Differences

This shows you the differences between two versions of the page.

--- multiple_regression_examples [2019/11/01 12:30] – [e.g.,] hkimscil
+++ multiple_regression_examples [2019/11/01 13:22] – [Backward elimination] hkimscil
@@ Line 611: / Line 611: @@
 ====== e.g., ======
+<code>
+library(ISLR)
+head(Carseats)
+str(Carseats)
+lm.full <- lm(Sales ~ . , data = Carseats)
+lm.null <- lm(Sales ~ 1 , data = Carseats)
+</code>
+<WRAP group>
+<WRAP half column>
 <code>
 > stepAIC(lm.full)
@@ Line 700: / Line 712: @@
 >
 </code>
+</WRAP>
+<WRAP half column>
+<code>
+> step(lm.full, direction="both")
+Start:  AIC=26.82
+Sales ~ CompPrice + Income + Advertising + Population + Price +
+    ShelveLoc + Age + Education + Urban + US
-<WRAP col2>
+              Df Sum of Sq     RSS    AIC
-The Carseats data set tracks sales information for car seats. It has 400 observations (each at a different store) and 11 variables:
+- Population   1      0.33  403.16  25.15
+- Education    1      1.19  404.02  26.00
+- Urban        1      1.23  404.06  26.04
+- US           1      1.57  404.40  26.38
+<none>                      402.83  26.82
+- Income       1     76.16  478.99  94.09
+- Advertising  1    127.14  529.97 134.54
+- Age          1    217.44  620.27 197.48
+- CompPrice    1    519.91  922.74 356.35
+- ShelveLoc    2   1053.20 1456.03 536.80
+- Price        1   1323.23 1726.06 606.85
-Sales: unit sales in thousands
+Step:  AIC=25.15
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + Urban + US
-CompPrice: price charged by competitor at each location
+              Df Sum of Sq     RSS    AIC
+- Urban        1      1.15  404.31  24.29
+- Education    1      1.36  404.52  24.49
+- US           1      1.89  405.05  25.02
+<none>                      403.16  25.15
++ Population   1      0.33  402.83  26.82
+- Income       1     75.94  479.10  92.18
+- Advertising  1    145.38  548.54 146.32
+- Age          1    218.52  621.68 196.38
+- CompPrice    1    521.69  924.85 355.27
+- ShelveLoc    2   1053.18 1456.34 534.89
+- Price        1   1323.51 1726.67 605.00
-Income: community income level in 1000s of dollars
+Step:  AIC=24.29
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + US
-Advertising: local ad budget at each location in 1000s of dollars
+              Df Sum of Sq     RSS    AIC
+- Education    1      1.44  405.76  23.72
+- US           1      1.85  406.16  24.12
+<none>                      404.31  24.29
++ Urban        1      1.15  403.16  25.15
++ Population   1      0.25  404.06  26.04
+- Income       1     76.64  480.96  91.73
+- Advertising  1    146.03  550.34 145.63
+- Age          1    217.59  621.91 194.53
+- CompPrice    1    526.17  930.48 355.69
+- ShelveLoc    2   1053.93 1458.25 533.41
+- Price        1   1322.80 1727.11 603.10
-Population: regional pop in thousands
+Step:  AIC=23.72
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + US
-Price: price for car seats at each site
+              Df Sum of Sq     RSS    AIC
+- US           1      1.63  407.39  23.32
+<none>                      405.76  23.72
++ Education    1      1.44  404.31  24.29
++ Urban        1      1.24  404.52  24.49
++ Population   1      0.41  405.35  25.32
+- Income       1     77.87  483.62  91.94
+- Advertising  1    145.30  551.06 144.15
+- Age          1    217.97  623.73 193.70
+- CompPrice    1    525.25  931.00 353.92
+- ShelveLoc    2   1056.88 1462.64 532.61
+- Price        1   1322.83 1728.58 601.44
-ShelveLoc: Bad, Good or Medium indicates quality of shelving location
+Step:  AIC=23.32
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age
-Age: age level of the population
+              Df Sum of Sq     RSS    AIC
+<none>                      407.39  23.32
++ US           1      1.63  405.76  23.72
++ Education    1      1.22  406.16  24.12
