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Type:Package
Title:Statistical Methods for Composite Material Data
Version:0.10.0
Date:2024-11-18
Depends:R (≥ 3.3)
Description:An implementation of the statistical methods commonly used for advanced composite materials in aerospace applications. This package focuses on calculating basis values (lower tolerance bounds) for material strength properties, as well as performing the associated diagnostic tests. This package provides functions for calculating basis values assuming several different distributions, as well as providing functions for non-parametric methods of computing basis values. Functions are also provided for testing the hypothesis that there is no difference between strength and modulus data from an alternate sample and that from a "qualification" or "baseline" sample. For a discussion of these statistical methods and their use, see the Composite Materials Handbook, Volume 1 (2012, ISBN: 978-0-7680-7811-4). Additional details about this package are available in the paper by Kloppenborg (2020, <doi:10.21105/joss.02265>).
URL:https://www.cmstatr.net/,https://github.com/cmstatr/cmstatr
BugReports:https://github.com/cmstatr/cmstatr/issues
License:AGPL-3
Encoding:UTF-8
LazyData:true
Imports:dplyr, generics, ggplot2, kSamples, MASS, purrr, rlang, stats,tibble, tidyr
Suggests:knitr, lintr, rmarkdown, spelling, testthat, vdiffr
RoxygenNote:7.3.2
VignetteBuilder:knitr
Language:en-US
Config/testthat/parallel:true
Config/testthat/edition:3
NeedsCompilation:no
Packaged:2024-11-19 04:04:50 UTC; stefan
Author:Stefan KloppenborgORCID iD [aut, cre], Billy Cheng [ctb], Ally Fraser [ctb], Jeffrey Borlik [ctb], Brice Langston [ctb], Comtek Advanced Structures, Ltd. [fnd]
Maintainer:Stefan Kloppenborg <stefan@kloppenborg.ca>
Repository:CRAN
Date/Publication:2024-11-19 07:50:08 UTC

cmstatr: Statistical Methods for Composite Material Data

Description

To learn more aboutcmstatr, start with the vignettes:browseVignettes(package = "cmstatr")

Author(s)

Maintainer: Stefan Kloppenborgstefan@kloppenborg.ca (ORCID)

Other contributors:

See Also

Useful links:

Anderson–Darling K-Sample Test

Description

This function performs an Anderson–Darling k-sample test. This is used todetermine if several samples (groups) share a common (unspecified)distribution.

Usage

ad_ksample(data = NULL, x, groups, alpha = 0.025)

Arguments

data

a data.frame

x

the variable in the data.frame on which to perform theAnderson–Darling k-Sample test (usually strength)

groups

a variable in the data.frame that defines the groups

alpha

the significance level (default 0.025)

Details

This function is a wrapper for thead.test function fromthe packagekSamples. The method "exact" is specified in the call toad.test. Refer to that package's documentation for details.

There is a minor difference in the formulation of the Anderson–Darlingk-Sample test in CMH-17-1G, compared with that in the Scholz andStephens (1987). This difference affects the test statistic and thecritical value in the same proportion, and therefore the conclusion ofthe test is unaffected. Whencomparing the test statistic generated by this function to that generatedby software that uses the CMH-17-1G formulation (such as ASAP, CMH17-STATS,etc.), the test statistic reported by this function will be greater bya factor of(k - 1), with a corresponding change in the criticalvalue.

For more information about the difference between this function andthe formulation in CMH-17-1G, see the vignette on the subject, whichcan be accessed by runningvignette("adktest")

Value

Returns an object of classadk. This object has the following fields:

References

F. W. Scholz and M. Stephens, “K-Sample Anderson–Darling Tests,” Journalof the American Statistical Association, vol. 82, no. 399. pp. 918–924,Sep-1987.

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

Examples

library(dplyr)carbon.fabric %>%  filter(test == "WT") %>%  filter(condition == "RTD") %>%  ad_ksample(strength, batch)#### Call:## ad_ksample(data = ., x = strength, groups = batch)#### N = 18          k = 3## ADK = 0.912     p-value = 0.95989## Conclusion: Samples come from the same distribution ( alpha = 0.025 )

Anderson–Darling test for goodness of fit

Description

Calculates the Anderson–Darling test statistic for a sample givena particular distribution, and determines whether to reject thehypothesis that a sample is drawn from that distribution.

Usage

anderson_darling_normal(data = NULL, x, alpha = 0.05)anderson_darling_lognormal(data = NULL, x, alpha = 0.05)anderson_darling_weibull(data = NULL, x, alpha = 0.05)

Arguments

data

a data.frame-like object (optional)

x

a numeric vector or a variable in the data.frame

alpha

the required significance level of the test.Defaults to 0.05.

Details

The Anderson–Darling test statistic is calculated for the distributiongiven by the user.

The observed significance level (OSL), or p-value, is calculated assumingthat the parametersof the distribution are unknown; these parameters are estimate from thedata.

The functionanderson_darling_normal computes the Anderson–Darlingtest statistic given a normal distribution with mean and standard deviationequal to the sample mean and standard deviation.

The functionanderson_darling_lognormal is the same asanderson_darling_normal except that the data is log transformedfirst.

The functionanderson_darling_weibull computes the Anderson–Darlingtest statistic given a Weibull distribution with shape and scale parametersestimated from the data using a maximum likelihood estimate.

The test statistic,A, is modified to account forthe fact that the parameters of the population are not known,but are instead estimated from the sample. This modification isa function of the sample size only, and is different for eachdistribution (normal/lognormal or Weibull). Several such modificationshave been proposed. This function uses the modification published inStephens (1974), Lawless (1982) and CMH-17-1G. Some other implementationsof the Anderson-Darling test, such as the implementation in thenortest package, use other modifications, such as the onepublished in D'Agostino and Stephens (1986). As such, the p-valuereported by this function may differ from the p-value reportedby implementations of the Anderson–Darling test that usedifferent modifiers. Only the unmodifiedtest statistic is reported in the result of this function, butthe modified test statistic is used to compute the OSL (p-value).

This function uses the formulae for observed significancelevel (OSL) published in CMH-17-1G. These formulae depend on the particulardistribution used.

The results of this function have been validated againstpublished values in Lawless (1982).

Value

an object of classanderson_darling. This object has the followingfields.

References

J. F. Lawless,Statistical models and methods for lifetime data.New York: Wiley, 1982.

"Composite Materials Handbook, Volume 1. Polymer MatrixComposites Guideline for Characterization of StructuralMaterials," SAE International, CMH-17-1G, Mar. 2012.

