---
title: "Attributes charts: p, np, c, u"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Attributes charts: p, np, c, u}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>",
                      fig.width = 7, fig.height = 4.2)
```

```{r setup, message = FALSE}
library(shewhartr)
```

When the quality characteristic is binary (defective / non-defective)
or a count of defects per unit, classical variables charts are the
wrong tool. Counts and proportions live on bounded supports and
follow Binomial / Poisson distributions; pretending they are normal
makes the chart limits wrong, sometimes badly.

| Data                                     | Distribution | Chart                  |
|------------------------------------------|--------------|------------------------|
| Proportion defective (variable n)        | Binomial     | `shewhart_p()`         |
| Number defective (constant n)            | Binomial     | `shewhart_np()`        |
| Defect count per unit (constant exposure)| Poisson      | `shewhart_c()`         |
| Defect count per unit (variable exposure)| Poisson      | `shewhart_u()`         |

## p chart with variable n

`claims_p` records 30 days of insurance-claim quality control. Each
day, a variable number of claims (`n`) is processed and a count of
errors (`defects`) is observed.

```{r}
fit <- shewhart_p(claims_p, defects = defects, n = n, index = day)
broom::tidy(fit)
```

Because `n` varies day-to-day, the limits also vary day-to-day:

```{r}
broom::augment(fit) |> head(10)
```

The default `limits = "3sigma"` uses the normal approximation
$\bar p \pm 3\sqrt{\bar p (1 - \bar p)/n_i}$. This is fine when
$n_i \bar p \gtrsim 5$ and $n_i (1-\bar p) \gtrsim 5$. For small
$n$ or extreme $\bar p$, switch to exact binomial limits:

```{r, eval = FALSE}
shewhart_p(claims_p, defects = defects, n = n, index = day,
           limits = "binomial")
```

## c chart and Poisson honesty

`pcb_solder` has 50 PCBs and a mean defect count of about 6. The
default 3-sigma c-chart works fine here:

```{r}
fit_c <- shewhart_c(pcb_solder, defects = defects, index = board)
broom::tidy(fit_c)
```

But if `c_bar` were small (say 2 or 3), the lower limit under the
normal approximation would be negative — which makes no sense for a
count. The package warns when this is likely:

```{r}
small_means <- data.frame(unit = 1:50, defects = rpois(50, lambda = 2))
suppressWarnings(
  fit_low <- shewhart_c(small_means, defects = defects, index = unit)
)
broom::tidy(fit_low)
```

For low-mean Poisson processes, use exact quantile limits:

```{r}
fit_low_exact <- shewhart_c(small_means, defects = defects, index = unit,
                            limits = "poisson")
broom::tidy(fit_low_exact)
```

George Box's advice — *don't transform if you can model the right
distribution* — applies. The exact Poisson limits use
$q(0.99865)$ and $q(0.00135)$ of $\mathrm{Poisson}(\bar c)$, the
same coverage probability as classical 3-sigma limits but without
the normal approximation.

## np chart for constant n

When subgroup size is constant, the np chart plots the *count* rather
than the proportion. Useful for direct interpretation when n is a
round number:

```{r}
fit_np <- shewhart_np(
  data.frame(day = 1:30, defects = rbinom(30, size = 200, prob = 0.04)),
  defects = defects,
  n       = 200,
  index   = day
)
broom::tidy(fit_np)
```

## u chart for variable exposure

When the inspection size differs (e.g. fabric rolls of different
length, machine-hours of different duration), the right chart is u —
defects per unit of exposure:

```{r}
set.seed(1)
df_u <- data.frame(
  roll    = 1:25,
  defects = rpois(25, lambda = 4 * runif(25, 0.5, 1.5)),
  m2      = runif(25, 0.5, 1.5)
)
fit_u <- shewhart_u(df_u, defects = defects, exposure = m2, index = roll)
broom::tidy(fit_u)
```

## References

- Montgomery, D. C. (2019). *Introduction to Statistical Quality
  Control* (8th ed.). Wiley. Chapter 7.
- Ryan, T. P. (2011). *Statistical Methods for Quality Improvement*
  (3rd ed.). Wiley. (On the inadequacy of 3-sigma limits for low-mean
  Poisson counts.)
- Box, G. E. P., Hunter, W. G., & Hunter, J. S. (2005). *Statistics
  for Experimenters* (2nd ed.). Wiley.