 127.089 10468.250 0.912 0.907
## AgeOfStore:Promotion 1 2651.785 195.514 10536.675 0.911 0.906
## MarketID 1 3277.325 36285.153 46626.314 0.608 0.590
## ---------------------------------------------------------------------------------------
##
## - MarketSize:Promotion
##
##
## Step 5 : AIC = 2642.776
## SalesInThousands ~ MarketID + MarketSize + AgeOfStore + Promotion + Week + MarketSize:AgeOfStore + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week + Promotion:Week
##
## ---------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------------
## MarketSize 1 2638.776 0.000 10414.641 0.912 0.908
## Week 1 2642.776 0.000 10414.641 0.912 0.908
## AgeOfStore 1 2642.776 0.000 10414.641 0.912 0.908
## AgeOfStore:Week 1 2640.804 0.684 10415.325 0.912 0.908
## Promotion:Week 1 2640.320 37.459 10452.101 0.912 0.908
## MarketSize:Week 1 2641.040 54.994 10469.636 0.912 0.908
## MarketSize:AgeOfStore 1 2644.225 132.812 10547.454 0.911 0.907
## AgeOfStore:Promotion 1 2647.039 202.069 10616.710 0.911 0.906
## Promotion 1 2662.786 598.077 11012.718 0.907 0.903
## MarketID 1 3297.090 39321.665 49736.307 0.582 0.567
## ---------------------------------------------------------------------------------------
##
## - MarketSize
##
##
## Step 6 : AIC = 2638.776
## SalesInThousands ~ MarketID + AgeOfStore + Promotion + Week + MarketSize:AgeOfStore + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week + Promotion:Week
##
## ---------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------------
## Week 1 2638.776 0.000 10414.641 0.912 0.908
## AgeOfStore 1 2638.776 0.000 10414.641 0.912 0.908
## AgeOfStore:Week 1 2636.804 0.684 10415.325 0.912 0.908
## Promotion:Week 1 2636.320 37.459 10452.101 0.912 0.908
## MarketSize:Week 1 2637.040 54.994 10469.636 0.912 0.908
## MarketSize:AgeOfStore 1 2640.225 132.812 10547.454 0.911 0.907
## AgeOfStore:Promotion 1 2643.039 202.069 10616.710 0.911 0.906
## Promotion 1 2658.786 598.077 11012.718 0.907 0.903
## MarketID 1 3346.602 45912.776 56327.417 0.526 0.511
## ---------------------------------------------------------------------------------------
##
## - Promotion:Week
##
##
## Step 7 : AIC = 2636.32
## SalesInThousands ~ MarketID + AgeOfStore + Promotion + Week + MarketSize:AgeOfStore + MarketSize:Week + AgeOfStore:Promotion + AgeOfStore:Week
##
## ---------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------------
## Week 1 2636.320 0.000 10452.101 0.912 0.908
## AgeOfStore 1 2636.320 0.000 10452.101 0.912 0.908
## AgeOfStore:Week 1 2634.332 0.301 10452.401 0.912 0.908
## MarketSize:Week 1 2634.754 59.341 10511.441 0.912 0.908
## MarketSize:AgeOfStore 1 2637.847 135.233 10587.334 0.911 0.907
## AgeOfStore:Promotion 1 2640.406 198.423 10650.523 0.910 0.907
## Promotion 1 2718.906 2331.554 12783.655 0.892 0.888
## MarketID 1 3343.072 45936.871 56388.971 0.526 0.513
## ---------------------------------------------------------------------------------------
##
## - AgeOfStore:Week
##
##
## Step 8 : AIC = 2634.332
## SalesInThousands ~ MarketID + AgeOfStore + Promotion + Week + MarketSize:AgeOfStore + MarketSize:Week + AgeOfStore:Promotion