M. A. Stephens, “EDF Statistics for Goodness of Fit and SomeComparisons,”Journal of the American Statistical Association, vol. 69, no. 347.pp. 730–737, 1974.

R. D’Agostino and M. Stephens, Goodness-of-Fit Techniques.New York: Marcel Dekker, 1986.

Examples

library(dplyr)carbon.fabric %>%  filter(test == "FC") %>%  filter(condition == "RTD") %>%  anderson_darling_normal(strength)## Call:## anderson_darling_normal(data = ., x = strength)#### Distribution:  Normal ( n = 18 )## Test statistic:  A = 0.9224776## OSL (p-value):  0.01212193  (assuming unknown parameters)## Conclusion: Sample is not drawn from a Normal distribution (alpha = 0.05)

Augment data with information from anmnr object

Description

Augment accepts anmnr object (returned from the functionmaximum_normed_residual()) and a dataset and adds the column.outlier to the dataset. The column.outlier is a logicalvector indicating whether each observation is an outlier.

When passing data intoaugment using thedata argument,the data must be exactly the data that was passed tomaximum_normed_residual.

Usage

## S3 method for class 'mnr'augment(x, data = x$data, ...)

Arguments

x

anmnr object created bymaximum_normed_residual()

data

adata.frame ortibble::tibble()containing the original data that was passed tomaximum_normed_residual

...

Additional arguments. Not used. Included only to match genericsignature.

Value

Whendata is supplied,augment returnsdata, but withone column appended. Whendata is not supplied,augmentreturns a newtibble::tibble() with the columnvalues containing the original values used bymaximum_normed_residaul plus one additional column. The additionalcolumn is:

See Also

maximum_normed_residual()

Examples

data <- data.frame(strength = c(80, 98, 96, 97, 98, 120))m <- maximum_normed_residual(data, strength)# augment can be called with the original dataaugment(m, data)##   strength .outlier## 1       80    FALSE## 2       98    FALSE## 3       96    FALSE## 4       97    FALSE## 5       98    FALSE## 6      120    FALSE# or augment can be called without the orignal data and it will be# reconstructedaugment(m)## # A tibble: 6 x 2##   values .outlier##    <dbl> <lgl>## 1     80 FALSE## 2     98 FALSE## 3     96 FALSE## 4     97 FALSE## 5     98 FALSE## 6    120 FALSE

Calculate basis values

Description

Calculate the basis value for a given data set. There are various functionsto calculate the basis values for different distributions.The basis value is the lower one-sided tolerance bound of a certainproportion of the population. For more information on tolerance bounds,see Meeker, et. al. (2017).For B-Basis, set the content of tolerance bound top=0.90 andthe confidence level toconf=0.95; for A-Basis, setp=0.99 andconf=0.95. While other tolerance boundcontents and confidence levels may be computed, they are infrequentlyneeded in practice.

These functions also perform some automated diagnostictests of the data prior to calculating the basis values. These diagnostictests can be overridden if needed.

Usage

basis_normal(  data = NULL,  x,  batch = NULL,  p = 0.9,  conf = 0.95,  override = c())basis_lognormal(  data = NULL,  x,  batch = NULL,  p = 0.9,  conf = 0.95,  override = c())basis_weibull(  data = NULL,  x,  batch = NULL,  p = 0.9,  conf = 0.95,  override = c())basis_pooled_cv(  data = NULL,  x,  groups,  batch = NULL,  p = 0.9,  conf = 0.95,  modcv = FALSE,  override = c())basis_pooled_sd(  data = NULL,  x,  groups,  batch = NULL,  p = 0.9,  conf = 0.95,  modcv = FALSE,  override = c())basis_hk_ext(  data = NULL,  x,  batch = NULL,  p = 0.9,  conf = 0.95,  method = c("optimum-order", "woodward-frawley"),  override = c())basis_nonpara_large_sample(  data = NULL,  x,  batch = NULL,  p = 0.9,  conf = 0.95,  override = c())basis_anova(data = NULL, x, groups, p = 0.9, conf = 0.95, override = c())

Arguments

data

a data.frame

x

the variable in the data.frame for which to find the basis value

batch

the variable in the data.frame that contains the batches.

p

the content of the tolerance bound. Should be 0.90 for B-Basisand 0.99 for A-Basis

conf

confidence level Should be 0.95 for both A- and B-Basis

override

a list of names of diagnostic tests to override,if desired. Specifying "all" will override all diagnostictests applicable to the current method.

groups

the variable in the data.frame representing the groups

modcv

a logical value indicating whether the modified CV approachshould be used. Only applicable to pooling methods.

method

the method for Hanson–Koopmans nonparametric basis values.should be "optimum-order" for B-Basis and "woodward-frawley"for A-Basis.

Details

data is an optional argument. Ifdata is given, it shouldbe adata.frame (or similar object). Whendata is specified, thevalue ofx is expected to be a variable withindata. Ifdata is not specified,x must be a vector.

Whenmodcv=TRUE is set, which is only applicable to thepooling methods,the data is first modified according to the modified coefficientof variation (CV)rules. This modified data is then used when both calculating thebasis values andalso when performing the diagnostic tests. The modified CV approachis a way ofadding extra variance to datasets with unexpectedly low variance.

basis_normal calculate the basis value by subtractingk timesthe standard deviation from the mean.k is given bythe functionk_factor_normal(). The equations inKrishnamoorthy and Mathew (2008) are used.basis_normal alsoperforms a diagnostic test for outliers (usingmaximum_normed_residual())and a diagnostic test for normality (usinganderson_darling_normal()).If the argumentbatch is given, this function also performsa diagnostic test for outliers withineach batch (usingmaximum_normed_residual())and a diagnostic test for between batch variability (usingad_ksample()). The argumentbatch is only usedfor these diagnostic tests.

basis_lognormal calculates the basis value in the same waythatbasis_normal does, except that the natural logarithm of thedata is taken.

basis_lognormal function also performsa diagnostic test for outliers (usingmaximum_normed_residual())and a diagnostic test for normality (usinganderson_darling_lognormal()).If the argumentbatch is given, this function also performsa diagnostic test for outliers withineach batch (usingmaximum_normed_residual())and a diagnostic test for between batch variability (usingad_ksample()). The argumentbatch is only usedfor these diagnostic tests.