++ Urban        1      1.19  406.20  24.15
++ Population   1      0.72  406.67  24.62
+- Income       1     76.68  484.07  90.30
+- Age          1    219.12  626.51 193.48
+- Advertising  1    234.03  641.42 202.89
+- CompPrice    1    523.83  931.22 352.01
+- ShelveLoc    2   1055.51 1462.90 530.68
+- Price        1   1324.42 1731.81 600.18
-Education: ed level at location
+Call:
+lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
+    ShelveLoc + Age, data = Carseats)
-Urban: Yes/No
+Coefficients:
+    (Intercept)        CompPrice           Income      Advertising
+.47523          0.09257          0.01578          0.11590
+          Price    ShelveLocGood  ShelveLocMedium              Age
+       -0.09532          4.83567          1.95199         -0.04613
-US: Yes/NoThe Carseats data set tracks sales information for car seats. It has 400 observations (each at a different store) and 11 variables:
+>
+</code>
+</WRAP>
+</WRAP>
-Sales: unit sales in thousands
+<WRAP group>
+<WRAP half column>
+<code>> stepAIC(lm.full, direction="backward")
+Start:  AIC=26.82
+Sales ~ CompPrice + Income + Advertising + Population + Price +
+    ShelveLoc + Age + Education + Urban + US
-CompPrice: price charged by competitor at each location
+              Df Sum of Sq     RSS    AIC
+- Population   1      0.33  403.16  25.15
+- Education    1      1.19  404.02  26.00
+- Urban        1      1.23  404.06  26.04
+- US           1      1.57  404.40  26.38
+<none>                      402.83  26.82
+- Income       1     76.16  478.99  94.09
+- Advertising  1    127.14  529.97 134.54
+- Age          1    217.44  620.27 197.48
+- CompPrice    1    519.91  922.74 356.35
+- ShelveLoc    2   1053.20 1456.03 536.80
+- Price        1   1323.23 1726.06 606.85
-Income: community income level in 1000s of dollars
+Step:  AIC=25.15
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + Urban + US
-Advertising: local ad budget at each location in 1000s of dollars
+              Df Sum of Sq     RSS    AIC
+- Urban        1      1.15  404.31  24.29
+- Education    1      1.36  404.52  24.49
+- US           1      1.89  405.05  25.02
+<none>                      403.16  25.15
+- Income       1     75.94  479.10  92.18
+- Advertising  1    145.38  548.54 146.32
+- Age          1    218.52  621.68 196.38
+- CompPrice    1    521.69  924.85 355.27
+- ShelveLoc    2   1053.18 1456.34 534.89
+- Price        1   1323.51 1726.67 605.00
-Population: regional pop in thousands
+Step:  AIC=24.29
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + US
-Price: price for car seats at each site
+              Df Sum of Sq     RSS    AIC
+- Education    1      1.44  405.76  23.72
+- US           1      1.85  406.16  24.12
+<none>                      404.31  24.29
+- Income       1     76.64  480.96  91.73
+- Advertising  1    146.03  550.34 145.63
+- Age          1    217.59  621.91 194.53
+- CompPrice    1    526.17  930.48 355.69
+- ShelveLoc    2   1053.93 1458.25 533.41
+- Price        1   1322.80 1727.11 603.10
-ShelveLoc: Bad, Good or Medium indicates quality of shelving location
+Step:  AIC=23.72
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + US
-Age: age level of the population
+              Df Sum of Sq     RSS    AIC
+- US           1      1.63  407.39  23.32
+<none>                      405.76  23.72
+- Income       1     77.87  483.62  91.94
+- Advertising  1    145.30  551.06 144.15
+- Age          1    217.97  623.73 193.70
+- CompPrice    1    525.25  931.00 353.92
+- ShelveLoc    2   1056.88 1462.64 532.61
+- Price        1   1322.83 1728.58 601.44
-Education: ed level at location
+Step:  AIC=23.32
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age
-Urban: Yes/No
+              Df Sum of Sq     RSS    AIC
+<none>                      407.39  23.32
