STAT3002 Week 2

 

2.2 Adding a Predictor to SLR

 

Auction Data cont’.  Recall the linear regression of Price vs Age :

 

> auc1.lm <- lm(Price ~ Age)

> summary(auc1.lm)

 

Call: lm(formula = Price ~ Age)

Residuals:

    Min     1Q Median    3Q   Max

 -485.3 -192.7  30.75 157.2 541.2

 

Coefficients:

                Value Std. Error   t value  Pr(>|t|)

(Intercept) -191.6576  263.8866    -0.7263    0.4733

        Age   10.4791    1.7900     5.8543    0.0000

 

Residual standard error: 273 on 30 degrees of freedom

Multiple R-Squared: 0.5332

F-statistic: 34.27 on 1 and 30 degrees of freedom, the p-value is 2.096e-006

 

Correlation of Coefficients:

    (Intercept)

Age -0.9831

 

The R2 = 0.5332, indicating that 53.3% of the variability in Price is explained by Age.

 

And the linear regression of Price vs Number of Bidders:

 

> summary(auc2.lm)

 

Call: lm(formula = Price ~ Bidders)

Residuals:

    Min     1Q Median    3Q   Max

 -516.3 -355.3 -29.49 302.8 688.2

 

Coefficients:

               Value Std. Error  t value Pr(>|t|)

(Intercept) 806.4049 230.6846     3.4957   0.0015

    Bidders  54.6362  23.2250     2.3525   0.0254

 

Residual standard error: 367.2 on 30 degrees of freedom

Multiple R-Squared: 0.1557

F-statistic: 5.534 on 1 and 30 degrees of freedom, the p-value is 0.0254

 

Correlation of Coefficients:

        (Intercept)

Bidders -0.9596

 

 

 

 

 

Are the predictors related?   It is difficult to tell…

 

Added Variable Plot (AVP)

 

 

 

 

For this regression R2 = 0.1557, indicating that 15.6% of variability in Price is explained by Bidders.  Is it reasonable to perform a regression on Price vs Age and bidders?

 

> auc.lm <- lm(Price ~ Age+Bidders)

> summary(auc.lm)

 

Call: lm(formula = Price ~ Age + Bidders)

Residuals:

    Min     1Q Median    3Q   Max

 -207.2 -117.8  16.49 102.7 213.5

 

Coefficients:

                 Value Std. Error    t value   Pr(>|t|)

(Intercept) -1336.7221   173.3561    -7.7108     0.0000

        Age    12.7362     0.9024    14.1140     0.0000

    Bidders    85.8151     8.7058     9.8573     0.0000

 

Residual standard error: 133.1 on 29 degrees of freedom

Multiple R-Squared: 0.8927

F-statistic: 120.7 on 2 and 29 degrees of freedom, the p-value is 8.771e-015

 

Correlation of Coefficients:

        (Intercept)     Age

    Age -0.8759           

Bidders -0.6701      0.2537

 

2.3 Introduction to Multiple Regression

 

Squid Data.  This Data was collected to study the size of squid eaten by sharks and tuna.  The covariates are characteristics of the beak or mouth of the squid.  They are:

 

The response is the width of the squid in pounds.

 

Weight Beak    Wing    Width

1.95     1.31     1.07     0.35

2.90     1.55     1.49     0.47

0.72     0.99     0.84     0.32

0.81     0.99     0.83     0.27

1.09     1.05     0.90     0.30

1.22     1.09     0.93     0.31

1.02     1.08     0.90     0.31

1.93     1.27     1.08     0.34

0.64     0.99     0.85     0.29

2.08     1.34     1.13     0.37

1.98     1.30     1.10     0.38

1.90     1.33     1.10     0.38

8.56     1.86     1.47     0.65

4.49     1.58     1.34     0.50

8.49     1.97     1.59     0.59

6.17     1.80     1.56     0.59

7.54     1.75     1.58     0.59

6.36     1.72     1.43     0.63

7.63     1.68     1.57     0.68

7.78     1.75     1.59     0.62

10.15   2.19     1.86     0.72

6.88     1.73     1.67     0.55

 

Regression Analysis

All 3 predictors:

 

> squid.lm <- lm(Weight ~ Beak+Wing+Width)

> anova(squid.lm)

Analysis of Variance Table

 

Response: Weight

 

Terms added sequentially (first to last)

          Df Sum of Sq  Mean Sq  F Value     Pr(F)

     Beak  1  199.1453 199.1453 399.0594 0.0000000

     Wing  1    0.1267   0.1267   0.2538 0.6205103

    Width  1    7.6701   7.6701  15.3698 0.0010027

Residuals 18    8.9827   0.4990 

 

 

> summary(squid.lm)

 

