Paired t-test
This variation of t-test is used for Paired Data
Paired Data
Two set of observations are paired if each observation in one set has exactly one corresponding observation is another set.
Examples:
- pre- and post-test scores on the same person
- measures in pairs at the same time or place
- outcome with or without a treatment - on same subject (cross-over study)
Using in R
library(openintro)
data(textbooks)
t.test(textbooks$diff, mu=x.bar.nul, alternative='two.sided')
or
t.test(textbooks$uclaNew, textbooks$amazNew, paired=T, alternative='two.sided')
Examples
Example: Bookstore vs Amazon
- two samples: local bookshop and amazon
- $\mu_\text{dif} = \mu_l - \mu_a$ - the mean of difference in the price
Test
- $H_0: \mu_\text{dif} = 0$ - there's no difference in the price
- $H_A: \mu_\text{dif} \ne 0$ - there's some difference
Calculations
- $\bar{X}_\text{dif} = 12.76$
- Standard Error: $\text{SE}_{\bar{X}_\text{dif}} = \cfrac{s_\text{dif}}{\sqrt{n_\text{dif}}} = 1.67$
- $T = \cfrac{\bar{X}_\text{dif}}{\text{SE}_{\bar{X}_\text{dif}}} = \cfrac{12.76}{1.67} = 7.59$
- $p = 6 \cdot 10^{-11}$, less than $\alpha = 0.05$, so we reject $H_0$
library(openintro)
data(textbooks)
hist(textbooks$diff, col='yellow')
n = length(textbooks$diff)
s = sd(textbooks$diff)
se = s / sqrt(n)
x.bar.nul = 0
x.bar.dif = mean(textbooks$diff)
t = (x.bar.dif - x.bar.nul) / se
t
p = pt(t, df=n-1, lower.tail=F) * 2
p
or
t.test(textbooks$diff, mu=x.bar.nul, alternative='two.sided')
Example 2
Let $\mu_d = \mu_0 - \mu_1$ be the difference between two methods
Our test:
- $H_0: \mu_d = 0, H_A: \mu_d \neq 0$
Say, we have:
- $\bar{X}_d = 6.854$
- $s_d = 11.056$
- $n = 398$
Test statistics:
- $\cfrac{\bar{X}_d - 0}{s_d / \sqrt{n}} = \cfrac{6.854}{11.056 / \sqrt{398}} \approx 12.37$
Then we compare it with $t_{397}$
- $p$-value is $2.9 \cdot 10^{29}$
And we conclude that the difference between the two methods is not 0
Sources