Baby Weights, Part I. (9.1, p 350)

The Child Health and Development Studies investigate a range of topics. One study considered all pregnancies between 1960 and 1967 among women in the Kaiser Foundation Health Plan in the San Francisco East Bay area. Here, we study the relationship between smoking and weight of the baby. The variable smoke is coded 1 if the mother is a smoker, and 0 if not. The summary table below shows the results of a linear regression model for predicting the average birth weight of babies, measured in ounces, based on the smoking status of the mother.

The variability within the smokers and non-smokers are about equal and the distributions are symmetric. With these conditions satisfied, it is reasonable to apply the model. (Note that we don’t need to check linearity since the predictor has only two levels.)

## 
## Call:
## lm(formula = babies$bwt ~ babies$parity)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -65.068 -11.068   0.396  10.932  57.860 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   120.0684     0.6005 199.942   <2e-16 ***
## babies$parity  -1.9287     1.1895  -1.621    0.105    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 18.22 on 1234 degrees of freedom
## Multiple R-squared:  0.002126,   Adjusted R-squared:  0.001317 
## F-statistic: 2.629 on 1 and 1234 DF,  p-value: 0.1052

A

Write the equation of the regression line.

\(birth weight = 120.07 - 1.93 * parity\)

B

Interpret the slope in this context, and calculate the predicted birth weight of babies born to smoker and non-smoker mothers.

Solution

The slope of -1.93 indicates that if the child is not the first born, the regression model will predict a weight at the time of birth of 1.93 ounces less than if the child is the first born child.

C

Is there a statistically significant relationship between the average birth weight and smoking?

Solution

There is not a statistically significant relationship between the average birth weight and parity.

The p-value of 0.1052 is greater than the significance level of 0.05.

Absenteeism, Part I. (9.4, p352)

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.

A

Write the equation of the regression line.

\(days absent = 18.93 - 9.11 * eth + 3.10 * sex + 2.15 * lrn\)

B

Interpret each one of the slopes in this context.

Solution

The slopes mean that the average number of days absent is predicted to decrease by 9.11 for not aboriginal students, increase by 3.10 for male students, and increaseby 2.15 for slow learners.

C

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.

The residual is -22.18 for the first observation in the dataset.

## [1] -22.18

Absenteeism, Part II. (9.8, p.357)

Exercise above 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?

Ethnicity should be removed from the model first because the negative R-squared value indicates a poor model.

Challenger disaster, Part I. (9.16, p. 380)

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.

A

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.

Solution

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.

It seems that as the temperature decreases, the amount of damaged O-rings increase.

B

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.

Solution

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 of the summary table are the intercept of 11.6630, which meants that at a temperature of 0 there would be an O-ring code of 11.6630. The p-value of 0 means that the relationship between temperature and O-ring failure is significant. The slope of -0.2162 means that for every degree the temperature increases, the O-ring failure probability decreases by 0.2162.

C

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

\(O-ring = 11.6630 - 0.2162 * temperture\)

Challenger disaster, Part II. (9.18, p. 381)

Exercise above introduced us to O-rings that were identifiedas 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.

A

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

\(log(\frac{\hat{p}}{1-\hat{p}}) = 11.6630 - 0.2162 * Temperature\)

Solution:

\(\hat{p}\) is the model-estimated probability that an O-ring will become damaged.

Using 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:

## [1] "51: 0.654029738456095" "53: 0.550922829727926" "55: 0.443245647107059"

B

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.

##  [1] "51: 0.654029738456095"  "53: 0.550922829727926" 
##  [3] "55: 0.443245647107059"  "57: 0.340649762822082" 
##  [5] "59: 0.251091386916725"  "61: 0.178697068887302" 
##  [7] "63: 0.123727017415509"  "65: 0.0839384318975581"
##  [9] "67: 0.0561256566847601" "69: 0.0371547886407227"
## [11] "71: 0.0244302390548414"

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.

Solution

The conditions for logistic regression are that each predictor \(x_i\) has to be linearly related to \(log(p_i)\) and each outcome \(Y_i\) is independent of eachother.