Question

Use the appropriate test to determine whether X1 can be dropped from the regression model given that X2 is retained. Use level of significance 0.05. Find the value of appropriate test statistic, the critical vale and the P-value. Plese show me how to use R to solve this.

1. 0.1537; 4.6679; 0.7014
2. 0.1537; 4.3245; 0.6523
3. 1.537; 4.6932; 0.4128
4. 2.5632; 4.6679; 0.3128

X1   X2   Y
190   130   35
176   174   81.7
205   134   42.5
210   191   98.3
230   165   52.7
192   194   82
220   143   34.5
235   186   95.4
240   139   56.7
230   188   84.4
200   175   94.3
218   156   44.3
220   190   83.3
210   178   91.4
208   132   43.5
225   148   51.7

Answer 1

After loading our date into R with the name of Test;

> Test <- Regression...Sheet1
> linearMod <- lm(Y ~ ., data=Test) # build linear regression model on full data
> print(linearMod)

Call:
lm(formula = Y ~ ., data = Test)

Coefficients:
(Intercept) X1 X2
-67.88436 -0.06419 0.90609

> summary(linearMod)

Call:
lm(formula = Y ~ ., data = Test)

Residuals:
Min 1Q Median 3Q Max
-15.172 -8.404 1.026 7.410 16.457

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -67.88436 40.58652 -1.673 0.118
X1 -0.06419 0.16391 -0.392 0.702
X2 0.90609 0.12337 7.344 5.63e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 11.2 on 13 degrees of freedom
Multiple R-squared: 0.8064,   Adjusted R-squared: 0.7766
F-statistic: 27.07 on 2 and 13 DF, p-value: 2.319e-05

Further breaking the components individually;

> modelCoeffs <- modelSummary$coefficients # model coefficients
> modelCoeffs
Estimate Std. Error t value Pr(>|t|)
(Intercept) -67.88435970 40.5865217 -1.6725838 1.182893e-01
X1 -0.06418911 0.1639142 -0.3916018 7.016961e-01
X2 0.90608862 0.1233709 7.3444291 5.628708e-06
> beta.estimate <- modelCoeffs["X1", "Estimate"] # get beta estimate for X1
> beta.estimate
[1] -0.06418911
> std.error <- modelCoeffs["X1", "Std. Error"] # get std.error for X1
> std.error
[1] 0.1639142
> t_value <- beta.estimate/std.error # calc t statistic
> t_value
[1] -0.3916018
> p_value <- 2*pt(-abs(t_value), df=nrow(Test)-ncol(Test)) # calc p Value
> p_value
[1] 0.7016961
> f_statistic <- linearMod$fstatistic[1] # fstatistic
> f_statistic
NULL
> f <- summary(linearMod)$fstatistic # parameters for model p-value calc
> f
value numdf dendf
27.07029 2.00000 13.00000
> model_p <- pf(f[1], f[2], f[3], lower=FALSE)
> model_p
value
2.318617e-05
>

As we can see the p-value is high for X1 and low value for t-statistic hence we can drop X1. Keeping variables that are not statistically significant can reduce the model’s precision.