install.packages(“lmtest”)
Consider the mtcars data set. Fit a model with mpg as the outcome that includes number of cylinders as a factor variable and weight as confounder. Give the adjusted estimate for the expected change in mpg comparing 8 cylinders to 4. *Answer: 6.07
Consider the mtcars data set. Fit a model with mpg as the outcome that includes number of cylinders as a factor variable and weight as a possible confounding variable. Compare the effect of 8 versus 4 cylinders on mpg for the adjusted and unadjusted by weight models. Here, adjusted means including the weight variable as a term in the regression model and unadjusted means the model without weight included. What can be said about the effect comparing 8 and 4 cylinders after looking at models with and without weight included?.
*Answer:Holding weight constant, cylinder appears to have less of an impact on mpg than if weight is disregarded.
## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.990794 1.8877934 18.005569 6.257246e-17
## factor(cyl)6 -4.255582 1.3860728 -3.070244 4.717834e-03
## factor(cyl)8 -6.070860 1.6522878 -3.674214 9.991893e-04
## wt -3.205613 0.7538957 -4.252065 2.130435e-04## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.663636 0.9718008 27.437347 2.688358e-22
## factor(cyl)6 -6.920779 1.5583482 -4.441099 1.194696e-04
## factor(cyl)8 -11.563636 1.2986235 -8.904534 8.568209e-10## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Likelihood ratio test
##
## Model 1: mpg ~ factor(cyl) + wt
## Model 2: mpg ~ factor(cyl) * wt
## #Df LogLik Df Chisq Pr(>Chisq)
## 1 5 -73.311
## 2 7 -70.741 2 5.1412 0.07649 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1*Answer: The p value is higher than 0.05 which means that we fail to reject the null hypothesis. The P-value is larger than 0.05. So, according to our criterion, we would fail to reject, which suggests that the interaction terms may not be necessary.
Consider the mtcars data set. Fit a model with mpg as the outcome that includes number of cylinders as a factor variable and weight inlcuded in the model as lm(mpg ~ I(wt * 0.5) + factor(cyl), data = mtcars) How is the wt coefficient interpretted?
nfit<-lm(mpg ~ I(wt * 0.5) + factor(cyl), data = mtcars)
summary(fit)$coef## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.990794 1.8877934 18.005569 6.257246e-17
## factor(cyl)6 -4.255582 1.3860728 -3.070244 4.717834e-03
## factor(cyl)8 -6.070860 1.6522878 -3.674214 9.991893e-04
## wt -3.205613 0.7538957 -4.252065 2.130435e-04*Answer:The estimated expected change in MPG per one ton increase in weight for a specific number of cylinders (4, 6, 8).
Consider the following data set x <- c(0.586, 0.166, -0.042, -0.614, 11.72) y <- c(0.549, -0.026, -0.127, -0.751, 1.344) Give the hat diagonal for the most influential point
x <- c(0.586, 0.166, -0.042, -0.614, 11.72)
y <- c(0.549, -0.026, -0.127, -0.751, 1.344)
new1<-lm(y~x)
hatvalues(new1)## 1 2 3 4 5
## 0.2286650 0.2438146 0.2525027 0.2804443 0.9945734*Answer:0.9946
Consider the following data set
x <- c(0.586, 0.166, -0.042, -0.614, 11.72) y <- c(0.549, -0.026, -0.127, -0.751, 1.344) Give the slope dfbeta for the point with the highest hat value.
## Influence measures of
## lm(formula = y ~ x) :
##
## dfb.1_ dfb.x dffit cov.r cook.d hat inf
## 1 1.0621 -3.78e-01 1.0679 0.341 2.93e-01 0.229 *
## 2 0.0675 -2.86e-02 0.0675 2.934 3.39e-03 0.244
## 3 -0.0174 7.92e-03 -0.0174 3.007 2.26e-04 0.253 *
## 4 -1.2496 6.73e-01 -1.2557 0.342 3.91e-01 0.280 *
## 5 0.2043 -1.34e+02 -149.7204 0.107 2.70e+02 0.995 **Answer: -134
Consider a regression relationship between Y and X with and without adjustment for a third variable Z. Which of the following is true about comparing the regression coefficient between Y and X with and without adjustment for Z.
*Answer:It is possible for the coefficient to reverse sign after adjustment. For example, it can be strongly significant and positive before adjustment and strongly significant and negative after adjustment.