---
title: "ECDF, QQ, and PP Plots"
output:
  html_document:
    toc: yes
    code_folding: show
    code_download: true
---

<link rel="stylesheet" href="stat4580.css" type="text/css" />
<!-- title based on Wilke's chapter -->

```{r setup, include = FALSE, message = FALSE}
source(here::here("setup.R"))
knitr::opts_chunk$set(collapse = TRUE, message = FALSE,
                      fig.height = 5, fig.width = 6, fig.align = "center")

set.seed(12345)
library(dplyr)
library(ggplot2)
library(lattice)
library(gridExtra)
source(here::here("datasets.R"))
```


## Cumulative Distribution and Survival Functions

The _empirical cumulative distribution function_ (ECDF) of a numeric
sample computes the proportion of the sample at or below a specified
value.
    
For the `yields` of the `barley` data:

```{r, class.source = "fold-hide"}
library(ggplot2)
data(barley, package = "lattice")
thm <- theme_minimal() +
    theme(text = element_text(size = 16)) +
    theme(panel.border =
          element_rect(color = "grey30",
                       fill = NA))
p <- ggplot(barley) +
    stat_ecdf(aes(x = yield)) +
    ylab("cumulative proportion") +
    thm
p
```

Flipping the axes produces an _empirical quantile plot_:

```{r, class.source = "fold-hide"}
p + coord_flip()
```

Both make it easy to look up:

* medians, quartiles, and other quantiles;

* the proportion of the sample below a particular value;

* the proportion above a particular value (one minus the proportion below).

An ECDF plot can also be constructed as a step function plot of the
_relative rank_ (rank over sample size) against the observed values:

```{r, class.source = "fold-hide"}
ggplot(barley) +
    geom_step(
        aes(x = yield,
            y = rank(yield) /
                length(yield))) +
    ylab("cumulative proportion") +
    thm
```

Reversing the relative ranks produces a plot of the _empirical survival
function_:

```{r}
ggplot(barley) +
    geom_step(
        aes(x = yield,
            y = rank(-yield) /
                length(yield))) +
    ylab("surviving proportion") +
    thm
```

Survival plots are often used for data representing time to failure
in engineering or time to death or disease recurrence in
medicine.

For a highly skewed distribution, such as the distribution of `price`
in the `diamonds` data, transforming the axis to a square root or log
scale may help.

```{r, fig.width = 8, class.source = "fold-hide"}
library(patchwork)
p1 <- ggplot(diamonds) +
    stat_ecdf(aes(x = price)) +
    ylab("cumulative proportion") +
    thm
p2 <- p1 + scale_x_log10()
p1 + p2
```

There is a downside: Interpolating on a non-linear scale is much harder.


## QQ Plots


### Basics

One way to assess how well a particular theoretical model describes a
data distribution is to plot data quantiles against theoretical quantiles.

This corresponds to transforming the empirical quantile plot
horizontal axis to the scale of the theoretical distribution.

The result is a plot of sample quantiles against theoretical
quantiles, and should be close to a 45-degree straight line if the
model fits the data well.

Such a plot is called a quantile-quantile plot, or a QQ plot for short.

Usually a QQ plot

* uses points rather than a step function, and

* 1/2 is subtracted from the ranks before calculating relative ranks (this
  makes the rank range more symmetric):

For the `barley` data:

```{r, class.source = "fold-hide"}
p <- ggplot(barley) +
    geom_point(
        aes(y = yield,
            x = qnorm((rank(yield) - 0.5) /
                      length(yield)))) +
    xlab("theoretical quantile") +
    ylab("sample quantile") +
    thm
p
```

For a location-scale family of models, like the normal family, a QQ
plot against standard normal quantiles should be close to a straight
line if the model is a good fit.

For the normal family the intercept will be the mean and the slope
will be the standard deviation.

Adding a line can help judge the quality of the fit:

```{r, class.source = "fold-hide"}
p + geom_abline(aes(intercept = mean(yield),
                    slope = sd(yield)),
                color = "red")
```

`ggplot` provides `geom_qq()` that makes this a little easier; base
graphics provides `qqnorm()` and `lattice` has `qqmath()`.


### Some Examples

The histograms and density estimates for the `duration` variable in
the `geyser` data set showed that the distribution is far from a
normal distribution, and the normal QQ plot shows this as well:

```{r, class.source = "fold-hide"}
data(geyser, package = "MASS")
ggplot(geyser) +
    geom_qq(aes(sample = duration)) +
    thm
```

Except for rounding the `parent` heights in the `Galton` data seemed
not too far from normally distributed:

```{r, class.source = "fold-hide"}
data(Galton, package = "HistData")
ggplot(Galton) +
    geom_qq(aes(sample = parent)) +
    thm
```

* Rounding interferes more with this visualization than with a histogram or a
  density plot.

* Rounding is more visible with this visualization than with a histogram or a
  density plot.

Another Gatlton dataset available in the `UsingR` package with less
rounding is `father.son`:

```{r, class.source = "fold-hide"}
data(father.son, package = "UsingR")
ggplot(father.son) +
    geom_qq(aes(sample = fheight)) +
    thm
```

The middle seems to be fairly straight, but the ends are somewhat wiggly.

