Question

Multiple Linear Regression: Statistics

y = (6.7, 14.15, 62.11, 7.8, 7.9, 8.1)

x1= (1.2, 4.5, 8.7, 3.3, 6.1, 7.2)

x2= (1.11, 7.5, 4.2, 9.1, 7.4, 8.0)

1). Construct individual 95% CIs on β1 and β2.

2).  Construct simultaneous 95% CIs on β1 and β2.

Answer 1

Answer:

Multiple Linear Regression: Statistics

y = (6.7, 14.15, 62.11, 7.8, 7.9, 8.1)

x1= (1.2, 4.5, 8.7, 3.3, 6.1, 7.2)

x2= (1.11, 7.5, 4.2, 9.1, 7.4, 8.0)

1). Construct individual 95% CIs on β1 and β2.

95% CIs on β1 = (-2.540837, 15.285902) and β2= (-11.915604 , 4.339309)

2).  Construct simultaneous 95% CIs on β1 and β2.

95% CIs on β1(-5.325072, 18.070137) and β2= (-14.454346,   6.878052)

If we want simultaneous confidence intervals for both the slopes, using the Bonferroni method with joint confidence level α, set the level equal to 1 – α / 2. In our case we set level as 0.975.

R used for computation:

R code:

y=c(6.7,14.15,62.11,7.8,7.9,8.1)

x1=c(1.2,4.5,8.7,3.3,6.1,7.2)

x2=c(1.11,7.5,4.2,9.1,7.4,8.0)

mydata <- data.frame(y,x1,x2)

model <-lm(y~x1+x2, data = mydata)

summary(model)

anova(model)

#Construct individual 95% CIs

confint(model,level=0.95)

#Construct simultanuous 95% CIs

confint(model,level=0.975)

R output:

Call:

lm(formula = y ~ x1 + x2, data = mydata)

Residuals:

      1       2       3       4       5       6

-5.167   5.460 14.155 12.818 -11.365 -15.902

Coefficients:

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

(Intercept)    8.425     19.368   0.435    0.693

x1             6.373      2.801   2.275    0.107

x2            -3.788      2.554 -1.483    0.235

Residual standard error: 16.36 on 3 degrees of freedom

Multiple R-squared: 0.6642,    Adjusted R-squared: 0.4404

F-statistic: 2.967 on 2 and 3 DF, p-value: 0.1946

> anova(model)

Analysis of Variance Table

Response: y

          Df Sum Sq Mean Sq F value Pr(>F)

x1         1 999.76 999.76 3.7342 0.1488

x2         1 589.06 589.06 2.2002 0.2346

Residuals 3 803.19 267.73              

> #Construct individual 95% CIs

> confint(model,level=0.95)

                 2.5 %    97.5 %

(Intercept) -53.211907 70.060994

x1           -2.540837 15.285902

x2          -11.915604 4.339309

> #Construct simultanuous 95% CIs

> confint(model,level=0.975)

                1.25 %   98.75 %

(Intercept) -72.465048 89.314136

x1           -5.325072 18.070137

x2          -14.454346 6.878052