+- Income       1     76.68  484.07  90.30
+- Age          1    219.12  626.51 193.48
+- Advertising  1    234.03  641.42 202.89
+- CompPrice    1    523.83  931.22 352.01
+- ShelveLoc    2   1055.51 1462.90 530.68
+- Price        1   1324.42 1731.81 600.18
-US: Yes/No
+Call:
-The Carseats data set tracks sales information for car seats. It has 400 observations (each at a different store) and 11 variables:
+lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
+    ShelveLoc + Age, data = Carseats)
-Sales: unit sales in thousands
+Coefficients:
+    (Intercept)        CompPrice           Income      Advertising            Price
+.47523          0.09257          0.01578          0.11590         -0.09532
+  ShelveLocGood  ShelveLocMedium              Age
+.83567          1.95199         -0.04613
-CompPrice: price charged by competitor at each location
+> </code>
+</WRAP>
-Income: community income level in 1000s of dollars
+<WRAP half column>
+<code>> step(lm.full, direction="backward")
+Start:  AIC=26.82
+Sales ~ CompPrice + Income + Advertising + Population + Price +
+    ShelveLoc + Age + Education + Urban + US
-Advertising: local ad budget at each location in 1000s of dollars
+              Df Sum of Sq     RSS    AIC
+- Population   1      0.33  403.16  25.15
+- Education    1      1.19  404.02  26.00
+- Urban        1      1.23  404.06  26.04
+- US           1      1.57  404.40  26.38
+<none>                      402.83  26.82
+- Income       1     76.16  478.99  94.09
+- Advertising  1    127.14  529.97 134.54
+- Age          1    217.44  620.27 197.48
+- CompPrice    1    519.91  922.74 356.35
+- ShelveLoc    2   1053.20 1456.03 536.80
+- Price        1   1323.23 1726.06 606.85
-Population: regional pop in thousands
+Step:  AIC=25.15
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + Urban + US
-Price: price for car seats at each site
+              Df Sum of Sq     RSS    AIC
+- Urban        1      1.15  404.31  24.29
+- Education    1      1.36  404.52  24.49
+- US           1      1.89  405.05  25.02
+<none>                      403.16  25.15
+- Income       1     75.94  479.10  92.18
+- Advertising  1    145.38  548.54 146.32
+- Age          1    218.52  621.68 196.38
+- CompPrice    1    521.69  924.85 355.27
+- ShelveLoc    2   1053.18 1456.34 534.89
+- Price        1   1323.51 1726.67 605.00
-ShelveLoc: Bad, Good or Medium indicates quality of shelving location
+Step:  AIC=24.29
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + US
-Age: age level of the population
+              Df Sum of Sq     RSS    AIC
+- Education    1      1.44  405.76  23.72
+- US           1      1.85  406.16  24.12
+<none>                      404.31  24.29
+- Income       1     76.64  480.96  91.73
+- Advertising  1    146.03  550.34 145.63
+- Age          1    217.59  621.91 194.53
+- CompPrice    1    526.17  930.48 355.69
+- ShelveLoc    2   1053.93 1458.25 533.41
+- Price        1   1322.80 1727.11 603.10
-Education: ed level at location
+Step:  AIC=23.72
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + US
-Urban: Yes/No
+              Df Sum of Sq     RSS    AIC
+- US           1      1.63  407.39  23.32
+<none>                      405.76  23.72
+- Income       1     77.87  483.62  91.94
+- Advertising  1    145.30  551.06 144.15
+- Age          1    217.97  623.73 193.70
+- CompPrice    1    525.25  931.00 353.92
+- ShelveLoc    2   1056.88 1462.64 532.61
+- Price        1   1322.83 1728.58 601.44
-US: Yes/No
+Step:  AIC=23.32
-The Carseats data set tracks sales information for car seats. It has 400 observations (each at a different store) and 11 variables:
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age
-Sales: unit sales in thousands
+              Df Sum of Sq     RSS    AIC
+<none>                      407.39  23.32