basis_weibull calculates the basis value for data distributedaccording to a Weibull distribution. The confidence level for thecontent requested is calculated using the conditional method, asdescribed in Lawless (1982) Section 4.1.2b. This has good agreementwith tables published in CMH-17-1G. Results differ between this functionand STAT17 by approximately 0.5\

basis_weibull function also performsa diagnostic test for outliers (usingmaximum_normed_residual())and a diagnostic test for normality (usinganderson_darling_weibull()).If the argumentbatch is given, this function also performsa diagnostic test for outliers withineach batch (usingmaximum_normed_residual())and a diagnostic test for between batch variability (usingad_ksample()). The argumentbatch is only usedfor these diagnostic tests.

basis_hk_ext calculates the basis value using the ExtendedHanson–Koopmans method, as described in CMH-17-1G and Vangel (1994).For nonparametric distributions, this function should be used for samplesup to n=28 for B-Basis and up ton=299 for A-Basis.This method uses a pair of order statistics to determine the basis value.CMH-17-1G suggests that for A-Basis, the first and last order statisticis used: this is called the "woodward-frawley" method in this package,after the paper in which this approach is described (as referencedby Vangel (1994)). For B-Basis, another approach is used whereby thefirst andj-th order statistic are used to calculate the basis value.In this approach, thej-th order statistic is selected to minimizethe difference between the tolerance limit (assuming that the orderstatistics are equal to the expected values from a standard normaldistribution) and the population quantile for a standard normaldistribution. This approach is described in Vangel (1994). This secondmethod (for use when calculating B-Basis values) is called"optimum-order" in this package.The results ofbasis_hk_ext have beenverified against example results from the program STAT-17. Agreement istypically well within 0.2%.

Note that the implementation ofhk_ext_z_j_opt changed aftercmstatrversion 0.8.0. This function is used internally bybasis_hk_extwhenmethod = "optimum-order". This implementation change may meanthat basis values computed using this method may change slightlyafter version 0.8.0. However, both implementations seem to be equallyvalid. See the included vignettefor a discussion of the differences between the implementation beforeand after version 0.8.0, as well as the factors given in CMH-17-1G.To access this vignette, run:vignette("hk_ext", package = "cmstatr")

basis_hk_ext also performsa diagnostic test for outliers (usingmaximum_normed_residual())and performs a pair of tests that the sample size and method selectedfollow the guidance described above.If the argumentbatch is given, this function also performsa diagnostic test for outliers withineach batch (usingmaximum_normed_residual())and a diagnostic test for between batch variability (usingad_ksample()). The argumentbatch is only usedfor these diagnostic tests.

basis_nonpara_large_sample calculates the basis valueusing the large sample method described in CMH-17-1G. This method usesa sum of binomials to determine the rank of the ordered statisticcorresponding with the desired tolerance limit (basis value). Resultsof this function have been verified against results of the STAT-17program.

basis_nonpara_large_sample also performsa diagnostic test for outliers (usingmaximum_normed_residual())and performs a test that the sample size is sufficiently large.If the argumentbatch is given, this function also performsa diagnostic test for outliers withineach batch (usingmaximum_normed_residual())and a diagnostic test for between batch variability (usingad_ksample()). The argumentbatch is only usedfor these diagnostic tests.

basis_anova calculates basis values using the ANOVA method.x specifies the data (normally strength) andgroupsindicates the group corresponding to each observation. This method isdescribed in CMH-17-1G, but when the ratio of between-batch meansquare to the within-batch mean square is less than or equalto one, the tolerance factor is calculated based on pooling the datafrom all groups. This approach is recommended by Vangel (1992)and by Krishnamoorthy and Mathew (2008), and is also implementedby the software CMH17-STATS and STAT-17.This function automatically performs a diagnostictest for outliers within each group(usingmaximum_normed_residual()) and a test for betweengroup variability (usingad_ksample()) as well as checkingthat the data contains at least 5 groups.This function has been verified against the results of the STAT-17 program.

basis_pooled_sd calculates basis values by pooling the data fromseveral groups together.x specifies the data (normally strength)andgroup indicates the group corresponding to each observation.This method is described in CMH-17-1G and matches the pooling methodimplemented in ASAP 2008.

basis_pooled_cv calculates basis values by pooling the data fromseveral groups together.x specifies the data (normally strength)andgroup indicates the group corresponding to each observation.This method is described in CMH-17-1G.

basis_pooled_sd andbasis_pooled_cv both automaticallyperform a number of diagnostic tests. Usingmaximum_normed_residual(), they check that there are nooutliers within each group and batch (provided thatbatch isspecified). They check the between batch variability usingad_ksample(). They check that there are no outliers withineach group (pooling all batches) usingmaximum_normed_residual(). They check for the normalityof the pooled data usinganderson_darling_normal().basis_pooled_sd checks for equality of variance of alldata usinglevene_test() andbasis_pooled_cvchecks for equality of variances of all data after transforming itusingnormalize_group_mean()usinglevene_test().

The object returned by these functions includes the named vectordiagnostic_results. This contains all of the diagnostic testsperformed. The name of each element of the vector corresponds with thename of the diagnostic test. The contents of each element will be"P" if the diagnostic test passed, "F" if the diagnostic test failed,"O" if the diagnostic test was overridden andNA if thediagnostic test was skipped (typically because an optionalargument was not supplied).

The objects produced by the diagnostic tests are included in the namedlistdiagnostic_obj. The name of each element in the list corresponds withthe name of the test. This can be useful when evaluating diagnostic testfailures.

The following list summarizes the diagnostic tests automaticallyperformed by each function.

Value

an object of classbasisThis object has the following fields:

References

J. F. Lawless, Statistical Models and Methods for Lifetime Data.New York: John Wiley & Sons, 1982.

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

M. Vangel, “One-Sided Nonparametric Tolerance Limits,”Communications in Statistics - Simulation and Computation,vol. 23, no. 4. pp. 1137–1154, 1994.

K. Krishnamoorthy and T. Mathew, Statistical Tolerance Regions: Theory,Applications, and Computation. Hoboken: John Wiley & Sons, 2008.

W. Meeker, G. Hahn, and L. Escobar, Statistical Intervals: A Guidefor Practitioners and Researchers, Second Edition.Hoboken: John Wiley & Sons, 2017.

M. Vangel, “New Methods for One-Sided Tolerance Limits for a One-WayBalanced Random-Effects ANOVA Model,” Technometrics, vol. 34, no. 2.Taylor & Francis, pp. 176–185, 1992.

