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

```{r, include = FALSE}
has_data <- requireNamespace("shewhartr", quietly = TRUE)
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 4.5,
                      eval = has_data)
```

```{r setup}
library(edetect)
```

Attribute charts monitor counts. The classical charts assume binomial or
Poisson counts and set three-sigma limits from a normal approximation. The
e-detector versions keep the counting model only as an upper bound on the
moment generating function, which is what the guarantee needs:

* the **p chart** assumes that each trial in a subgroup has a conditional
  success probability (given the earlier trials) at most, at least or equal to
  the centre; beyond that, dependence within a subgroup is allowed;
* the **c and u charts** assume that the counts are *sub-Poisson*: their moment
  generating function is at most that of a Poisson variable with the declared
  rate per unit exposure. Poisson and underdispersed counts qualify;
  overdispersed counts do not, and for those a bounded class with a declared
  maximum count is the honest choice.

## p chart: daily error rates of insurance claims

Each day a variable number of claims is processed and the claims with errors
are counted. The error rate is 5% for the first 22 days and 9% from day 23. The
first fifteen days serve as Phase I.

```{r claims}
claims <- shewhartr::claims_p
head(claims, 3)

p_chart <- edetect_p(x = claims$defects[16:30], n = claims$n[16:30],
                     phase1 = data.frame(x = claims$defects[1:15], n = claims$n[1:15]),
                     arl = 370, direction = "up")
edetect_alarm(p_chart)
autoplot(p_chart)
```

The alarm came at the eleventh Phase II day (day 26), three days after the
rate rose, and the change is estimated at the eighth Phase II day, that is
day 23. The Phase I estimate of the rate (`r round(p_chart$design$center, 4)`)
is recorded as an assumption of the chart.

```{r claims-report}
edetect_report(p_chart)
```

## c chart: solder defects on circuit boards

Fifty boards are inspected and the defective solder joints on each are
counted; the process is stable with a mean of about 6. The first twenty boards
serve as Phase I. A stable process should produce no alarm, and the evidence
should stay far from the threshold.

```{r pcb}
defects <- shewhartr::pcb_solder$defects
c_chart <- edetect_c(defects[21:50], phase1 = defects[1:20], arl = 370, direction = "both")
edetect_alarm(c_chart)
max(tidy(c_chart)$evidence)
autoplot(c_chart)
```

## u chart: defects per board in samples of unequal size

When boards are inspected in samples of two to four, the count per sample is
monitored per board inspected: the exposure `n` is the number of boards. The
rate per board is declared from the specification (6 defects per board).

```{r u}
set.seed(1)
sizes <- sample(2:4, 20, replace = TRUE)
sizes <- sizes[cumsum(sizes) <= 50]
sample_id <- rep(seq_along(sizes), sizes)
samples <- data.frame(n = sizes,
                      x = as.integer(tapply(defects[seq_along(sample_id)], sample_id, sum)))
head(samples, 4)

u_chart <- edetect_u(samples$x, n = samples$n, center = 6, arl = 370, direction = "both")
edetect_alarm(u_chart)
autoplot(u_chart)
```

## What the alarm means, and what it does not

An alarm says that the evidence against the pre-change class reached the
target ARL. It does not say that the process changed: with a target ARL of 370
a stable process produces, on average, one alarm every 370 observations or
more. A missing alarm does not say that the process is stable either; it says
that the evidence has not accumulated yet, which for small changes can take
long. `edetect_arl()` measures both sides for a design before monitoring (see
the [run-lengths](run-lengths.html) vignette).