Chapter 8 - Multiple and Logistic Regression

Graded: 8.2, 8.4, 8.8, 8.16, 8.18

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.
       avg_birth_weight <- 120.07 - 1.93 * parity

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

        Every child who is not a first born will predicted to be have 1.93 less than birth weight of the first child

        First born: parity = 0; the baby’s weight will be 120.07 ounces.

        Others: parity = 1; the baby’s weight will be 118.14 ounces.

  1. Is there a statistically significant relationship between the average birth weight and parity?       
    P-value is greater than 0.05, So there is no statistically significant relationship between avarage birth weight and parity.
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 SouthWales, 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.
           Absenteeism = 18.93 - 9.11 * (ethnic background) + 3.10 * (sex) + 2.15 * (learner status)

  2. Interpret each one of the slopes in this context.
           eth <- The slope of ethic indicate that it is 9.11 decrease in predicted absenteeism when children are no aboriginal.

            sex <- The slope of sex indicate that it is 3.10 increase in predicated absenteeism when children are male.

            lrn <- The slope of lrn indicate that it is 2.15 increas in predicated absenteeism when children are in slow learner.

  1. 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 # aboriginal
sex <- 1 # male
lrn <- 1 # slow learner

Absenteeism <- 18.93 - 9.11 * eth + 3.1 * sex + 2.15 * lrn

residual <- 2 - Absenteeism
residual
## [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.
n <- 146 #observations
k <- 3   #predictor variable
var_residual <- 240.57 #variance of the residuals
var_students <- 264.17 #variance for all students

R2 <- 1 - (var_residual / var_students)  #R2

adjusted_R2 <- 1 - (var_residual / var_students) * ( (n-1) / (n-k-1) ) #Adjusted R2
adjusted_R2
## [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 first step of the backwards elimination process. Which, if any, variable should be removed from the model first?
       The Adjusted R2=0.0723 is better, so the lrn variable should be removed from the model first.

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.
           
temperature <- c(53,57,58,63,66,67,67,67,68,69,70,70,70,70,72,73,75,75,76,76,78,79,81)
damaged <- c(5,1,1,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0)
undamaged <- c(1,5,5,5,6,6,6,6,6,6,5,6,5,6,6,6,6,5,6,6,6,6,6)
shuttle_mission <- data.frame(temperature, damaged, undamaged)

plot(shuttle_mission)

      It looks like the higher damage O-rings when lower temperatures were recorded.

  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 fit to these data. A summary of this model is given below. Describe the key components of this summary table in words.

       The key components are the Intercept and the Temperature values. The intercepts is the value of damage o-ring when temperature is 0. And the temperature values is the slope of increasing temperature per each degrees.

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

             loge(p/1-p)=11.6630-0.2162×Temperature

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

             p value for temperature is 0. The estimate is O-ring damages through temperature, So that concerns regarding O-rings are justi???ed.

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 takeoff 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.

  1. The data provided in the previous exercise are shown in the plot.The logistic model fit to these data may be written as loge(p/1???p)=11.6630???0.2162×Temperature 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:
    ^p57 = 0.341 ^p59 = 0.251 ^p61 = 0.179 ^p63 = 0.124
    ^p65 = 0.084 ^p67 = 0.056 ^p69 = 0.037 ^p71 = 0.024
temp <- c(51,53,55)

model <- function(temp)
{
  damage <- 11.6630 - 0.2162 * temp
  p <- exp(damage) / (1 + exp(damage))
  return (round(p*100,2))
}

model(temp)
## [1] 65.40 55.09 44.32
  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)

temp <- 51:71

model_estimate <- data.frame(temp,model(temp))

ggplot(model_estimate,aes(x = temp, y = model(temp))) + geom_smooth()
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'


(c) 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 logistic regression appears to be a linear, but the data points do not show as they are linear. So the assumption is each point is independent of one another in order to have a valid model.