See Also

hk_ext_z_j_opt()

k_factor_normal()

transform_mod_cv()

maximum_normed_residual()

anderson_darling_normal()

anderson_darling_lognormal()

anderson_darling_weibull()

ad_ksample()

normalize_group_mean()

Examples

library(dplyr)# A single-point basis value can be calculated as follows# in this example, three failed diagnostic tests are# overridden.res <- carbon.fabric %>%  filter(test == "FC") %>%  filter(condition == "RTD") %>%  basis_normal(strength, batch,               override = c("outliers",                            "outliers_within_batch",                            "anderson_darling_normal"))print(res)## Call:## basis_normal(data = ., x = strength, batch = batch,##     override = c("outliers", "outliers_within_batch",##    "anderson_darling_normal"))#### Distribution:  Normal ( n = 18 )## The following diagnostic tests were overridden:##     `outliers`,##     `outliers_within_batch`,##     `anderson_darling_normal`## B-Basis:   ( p = 0.9 , conf = 0.95 )## 76.94656print(res$diagnostic_obj$between_batch_variability)## Call:## ad_ksample(x = x, groups = batch, alpha = 0.025)#### N = 18           k = 3## ADK = 1.73       p-value = 0.52151## Conclusion: Samples come from the same distribution ( alpha = 0.025 )# A set of pooled basis values can also be calculated# using the pooled standard deviation method, as follows.# In this example, one failed diagnostic test is overridden.carbon.fabric %>%  filter(test == "WT") %>%  basis_pooled_sd(strength, condition, batch,                  override = c("outliers_within_batch"))## Call:## basis_pooled_sd(data = ., x = strength, groups = condition,##                 batch = batch, override = c("outliers_within_batch"))#### Distribution:  Normal - Pooled Standard Deviation ( n = 54, r = 3 )## The following diagnostic tests were overridden:##     `outliers_within_batch`## B-Basis:   ( p = 0.9 , conf = 0.95 )## CTD  127.6914## ETW  125.0698## RTD  132.1457

Calculate the modified CV from the CV

Description

This function calculates the modified coefficient of variation (CV)based on a (unmodified) CV.The modified CV is calculated based on the rules in CMH-17-1G. Thoserules are:

Usage

calc_cv_star(cv)

Arguments

cv

The CV to modify

Value

The value of the modified CV

References

"Composite Materials Handbook, Volume 1. Polymer Matrix CompositesGuideline for Characterization of Structural Materials,"SAE International, CMH-17-1G, Mar. 2012.

See Also

cv()

Examples

# The modified CV for values of CV smaller than 4% is 6%calc_cv_star(0.01)## [1] 0.06# The modified CV for values of CV larger than 8% is unchangedcalc_cv_star(0.09)## [1] 0.09

Sample data for a generic carbon fabric

Description

Datasets containing sample data that is typical of a generic carbonfabric prepreg. This data is used in several examples within thecmstatr package. This data is fictional and shouldonly be used for learning how to use this package.

Usage

carbon.fabriccarbon.fabric.2

Format

An object of classdata.frame with 216 rows and 5 columns.

An object of classdata.frame with 177 rows and 9 columns.

Produce basis summary statistics for each (environmental) condition

Description

Produces adata.frame containing the sample size and mean for eachcondition. If a reference condition (e.g. "RTD") is specified, the ratioof each condition mean value to the mean value for the reference conditionis also calculated. If abasis object returned by one of thebasis_pooled functions is given as an argument, this table also containsthe basis value for each condition.

Usage

condition_summary(data, ...)## S3 method for class 'data.frame'condition_summary(data, x, condition, ref_condition = NULL, ...)## S3 method for class 'basis'condition_summary(data, ref_condition = NULL, ...)

Arguments

data

adata.frame or abasis object

...

not used

x

the column name of the variable of interest (usually strength)

condition

the column name for the condition

ref_condition

acharacter representing the reference condition

Value

adata.frame

Examples

library(dplyr)carbon.fabric.2 %>%  filter(test == "WT") %>%  condition_summary(strength, condition, "RTD")##   condition  n     mean mean_fraction## 1       CTD 19 135.4719     0.9702503## 2       RTD 28 139.6257     1.0000000## 3       ETW 18 134.1009     0.9604312## 4      ETW2 21 130.1545     0.9321673carbon.fabric %>%  filter(test == "FT") %>%  basis_pooled_sd(strength, condition, batch) %>%  condition_summary("RTD")##   condition  n     mean mean_fraction    basis basis_fraction## 1       RTD 18 127.6211     1.0000000 116.8894      1.0000000## 2       ETW 18 117.8080     0.9231072 107.0762      0.9160476## 3       CTD 18 125.9629     0.9870063 115.2311      0.9858133

Calculate the coefficient of variation

Description

The coefficient of variation (CV) is the ratio of the standarddeviation to the mean of a sample. This function takes a vectorof data and calculates the CV.

Usage

cv(x, na.rm = FALSE)

Arguments

x

a vector

na.rm

logical. Should missing values be removed?

Value

The calculated CV

Examples

set.seed(15)  # make this example reproduciblex <- rnorm(100, mean = 100, sd = 5)cv(x)## [1] 0.04944505# the cv function can also be used within a call to dplyr::summariselibrary(dplyr)carbon.fabric %>%filter(test == "WT") %>%  group_by(condition) %>%  summarise(mean = mean(strength), cv = cv(strength))## # A tibble: 3 x 3##   condition  mean     cv##   <chr>     <dbl>  <dbl>## 1 CTD        137. 0.0417## 2 ETW        135. 0.0310## 3 RTD        142. 0.0451

Equivalency based on change in mean value

Description

Checks for change in the mean value between a qualification data set anda sample. This is normally used to check for properties such as modulus.This function is a wrapper for a two-sample t–test.

Usage

equiv_change_mean(  df_qual = NULL,  data_qual = NULL,  n_qual = NULL,  mean_qual = NULL,  sd_qual = NULL,  data_sample = NULL,  n_sample = NULL,  mean_sample = NULL,  sd_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 ofdf_qual that contains this data. Defaults to NULL

n_qual

the number of observations in the qualification data to whichthe sample is being compared for equivalency

mean_qual

the mean from the qualification data to which the sampleis being compared for equivalency

sd_qual

the standard deviation from the qualification data to whichthe sample is being compared for equivalency

data_sample

a vector of observations from the sample being comparedfor equivalency

n_sample

the number of observations in the sample being compared forequivalency

mean_sample

the mean of the sample being compared for equivalency

sd_sample

the standard deviation of the sample being compared forequivalency

alpha

the acceptable probability of a Type I error

modcv

a logical value indicating whether the modified CV approachshould be used. Defaults toFALSE

Details

There are several optional arguments to this function. Either (but not both)data_sample or all ofn_sample,mean_sample andsd_sample must be supplied. And, either (but not both)data_qual(and alsodf_qual ifdata_qual is a column name and not avector) or all ofn_qual,mean_qual andsd_qual mustbe supplied. If these requirements are violated, warning(s) or error(s) willbe issued.

