+ +

+ + + + + + + +

All exercises taken from the OpenIntro Statistics textbook, \(4^{th}\) edition, Chapter 9.

9.4 Absenteeism in Australian schools

Hereâs a way to write the equation:

\[\widehat{days} = 18.93 - (9.11\times{ethnicity}) + (3.10 \times sex) + (2.15 \times learner~status)\]

Letâs go through these one-by-one. I should note that I think the labeling of these variables encode racial and gender biases in some quiet ways that are irresponsible:
+

\(b_{eth}\): On average, not-aboriginal students are estimated to be absent 9.11 days less than aboriginal students.
+
\(b_{sex}\): On average, male students are estimated to be absent 3.10 days more than female students.
+
\(b_{lrn}\): On average, students classified as âslow learnersâ are estimated to be absent 2.15 days more than those classified as âaverage learners.â

A residual for some observation \(i\) is the observed outcome minus the fitted value (\(y_i - \hat{y_i}\)). I can calculate \(\hat{y_i}\) by plugging the observed predictor values into the regression equation above in part (a):

18.93 - (9.11*0) + (3.10 * 1) + (2.15 * 1)

## [1] 24.18

The observed outcome for this student (\(y_i\)) was \(2\) days absent. So the residual is \(2-24.18=-22.18\).

Formulas for this appear below. Note that I denote the variance as \(\sigma^2\), the residuals as \(e\), and the outcome \(y\).

\[R^2 = 1-\frac{\sigma^2_e}{\sigma^2_y}= 1-\frac{240.57}{264.17} = 0.0893\] \[R^2_{adj} = 1-\frac{ \frac{\sigma^2_e}{(n-p-1)} }{\frac{\sigma^2_y}{n-1}}= 1-\frac{ \frac{240.57}{146-3-1}} {\frac{264.17}{146-1}} = 0.0701\]

9.16 O-rings

The damaged O-rings almost all occurred at the lower launch-time temperatures, with the lowest launch temperature accounting for nearly half of the total number of damaged O-rings.
The model suggests that lower launch-time temperatures result in a higher probability of O-ring damage. The coefficient of the âTemperatureâ term is negative with a very small (proportionally speaking) standard error. It is statistically significant with a p-value near 0 (\(H_0:~\beta_{temp} = 0\), \(H_A:~\beta_{temp}\neq 0\)), indicating that the data provide evidence that the coefficient is likely different from 0. By exponentiating the coefficient (see the R-code below), we see that a one degree farenheit increase in launch-time temperature is associated with 81% as large odds of O-ring damage occurring. In other words, the model indicates that higher launch temperatures associate with reduced odds of O-ring damage.

exp(-.2162)

## [1] 0.8055742

The corresponding logistic model where \(\hat{p}_{damage}\) represents the probability of a damaged O-ring:
+\[log(\frac{\hat{p}_{damage}}{1-\hat{p}_{damage}}) = 11.663 - 0.2162\times~Temperature\]
+
Given the high stakes in terms of human lives and vast costs involved, concerns about the relationship between O-rings and launch-time temperature seem more than justified from this data. The significant negative association between temperature and O-ring damage suggest increased potential for failures at low launch temperatures. That said, several limitations of the data, modeling strategy, and estimates should be kept in mind. See my answer to part (c) of 9.18 below for more on this.

9.18 More O-rings

Letâs do this in R. Note that weâll need to plug in the values for temperature and do some algebra with the natural logarithm parts of the formula (on the left hand side above) to find the predicted probability of O-ring damage. Iâll solve it by writing a little function probs that takes fitted values from the model and runs them through the inverse logistic function to return probabilities (see the textbook for some of the algebraic details here). I can test my function on some of the example model-estimated probabilities provided in the textbook:

probs <- function(x){
+  p.hat <- exp(11.663-(0.2162*x) ) 
+  pred <- p.hat / (1 + p.hat) # inverse logit
+  return(round(pred, 3))
+}
+
+## examples
+probs(57)

## [1] 0.341

probs(65)

## [1] 0.084

Both of those look good, so now I can plug in the values the problem asks me to solve for:

vals <- c(51, 53, 55)
+
+probs(vals)

## [1] 0.654 0.551 0.443

Iâll use my little function above to build a data frame with the predicted values and plot everything in ggplot.

Note that the question asks for a âsmooth curveâ fit to the dots. There are many ways to do this in ggplot. I demonstrate one here using geom_smooth() that fits a quadratic function (\(y = x^2\)) to the points. You might experiment with the different options for geom_smooth() or, for a simpler solution, just try geom_line() (with no arguments) instead.

temp =  seq(51, 71, by=2) # This creates a vector from 51 to 71, counting by twos 
+
+preds <- data.frame( # I'll want the data frame for ggplot
+  temp, 
+  pred.probability = probs(temp) # store the probabilities as another vector
+)
+
+library(ggplot2)
+
+ggplot(data=preds, aes(x=temp, y=pred.probability)) + 
+  geom_point(color="purple") + # Plot the points
+  geom_smooth(color="orange", # Add a smooth line
+              method="glm", # Create a line fit to the data
+              formula = y ~ poly(x, 2), # Using this formula
+              se=FALSE) # Don't show standard errors

I have several concerns about this application of logistic regression to the problem and data at hand. First, this is a fairly small observational dataset with a lot of potential confounders and threats to the assumptions necessary to identify the model. For instance, the textbook is unclear whether each mission was treated as an independent trial or each O-ring was treated as an independent trial. Either assumption is problematic. The O-rings within any given mission are probably more similar to each other than to the O-rings on other missions. In addition, it is possible that the O-ring production or installation procedures may have changed across the missions over time. Likewise any of the flight and/or launch procedures may have varied in subtle ways correlated (or not) with the temperature and/or the O-ring outcomes. Any such clustered or time-dependent structures lurking in the data could lead to unobserved bias in the estimates when we model each set of mission-specific or O-ring-specific outcomes as independent events without accounting for these added sources of covariance/clustering.

Furthermore, if the model treats each O-ring as an indpendent trial, about 50% of the observed failures occurred in a single missionâthe mission with the lowest observed launch-time temperature. The result is that this one mission with its one launch temperature could drive the model results disproportionately (it generates observations that exert âhigh leverageâ on the model fit to the data). Without knowing ahead of time that temperature was a likely explanation (as compared against any of the other infinite details of that one mission), itâs hard to see how NASA analysts necessarily should have drawn this conclusion on the basis of evidence like this.

+ + + +

Chapter 9 Textbook exercises

Solutions to even-numbered questions
+Statistics and statistical programming
+Northwestern University
+MTS 525

Aaron Shaw

November 11, 2020

9.4 Absenteeism in Australian schools

9.16 O-rings

9.18 More O-rings

Chapter 9 Textbook exercises

Solutions to even-numbered questions +Statistics and statistical programming +Northwestern University +MTS 525

Aaron Shaw

November 11, 2020

9.4 Absenteeism in Australian schools

9.16 O-rings

9.18 More O-rings

Solutions to even-numbered questions
+Statistics and statistical programming
+Northwestern University
+MTS 525