+- Income       1     76.68  484.07  90.30
+- Age          1    219.12  626.51 193.48
+- Advertising  1    234.03  641.42 202.89
+- CompPrice    1    523.83  931.22 352.01
+- ShelveLoc    2   1055.51 1462.90 530.68
+- Price        1   1324.42 1731.81 600.18
-CompPrice: price charged by competitor at each location
+Call:
+lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
+    ShelveLoc + Age, data = Carseats)
-Income: community income level in 1000s of dollars
+Coefficients:
+    (Intercept)        CompPrice           Income      Advertising            Price
+.47523          0.09257          0.01578          0.11590         -0.09532
+  ShelveLocGood  ShelveLocMedium              Age
+.83567          1.95199         -0.04613
+</code>
+</WRAP>
+</WRAP>
+===== Pick the model from stepAIC =====
+<code>
+lm.fit.01 <- lm(formula = Sales ~
+                 CompPrice + Income +
+                 Advertising + Price +
+                 ShelveLoc + Age,
+                 data = Carseats)
+summary(lm.fit.01)
+</code>
-Advertising: local ad budget at each location in 1000s of dollars
+<code>
+> lm.fit.01 <- lm(formula = Sales ~ CompPrice +
+         Income + Advertising + Price +
+         ShelveLoc + Age, data = Carseats)
+> summary(lm.fit.01)
-Population: regional pop in thousands
+Call:
+lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
+    ShelveLoc + Age, data = Carseats)
-Price: price for car seats at each site
+Residuals:
+    Min      1Q  Median      3Q     Max
+-2.7728 -0.6954  0.0282  0.6732  3.3292
-ShelveLoc: Bad, Good or Medium indicates quality of shelving location
+Coefficients:
+                 Estimate Std. Error t value Pr(>|t|)
+(Intercept)      5.475226   0.505005   10.84   <2e-16 ***
+CompPrice        0.092571   0.004123   22.45   <2e-16 ***
+Income           0.015785   0.001838    8.59   <2e-16 ***
+Advertising      0.115903   0.007724   15.01   <2e-16 ***
+Price           -0.095319   0.002670  -35.70   <2e-16 ***
+ShelveLocGood    4.835675   0.152499   31.71   <2e-16 ***
+ShelveLocMedium  1.951993   0.125375   15.57   <2e-16 ***
+Age             -0.046128   0.003177  -14.52   <2e-16 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-Age: age level of the population
+Residual standard error: 1.019 on 392 degrees of freedom
+Multiple R-squared:  0.872,	Adjusted R-squared:  0.8697
+F-statistic: 381.4 on 7 and 392 DF,  p-value: < 2.2e-16
+</code>
-Education: ed level at location
+Compare the fitted model to full model
-Urban: Yes/No
+<code>
+anova(lm.full, lm.fit.01)
+</code>
-US: Yes/No
+<code>> anova(lm.full, lm.fit.01)
-The Carseats data set tracks sales information for car seats. It has 400 observations (each at a different store) and 11 variables:
+Analysis of Variance Table
-Sales: unit sales in thousands
+Model 1: Sales ~ CompPrice + Income + Advertising + Population + Price +
+    ShelveLoc + Age + Education + Urban + US
+Model 2: Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age
+  Res.Df    RSS Df Sum of Sq      F Pr(>F)
+    388 402.83
+    392 407.39 -4   -4.5533 1.0964  0.358
+> </code>
+===== Backward elimination =====
+<code>
+drop1(lm.full, test = "F")
+</code>
-CompPrice: price charged by competitor at each location
+<code>
+> drop1(lm.full, test = "F")
+Single term deletions
-Income: community income level in 1000s of dollars
+Model:
+Sales ~ CompPrice + Income + Advertising + Population + Price +
+    ShelveLoc + Age + Education + Urban + US
+            Df Sum of Sq     RSS    AIC   F value    Pr(>F)
+<none>                    402.83  26.82
+CompPrice    1    519.91  922.74 356.35  500.7659 < 2.2e-16 ***
+Income       1     76.16  478.99  94.09   73.3537  2.58e-16 ***
+Advertising  1    127.14  529.97 134.54  122.4571 < 2.2e-16 ***
+Population   1      0.33  403.16  25.15    0.3149    0.5750