This function uses a two-sample t-test to determine if there is a differencein the mean value of the qualification data and the sample. A pooledstandard deviation is used in the t-test. The procedure is per CMH-17-1G.

Ifmodcv is TRUE, the standard deviation used to calculate thethresholds will be replaced with a standard deviation calculatedusing the Modified Coefficient of Variation (CV) approach.The Modified CV approach is a way of adding extra variance to thequalification data in the case that the qualification data has lessvariance than expected, which sometimes occurs when qualification testingis performed in a short period of time.Using the Modified CV approach, the standard deviation is calculated bymultiplyingCV_star * mean_qual wheremean_qual is either thevalue supplied or the value calculated bymean(data_qual) andCV* is determined usingcalc_cv_star().

Note that the modified CV option should only be used if that data passes theAnderson–Darling test.

Value

References

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

See Also

calc_cv_star()

stats::t.test()

Examples

equiv_change_mean(alpha = 0.05, n_sample = 9, mean_sample = 9.02,                  sd_sample = 0.15785, n_qual = 28, mean_qual = 9.24,                  sd_qual = 0.162, modcv = TRUE)## Call:## equiv_change_mean(n_qual = 28, mean_qual = 9.24, sd_qual = 0.162,##                   n_sample = 9, mean_sample = 9.02, sd_sample = 0.15785,##                   alpha = 0.05,modcv = TRUE)#### For alpha = 0.05## Modified CV used##                   Qualification        Sample##           Number        28               9##             Mean       9.24             9.02##               SD      0.162           0.15785##           Result               PASS##    Passing Range       8.856695 to 9.623305

Test for decrease in mean or minimum individual

Description

This test is used when determining if a new process ormanufacturing location produces material properties that are"equivalent" to an existing dataset, and hence the existingbasis values are applicable to the new dataset. This test is alsosometimes used for determining if a new batch of material is acceptable.This function determines thresholds based on both minimumindividual and mean, and optionally evaluates a sample against thosethresholds. The joint distribution between the sample meanand 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 aprobability of\alpha of falsely concluding that there is adifference in mean or variance. It is assumed that both the originaland new populations are normally distributed.According to Vangel (2002), this test provides improved power comparedwith a test of mean and standard deviation.

Usage

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 ofdf_qual that contains this data. Defaults to NULL

mean_qual

(optional) the mean from the "qualification" data to whichequivalency 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 forwhich equivalency is being tested. Defaults to NULL

n_sample

(optional) the number of observations in the sample forwhich 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 CVshould be used. Defaults to FALSE, in which case the standard deviationsupplied (or calculated fromdata_qual) will be used directly.

Details

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

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

There are several optional arguments to this function. However, you can'tomit all of the optional arguments. You must supply eitherdata_sample orn_sample, but not both. You must also supplyeitherdata_qual (anddf_qual ifdata_qual is avariable name and not a vector) or bothmean_qual andsd_qual,but if you supplydata_qual (and possiblydf_qual) you shouldnot supply eithermean_qual orsd_qual (and visa-versa). Thisfunction will issue a warning or error if you violate any of these rules.

Ifmodcv is TRUE, the standard deviation used to calculate thethresholds will be replaced with a standard deviation calculatedusing the Modified Coefficient of Variation (CV) approach.The Modified CV approach is a way of adding extra variance to thequalification data in the case that the qualification data has lessvariance than expected, which sometimes occurs when qualification testingis performed in a short period of time.Using the Modified CV approach, the standard deviation is calculated bymultiplyingCV_star * mean_qual wheremean_qual is either thevalue supplied or the value calculated bymean(data_qual) andCV* is the value computed bycalc_cv_star().

Value

Returns an object of classequiv_mean_extremum. This object is a listwith the following named elements:

References

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

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

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

See Also

k_equiv()

calc_cv_star()

Examples

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

Jittered points showing (possibly multiple) failure modes

Description

Thegeom_jitter_failure_mode is very similar toggplot2::geom_jitter() except that a failure mode variable specifiedas the color and/or shape aesthetic is parsed to separate multiplefailure modes and plot them separately. For example, if an observationhas the failure mode "LAT/LAB", two points will be plotted, one with thefailure mode "LAT" and the second with the failure mode "LAB".

Usage

geom_jitter_failure_mode(  mapping = NULL,  data = NULL,  stat = "identity",  position = "jitter",  ...,  width = NULL,  height = NULL,  na.rm = FALSE,  show.legend = NA,  inherit.aes = TRUE,  sep = "[/, ]+")

Arguments

mapping

Set of aesthetic mapping created byaes().Seeggplot2::geom_jitter() for additional details.

data

The data to be displayed by this layer.Seeggplot2::geom_jitter() for additional details.

stat

The statistical transformation to use on the data for this layer.Seeggplot2::geom_jitter() for additional details.

position

A position adjustment to use on the data for this layer.Seeggplot2::geom_jitter() for additional details.

...

Other arguments passed on tolayer()'sparams argument.Seeggplot2::geom_jitter() for additional details.

width

The amount of horizontal jitter.Seeggplot2::geom_jitter() for additional details.

height

The amount of vertical jitter.Seeggplot2::geom_jitter() for additional details.

na.rm

If FALSE, the default, missing values are removed with warning.Seeggplot2::geom_jitter() for additional details.

show.legend

NA, the default, indicates that nay aesthetics aremapped.Seeggplot2::geom_jitter() for additional details.

inherit.aes

if FALSE, overrides the default aesthetics.Seeggplot2::geom_jitter() for additional details.

sep

A regular expression indicating the character(s) separatingmultiple failure modes. By default"[/, ]+"

Details

The variable specified for the aestheticsshape andcolor are passedto the functionseparate_failure_modes() to parse the failure modes andseparate multiple failure modes separated by character(s) specified inthe regular expression given in the parametersep. By default, multiplefailure modes are expected to be separated by spaces, commas or forwardslashes, but this can be overridden.

If bothshape andcolor aesthetics are specified, both must be identical.

See Also

separate_failure_modes()

ggplot2::geom_jitter()

Examples

library(dplyr)library(ggplot2)carbon.fabric.2 %>%  filter(test == "WT") %>%  ggplot(aes(x = condition, y = strength)) +  geom_boxplot() +  geom_jitter_failure_mode(aes(color = failure_mode, shape = failure_mode))

Glance at aadk (Anderson–Darling k-Sample) object

Description

Glance accepts an object of typeadk and returns atibble::tibble() withone row of summaries.

