---
title: "Matched pairs"
editor: 
  markdown: 
    wrap: 72
---

## Matched pairs

Some data:

![](Screenshot_2019-04-26_13-41-29.png){height="250"}

## Matched pairs 1/2

-   Data are comparison of 2 drugs for effectiveness at reducing pain.

    -   12 subjects (cases) were arthritis sufferers
    -   Response is #hours of pain relief from each drug.

-   In reading example, each child tried only one reading method.

-   But here, each subject tried out both drugs, giving us two
    measurements.

-   Possible because, if you wait long enough, one drug has no influence
    over effect of other.

## Matched pairs 2/2

-   Advantage: focused comparison of drugs. Compare one drug with
    another on same person, removes a lot of variability due to
    differences between people.

-   Matched pairs, requires different analysis.

-   Design: randomly choose 6 of 12 subjects to get drug A first, other
    6 get drug B first.

## Packages

```{r}
library(tidyverse)
library(smmr) # for a sign test later
```

## Reading the data

Values aligned in columns:

\footnotesize

```{r inference-4b-R-1}
my_url <- "http://ritsokiguess.site/datafiles/analgesic.txt"
pain <- read_table(my_url)
pain
```

\normalsize

## Paired $t$-test

```{r inference-4b-R-3}
with(pain, t.test(druga, drugb, paired = TRUE))
```

-   P-value is 0.053.
-   Not quite evidence of difference between drugs.

## t-testing the differences

-   Or, calculate the differences yourself and then do a
    1-sample t-test on them.

\footnotesize

```{r inference-4b-R-4}
pain %>% mutate(diff = druga - drugb) -> pain
pain
```

\normalsize

## t-test on the differences

-   then throw them into t.test, testing that the mean is zero, with
    same result as before:

```{r inference-4b-R-5}
with(pain, t.test(diff, mu = 0))
```

-   Same P-value (0.053) and conclusion.

## Assessing normality

-   1-sample and 2-sample t-tests assume (each) group normally
    distributed.
-   Matched pairs analyses assume (theoretically) that differences
    normally distributed.
-   How to assess normality? A normal quantile plot.

## The normal quantile plot (of differences)

```{r inference-4b-R-6, fig.height=4}
ggplot(pain,aes(sample=diff))+stat_qq()+stat_qq_line()
```

-   Points should follow the straight line. Bottom left one way off, so
    normality questionable here: outlier.

## What to do instead?

-   Matched pairs $t$-test based on one sample of differences
-   the differences not normal (enough)
-   so do *sign test* on differences, null median 0:

```{r inference-4b-R-7}
sign_test(pain, diff, 0)
```

## Did we need to worry about that outlier?

Bootstrap sampling distribution of sample mean differences:

```{r}
#| fig-height: 4
tibble(sim = 1:10000) %>% 
  rowwise() %>% 
  mutate(my_sample = list(sample(pain$diff, replace = TRUE))) %>% 
  mutate(my_mean = mean(my_sample)) %>% 
  ggplot(aes(sample = my_mean)) + stat_qq() + stat_qq_line()
```

Yes we did; this is clearly skewed left and not normal.

## Comments

-   no evidence of any difference between drugs (P-value 0.1460)
-   in $t$-test, the low outlier difference pulled mean difference
    downward and made it look more negative than it should have been
-   therefore, there really isn't any difference between the drugs.