##
## ---------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------------
## Week 1 2634.332 0.000 10452.401 0.912 0.908
## AgeOfStore 1 2634.332 0.000 10452.401 0.912 0.908
## MarketSize:Week 1 2632.755 59.067 10511.468 0.912 0.908
## MarketSize:AgeOfStore 1 2635.881 135.767 10588.168 0.911 0.907
## AgeOfStore:Promotion 1 2638.434 198.812 10651.214 0.910 0.907
## Promotion 1 2717.043 2335.311 12787.712 0.892 0.888
## MarketID 1 3342.293 46096.871 56549.273 0.524 0.513
## ---------------------------------------------------------------------------------------
##
## - MarketSize:Week
##
##
## Step 9 : AIC = 2632.755
## SalesInThousands ~ MarketID + AgeOfStore + Promotion + Week + MarketSize:AgeOfStore + AgeOfStore:Promotion
##
## ---------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------------
## AgeOfStore 1 2632.755 0.000 10511.468 0.912 0.908
## Week 1 2631.440 16.766 10528.234 0.911 0.908
## MarketSize:AgeOfStore 1 2634.263 135.509 10646.977 0.910 0.907
## AgeOfStore:Promotion 1 2636.753 197.337 10708.805 0.910 0.907
## Promotion 1 2714.715 2326.060 12837.528 0.892 0.888
## MarketID 1 3428.028 59160.616 69672.084 0.414 0.403
## ---------------------------------------------------------------------------------------
##
## - Week
##
##
## Step 10 : AIC = 2631.44
## SalesInThousands ~ MarketID + AgeOfStore + Promotion + MarketSize:AgeOfStore + AgeOfStore:Promotion
##
## ---------------------------------------------------------------------------------------
## Variable DF AIC Sum Sq RSS R-Sq Adj. R-Sq
## ---------------------------------------------------------------------------------------
## AgeOfStore 1 2631.440 0.000 10528.234 0.911 0.908
## MarketSize:AgeOfStore 1 2632.950 135.772 10664.006 0.910 0.907
## AgeOfStore:Promotion 1 2635.345 195.324 10723.558 0.910 0.907
## Promotion 1 2713.173 2322.989 12851.223 0.892 0.888
## MarketID 1 3426.029 59143.949 69672.184 0.414 0.404
## ---------------------------------------------------------------------------------------
##
##
## No more variables to be removed.
##
## Variables Removed:
##
## - MarketID:MarketSize
## - MarketID:Promotion
## - MarketID:AgeOfStore
## - MarketID:Week
## - MarketSize:Promotion
## - MarketSize
## - Promotion:Week
## - AgeOfStore:Week
## - MarketSize:Week
## - Week
##
##
## Final Model Output
## ------------------
##
## Model Summary
## --------------------------------------------------------------
## R 0.955 RMSE 5.049
## R-Squared 0.911 Coef. Var 9.492
## Adj. R-Squared 0.908 MSE 25.492
## Pred R-Squared 0.904 MAE 3.975
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ------------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ------------------------------------------------------------------------
## Regression 108378.252 16 6773.641 265.715 0.0000
## Residual 10528.234 413 25.492
## Total 118906.487 429
## ------------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------------------------
## (Intercept) 40.142 1.126 35.666 0.000 37.930 42.355
## MarketID2 23.286 2.027 0.295 11.488 0.000 19.302 27.271
## MarketID3 51.166 1.267 1.095 40.393 0.000 48.676 53.656
## MarketID4 17.982 1.561 0.258 11.517 0.000 14.913 21.052
## MarketID5 16.311 1.101 0.306 14.815 0.000 14.146 18.475
## MarketID6 1.833 1.087 0.034 1.686 0.092 -0.304 3.969
## MarketID7 10.631 1.072 0.203 9.914 0.000 8.523 12.739
## MarketID8 12.690 1.159 0.219 10.947 0.000 10.411 14.969
## MarketID9 17.443 1.222 0.267 14.269 0.000 15.040 19.846
## MarketID10 20.343 1.220 0.441 16.669 0.000 17.944 22.742
## AgeOfStore -0.148 0.083 -0.060 -1.781 0.076 -0.311 0.015
## Promotion2 -9.405 0.999 -0.270 -9.411 0.000 -11.369 -7.441
## Promotion3 -6.603 1.073 -0.186 -6.153 0.000 -8.712 -4.493
## AgeOfStore:MarketSizeMedium 0.120 0.089 0.049 1.354 0.177 -0.054 0.295
## AgeOfStore:MarketSizeSmall 0.287 0.125 0.072 2.295 0.022 0.041 0.532
## AgeOfStore:Promotion2 -0.065 0.094 -0.022 -0.693 0.489 -0.250 0.119
## AgeOfStore:Promotion3 0.181 0.094 0.064 1.921 0.055 -0.004 0.365
## ---------------------------------------------------------------------------------------------------------
##
## Call:
## lm(formula = paste(response, "~", paste(preds, collapse = " + ")),
## data = l)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.7685 -3.2042 -0.1668 3.3501 15.2276
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 40.14225 1.12551 35.666 < 2e-16 ***
## MarketID2 23.28624 2.02700 11.488 < 2e-16 ***
## MarketID3 51.16590 1.26670 40.393 < 2e-16 ***
## MarketID4 17.98249 1.56133 11.517 < 2e-16 ***
## MarketID5 16.31058 1.10097 14.815 < 2e-16 ***
## MarketID6 1.83287 1.08688 1.686 0.0925 .
## MarketID7 10.63137 1.07240 9.914 < 2e-16 ***
## MarketID8 12.68999 1.15922 10.947 < 2e-16 ***
## MarketID9 17.44284 1.22247 14.269 < 2e-16 ***
## MarketID10 20.34276 1.22037 16.669 < 2e-16 ***
## AgeOfStore -0.14774 0.08294 -1.781 0.0756 .
## Promotion2 -9.40500 0.99936 -9.411 < 2e-16 ***
## Promotion3 -6.60275 1.07312 -6.153 1.8e-09 ***
## AgeOfStore:MarketSizeMedium 0.12044 0.08898 1.354 0.1766
## AgeOfStore:MarketSizeSmall 0.28683 0.12498 2.295 0.0222 *
## AgeOfStore:Promotion2 -0.06502 0.09386 -0.693 0.4889
## AgeOfStore:Promotion3 0.18061 0.09403 1.921 0.0555 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.049 on 413 degrees of freedom
## Multiple R-squared: 0.9115, Adjusted R-squared: 0.908
## F-statistic: 265.7 on 16 and 413 DF, p-value: < 2.2e-16
Scoring the validation data set
- Scoring the validation data set let’s us calculate the variable ‘predicted’ sales.
- It helps us answer how strong is the model.
- It shows us the by what proportion (validation$diff) were the estimation of sales off from the actual sales.
- The histogram tells that over 70% of the model is below par by 0-10% and another 25% of the data is below par by 20%.
- The summary of the difference between predicted and actual sales tells us that on an average prediction is off by $4000.
- The maximum difference is $15.384.
- The R square of the validation data tells that the model explains 91 percent of the variance in the data set.
- The mean sales predicted is 55.675 versus mean of the actual sales at 53.46.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.001237 0.031558 0.065261 0.094147 0.126537 0.575793
## Min. 1st Qu. Median Mean 3rd Qu. Max.
##
```