Glance does not do any calculations: it just gathers the results in atibble.

Usage

## S3 method for class 'adk'glance(x, ...)

Arguments

x

anadk object

...

Additional arguments. Not used. Included only to match genericsignature.

Value

A one-rowtibble::tibble() with the followingcolumns:

See Also

ad_ksample()

Examples

x <- c(rnorm(20, 100, 5), rnorm(20, 105, 6))k <- c(rep(1, 20), rep(2, 20))a <- ad_ksample(x = x, groups = k)glance(a)## A tibble: 1 x 7##   alpha     n     k sigma    ad       p reject_same_dist##   <dbl> <int> <int> <dbl> <dbl>   <dbl> <lgl>## 1 0.025    40     2 0.727  4.37 0.00487 TRUE

Glance at ananderson_darling object

Description

Glance accepts an object of typeanderson_darling andreturns atibble::tibble() withone row of summaries.

Glance does not do any calculations: it just gathers the results in atibble.

Usage

## S3 method for class 'anderson_darling'glance(x, ...)

Arguments

x

ananderson_darling object

...

Additional arguments. Not used. Included only to match genericsignature.

Value

A one-rowtibble::tibble() with the followingcolumns:

See Also

anderson_darling()

Examples

x <- rnorm(100, 100, 4)ad <- anderson_darling_weibull(x = x)glance(ad)## # A tibble: 1 x 6##   dist        n     A        osl alpha reject_distribution##   <chr>   <int> <dbl>      <dbl> <dbl> <lgl>## 1 Weibull   100  2.62 0.00000207  0.05 TRUE

Glance at a basis object

Description

Glance accepts an object of type basis and returns atibble::tibble() withone row of summaries for each basis value.

Glance does not do any calculations: it just gathers the results in atibble.

Usage

## S3 method for class 'basis'glance(x, include_diagnostics = FALSE, ...)

Arguments

x

a basis object

include_diagnostics

a logical value indicating whether to includecolumns for diagnostic tests. Default FALSE.

...

Additional arguments. Not used. Included only to match genericsignature.

Details

For the pooled basis methods (basis_pooled_cv andbasis_pooled_sd), thetibble::tibble()returned byglance will have one row for each group included inthe pooling. For all other basis methods, the resultingtibblewill have a single row.

Ifinclude_diagnostics=TRUE, there will be additional columnscorresponding with the diagnostic tests performed. These column(s) willbe of type character and will contain a "P" if the diagnostic testpassed, a "F" if the diagnostic test failed, an "O" if the diagnostictest was overridden orNA if the test was not run (typicallybecause an optional argument was not passed to the function thatcomputed the basis value).

Value

Atibble::tibble() with the followingcolumns:

See Also

basis()

Examples

set.seed(10)x <- rnorm(20, 100, 5)b <- basis_normal(x = x)glance(b)## # A tibble: 1 x 7##       p  conf distribution modcv     n r     basis##   <dbl> <dbl> <chr>        <lgl> <int> <lgl> <dbl>## 1   0.9  0.95 Normal       FALSE    20 NA     92.0glance(b, include_diagnostics = TRUE)## # A tibble: 1 x 11##        p  conf distribution modcv     n r     basis outliers_within…##    <dbl> <dbl> <chr>        <lgl> <int> <lgl> <dbl> <chr>##  1   0.9  0.95 Normal       FALSE    20 NA     92.0 NA## # … with 3 more variables: between_batch_variability <chr>,## #   outliers <chr>, anderson_darling_normal <chr>

Glance at aequiv_change_mean object

Description

Glance accepts an object of typeequiv_change_meanand returns atibble::tibble() withone row of summaries.

Glance does not do any calculations: it just gathers the results in atibble.

Usage

## S3 method for class 'equiv_change_mean'glance(x, ...)

Arguments

x

aequiv_change_mean object returned fromequiv_change_mean()

...

Additional arguments. Not used. Included only to match genericsignature.

Value

A one-rowtibble::tibble() with the followingcolumns:

See Also

equiv_change_mean()

Examples

x0 <- rnorm(30, 100, 4)x1 <- rnorm(5, 91, 7)eq <- equiv_change_mean(data_qual = x0, data_sample = x1, alpha = 0.01)glance(eq)## # A tibble: 1 x 14##   alpha n_sample mean_sample sd_sample n_qual mean_qual sd_qual modcv##   <dbl>    <int>       <dbl>     <dbl>  <int>     <dbl>   <dbl> <lgl>## 1  0.01        5        85.8      9.93     30      100.    3.90 FALSE## # ... with 6 more variables: sp <dbl>, t0 <dbl>, t_req <dbl>,## #   threshold_min <dbl>, threshold_max <dbl>, result <chr>

Glance at anequiv_mean_extremum object

Description

Glance accepts an object of typeequiv_mean_extremum and returns atibble::tibble() withone row of summaries.

Glance does not do any calculations: it just gathers the results in atibble.

Usage

## S3 method for class 'equiv_mean_extremum'glance(x, ...)

Arguments

x

an equiv_mean_extremum object returned fromequiv_mean_extremum()

...

Additional arguments. Not used. Included only to match genericsignature.

Value

A one-rowtibble::tibble() with the followingcolumns:

See Also

equiv_mean_extremum()

Examples

x0 <- rnorm(30, 100, 4)x1 <- rnorm(5, 91, 7)eq <- equiv_mean_extremum(data_qual = x0, data_sample = x1, alpha = 0.01)glance(eq)## # A tibble: 1 x 9##   alpha n_sample modcv threshold_min_indiv threshold_mean##   <dbl>    <int> <lgl>               <dbl>          <dbl>## 1  0.01        5 FALSE                86.2           94.9## # ... with 4 more variables: result_min_indiv <chr>, result_mean <chr>,## #   min_sample <dbl>, mean_sample <dbl>

Glance at alevene object

Description

Glance accepts an object of typelevene and returns atibble::tibble() withone row of summaries.

Glance does not do any calculations: it just gathers the results in atibble.

Usage

## S3 method for class 'levene'glance(x, ...)

Arguments

x

alevene object returned fromlevene_test()

...

Additional arguments. Not used. Included only to match genericsignature.