How can you calibrate your judgment?


### Calibrating the Variability

One approach is to use simulation, sometimes called a _graphical
bootstrap_.

The `nboot()` function will simulate `R` samples from a normal
distribution that match a variable `x` on sample size, sample mean,
and sample SD.

The result is returned in a data frame suitable for plotting:

```{r, message = FALSE}
library(dplyr)
nsim <- function(n, m = 0, s = 1) {
    z <- rnorm(n)
    m + s * ((z - mean(z)) / sd(z))
}

nboot <- function(x, R) {
    n <- length(x)
    m <- mean(x)
    s <- sd(x)
    sim <- function(i) {
        xx <- sort(nsim(n, m, s))
        p <- (seq_along(x) - 0.5) / n
        data.frame(x = xx, p = p, sim = i)
    }
    bind_rows(lapply(1 : R, sim))
}
```

Plotting these as lines shows the variability in shapes we can expect
when sampling from the theoretical normal distribution:

```{r, class.source = "fold-hide"}
gb <- nboot(father.son$fheight, 50)
ggplot() +
    geom_line(aes(x = qnorm(p),
                  y = x,
                  group = sim),
              color = "gray", data = gb) +
    thm
```

We can then insert this simulation behind our data to help calibrate
the visualization:

```{r, class.source = "fold-hide"}
ggplot(father.son) +
    geom_line(aes(x = qnorm(p),
                  y = x,
                  group = sim),
              color = "gray", data = gb) +
    geom_qq(aes(sample = fheight)) +
    thm
```


### Scalability

For large sample sizes, such as `price` from the `diamonds` data,
overplotting will occur:

```{r, class.source = "fold-hide"}
ggplot(diamonds) +
    geom_qq(aes(sample = price)) +
    thm
```

This can be alleviated by using a grid of quantiles:

```{r, class.source = "fold-hide"}
nq <- 100
p <- ((1 : nq) - 0.5) / nq
ggplot() +
    geom_point(aes(x = qnorm(p),
                   y = quantile(diamonds$price, p))) +
    thm
```

A more reasonable model might be an exponential distribution:

```{r}
ggplot() +
    geom_point(aes(x = qexp(p), y = quantile(diamonds$price, p))) +
    thm
```


### Comparing Two Distributions

The QQ plot can also be used to compare two distributions based on a
sample from each.

If the samples are the same size then this is just a plot of the
ordered sample values against each other.

Choosing a fixed set of quantiles allows samples of unequal size to be
compared.

Using a small set of quantiles we can compare the distributions of
waiting times between eruptions of Old Faithful from the two different
data sets we have looked at:

```{r, class.source = "fold-hide"}
nq <- 31
p <- (1 : nq) / nq - 0.5 / nq
wg <- geyser$waiting
wf <- faithful$waiting
ggplot() +
    geom_point(aes(x = quantile(wg, p),
                   y = quantile(wf, p))) +
    thm
```

Adding a 45-degree line:

```{r, class.source = "fold-hide"}
ggplot() +
    geom_abline(intercept = 0, slope = 1, lty = 2) +
    geom_point(aes(x = quantile(wg, p), y = quantile(wf, p))) +
    thm
```


## PP Plots

The PP plot for comparing a sample to a theoretical model plots the
theoretical proportion less than or equal to each observed value
against the actual proportion.

For a theoretical cumulative distribution function $F$ this means plotting

$$
F(x_i) \sim p_i
$$

with

$$
p_i = \frac{r_i - 1/2}{n}
$$

where $r_i$ is the $i$-th observation's rank.

For the `fheight` variable in the `father.son` data:

```{r, class.source = "fold-hide"}
m <- mean(father.son$fheight)
s <- sd(father.son$fheight)
ggplot(father.son) +
    geom_point(aes(x = (rank(fheight) - 0.5) /
                       length(fheight),
                   y = pnorm(fheight, m, s))) +
    thm
```

* The values on the vertical axis are the _probability integral
  transform_ of the data for the theoretical distribution.

* If the data are a sample from the theoretical distribution then
  these transforms would be uniformly distributed on $[0, 1]$.

* The PP plot is a QQ plot of these transformed values against a
  uniform distribution.

* The PP plot goes through the points $(0, 0)$ and $(1, 1)$ and so is
  much less variable in the tails.

Using the simulated data:

```{r, class.source = "fold-hide"}
pp <- ggplot() +
    geom_line(aes(x = p,
                  y = pnorm(x, m, s),
                  group = sim),
              color = "gray",
              data = gb) +
    thm
pp
```

Adding the `father.son` data:

```{r, class.source = "fold-hide"}
pp +
    geom_point(aes(x = (rank(fheight) - 0.5) /
                       length(fheight),
                   y = pnorm(fheight, m, s)),
               data = (father.son))
```

The PP plot is also less sensitive to deviations in the tails.

