---
title: "Normal quantile plots"
---

## The normal quantile plot

-   see that normal distributions of data (or being normal enough)
    important
-   only tools we have to assess this are histograms and maybe boxplots
-   a better tool is **normal quantile plot**:
    -   plot data against what you expect if data actually normal
    -   look for points to follow a straight line, at least approx
-   `ggplot` code: `aes` `sample`; geoms `stat_qq` and `stat_qq_line`

## Packages

The usual:

```{r}
library(tidyverse)
```

## Kids learning to read

```{r inference-4a-R-1, echo=FALSE, message=FALSE}
my_url <- "http://ritsokiguess.site/datafiles/drp.txt"
kids <- read_delim(my_url," ")
glimpse(kids)
```

```{r inference-4a-R-2}
ggplot(kids, aes(x = group, y = score)) + geom_boxplot()
```

Each group looks more or less normal, or at least close to symmetric.

## Get the groups separately

```{r inference-4a-R-3}
kids %>% filter(group == "t") -> treatment
kids %>% filter(group == "c") -> control
treatment
```

to check

```{r inference-4a-R-4}
treatment %>% count(group)
control %>% count(group)
```

## The treatment group

```{r inference-4a-R-5, fig.height=4.5}
ggplot(treatment, aes(sample = score)) + 
  stat_qq() + stat_qq_line()
```

only problem here is lowest value a little too low (mild outlier).

## Control group

```{r inference-4a-R-6, fig.height=4}
ggplot(control, aes(sample = score)) + 
  stat_qq() + stat_qq_line()
```

This time, highest value a little too high, but again, no real problem
with normality.

## Facetting more than one sample

Use the whole data set and facet by groups

```{r inference-4a-R-7, fig.height=4.5}
ggplot(kids, aes(sample = score)) + 
  stat_qq() + stat_qq_line() + facet_wrap(~group)
```

## Blue Jays attendances, skewed to right

```{r inference-4a-R-8, echo=FALSE, message=FALSE}
jays <- read_csv("jays15-home.csv")
```

```{r inference-4a-R-9}
ggplot(jays, aes(x = attendance)) + geom_histogram(bins = 6)
```

## On a normal quantile plot

```{r inference-4a-R-10, fig.height=3.5}
ggplot(jays, aes(sample = attendance)) + 
  stat_qq() + stat_qq_line()
```

-   Attendances at low end too bunched up: skewed to right.
-   Right-skewness can also show up as highest values being too high, or
    as a curved pattern in the points.

## More normal quantile plots

-   How straight does a normal quantile plot have to be?
-   There is randomness in real data, so even a normal quantile plot
    from normal data won't look perfectly straight.
-   With a small sample, can look not very straight even from normal
    data.
-   Looking for systematic departure from a straight line; random
    wiggles ought not to concern us.
-   Look at some examples where we know the answer, so that we can see
    what to expect.

## Normal data, large sample

```{r set-seed, echo=F}
set.seed(457299)
```

```{r inference-4a-R-11, fig.height=4.5}
d <- tibble(x=rnorm(200))
ggplot(d, aes(x=x)) + geom_histogram(bins=10)
```

## The normal quantile plot

```{r inference-4a-R-12, fig.height=4.5}
ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()
```

## Normal data, small sample

```{r inference-4a-R-13, echo=F}
set.seed(457299)
```

-   Not so convincingly normal, but not obviously skewed:

```{r normal-small, fig.height=4.5}
d <- tibble(x=rnorm(20))
ggplot(d, aes(x=x)) + geom_histogram(bins=5)
```

## The normal quantile plot

Good, apart from the highest and lowest points being slightly off. I'd
call this good:

```{r inference-4a-R-14, fig.height=4.5}
ggplot(d, aes(sample=x)) + stat_qq() + stat_qq_line()
```

## Chi-squared data, *df* = 10

Somewhat skewed to right:

```{r inference-4a-R-15, fig.height=4.5}
d <- tibble(x=rchisq(100, 10))
ggplot(d,aes(x=x)) + geom_histogram(bins=10)
```

## The normal quantile plot

Somewhat opening-up curve:

```{r inference-4a-R-16, fig.height=4.5}
ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()
```

## Chi-squared data, df = 3

Definitely skewed to right:

```{r chisq-small-df, fig.height=4.5}
d <- tibble(x=rchisq(100, 3))
ggplot(d, aes(x=x)) + geom_histogram(bins=10)
```

## The normal quantile plot

Clear upward-opening curve:

```{r inference-4a-R-17, fig.height=4.5}
ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()
```

## t-distributed data, df = 3

Long tails (or a very sharp peak):

```{r t-small, fig.height=4.5}
d <- tibble(x=rt(300, 3))
ggplot(d, aes(x=x)) + geom_histogram(bins=15)
```

## The normal quantile plot

Low values too low and high values too high for normal.

```{r inference-4a-R-18, fig.height=4.5}
ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()
```

## Summary

On a normal quantile plot:

-   points following line (with some small wiggles): normal.
-   kind of deviation from a straight line indicates kind of
    nonnormality:
    -   a few highest point(s) too high and/or lowest too low: outliers
    -   else see how points at each end off the line:

|                | High points |              |
|----------------|-------------|--------------|
| **Low points** | **Too low** | **Too high** |
| **Too low**    | Skewed left | Long tails   |
| **Too high**   | Short tails | Skewed right |

-   short-tailed distribution OK for $t$ (mean still good), but others
    problematic (depending on sample size).