p-Values for Equivalency
In cmstatrExt: More Statistical Methods for Composite Material Data

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Introduction

The dual acceptance criteria used for composite materials accept or reject a new lot of material (or a process change) based on the sample minimum and sample mean from the new lot of material. Acceptance limits are normally set such that under the null hypothesis, there is an equal probability of rejecting the lot due to the minimum and rejecting the lot due to the mean. These acceptance limits are set so that the probability of rejecting the lot (due to either the minimum or mean) under the null hypothesis is $\alpha$. If we eliminate the constraint that there is an equal probability of rejecting a lot due to the minimum or the mean, there is not longer unique values for the acceptance limits: instead, we can calculate a p-value from the sample minimum and the sample mean and compare this p-value with the selected value of $\alpha$.

The cmstatrExt package provides functions for computing acceptance limits, p-values and curves indicating all values of the minimum and mean that result in the same p-value. This vignette demonstrates this functionality. The "two-sample" method in which only the sample statistics for the qualification data are known.

Caution: If the true mean of the population from which the acceptance sample is drawn is higher than the population mean for the qualification distribution, then using the p-value method here may declare an acceptance sample as equivalent even if the standard deviation is larger. This is due to the fact that this statistical test is a one-sided test. Similarly, if the acceptance population has a much lower standard deviation than the qualification population, this test may allow for an undesirable decrease in mean. As such, considerable judgement is required when using this method.

In this vignette, we'll use the cmstatrExt package. We'll also use the tidyverse package for data manipulation and graphing. Finally, we'll use one of the example data sets from the cmstatr package.

library(cmstatrExt)
library(tidyverse)
library(cmstatr)

Example Data

As an example, we'll use the RTD warp tension strength from the carbon.fabric.2 example data set from the cmstatr package. This data is as follows:

dat <- carbon.fabric.2 %>%
  filter(condition == "RTD" & test == "WT")
dat

From this sample, we can calculate the following summary statistics for the strength:

qual <- dat %>%
  summarise(n = n(), mean = mean(strength), sd = sd(strength))
qual

Acceptance Limits

We can calculate the acceptance factors acceptance sample size of 8 and $alpha=0.05$ using the cmstatrExt package as follows:

k <- k_equiv_two_sample(0.05, qual$n, 8)
k

These factors can be transformed into limits using the following equations:

$$ W_{indiv} = \bar{x}{qual} - k_1 s{qual} \ W_{mean} = \bar{x}{qual} - k_2 s{qual} $$

Implementing this in R:

acceptance_limits <- qual$mean - k * qual$sd
acceptance_limits

So, if an acceptance sample has a minimum individual less than r format(acceptance_limits[1], digits = 4) or a mean less than r format(acceptance_limits[2], digits = 4), we would reject it.

p-Value

You might ask what happens if there's one low value in the acceptance sample that's below the acceptance limit for minimum individual, but the mean is well above the limit. The naive response would be to reject the sample. But, the acceptance limits that we just calculated are based on setting an equal probability of rejecting a sample based on the minimum and the mean under the null hypothesis --- there are other pairs of minimum and mean values that have the same p-value as the acceptance limits that we calculated.

In order to use the p-value function from the cmstatrExt package, we need to apply the following transformation:

$$ t_1 = \frac{\bar{x}{qual} - x{acceptance\,(1)}}{s_{qual}} \ t_2 = \frac{\bar{x}{qual} - \bar{x}{acceptance}}{s_{qual}} $$

As a demonstration, let's first calculate the p-value of the acceptance limits. We should get $p=\alpha$.

p_equiv_two_sample(
  n = qual$n,
  m = 8,
  t1 = (qual$mean - acceptance_limits[1]) / qual$sd,
  t2 = (qual$mean - acceptance_limits[2]) / qual$sd
)

This value is very close to $\alpha=0.05$ --- within expected numeric precision.

Now, let's consider the case where the sample minimum is 116 and the mean is 138. The sample minimum is below the acceptance limit (116 < 120), but the sample mean is well above the acceptance limit (138 > 134). Let's calculate the p-value for this case:

p_equiv_two_sample(
  n = qual$n,
  m = 8,
  t1 = (qual$mean - 116) / qual$sd,
  t2 = (qual$mean - 138) / qual$sd
)

Since this value is well above the selected value of $\alpha=0.05$, we would accept this sample. This sort of analysis can be useful during site- or process-equivalency programs, or for MRB activities.

Curves of Constant p-Values

The cmstatrExt package provides a function that produces a data.frame containing values of $t_1$ and $t_2$ that result in the same p-value. We can create such a data.frame for p-values of 0.05 as follows:

curve <- iso_equiv_two_sample(qual$n, 8, 0.05, 4, 1.5, 10)
curve

We can plot this curve using ggplot2, which is part of the tidyverse package:

curve %>%
  ggplot(aes(x = t1, y = t2)) +
  geom_path() +
  ggtitle("Acceptance criteria for alpha=0.05")

When you plot this, make sure to use geom_path and not geom_line. The former will plot the points in the order given; the latter will plot the points in ascending order of the x variable, which can cause problems in the vertical portion of the graph.

Let's overlay the acceptance limits calculated by the k_equiv_two_sample function as well as the values of t_1 and t_2 from the sample that we discussed in the previous section.

curve %>%
  ggplot(aes(x = t1, y = t2)) +
  geom_path() +
  geom_hline(yintercept = k[2], color = "red") +
  geom_vline(xintercept = k[1], color = "red") +
  geom_point(data = data.frame(
    t1 = (qual$mean - 116) / qual$sd,
    t2 = (qual$mean - 138) / qual$sd
  ),
  shape = "*", size = 5) +
  ggtitle("Acceptance criteria for alpha=0.05")

Or better yet, we can transform this back into engineering units:

curve %>%
  mutate(x_min = qual$mean - t1 * qual$sd,
         x_mean = qual$mean - t2 * qual$sd) %>%
  ggplot(aes(x = x_min, y = x_mean)) +
  geom_path() +
  geom_hline(yintercept = acceptance_limits[2], color = "red") +
  geom_vline(xintercept = acceptance_limits[1], color = "red") +
  geom_point(data = data.frame(
    x_min = 116,
    x_mean = 138
  ),
  shape = "*", size = 5) +
  ggtitle("Acceptance criteria for alpha=0.05")