Paired and Two Sample t-tests

Edward A. Roualdes

Recap

Recap: \(t\)-distribution

So far, we have used the t-distribution for confidence intervals or hypothesis tests of one mean (proportions are means).

t.test(df$x, mu = some_value, conf.level = 0.95)

Paired Data

Paired Data, definition

Paired data are somehow intimately connected.

Two sets of observations are paired if each observation in one set has a special correspondence or connection with exactly one observation in the other data set.

Paired Data, by example

Tell me whether or not these data are paired.

two website’s price for the same book
eye sight ratings by person
jumping spider vertical (jump height)
upper verse lower bird beak lengths
weights of male and female babies

Paired Data, t-test

If the data are paired, their difference has direct and interpretable meaning both in English and in statistics; \(X_{i,diff} = X_{i,a} - X_{i,b}\) has meaning. Therefore

\[\bar{X}_{diff} \quad \text{ and } \quad s_{\bar{X}_{diff}}\]

are simply fancy ways to write new random variables.

An Example

Paired Data, confidence interval

Are textbooks actually cheaper online? Compare the price of textbooks at the University of California, Los Angeles’ (UCLA’s) bookstore and prices at Amazon.com. Seventy-three UCLA courses were randomly sampled in Spring 2010.

books <- read.csv("https://raw.githubusercontent.com/roualdes/data/master/books.csv")
## look at data in RStudio
## what plot should we make?

Paired Data, confidence interval

Plot the data!

suppressMessages(library(ggplot2))
ggplot(books, aes(uclaNew, amazNew),
       xlab="UCLA price ($)", ylab="Amazon price ($)") +
    geom_point() +
    geom_abline(intercept=0, slope=1)

Paired Data, confidence interval

Calculate and interpret a 95% confidence interval of the difference in Amazon.com versus UCLA’s book prices.

Paired Data, confidence interval

d <- books$amazNew - books$uclaNew # create new variable
# now the same as before
t.test(d, conf.level = 0.95)


    One Sample t-test

data:  d
t = -7.6488, df = 72, p-value = 6.928e-11
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -16.087652  -9.435636
sample estimates:
mean of x 
-12.76164

Paired Data, confidence interval

We are \(95\)% confident that the true mean difference in price between Amazon.com and UCLA’s books is between -16.09 and -9.44.

Paired Data, hypothesis test

Set up, evaluate, and conclude in context a hypothesis test at \(\alpha = 0.05\).

Paired Data, hypothesis test

The natural hypotheses are

\[ H_0: \mu_{diff} = 0 \text{ versus } H_1: \mu_{diff} \ne 0. \]

Paired Data, hypothesis test

d <- books$amazNew - books$uclaNew # create new variable
t.test(d, mu = 0, conf.level = 0.95)


    One Sample t-test

data:  d
t = -7.6488, df = 72, p-value = 6.928e-11
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -16.087652  -9.435636
sample estimates:
mean of x 
-12.76164

Paired Data, hypothesis test

Becuase p-value\(<0.0001 < \alpha = 0.05\), we reject \(H_0\). There is insufficient evidence to claim that Amazon.com and UCLA’s book prices are the same.

Two Sample t-test

Two sample t-tests estimate the difference between two population means from two independent samples of data. We estimate \(\mu_a - \mu_b\) with the point estimator \(\bar{X}_a - \bar{X}_b\).

Two Sample t-test, confidence interval

Confidence intervals for two sample t-tests follow the same pattern as before,

\[ (\bar{X}_a - \bar{X}_b) \pm t^*_{df} \cdot s_{\bar{X}_a - \bar{X}_b}. \]

Two Sample t-test, hypothesis tests

Test statistics for two sample t-tests follow the same pattern as before, and p-values are exactly the same.

Two Sample t-test, example

Consider the data set ape::carnivora. Calculate a \(98\)% confidence interval for the difference in mean longevity between the two SuperFamilies Caniformia and Feliformia.

suppressMessages(library(ape))
data(carnivora)
# ?carnivora
# look at importing data

Two Sample t-test, confidence interval

A \(98\)% CI, difference in longevity by Caniformia and Feliformia.

t.test(LY~SuperFamily, data=carnivora,
       conf.level=0.98)


    Welch Two Sample t-test

data:  LY by SuperFamily
t = 1.0243, df = 37.394, p-value = 0.3123
alternative hypothesis: true difference in means between group Caniformia and group Feliformia is not equal to 0
98 percent confidence interval:
 -28.13845  69.13511
sample estimates:
mean in group Caniformia mean in group Feliformia 
                192.4583                 171.9600

Two Sample t-test, confidence interval

We are \(98\)% confidence that the population difference in mean longevity between the SuperFamlies Caniformia and Feliformia is between -28.1 and 69.1.

Two Sample t-test, hypothesis test

Set up, evaluate, and conclude in context a hypothesis test at \(\alpha = 0.02\).

Two Sample t-test, hypothesis test

The natural hypotheses are

\[ H_0: \mu_C = \mu_F \text{ versus } H_1: \mu_C \ne \mu_F \]

and \(\alpha = 0.02\).

Two Sample t-test, hypothesis test

Because p-value \(=0.31 > \alpha = 0.02\), we fail to reject \(H_0\). There is insufficient evidence to claim that the true difference in mean longevity between Caniformia and Feliformia is different.

Take Away

Overall things have stayed pretty much the same: confidence intervals, hypothesis tests, and interpretations. Now we have new types of data we can work with.

Paired data
- Two variables are intimately connected \(\Rightarrow\) their difference has meaning
- create one variable from the two \(\Rightarrow\) one sample t-test
Two sample data
- Two variables are independent
- Point estimate is difference of means
- Standard error follows from independence