Author: Stefan Kloppenborg
Date: 2024-05-06

# p-Values for Equivalency

#### Stefan Kloppenborg

#### 2024-05-06

# 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
    #>    test condition batch panel thickness nplies strength modulus failure_mode
    #> 1    WT       RTD     A     1     0.113     14  129.224   8.733          LAB
    #> 2    WT       RTD     A     1     0.112     14  144.702   8.934      LAT,LWB
    #> 3    WT       RTD     A     1     0.113     14  137.194   8.896          LAB
    #> 4    WT       RTD     A     1     0.113     14  139.728   8.835      LAT,LWB
    #> 5    WT       RTD     A     2     0.113     14  127.286   9.220          LAB
    #> 6    WT       RTD     A     2     0.111     14  129.261   9.463          LAT
    #> 7    WT       RTD     A     2     0.112     14  130.031   9.348          LAB
    #> 8    WT       RTD     B     1     0.111     14  140.038   9.244      LAT,LGM
    #> 9    WT       RTD     B     1     0.111     14  132.880   9.267          LWT
    #> 10   WT       RTD     B     1     0.113     14  132.104   9.198          LAT
    #> 11   WT       RTD     B     2     0.114     14  137.618   9.179      LAT,LAB
    #> 12   WT       RTD     B     2     0.113     14  139.217   9.123          LAB
    #> 13   WT       RTD     B     2     0.113     14  134.912   9.116          LAT
    #> 14   WT       RTD     B     2     0.111     14  141.558   9.434    LAB / LAT
    #> 15   WT       RTD     C     1     0.108     14  150.242   9.451          LAB
    #> 16   WT       RTD     C     1     0.109     14  147.053   9.391          LGM
    #> 17   WT       RTD     C     1     0.111     14  145.001   9.318      LAT,LWB
    #> 18   WT       RTD     C     1     0.113     14  135.686   8.991    LAT / LAB
    #> 19   WT       RTD     C     1     0.112     14  136.075   9.221          LAB
    #> 20   WT       RTD     C     2     0.114     14  143.738   8.803      LAT,LGM
    #> 21   WT       RTD     C     2     0.113     14  143.715   8.893      LAT,LAB
    #> 22   WT       RTD     C     2     0.113     14  147.981   8.974      LGM,LWB
    #> 23   WT       RTD     C     2     0.112     14  148.418   9.118      LAT,LWB
    #> 24   WT       RTD     C     2     0.113     14  135.435   9.217      LAT/LAB
    #> 25   WT       RTD     C     2     0.113     14  146.285   8.920      LWT/LWB
    #> 26   WT       RTD     C     2     0.111     14  139.078   9.015          LAT
    #> 27   WT       RTD     C     2     0.112     14  146.825   9.036      LAT/LWT
    #> 28   WT       RTD     C     2     0.110     14  148.235   9.336      LWB/LAB

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
    #>    n     mean       sd
    #> 1 28 139.6257 6.716047

# 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
    #> [1] 2.9462891 0.7972005

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
    #> [1] 119.8383 134.2717

So, if an acceptance sample has a minimum individual less than 119.8 or a mean less than 134.3, 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
    )
    #> [1] 0.05003139

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
    )
    #> [1] 0.2771053

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
    #>          t1        t2
    #> 1  4.000000 0.6876226
    #> 2  3.870890 0.6892186
    #> 3  3.742796 0.6924106
    #> 4  3.616481 0.6971987
    #> 5  3.492200 0.7027848
    #> 6  3.372239 0.7115630
    #> 7  3.258124 0.7235332
    #> 8  3.152269 0.7394934
    #> 9  3.056580 0.7594437
    #> 10 2.975250 0.7849802
    #> 11 2.880451 0.8321028
    #> 12 2.825808 0.8732404
    #> 13 2.782601 0.9266275
    #> 14 2.749561 0.9895908
    #> 15 2.726687 1.0602149
    #> 16 2.713980 1.1365848
    #> 17 2.706355 1.2156679
    #> 18 2.701272 1.2961076
    #> 19 2.701272 1.3776646
    #> 20 2.698730 1.4591417

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")