+Price        1   1323.23 1726.06 606.85 1274.5022 < 2.2e-16 ***
+ShelveLoc    2   1053.20 1456.03 536.80  507.2079 < 2.2e-16 ***
+Age          1    217.44  620.27 197.48  209.4333 < 2.2e-16 ***
+Education    1      1.19  404.02  26.00    1.1450    0.2853
+Urban        1      1.23  404.06  26.04    1.1831    0.2774
+US           1      1.57  404.40  26.38    1.5094    0.2200
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+>
+> </code>
-Advertising: local ad budget at each location in 1000s of dollars
+<code>
+drop1(update(lm.full, ~ . -Population), test = "F")
+</code>
-Population: regional pop in thousands
+<code>
+> drop1(update(lm.full, ~ . -Population), test = "F")
+Single term deletions
-Price: price for car seats at each site
+Model:
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + Urban + US
+            Df Sum of Sq     RSS    AIC   F value    Pr(>F)
+<none>                    403.16  25.15
+CompPrice    1    521.69  924.85 355.27  503.3686 < 2.2e-16 ***
+Income       1     75.94  479.10  92.18   73.2717 2.652e-16 ***
+Advertising  1    145.38  548.54 146.32  140.2694 < 2.2e-16 ***
+Price        1   1323.51 1726.67 605.00 1277.0276 < 2.2e-16 ***
+ShelveLoc    2   1053.18 1456.34 534.89  508.0927 < 2.2e-16 ***
+Age          1    218.52  621.68 196.38  210.8411 < 2.2e-16 ***
+Education    1      1.36  404.52  24.49    1.3122    0.2527
+Urban        1      1.15  404.31  24.29    1.1132    0.2920
+US           1      1.89  405.05  25.02    1.8262    0.1774
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+>
+> </code>
-ShelveLoc: Bad, Good or Medium indicates quality of shelving location
+<code>
+drop1(update(lm.full, ~. - Population -Urban), test="F")
+</code>
-Age: age level of the population
+<code>
+> drop1(update(lm.full, ~. - Population -Urban), test="F")
+Single term deletions
-Education: ed level at location
+Model:
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + Education + US
+            Df Sum of Sq     RSS    AIC   F value Pr(>F)
+<none>                    404.31  24.29
+CompPrice    1    526.17  930.48 355.69  507.5374 <2e-16 ***
+Income       1     76.64  480.96  91.73   73.9307 <2e-16 ***
+Advertising  1    146.03  550.34 145.63  140.8593 <2e-16 ***
+Price        1   1322.80 1727.11 603.10 1275.9661 <2e-16 ***
+ShelveLoc    2   1053.93 1458.25 533.41  508.3098 <2e-16 ***
+Age          1    217.59  621.91 194.53  209.8897 <2e-16 ***
+Education    1      1.44  405.76  23.72    1.3930 0.2386
+US           1      1.85  406.16  24.12    1.7848 0.1823
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+>
+>
+</code>
-Urban: Yes/No
+<code>
+drop1(update(lm.full, ~. - Population -Urban -Education), test="F")
+</code>
-US: Yes/No
+<code>
-The Carseats data set tracks sales information for car seats. It has 400 observations (each at a different store) and 11 variables:
+> drop1(update(lm.full, ~. - Population -Urban -Education), test="F")
+Single term deletions
-Sales: unit sales in thousands
+Model:
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age + US
+            Df Sum of Sq     RSS    AIC   F value Pr(>F)
+<none>                    405.76  23.72
+CompPrice    1    525.25  931.00 353.92  506.1421 <2e-16 ***
+Income       1     77.87  483.62  91.94   75.0336 <2e-16 ***
+Advertising  1    145.30  551.06 144.15  140.0181 <2e-16 ***
+Price        1   1322.83 1728.58 601.44 1274.7123 <2e-16 ***
+ShelveLoc    2   1056.88 1462.64 532.61  509.2202 <2e-16 ***
+Age          1    217.97  623.73 193.70  210.0409 <2e-16 ***
+US           1      1.63  407.39  23.32    1.5693 0.2111
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+>
+</code>
-CompPrice: price charged by competitor at each location
+<code>
+drop1(update(lm.full, ~. - Population -Urban -Education -US), test="F")
+</code>
