Lesson 23: Worksheet

Author

Jane Doe

You are going to work with data collected in the 1970s to examine the effects of smoking and exposure to second-hand smoke on pulmonary functions in youths. The data frame is FEV and is included in the {math300} package.

The response variable that we will study is also called FEV, standing for the “forced expiratory volume” measured in the participants in the study. In general, higher forced expiratory volume is considered a sign of better respiratory health.

Task 1

Build a model using the FEV data frame and the model specification FEV ~ smoker and pipe it through conf_interval(). Then answer these questions:

  1. One of the model coefficients is called smokersmoker. This looks like a typo, but it is not. Explain why the variable is smoker but the name of the coefficient is smokersmoker. (Hint: Look at the documentation for FEV.)
ANSWER

smoker is a categorical variable with two levels: “smoker” and “not.” The label smokersmoker refers to the smoker level of the smoker variable.

  1. The numerical value of the smokersmoker coefficient is 0.71. Based on this, is smoking associated with higher or lower FEV.
ANSWER

smokersmoker quantifies how much FEV differs for the smoker level compared to the not level. A positive coefficient means that smoking is associated with higher FEV.

  1. What are the units of FEV? What are the units of the smokersmoker coefficient.
ANSWER

The units of the response variable, FEV, are liters. Since smoker is categorical, smokersmoker also has units of liters.

Task 2

  1. What is the width of the confidence interval on smokersmoker?
ANSWER

You can find the length of the confidence interval simply by subtracting the .lwr limit of the interval from the .upr limit. Here that’s 0.927 - 0.494 = 0.433.

Task 3.

Just for pedagogical purposes, we are going to explore how the width of the confidence interval would change if we had more or less data. You already have heard the theoretical relationship of the width of the confidence interval as a function of sample size \(n\).

  1. What is the size of the sample contained in FEV?
ANSWER 
nrow(FEV)
[1] 654
  1. Using the theoretical relationship with \(n\), what do you think the width of the confidence interval would be if only \(n=150\) rows of data were available?
ANSWER

150 is about one-quarter the sample size of FEV. So a confidence interval calculated on a sample of \(n=150\) will be about \(\sqrt{4}\) times larger. That is, the sample size 150 will lead to a confidence interval about twice as wide as the confidence interval from the full sample.

  1. We can easily simulate working with a sample of \(n=150\). To do this, fit a model (and calculate the confidence interval on smokersmoker), but rather than using the argument data=FEV use this instead: data=sample(FEV, size=150). Compare the width of confidence interval you get in this way to your theoretical prediction in (2).

  2. This will be surprising, but we can actually simulate what would happen if we had a larger sample size. (This is just a simulation, and just for pedagogical purposes. This is not a way to collect a genuine sample of a larger size.)

To create a (simulated) sample of size, say, \(n=2500\) set the data argument to lm() this way: data=resample(FEV, size=2500). (NOTE: The function being used here is not sample() but the closely related resample(), with an re in front.)

Calculate the width of the (simulated) confidence interval on smokersmoker for the sample size of 2500.

ANSWER 
lm(FEV ~ smoker, data=resample(FEV, size=2500)) |> conf_interval()
# A tibble: 2 × 4
  term          .lwr .coef  .upr
  <chr>        <dbl> <dbl> <dbl>
1 (Intercept)  2.54  2.58  2.61 
2 smokersmoker 0.561 0.672 0.783

The width of the confidence interval on `smokersmoker is about 0.20.

Task 3

Were you surprised to see in Task 1 that smoking is associated with a higher FEV than non-smoking? Since a higher FEV is considered healthier, does this mean that smoking is healthy? The answer is “no,” but let’s consider it from the perspective of accuracy versus precision.

The confidence interval on smokersmoker in the model FEV ~ smoker was 0.50 to 0.93 liters. This precision is good enough to rightfully claim that the smokersmoker coefficient is not zero or negative.

But precision is different from accuracy. One of the major potential determinants of FEV is age.

  1. Build a model FEV ~ age and construct the confidence interval on age. Explain whether your model is consistent or not with the idea that FEV depends on age.
ANSWER 
lm(FEV ~ age, data=FEV) |> conf_interval()
# A tibble: 2 × 4
  term         .lwr .coef  .upr
  <chr>       <dbl> <dbl> <dbl>
1 (Intercept) 0.279 0.432 0.585
2 age         0.207 0.222 0.237

The age coefficient is about 0.2 liters per year. In other words, FEV increases with age.

  1. It also happens that smoking is associated with age. The younger kids don’t smoke. We can demonstrate this with a model of smoker ~ age. Since smoker is a categorical variable, we need to convert it to a zero-one variable before fitting the model. Here’s a chunk to do so, assigning the smokers to have a value of 1:
FEV |> 
  mutate(smoke = zero_one(smoker, one="smoker")) |>
  lm(smoke ~ age, data = _) |>
  conf_interval()
# A tibble: 2 × 4
  term           .lwr   .coef    .upr
  <chr>         <dbl>   <dbl>   <dbl>
1 (Intercept) -0.381  -0.308  -0.234 
2 age          0.0338  0.0410  0.0481

Interpret the coefficient as a rate of probability: how the probability that a participant smokes changes per year of age.

  1. Is the age coefficient consistent with the idea that older kids are more likely to smoke?

  2. Now the point about accuracy and precision being different things. FEV increases with age and so does the probability of being a smoker. That means that smoker is also related to age. In fact, to some extent smoker is a proxy for age. As an exercise, draw on paper a DAG where age influences FEV, and age influences smoking status, and also smoking status influences FEV.

For the DAG just described, an accurate model to estimate the direct effect of smoking on FEV is FEV ~ age + smoker. Fit this model and use the confidence interval on smoker to make an accurate statement about smoking and FEV.

ANSWER 
lm(FEV ~ age + smoker, data = FEV) |>
  conf_interval()
# A tibble: 3 × 4
  term           .lwr  .coef    .upr
  <chr>         <dbl>  <dbl>   <dbl>
1 (Intercept)   0.207  0.367  0.527 
2 age           0.215  0.231  0.247 
3 smokersmoker -0.368 -0.209 -0.0504

The confidence interval on smokersmoker is entirely negative. Negative means that smoking is associated with smaller FEV. This sort of thing would be summarized as, “After adjusting for age, smoking is associated with smaller FEV.”