Using the seatpos data, fit the linear regression of hipcenter on all of the oth

Question

Using the seatpos data, fit the linear regression of hipcenter on all of the other variables.

(a) Produce a summary of the regression results.

(b) Do any variables appear to be signi cant based on the individual t-tests for their
coeficients? What about based on the overall F-test (for all of the variables together)?

(c) Compute the variance inflation factors (VIFs) for the variables. Using the threshold of 10 to determine if a VIF indicates a problem of collinearity, which variables have a VIF
indicating a possible problem?

(d) Reduce the model by removing all variables that had VIFs you identified as problematic in the previous part. Produce a summary of the regression results.

(e) For the model of the previous part, do any variables appear to be significant based on the individual t-tests for their coecients? What about based on the overall F-test (for
all of the variables together)?

(f) Compute the VIFs for the reduced set of variables. (Have they changed?) Again using the threshold of 10, which variables have a VIF indicating a possible problem?

Install seatpos data package in R., or download data at: https://cran.r-project.org/web/packages/faraway/index.html

Explanation / Answer

The complete R snippet is as follows

library(faraway)

data.df <- data.frame(seatpos)

# fit the model
fit <- lm(hipcenter~.,,data = data.df)
summary(fit)

# get the vif values
vif(fit)

# all variables where vif is greater than 2 are problematic
sqrt(vif(fit)) > 2

## htshoes , ht , seated leg are all problematic

# refit the model
myformula <- hipcenter~ Age + Weight +Thigh

# fit the model
fit1 <- lm(myformula,,data = data.df)
summary(fit1)

The results are

> summary(fit)

Call:
lm(formula = hipcenter ~ ., data = data.df)

Residuals:
Min 1Q Median 3Q Max
-73.827 -22.833 -3.678 25.017 62.337

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 436.43213 166.57162 2.620 0.0138 *
Age 0.77572 0.57033 1.360 0.1843
Weight 0.02631 0.33097 0.080 0.9372
HtShoes -2.69241 9.75304 -0.276 0.7845
Ht 0.60134 10.12987 0.059 0.9531
Seated 0.53375 3.76189 0.142 0.8882
Arm -1.32807 3.90020 -0.341 0.7359
Thigh -1.14312 2.66002 -0.430 0.6706
Leg -6.43905 4.71386 -1.366 0.1824
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 37.72 on 29 degrees of freedom
Multiple R-squared: 0.6866,   Adjusted R-squared: 0.6001
F-statistic: 7.94 on 8 and 29 DF, p-value: 1.306e-05 # none of the variables are significant as the p values are not lessthan 0.05 , however the overall model is signficant which tells that the model is effected by multicollinearity . Lets find varaince inflaiton factor

> vif(fit)
Age Weight HtShoes Ht Seated Arm Thigh
1.997931 3.647030 307.429378 333.137832 8.951054 4.496368 2.762886
Leg
6.694291
> sqrt(vif(fit)) > 2
Age Weight HtShoes Ht Seated Arm Thigh Leg
FALSE FALSE TRUE TRUE TRUE TRUE FALSE TRUE

> myformula <- hipcenter~ Age + Weight +Thigh
> fit1 <- lm(myformula,,data = data.df)
> summary(fit1)

Call:
lm(formula = myformula, data = data.df)

Residuals:
Min 1Q Median 3Q Max
-84.764 -26.436 2.596 20.809 84.995

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 126.7917 69.6700 1.820 0.07759 .
Age 1.0654 0.4438 2.401 0.02198 *
Weight -0.7679 0.2315 -3.316 0.00218 **
Thigh -5.4259 2.1400 -2.535 0.01599 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 41.29 on 34 degrees of freedom
Multiple R-squared: 0.5597,   Adjusted R-squared: 0.5208
F-statistic: 14.41 on 3 and 34 DF, p-value: 3.194e-06

when we refit the model by accounting for vifs we get the variales as significant

Please note that we can answer only 4 subparts of a question at a time , as per the answering guidelines.

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