Value

A one-rowtibble::tibble() with the followingcolumns:

See Also

levene_test()

Examples

df <- data.frame(  groups = c(rep("A", 5), rep("B", 6)),  strength = c(rnorm(5, 100, 6), rnorm(6, 105, 7)))levene_result <- levene_test(df, strength, groups)glance(levene_result)## # A tibble: 1 x 7##   alpha modcv     n     k      f     p reject_equal_variance##   <dbl> <lgl> <int> <int>  <dbl> <dbl> <lgl>## 1  0.05 FALSE    11     2 0.0191 0.893 FALSE

Glance at amnr (maximum normed residual) object

Description

Glance accepts an object of typemnr and returns atibble::tibble() withone row of summaries.

Glance does not do any calculations: it just gathers the results in atibble.

Usage

## S3 method for class 'mnr'glance(x, ...)

Arguments

x

Anmnr object

...

Additional arguments. Not used. Included only to match genericsignature.

Value

A one-rowtibble::tibble() with the followingcolumns:

See Also

maximum_normed_residual()

Examples

x <- c(rnorm(20, 100, 5), 10)m <- maximum_normed_residual(x = x)glance(m)## # A tibble: 1 x 4##     mnr alpha  crit n_outliers##   <dbl> <dbl> <dbl>      <dbl>## 1  4.23  0.05  2.73          1

Calculate values related to Extended Hanson–Koopmans tolerance bounds

Description

Calculates values related to Extended Hanson–Koopmans tolerance boundsas described by Vangel (1994).

Usage

hk_ext_z(n, i, j, p, conf)hk_ext_z_j_opt(n, p, conf)

Arguments

n

the sample size

i

the first order statistic (1 <= i < j)

j

the second order statistic (i < j <= n)

p

the content of the tolerance bound (normally 0.90 or 0.99)

conf

the confidence level (normally 0.95)

Details

Hanson (1964) presents a nonparametric method for determiningtolerance bounds based on consecutive order statistics.Vangel (1994) extends this method using non-consecutive order statistics.

The extended Hanson–Koopmans method calculates a tolerance bound(basis value) based on two order statistics and a weighting valuez. The value ofz is based on the sample size, whichorder statistics are selected, the desired content of the tolerancebond and the desired confidence level.

The functionhk_ext_z calculates the weighting variablezbased on selected order statisticsi andj. Based on thisvaluez, the tolerance bound can be calculated as:

S = z X_{(i)} + (1 - z) X_{(j)}

WhereX_{(i)} andX_{(j)} are thei-thandj-th ordered observation.

The functionhk_ext_z_j_opt determines the value ofj andthe corresponding value ofz, assumingi=1. The valueofj is selected such that the computed tolerance limit isnearest to the desired population quantile for a standard normaldistribution when the order statistics are equal to the expectedvalue of the order statistics for the standard normal distribution.

Value

Forhk_ext_z, the return value is a numeric value representingthe parameter z (denoted as k in CMH-17-1G).

Forhk_ext_z_j_opt, the return value is named list containingz andk. The former is the value of z, as defined byVangel (1994), and the latter is the corresponding order statistic.

References

M. Vangel, “One-Sided Nonparametric Tolerance Limits,”Communications in Statistics - Simulation and Computation,vol. 23, no. 4. pp. 1137–1154, 1994.

D. L. Hanson and L. H. Koopmans,“Tolerance Limits for the Class of Distributions with IncreasingHazard Rates,” The Annals of Mathematical Statistics,vol. 35, no. 4. pp. 1561–1570, 1964.

See Also

basis_hk_ext()

Examples

# The factors from Table 1 of Vangel (1994) can be recreated# using the hk_ext_z function. For the sample size n=21,# the median is the 11th ordered observation. The factor# required for calculating the tolerance bound with a content# of 0.9 and a confidence level of 0.95 based on the median# and first ordered observation can be calculated as follows.hk_ext_z(n = 21, i = 1, j = 11, p = 0.9, conf = 0.95)## [1] 1.204806# The hk_ext_z_j_opt function can be used to refine this value# of z by finding an optimum value of j, rather than simply# using the median. Here, we find that the optimal observation# to use is the 10th, not the 11th (which is the median).hk_ext_z_j_opt(n = 21, p = 0.9, conf = 0.95)## $z## [1] 1.217717#### $j## [1] 10

k-factors for determining acceptance based on sample mean and an extremum

Description

k-factors for determining acceptance based on sample mean and an extremum

Usage

k_equiv(alpha, n)

Arguments

alpha

the acceptable probability of a type I error

n

the number of observations in the sample to test

Details

The k-factors returned by this function are used for determiningwhether to accept a new dataset.

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

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

According to Vangel (2002), the use of mean and extremum for this purposeis more powerful than the use of mean and standard deviation.

The results of this function match those published by Vangel within0.05\by Vangel are identical to those published in CMH-17-1G.

This function uses numerical integration and numerical optimization tofind values of the factorsk_1 andk_2 based on Vangel'ssaddle point approximation.

The valuen refers to the number of observations in the samplebeing compared with the original population (the qualification sample isusually assumed to be equal to the population statistics).

The value ofalpha is the acceptable probability of a type I error.Normally, this is set to 0.05 for material or process equivalency and 0.01when setting lot acceptance limits. Though, in principle, this parametercan be set to any number between 0 and 1. This function, however, has onlybeen validated in the range of1e-5 \le alpha \le 0.5.

Value

a vector with elements c(k1, k2). k1 is for testing the sampleextremum. k2 is for testing the sample mean

References

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

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

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

See Also

equiv_mean_extremum()

Examples

qual_mean <- 100qual_sd <- 3.5k <- k_equiv(0.01, 5)print("Minimum Individual Acceptance Limit:")print(qual_mean - qual_sd * k[1])print("Minimum Average Acceptance Limit:")print(qual_mean - qual_sd * k[2])## [1] "Minimum Individual Acceptance Limit:"## [1] 89.24981## [1] "Minimum Average Acceptance Limit:"## [1] 96.00123

Calculate k factor for basis values (kB,kA) with normaldistribution

Description

The factors returned by this function are used when calculating basisvalues (one-sided confidence bounds) when the data are normallydistributed. The basis value willbe equal to\bar{x} - k s,where\bar{x} is the sample mean,s is the sample standard deviation andk is the resultof this function.This function is internally used bybasis_normal() whencomputing basis values.