-Income: community income level in 1000s of dollars
+<code>
+> drop1(update(lm.full, ~. - Population -Urban -Education -US), test="F")
+Single term deletions
-Advertising: local ad budget at each location in 1000s of dollars
+Model:
+Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
+    Age
+            Df Sum of Sq     RSS    AIC  F value    Pr(>F)
+<none>                    407.39  23.32
+CompPrice    1    523.83  931.22 352.01  504.047 < 2.2e-16 ***
+Income       1     76.68  484.07  90.30   73.784 < 2.2e-16 ***
+Advertising  1    234.03  641.42 202.89  225.192 < 2.2e-16 ***
+Price        1   1324.42 1731.81 600.18 1274.400 < 2.2e-16 ***
+ShelveLoc    2   1055.51 1462.90 530.68  507.822 < 2.2e-16 ***
+Age          1    219.12  626.51 193.48  210.848 < 2.2e-16 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+> </code>
-Population: regional pop in thousands
-Price: price for car seats at each site
+<WRAP group>
+<WRAP half column>
+<code>lm.fit.be <- lm(Sales ~
+            CompPrice + Income +
+            Advertising + Price +
+            ShelveLoc + Age, data = Carseats)
+summary(lm.fit.be)
+</code>
-ShelveLoc: Bad, Good or Medium indicates quality of shelving location
+<code>
+> lm.fit.be <- lm(Sales ~ CompPrice +
+        Income + Advertising +
+        Price + ShelveLoc +
+        Age, data = Carseats)
+> summary(lm.fit.be)
-Age: age level of the population
+Call:
+lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
+    ShelveLoc + Age, data = Carseats)
-Education: ed level at location
+Residuals:
+    Min      1Q  Median      3Q     Max
+-2.7728 -0.6954  0.0282  0.6732  3.3292
-Urban: Yes/No
+Coefficients:
+                 Estimate Std. Error t value Pr(>|t|)
+(Intercept)      5.475226   0.505005   10.84   <2e-16 ***
+CompPrice        0.092571   0.004123   22.45   <2e-16 ***
+Income           0.015785   0.001838    8.59   <2e-16 ***
+Advertising      0.115903   0.007724   15.01   <2e-16 ***
+Price           -0.095319   0.002670  -35.70   <2e-16 ***
+ShelveLocGood    4.835675   0.152499   31.71   <2e-16 ***
+ShelveLocMedium  1.951993   0.125375   15.57   <2e-16 ***
+Age             -0.046128   0.003177  -14.52   <2e-16 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-US: Yes/No
+Residual standard error: 1.019 on 392 degrees of freedom
+Multiple R-squared:  0.872,	Adjusted R-squared:  0.8697
+F-statistic: 381.4 on 7 and 392 DF,  p-value: < 2.2e-16
+>
+</code>
 </WRAP>
+<WRAP half column>
+<code>
+lm.fit.01 <- lm(formula = Sales ~
+                 CompPrice + Income +
+                 Advertising + Price +
+                 ShelveLoc + Age,
+                 data = Carseats)
+summary(lm.fit.01)
+</code>
+<code>
+> lm.fit.01 <- lm(formula = Sales ~ CompPrice +
+         Income + Advertising + Price +
+         ShelveLoc + Age, data = Carseats)
+> summary(lm.fit.01)
+Call:
+lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
+    ShelveLoc + Age, data = Carseats)
+Residuals:
+    Min      1Q  Median      3Q     Max
+-2.7728 -0.6954  0.0282  0.6732  3.3292
+Coefficients:
+                 Estimate Std. Error t value Pr(>|t|)
+(Intercept)      5.475226   0.505005   10.84   <2e-16 ***
+CompPrice        0.092571   0.004123   22.45   <2e-16 ***
+Income           0.015785   0.001838    8.59   <2e-16 ***
+Advertising      0.115903   0.007724   15.01   <2e-16 ***
+Price           -0.095319   0.002670  -35.70   <2e-16 ***
+ShelveLocGood    4.835675   0.152499   31.71   <2e-16 ***
+ShelveLocMedium  1.951993   0.125375   15.57   <2e-16 ***
+Age             -0.046128   0.003177  -14.52   <2e-16 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+Residual standard error: 1.019 on 392 degrees of freedom
+Multiple R-squared:  0.872,	Adjusted R-squared:  0.8697
+F-statistic: 381.4 on 7 and 392 DF,  p-value: < 2.2e-16
+</code>
+</WRAP>
+</WRAP>