# equiv_mean_extremum: Test for decrease in mean or minimum individual In cmstatr: Statistical Methods for Composite Material Data

## Description

This test is used when determining if a new process or manufacturing location produces material properties that are "equivalent" to an existing dataset, and hence the existing basis values are applicable to the new dataset. This test is also sometimes used for determining if a new batch of material is acceptable. This function determines thresholds based on both minimum individual and mean, and optionally evaluates a sample against those thresholds. The joint distribution between the sample mean and sample minimum is used to generate these thresholds. When there is no true difference between the existing ("qualification") and the new population from which the sample is obtained, there is a probability of α of falsely concluding that there is a difference in mean or variance. It is assumed that both the original and new populations are normally distributed. According to Vangel (2002), this test provides improved power compared with a test of mean and standard deviation.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```equiv_mean_extremum( df_qual = NULL, data_qual = NULL, mean_qual = NULL, sd_qual = NULL, data_sample = NULL, n_sample = NULL, alpha, modcv = FALSE ) ```

## Arguments

 `df_qual` (optional) a data.frame containing the qualification data. Defaults to NULL. `data_qual` (optional) a vector of observations from the "qualification" data to which equivalency is being tested. Or the column of `df_qual` that contains this data. Defaults to NULL `mean_qual` (optional) the mean from the "qualification" data to which equivalency is being tested. Defaults to NULL `sd_qual` (optional) the standard deviation from the "qualification" data to which equivalency is being tested. Defaults to NULL `data_sample` (optional) a vector of observations from the sample for which equivalency is being tested. Defaults to NULL `n_sample` (optional) the number of observations in the sample for which equivalency will be tested. Defaults to NULL `alpha` the acceptable probability of a type I error `modcv` (optional) a boolean value indicating whether a modified CV should be used. Defaults to FALSE, in which case the standard deviation supplied (or calculated from `data_qual`) will be used directly.

## Details

This function is used to determine acceptance limits for a sample mean and sample minimum. These acceptance limits are often used to set acceptance limits for material strength for each lot of material, or each new manufacturing site. When a sample meets the criteria that its mean and its minimum are both greater than these limits, then one may accept the lot of material or the new manufacturing site.

This procedure is used to ensure that the strength of material processed at a second site, or made with a new batch of material are not degraded relative to the data originally used to determine basis values for the material. For more information about the use of this procedure, see CMH-17-1G or PS-ACE 100-2002-006.

There are several optional arguments to this function. However, you can't omit all of the optional arguments. You must supply either `data_sample` or `n_sample`, but not both. You must also supply either `data_qual` (and `df_qual` if `data_qual` is a variable name and not a vector) or both `mean_qual` and `sd_qual`, but if you supply `data_qual` (and possibly `df_qual`) you should not supply either `mean_qual` or `sd_qual` (and visa-versa). This function will issue a warning or error if you violate any of these rules.

If `modcv` is TRUE, the standard deviation used to calculate the thresholds will be replaced with a standard deviation calculated using the Modified Coefficient of Variation (CV) approach. The Modified CV approach is a way of adding extra variance to the qualification data in the case that the qualification data has less variance than expected, which sometimes occurs when qualification testing is performed in a short period of time. Using the Modified CV approach, the standard deviation is calculated by multiplying `CV_star * mean_qual` where `mean_qual` is either the value supplied or the value calculated by `mean(data_qual)` and CV* is the value computed by `calc_cv_star()`.

## Value

Returns an object of class `equiv_mean_extremum`. This object is a list with the following named elements:

• `call` the expression used to call this function

• `alpha` the value of alpha passed to this function

• `n_sample` the number of observations in the sample for which equivalency is being checked. This is either the value `n_sample` passed to this function or the length of the vector `data_sample`.

• `k1` the factor used to calculate the minimum individual threshold. The minimum individual threshold is calculated as Wmin = qual_mean - k1 * qual_sd

• `k2` the factor used to calculate the threshold for mean. The threshold for mean is calculated as Wmean = qual_mean - k2 * qual_sd

• `modcv` logical value indicating whether the acceptance thresholds are calculated using the modified CV approach

• `cv` the coefficient of variation of the qualification data. This value is not modified, even if `modcv=TRUE`

• `cv_star` The modified coefficient of variation. If `modcv=FALSE`, this will be `NULL`

• `threshold_min_indiv` The calculated threshold value for minimum individual

• `threshold_mean` The calculated threshold value for mean

• `result_min_indiv` a character vector of either "PASS" or "FAIL" indicating whether the data from `data_sample` passes the test for minimum individual. If `data_sample` was not supplied, this value will be `NULL`

• `result_mean` a character vector of either "PASS" or "FAIL" indicating whether the data from `data_sample` passes the test for mean. If `data_sample` was not supplied, this value will be `NULL`

• `min_sample` The minimum value from the vector `data_sample`. if `data_sample` was not supplied, this will have a value of `NULL`

• `mean_sample` The mean value from the vector `data_sample`. If `data_sample` was not supplied, this will have a value of `NULL`

## References

M. G. Vangel. Lot Acceptance and Compliance Testing Using the Sample Mean and an Extremum, Technometrics, vol. 44, no. 3. pp. 242–249. 2002.

“Composite Materials Handbook, Volume 1. Polymer Matrix Composites Guideline for Characterization of Structural Materials,” SAE International, CMH-17-1G, Mar. 2012.

Federal Aviation Administration, “Material Qualification and Equivalency for Polymer Matrix Composite Material Systems,” PS-ACE 100-2002-006, Sep. 2003.

`k_equiv()`
`calc_cv_star()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```equiv_mean_extremum(alpha = 0.01, n_sample = 6, mean_qual = 100, sd_qual = 5.5, modcv = TRUE) ## ## Call: ## equiv_mean_extremum(mean_qual = 100, sd_qual = 5.5, n_sample = 6, ## alpha = 0.01, modcv = TRUE) ## ## Modified CV used: CV* = 0.0675 ( CV = 0.055 ) ## ## For alpha = 0.01 and n = 6 ## ( k1 = 3.128346 and k2 = 1.044342 ) ## Min Individual Sample Mean ## Thresholds: 78.88367 92.95069 ```