Call: lm(formula = Weight ~ Beak + Wing + Width)

Residuals:

    Min      1Q  Median     3Q    Max

 -1.421 -0.5198 0.09608 0.4467 0.9206

 

Coefficients:

               Value Std. Error  t value Pr(>|t|)

(Intercept)  -6.7916   0.7801    -8.7059   0.0000

       Beak   4.1860   2.0355     2.0564   0.0545

       Wing  -1.3270   2.0965    -0.6330   0.5347

      Width  14.0463   3.5828     3.9204   0.0010

 

Residual standard error: 0.7064 on 18 degrees of freedom

Multiple R-Squared: 0.9584

F-statistic: 138.2 on 3 and 18 degrees of freedom, the p-value is 1.29e-012

 

Correlation of Coefficients:

      (Intercept)    Beak    Wing

 Beak -0.4758                   

 Wing -0.0877     -0.7073       

Width  0.5440     -0.4451 -0.2846

 

 

Two predictors:

 

>squid.lm2 <- lm(Weight ~ Beak+Wing)

> summary(squid.lm2)

 

Call: lm(formula = Weight ~ Beak + Wing)

Residuals:

    Min      1Q Median     3Q   Max

 -2.147 -0.7121 0.1499 0.6422 1.497

 

Coefficients:

              Value Std. Error t value Pr(>|t|)

(Intercept) -8.4556  0.8675    -9.7474  0.0000

       Beak  7.7377  2.4157     3.2031  0.0047

       Wing  1.0125  2.6635     0.3802  0.7080

 

Residual standard error: 0.9362 on 19 degrees of freedom

Multiple R-Squared: 0.9229

F-statistic: 113.7 on 2 and 19 degrees of freedom, the p-value is 2.681e-011

 

Correlation of Coefficients:

     (Intercept)    Beak

Beak -0.3109           

Wing  0.0835     -0.9715

> anova(squid.lm2)

Analysis of Variance Table

 

Response: Weight

 

Terms added sequentially (first to last)

          Df Sum of Sq  Mean Sq  F Value     Pr(F)

     Beak  1  199.1453 199.1453 227.2154 0.0000000

     Wing  1    0.1267   0.1267   0.1445 0.7080468

Residuals 19   16.6528   0.8765

 

One Predictor:

 

> squid.lm1 <- lm(Weight ~ Beak)

> summary(squid.lm3)

 

Call: lm(formula = Weight ~ Beak)

Residuals:

    Min      1Q  Median     3Q   Max

 -1.993 -0.7324 0.09125 0.6395 1.615

 

Coefficients:

               Value Std. Error  t value Pr(>|t|)

(Intercept)  -8.4831   0.8457   -10.0303   0.0000

       Beak   8.6299   0.5601    15.4068   0.0000

 

Residual standard error: 0.916 on 20 degrees of freedom

Multiple R-Squared: 0.9223

F-statistic: 237.4 on 1 and 20 degrees of freedom, the p-value is 1.468e-012

 

Correlation of Coefficients:

     (Intercept)

Beak -0.973 

> anova(squid.lm1)

Analysis of Variance Table

 

Response: Weight

 

Terms added sequentially (first to last)

          Df Sum of Sq  Mean Sq  F Value         Pr(F)

     Beak  1  199.1453 199.1453 237.3686 1.467826e-012

Residuals 20   16.7794   0.8390

 

> summary(squid.lm1)

 

Call: lm(formula = Weight ~ Beak)

Residuals:

    Min      1Q  Median     3Q   Max

 -1.993 -0.7324 0.09125 0.6395 1.615

 

Coefficients:

               Value Std. Error  t value Pr(>|t|)

(Intercept)  -8.4831   0.8457   -10.0303   0.0000

       Beak   8.6299   0.5601    15.4068   0.0000

 

Residual standard error: 0.916 on 20 degrees of freedom

Multiple R-Squared: 0.9223

F-statistic: 237.4 on 1 and 20 degrees of freedom, the p-value is 1.468e-012

 

Correlation of Coefficients:

     (Intercept)

Beak -0.973  

 

Variables in a different Order

 

> squid.lm3 <- lm(Weight ~ Width+Beak+Wing)

> anova(squid.lm3)

Analysis of Variance Table

 

Response: Weight

 

Terms added sequentially (first to last)

          Df Sum of Sq  Mean Sq  F Value     Pr(F)

    Width  1  204.1576 204.1576 409.1034 0.0000000

     Beak  1    2.5845   2.5845   5.1790 0.0353166

     Wing  1    0.1999   0.1999   0.4006 0.5347209

Residuals 18    8.9827   0.4990 

 

 

 

Added Variable Plots

 

 

Check Assumptions!  Do the residuals look normally distributed?