A compromise between the QQ and PP plots uses the _arcsine square root_
variance-stabilizing transformation, which makes the variability
approximately constant across the range of the plot:

```{r, class.source = "fold-hide"}
vpp <- ggplot() +
    geom_line(aes(x = asin(sqrt(p)),
                  y = asin(sqrt(pnorm(x, m, s))),
                  group = sim),
              color = "gray", data = gb) +
    thm
vpp
```

Adding the data:

```{r, class.source = "fold-hide"}
vpp +
    geom_point(
        aes(x = asin(sqrt((rank(fheight) - 0.5) /
                          length(fheight))),
            y = asin(sqrt(pnorm(fheight, m, s)))),
        data = (father.son))
```


## Plots For Assessing Model Fit

Both QQ and PP plots can be used to asses how well a theoretical
family of models fits your data, or your residuals.

To use a PP plot you have to estimate the parameters first.

For a location-scale family, like the normal distribution family, you
can use a QQ plot with a standard member of the family.

Some other families can use other transformations that lead to
straight lines for family members.

The Weibull family is widely used in reliability modeling; its
CDF is

$$ F(t) = 1 - \exp\left\(-\left(\frac{t}{b}\right)^a\right\)$$

The logarithms of Weibull random variables form a location-scale
family.

Special paper used to be available for [Weibull probability
plots](https://web.archive.org/web/20130207071825/https://weibull.com/hotwire/issue8/relbasics8.htm).

A Weibull QQ plot for `price` in the `diamonds` data:

```{r, class.source = "fold-hide"}
n <- nrow(diamonds)
p <- (1 : n) / n - 0.5 / n
ggplot(diamonds) +
    geom_point(aes(x = log10(qweibull(p, 1, 1)), y = log10(sort(price)))) +
    thm
```

The lower tail does not match a Weibull distribution.

Is this important?

In engineering applications it often is.

In selecting a reasonable model to capture the shape of this
distribution it may not be.

QQ plots are helpful for understanding departures from a theoretical
model.

No data will fit a theoretical model perfectly.

Case-specific judgment is needed to decide whether departures are
important.

> George Box: All models are wrong but some are useful.


## Some References

Adam Loy, Lendie Follett, and Heike Hofmann (2016), "Variations of Q–Q
plots: The power of our eyes!", _The American Statistician_;
([preprint](https://arxiv.org/abs/1503.02098)).

John R. Michael (1983), "The stabilized probability plot," _Biometrika_
[JSTOR](https://www.jstor.org/stable/2335939?seq=1#page_scan_tab_contents).

M. B. Wilk and R. Gnanadesikan (1968), "Probability plotting methods
for the analysis of data," _Biometrika_
[JSTOR](https://www.jstor.org/stable/2334448?seq=1#page_scan_tab_contents).

Box, G. E. P. (1979), "Robustness in the strategy of scientific model
building", in Launer, R. L.; Wilkinson, G. N., _Robustness in
Statistics_, Academic Press, pp. 201–236.

Thomas Lumley (2019), ["What have I got against the Shapiro-Wilk
test?"](https://notstatschat.rbind.io/2019/02/09/what-have-i-got-against-the-shapiro-wilk-test/)


## Readings

Chapter [_Visualizing distributions: Empirical cumulative distribution
  functions and q-q
  plots_](https://clauswilke.com/dataviz/ecdf-qq.html) in
  [_Fundamentals of Data
  Visualization_](https://clauswilke.com/dataviz/).


## Exercises

1. The data set `heights` in package `dslabs` contains self-reported
    heights for a number of female and male students. The plot below
    shows the empirical CDF for male heights:

    ```{r, echo = FALSE, message = FALSE}
    library(dplyr)
    library(ggplot2)
    data(heights, package = "dslabs")
    thm <- theme_minimal() + theme(text = element_text(size = 16))
    filter(heights, sex == "Male") |>
        ggplot(aes(x = height)) +
        stat_ecdf() +
        labs(y = "cumulative frequency",
             x = "height (inches)") +
        thm
    ```

    Based on the plot, what percentage of males are taller than 75 inches?

    a. 100%
    b. 15%
    c. 20%
    d. 3%


2. Consider the following normal QQ plots.


    ```{r, fig.height = 4, fig.width = 8, message = FALSE}
    library(dplyr)
    library(ggplot2)
    library(patchwork)
    thm <- theme_minimal() + theme(text = element_text(size = 16))
    data(heights, package = "dslabs")
    set.seed(12345)
    p1 <- ggplot(NULL, aes(sample = rnorm(200))) +
        geom_qq() +
        labs(title = "Sample 1",
             x = "theoretical quantile",
             y = "sample quantile") +
        thm
    p2 <- ggplot(faithful, aes(sample = eruptions)) +
        geom_qq() +
        labs(title = "Sample 2",
             x = "theoretical quantile",
             y = "sample quantile") +
        thm
    p1 + p2
    ```

    Would a normal distribution be a reasonable model for either of
    these samples?

    a. No for Sample 1. Yes for Sample 2.
    b. Yes for Sample 1. No for Sample 2.
    c. Yes for both.
    d. No for both.
