Data 606_Chapter 8_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 first 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.

baby.b = 120.07 #intercept 
baby.m = -1.93 # parity coefficient
paste0("y = ", round(baby.b, 2), " + ", round(baby.m, 2), " * x + ")

## [1] "y = 120.07 + -1.93 * x + "

2. Interpret the slope in this context, and calculate the predicted birth weight of first borns and others.

baby.x1 = 0
baby.x2 = 1
baby.y1 = baby.m * baby.x1 + baby.b
baby.y2 = baby.m * baby.x2 + baby.b
baby.y1

## [1] 120.07

baby.y2

## [1] 118.14

** On average, birth weight is 1.93oz lower for children born after the first. The predicted birth weight of first-born children is 120.07oz, and of non-first-born children is 118.14oz. **

3. Is there a statistically significant relationship between the average birth weight and parity?

** p-value is .1052, so there is not sufficient evidence to dismiss the null hypothesis at the 95% or 90% confidence levels.**

8.4 Absenteeism, Part I. 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.

absent = 18.93 - 9.11 * eth + 3.10 * sex + 2.15 * lrn

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

The strongest predictor of absenteeism in this model is aboriginal ethnicity, followed by male sex, then slow learning status. Ceteris paribus, on average non-aboriginal studnets miss 9.11 fewer days; male students miss 3.10 more days; slow learners miss 2.15 more days.

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

eth = 0
sex = 1
lrn = 1
days = 2
absent = 18.93 - 9.11 * eth + 3.10 * sex + 2.15 * lrn
days - absent

## [1] -22.18

The residual is -22.18 - the model overestimated number of days absent by 22 days.

4. 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.

baby.n <- 146 
baby.k <- 3
baby.s.e <- 240.57
baby.s.y <- 264.17

baby.R2 <- 1 - (baby.s.e / baby.s.y)
baby.R2.adj <- 1 - (baby.s.e * (baby.n - 1)) / ((baby.s.y) * (baby.n - baby.k - 1))
paste0("R2 = ", round(baby.R2, 4), " and Adjusted R2 = ", round(baby.R2.adj, 4))

## [1] "R2 = 0.0893 and Adjusted R2 = 0.0701"

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 first step of the backwards elimination process. Which, if any, variable should be removed from the model first?

We should remove learner status first. When it is not included in the model, the adjusted R^2 is .0723, which is higher than including all three predictors, removing sex, and removing ethnicity.

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 flight, 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.

The largest number of damaged O-rings occured on the shuttle mission with the lowest temperature. 3 out of 13 missions with temperatures 70 or above suffered a single damaged O-ring on the launch. 1 of the 7 missions with temperatures in the 60s suffered a single damaged O-ring on the launch. All three of the missions with temperatures in the 50s suffered damaged O-rings on the launch.

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

There’s a negative relationship between temperature and damage to O-rings, and the p-value below .05 means that there is enough evidence to dismiss the null hypothesis.

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

log(p.hat / (1 - p.hat)) = 11.6630 - .2162 * temp

4. Based on the model, do you think concerns regarding O-rings are justified? Explain.

Yes, the slope of -.2162 means that the probability of ring damage diminishes as tempertatures rise. As mentioned above, the p-value tells us this is not due to chance.

8.18 Challenger disaster, Part II. Exercise 8.16 introduced us to O-rings that were identified 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.

The data provided in the previous exercise are shown in the plot. The logistic model fit to these data may be written as … 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:

temp <- c(51, 53, 55)
p.hat <- exp(11.6630 - .2162 * temp) / (1 + exp(11.6630 - .2162 * temp))
p.hat

## [1] 0.6540297 0.5509228 0.4432456

2. 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.

temp <- seq(from = 51, to = 81)
p.hat <- exp(11.6630 - .2162 * temp) / (1 + exp(11.6630 - .2162 * temp))
oring <- as.data.frame(cbind(temp, p.hat))
ggplot(oring, aes(x = temp, y = p.hat)) + 
  geom_point() +
  geom_smooth(method = "loess")

3. 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.

The sample size is 23 launches, which is low. We’re only lookin at one explanatory variable (temperature) when there could be others worth exploring. As for that variable, it treats damage of O-rings as a binary, which is fine so long as the threshold between damaged / undamaged is well understood. As this is a time series, it’s possible that treating the performance of O-rings in each launch as independent events is not a good idea.