Paired Data: One Quantitative Variable

Chapter 7

Introduction The paired datasets in this chapter have one pair

of quantitative response values for each observational unit.

This allows for a built-in comparison. Studies with paired data remove individual

variability by looking at the difference score for each individual.

Reducing variability in data improves inferences: Narrower confidence intervals Smaller p-values when the null hypothesis is

false

Introduction Our data that we will analyze will just be a

single quantitative variable. So things like mean and standard

deviation are important to look at, but really nothing new for descriptive statistics.

Section 7.1: Simulation-based method Section 7.2: Theory-based method

Section 7.1: Simulation-Based Approach for

Analyzing Paired Data

Example 7.1: Rounding First Base

First Base Imagine you’ve hit a

line drive and are trying to reach second base.

Does the path that you take to “round” first base make much a difference? Narrow angle Wide angle

First Base Hollander and Wolfe (1999) report on a Master’s

Thesis by Woodward (1970) that investigates base running strategies.

Woodward timed 22 different runners from a spot 35 feet past home to a spot 15 feet before second.

Each runner used each strategy (paired design), with a rest between.

This paired design controls for the runner-to-runner variability.

He used random assignment to decide which path each runner should do first.

First Base Times for the first 10 runners

Dotplots of times for all 22 runners

Subject 1 2 3 4 5 6 7 8 9 10  narrow angle

5.50 5.70 5.60 5.50 5.85 5.55 5.40 5.50 5.15 5.80 …

wide angle

5.55 5.75 5.50 5.40 5.70 5.60 5.35 5.35 5.00 5.70 …

First Base There is a lot of overlap in the distributions

and a fair bit of variability

Difficult to detect a difference between the methods when there’s a lot of variation

Mean SD

Narrow 5.534 0.260

Wide 5.459 0.273

First Base

What are the observational units in this study? The runners (22 total)

What variables are recorded? What are their types and roles? Explanatory variable: base running method: wide or

narrow angle (categorical) Response variable: time for middle of the route from

home plate to second base (quantitative) Is this an observational study or an experiment?

Randomized experiment since the explanatory variable was randomly applied to determined which method each runner used first

First Base These data are clearly paired. The paired response variable is time

difference in running between the two methods (narrow angle – wide angle).

Could we do wide angle – narrow angle?

First Base Differences for the first 10 runners

A dotplot of the differences for all 22 runners.

Subject 1 2 3 4 5 6 7 8 9 10  narrow angle

5.50 5.70 5.6 5.50 5.85 5.55 5.40 5.50 5.15 5.80 …

wide angle

5.55 5.75 5.5 5.40 5.70 5.60 5.35 5.35 5.00 5.70 …

diff -0.05 -0.05 0.1 0.1 0.15 -0.05 0.05 0.15 0.15 0.10 …

First Base The distribution is a bit skewed left, but

not too bad. Mean difference is d = 0.075 seconds Standard deviation is SDd = 0.0883 sec Standard deviation (0.0883) is smaller

than the original standard deviations of the running times (0.260 and 0.273).

First Base The original dotplots with each

observation paired between the base running strategies.

What do you notice?

First Base

Is the average difference of d = 0.075 seconds significantly different from 0?

The parameter of interest is, µd, is the population average difference in running times by some population of runners when using the narrow angle and the wide angle. (narrow – wide)

First BaseThe hypotheses: 

H0: µd = 0 On average, the mean of the differences

between the running times (narrow – wide) is 0. Ha: µd 0

On average, the mean of the differences in running times (narrow – wide) is not 0.

If the parameter of interest is the population average difference, then the corresponding statistic is the sample average difference.

First Base

How can simulation-based methods find an approximate p-value? The null basically says the running path

doesn’t matter --- the times, on average, will be the same for the two methods.

So we can use our same data set and randomly decide which time goes with the narrow and wide methods and compute a mean difference. (Notice we don’t break our pairs.)

We can repeat this process many times to develop a null distribution.

First Base The results of random swaps for the first

10 runners (done with coin flips) An average difference of d = -0.025

seconds Repeat many times to construct a null

distribution This models no connection between times

and the strategy used

Subject 1 2 3 4 5 6 7 8 9 10  narrow angle

5.55 5.70 5.50 5.50 5.70 5.60 5.40 5.50 5.15 5.70 …

wide angle

5.50 5.75 5.60 5.40 5.85 5.55 5.35 5.35 5.00 5.80 …

diff 0.05 -0.05 -0.1 0.1 -0.15 0.05 0.05 0.15 0.15 -0.1 …

First Base Mean differences from 1000 repetitions Describe the shape of the distribution. The distribution appears to be centered at

about 0. Does that make sense?

First Base Using the null distribution is the observed

average from the study of 0.075 out in the tail?

First Base Only 2 of the 1000 repetitions of random

swappings gave a value at least as extreme as 0.075

First Base We can also standardize 0.075 by dividing

by the applet’s estimate of the SD ≈ 0.024 to see we are standard deviations above zero.

First Base Based on the p-value and standardized

statistic we have very strong evidence against the null hypothesis.

We can draw a cause-and-effect conclusions since the researcher used random assignment of the two base running methods for each runner.

There was not a lot of information about how these 22 runners were selected to decide if we can generalize to a larger population.

First Base Approximate a 95% confidence interval for

d: 0.075 ± 2(0.024) seconds (0.027, 0.124) seconds

What does this mean? We are 95% confident that, on average, the

narrow angle route takes 0.027 to 0.124 seconds longer than the wide angle route

Let’s try this out with the applet.

First BaseAlternative Analysis

What do you think would happen if we wrongly analyzed the data using a 2 independent samples procedure?

I.e. the researcher selected 22 runners to use the wide method and an independent sample of 22 other runners to use the narrow method, obtaining the same 44 times as in the actual study.

Would the p-value stay the same, increase, or decrease?

First BaseUsing the Two Means applet (which does an independent test) we get a p-value of 0.1830

Does it make sense that this p-value is larger than the one we obtained earlier?

Exercise and Heart Rate

Exploration 7.1

Section 7.2: Theory-based methods for paired data.

 H0: µd = 0 Path doesn’t matter

Ha: µd 0 Path does matter

narrow - wide

d = 0.075 seconds SDd = 0.0883

seconds n = 22

First Base Our null distribution was centered at zero

and fairly bell-shaped. This can all be predicted (along with the

variability) using theory-based methods.

To do this, our sample size should be at least 20.

Theory-based test We can do theory-based methods with the

applet we used last time or the theory-based applet.

With the applet we used last time, we need to calculate the t-statistic:

With the theory-based applet, we just need to enter the summary statistics and use a test for a one mean.

Theory-based results

First Base

The theory-based model gives slightly different results, but we come to the same conclusion. Which base running path used does make a difference in the average times (we can see that with our small p-value).

We estimate the narrow angle path will take between 0.036 to 0.114 seconds longer, on average, to complete than the wide angle path.

Exploration 7.2Comparing Auction Formats We will compare:

Dutch auction the item for sale starts at a very high price and is lowered gradually until someone finds the price low enough to buy.

First-price sealed bid auction each bidder summits a single sealed bid before a particular deadline. After the deadline, the person with the highest bid wins.