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 \(\alpha\) 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.
equiv_mean_extremum( df_qual = NULL, data_qual = NULL, mean_qual = NULL, sd_qual = NULL, data_sample = NULL, n_sample = NULL, alpha, modcv = FALSE )
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

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 
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 \(W_{min} = qual\,mean  k_1 \cdot qual\,sd\)
k2
the factor used to calculate the threshold for mean. The threshold for mean is calculated as \(W_{mean} = qual\,mean  k_2 \cdot qual\,sd\)
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
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 CMH171G or PSACE 1002002006.
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 visaversa). 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
.
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, CMH171G, Mar. 2012.
Federal Aviation Administration, “Material Qualification and Equivalency for Polymer Matrix Composite Material Systems,” PSACE 1002002006, Sep. 2003.
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## ## 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