Chapter 8 - Multiple and Logistic Regression
Practice: 8.1, 8.3, 8.7, 8.15, 8.17
Graded: 8.2, 8.4, 8.8, 8.16, 8.18

Start of homework

8.2 Baby weights, Part II.

Exercise 8.1 introduces a data set on birth weight of babies. Another variable we consider is parity, which is 0 if the child is the ???rst born, and 1 otherwise. The summary table below shows the results of a linear regression model for predicting the average birth weight of babies, measured in ounces, from parity.

  1. Write the equation of the regression line.

\[ \begin{aligned} \widehat{weight} &= \hat{\beta}_0 + \hat{\beta}_1 \times parity \\ &= 120.07 + -1.93 \times parity\end{aligned} \]

  1. Interpret the slope in this context, and calculate the predicted birth weight of ???rst borns and others.

first born = 120.07

others =

120.07-1.93
## [1] 118.14
  1. Is there a statistically signi???cant relationship between the average birth weight and parity?

No. P-value exceeds usual .05 threshold.

8.4 Absenteeism.

Researchers interested in the relationship between absenteeism from school and certain demographic characteristics of children collected data from 146 randomly sampled students in rural New South Wales, Australia, in a particular school year. Below are three observations from this data set.

The summary table below shows the results of a linear regression model for predicting the average number of days absent based on ethnic background (eth: 0 - aboriginal, 1 - not aboriginal), sex (sex: 0 - female, 1 - male), and learner status (lrn: 0 - average learner, 1 - slow learner).

  1. Write the equation of the regression line.

\[ \begin{aligned} \widehat{days\_absent} &= \hat{\beta}_0 + \hat{\beta}_1 \times eth + \hat{\beta}_2 \times sex + \hat{\beta}_3 \times lrn\\ &= 18.93 + -9.11 \times eth + 3.10 \times sex + 2.15 \times lrn\end{aligned} \]

  1. Interpret each one of the slopes in this context.

Ethnicity looks to have a significant correlation with absenteeism, with aboriginal students missing on average more than nine days of school than non-aborginals. Sex and learner status both have p-values above .05, but data hints that males and slow learners, respectively, may miss more days.

  1. Calculate the residual for the ???rst observation in the data set: a student who is aboriginal, male, a slow learner, and missed 2 days of school.

Expected:

18.93 + -9.11 * 0 + 3.10 * 1 + 2.15 * 1
## [1] 24.18

Residual:

2-24.18
## [1] -22.18
  1. The variance of the residuals is 240.57, and the variance of the number of absent days for all students in the data set is 264.17. Calculate the R2 and the adjusted R2. Note that there are 146 observations in the data set.

\[ \begin{aligned} R^2 &= 1 - \frac{S^2\_residuals}{S^2\_absent}\\ \end{aligned} \]

R^2:

1-(240.57/264.17)
## [1] 0.08933641

Adjusted R^2:

#adjusted = mult by (n-1)/(n-k-1)
1-(240.57/264.17) * (145/142)
## [1] 0.07009704

8.8 Absenteeism, Part II.

Exercise 8.4 considers a model that predicts the number of days absent using three predictors: ethnic background (eth), gender (sex), and learner status (lrn). The table below shows the adjusted R-squared for the model as well as adjusted R-squared values for all models we evaluate in the ???rst step of the backwards elimination process.

Which, if any, variable should be removed from the model ???rst?

The model improves if learner status is removed - R^2 is higher than the full model.

8.16 Challenger disaster, Part I.

On January 28, 1986, a routine launch was anticipated for the Challenger space shuttle. Seventy-three seconds into the ???ight, disaster happened: the shuttle broke apart, killing all seven crew members on board. An investigation into the cause of the disaster focused on a critical seal called an O-ring, and it is believed that damage to these O-rings during a shuttle launch may be related to the ambient temperature during the launch. The table below summarizes observational data on O-rings for 23 shuttle missions, where the mission order is based on the temperature at the time of the launch. Temp gives the temperature in Fahrenheit, Damaged represents the number of damaged O-rings, and Undamaged represents the number of O-rings that were not damaged.

  1. Each column of the table above represents a different shuttle mission. Examine these data and describe what you observe with respect to the relationship between temperatures and damaged O-rings.

At first glance, the damaged o-rings tend to occur at lower temperatures. But it could also be that that they got better at building or configuring o-rings on later missions.

  1. Failures have been coded as 1 for a damaged O-ring and 0 for an undamaged O-ring, and a logistic regression model was ???t to these data. A summary of this model is given below. Describe the key components of this summary table in words.

The y-intercept implies that at 0 degrees, there would be more than 11 o-ring failures. With each rise in temperature, the number of damaged o-rings falls by .216. In practice, this means there would be no damaged o-rings at a certain temperature.

  1. Write out the logistic model using the point estimates of the model parameters.

\[ \begin{aligned} \widehat{Damaged\_o-rings} &= \hat{\beta}_0 + \hat{\beta}_1 \times temp\\ &= 11.6630 + -.2162 \times temp\end{aligned} \]

  1. Based on the model, do you think concerns regarding O-rings are justi???ed? Explain.

Maybe. Need to know more about advances in o-ring design and installation. It might be more likely that they’ve just started making better o-rings over time.

8.18

Challenger disaster, Part II. Exercise 8.16 introduced us to O-rings that were identi???ed as a plausible explanation for the breakup of the Challenger space shuttle 73 seconds into takeo??? in 1986. The investigation found that the ambient temperature at the time of the shuttle launch was closely related to the damage of O-rings, which are a critical component of the shuttle. See this earlier exercise if you would like to browse the original data.

where ^ p is the model-estimated probability that an O-ring will become damaged. Use the model to calculate the probability that an O-ring will become damaged at each of the following ambient temperatures: 51, 53, and 55 degrees Fahrenheit. The model-estimated probabilities for several additional ambient temperatures are provided below, where subscripts indicate the temperature:

p_51 =

p_o <- function(temp) {
o_damage <- (11.6630 - (.2162 * temp))
p <- exp(o_damage)/(1+exp(o_damage))
return(p)
}

p_o(51)
## [1] 0.6540297
p_o(53)
## [1] 0.5509228
p_o(55)
## [1] 0.4432456
#check
p_o(71)
## [1] 0.02443024
  1. Add the model-estimated probabilities from part (a) on the plot, then connect these dots using a smooth curve to represent the model-estimated probabilities.
library(ggplot2)

temps <- 51:71

me_prob <- data.frame(temps,p_o(temps))

ggplot(me_prob,aes(x = temps, y = p_o(temps))) + geom_smooth()

  1. Describe any concerns you may have regarding applying logistic regression in this application, and note any assumptions that are required to accept the model’s validity.

As implied earlier, I’m concerned about the independence of observations. If the numbering on the shuttle missions is sequential, there’s a large possibility that improvements in space travel technology could be the cause of improved o-ring survival, not temperature.