Case C→Q

Recall that in the Exploratory Data Analysis unit, examining the relationship between X and Y in this situation amounts, in practice, to:

• Comparing the distributions of the (quantitative) response Y for each value (category) of the explanatory X.

To do that, we used

• side-by-side boxplots (each representing the distribution of Y in one of the groups defined by X),
• and supplemented the display with the corresponding descriptive statistics.

EXAMPLE: GPA and Year in College

Suppose that our variable of interest is the GPA of college students in the United States. From Unint 4A, we know that since GPA is quantitative, we will conduct inference on μ, the (population) mean GPA among all U.S. college students.

Since this section is about relationships, let’s assume that what we are really interested in is not simply GPA, but the relationship between:

• X : year in college (1 = freshmen, 2 = sophomore, 3 = junior, 4 = senior) and
• Y : GPA

In other words, we want to explore whether GPA is related to year in college.

The way to think about this is that the population of U.S. college students is now broken into 4 sub-populations: freshmen, sophomores, juniors and seniors. Within each of these four groups, we are interested in the GPA.

The inference must therefore involve the 4 sub-population means:

• μ₁ : mean GPA among freshmen in the United States.
• μ₂ : mean GPA among sophomores in the United States
• μ₃ : mean GPA among juniors in the United States
• μ₄ : mean GPA among seniors in the United States

It makes sense that the inference about the relationship between year and GPA has to be based on some kind of comparison of these four means.

If we infer that these four means are not all equal (i.e., that there are some differences in GPA across years in college) then that’s equivalent to saying GPA is related to year in college. Let’s summarize this example with a figure:

EXAMPLE:

Suppose we are conducting a clinical trial. Participants are randomized into two independent subpopulations:

• those who are given a drug and
• those who are given a placebo.

Each individual appears in only one of these two groups and individuals are not matched or paired in any way. Thus the two samples or groups are independent. We can say those given the drug are independent from those given the placebo.

Recall: By randomly assigning individuals to the treatment we control for both known and unknown lurking variables.

EXAMPLE:

Suppose the Highway Patrol wants to study the reaction times of drivers with a blood alcohol content of half the legal limit in their state.

An observational study was designed which would also serve as publicity on the topic of drinking and driving. At a large event where enough alcohol would be consumed to obtain plenty of potential study participants, officers set up an obstacle course and provided the vehicles. (Other considerations were also implemented to keep the car and track conditions consistent for each participant.)

Volunteers were recruited from those in attendance and given a breathalyzer test to determine their blood alcohol content. Two types of volunteers were chosen to participate:

• Those with a blood alcohol content of zero – as measured by the breathalyzer – of which 10 were chosen to drive the course.
• Those with a blood alcohol content within a small range of half the legal limit (in Florida this would be around 0.04%) – of which 9 were chosen.

Here also, we have two independent groups – even if originally they were taken from the same sample of volunteers – each individual appears in only one of the two groups, the comparison of the reaction times is a comparison between two independent groups.

However, in this study, there was NO random assignment to the treatment and so we would need to be much more concerned about the possibility of lurking variables in this study compared to one in which individuals were randomized into one of these two groups.

In this design, then, the same individual or a matched pair of individuals is used to make two measurements of the response – one for each of the two levels of the categorical explanatory variable.

Advantages of a paired sample approach include:

• Reduced measurement error since the variance within subjects is typically smaller than that between subjects
• Requires smaller number of subjects to achieve the same power than independent sample methods.

Disadvantages of a paired sample approach include: 

• An order effect based upon which treatment individuals received first.
• A carryover effect such as a drug remaining in the system.
• Testing effect such as particpants learning the obstacle course in the first run improving their performance in the 2nd.

EXAMPLE:

Suppose we are conducting a study on a pain blocker which can be applied to the skin and are comparing two different dosage levels of the solution which in this study will be applied to the forearm.

For each participant both solutions are applied with the following protocol:

• Which drug is applied to which arm is random.
• Patients and clinical staff are blind to the two treatment applications.
• Pain tolerance is measured on both arms using the same standard test with the order of testing randomized.

Here we have dependent samples since the same patient appears in both dosage groups.

Again, randomization is employed to help minimize other issues related to study design such as an order or testing effect.

EXAMPLE:

Suppose the department of motor vehicles wants to check whether drivers are impaired after drinking two beers.

The reaction times (measured in seconds) in an obstacle course are measured for 8 randomly selected drivers before and then after the consumption of two beers.

We have a matched-pairs design, since each individual was measured twice, once before and once after.

In matched pairs, the comparison between the reaction times is done for each individual.