Usage

k_factor_normal(n, p = 0.9, conf = 0.95)

Arguments

n

the number of observations (i.e. coupons)

p

the desired content of the tolerance bound.Should be 0.90 for B-Basis and 0.99 for A-Basis

conf

confidence level. Should be 0.95 for both A- and B-Basis

Details

This function calculates the k factors used when determining A- andB-Basis values for normally distributed data. To getkB, setthe content of the tolerance bound top = 0.90 andthe confidence level toconf = 0.95. To getkA, setp = 0.99 andconf = 0.95. While other tolerance boundcontents and confidence levels may be computed, they are infrequentlyneeded in practice.

The k-factor is calculated using equation 2.2.3 ofKrishnamoorthy and Mathew (2008).

This function has been validated against thekB tables inCMH-17-1G for each value ofn fromn = 2 ton = 95.It has been validated against thekA tables in CMH-17-1G for eachvalue ofn fromn = 2 ton = 75. Larger values ofnalso match the tables in CMH-17-1G, but Remits warnings that "full precision may not have been achieved." Whenvalidating the results of this function against the tables in CMH-17-1G,the maximum allowable difference between the two is 0.002. The tables inCMH-17-1G give values to three decimal places.

For more information about tolerance bounds in general, seeMeeker, et. al. (2017).

Value

the calculated factor

References

K. Krishnamoorthy and T. Mathew, Statistical Tolerance Regions: Theory,Applications, and Computation. Hoboken: John Wiley & Sons, 2008.

W. Meeker, G. Hahn, and L. Escobar, Statistical Intervals: A Guidefor Practitioners and Researchers, Second Edition.Hoboken: John Wiley & Sons, 2017.

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

See Also

basis_normal()

Examples

kb <- k_factor_normal(n = 10, p = 0.9, conf = 0.95)print(kb)## [1] 2.35464# This can be used to caclulate the B-Basis if# the sample mean and sample standard deviation# is known, and data is assumed to be normally# distributedsample_mean <- 90sample_sd <- 5.2print("B-Basis:")print(sample_mean - sample_sd * kb)## [1] B-Basis:## [1] 77.75587

Levene's Test (Median) for Equality of Variance

Description

This function performs the Levene's test for equality of variance usingthe median. This is also known as the Brown-Forsythe test.

Usage

levene_test(data = NULL, x, groups, alpha = 0.05, modcv = FALSE)

Arguments

data

a data.frame

x

the variable in the data.frame or a vector on which to perform theLevene's test (usually strength)

groups

a variable in the data.frame that defines the groups

alpha

the significance level (default 0.05)

modcv

a logical value indicating whether the modified CV approachshould be used.

Details

This function performs the Levene's test for equality of variance usingmedian (also known as the Brown-Forsythe test). Thedata is transformed as follows:

w_{ij} = \left| x_{ij} - m_i \right|

Wherem_i is median of theith group. An F-Test is thenperformed on the transformed data.

Whenmodcv=TRUE, the data from each group is first transformedaccording to the modified coefficient of variation (CV) rules beforeperforming Levene's test.

Value

Returns an object of classlevene. This object has the following fields:

References

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

NIST/SEMATECH e-Handbook of Statistical Methods,https://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm, 2024.

Brown, M. B. and Forsythe, A. B. (1974), Journal of the AmericanStatistical Association, 69, pp. 364-367.

See Also

calc_cv_star()

transform_mod_cv()

Examples

library(dplyr)carbon.fabric.2 %>%  filter(test == "FC") %>%  levene_test(strength, condition)#### Call:## levene_test(data = ., x = strength, groups = condition)#### n = 91          k = 5## F = 3.883818    p-value = 0.00600518## Conclusion: Samples have unequal variance ( alpha = 0.05 )

Detect outliers using the maximum normed residual method

Description

This function detects outliers using the maximum normed residualmethod described in CMH-17-1G. This method identifies a valueas an outlier if the absolute difference between the value andthe sample mean divided by the sample standard deviationexceeds a critical value.

Usage

maximum_normed_residual(data = NULL, x, alpha = 0.05)

Arguments

data

a data.frame

x

the variable in the data.frame for which to find the MNRor a vector ifdata=NULL. This must include at least 3observations.

alpha

the significance level for the test. Defaults to 0.05

Details

data is an optional argument. Ifdata is given, itshould be adata.frame (or similar object). Whendata is specified, thevalue ofx is expected to be a variable withindata. Ifdata is not specified,x must be a vector.

The maximum normed residual test is a test for outliers. The test statisticis given in CMH-17-1G. Outliers are identified in the returned object.

The maximum normed residual test statistic is defined as:

MNR = max \frac{\left| x_i - \bar{x} \right|}{s}

When the value of the MNR test statistic exceeds the critical valuedefined in Section 8.3.3.1 of CMH-17-1G, the corresponding valueis identified as an outlier. It is then removed from the sample, andthe test statistic is computed again and compared with the criticalvalue corresponding with the new sample. This process is repeated untilno values are identified as outliers.

Value

an object of classmnrThis object has the following fields:

References

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

Examples

library(dplyr)carbon.fabric.2 %>%  filter(test=="FC" & condition=="ETW2" & batch=="A") %>%  maximum_normed_residual(strength)## Call:## maximum_normed_residual(data = ., x = strength)#### MNR =  1.958797  ( critical value = 1.887145 )#### Outliers ( alpha = 0.05 ):##   Index  Value##       6  44.26carbon.fabric.2 %>%  filter(test=="FC" & condition=="ETW2" & batch=="B") %>%  maximum_normed_residual(strength)## Call:## maximum_normed_residual(data = ., x = strength)#### MNR =  1.469517  ( critical value = 1.887145 )#### No outliers detected ( alpha = 0.05 )

Create a plot of nested sources of variation

Description

Creates a plot showing the breakdown of variation within a sample. Thisfunction usesggplot2 internally.

Usage

nested_data_plot(  dat,  x,  groups = c(),  stat = "mean",  ...,  y_gap = 1,  divider_color = "grey50",  point_args = list(),  dline_args = list(),  vline_args = list(),  hline_args = list(),  label_args = list(),  connector_args = list())

Arguments

dat

adata.frame or similar object

x

the variable withindat to plot. Most often this would be astrength or modulus variable.

groups

a vector of variables to group the data by

stat

a function for computing the central location for each group.This is normally "mean" but could be "median" or anotherfunction.

...

extra options. See Details.

y_gap

the vertical gap between grouping variables

divider_color

the color of the lines between grouping variables.OrNULL to omit these lines.

point_args

arguments to pass toggplot2::geom_point when plottingindividual data points.

dline_args

arguments to pass toggplot2::geom_segment when plottingthe horizontal lines between data points.

vline_args

arguments to pass toggplot2::geom_segment when plottingvertical lines

hline_args

arguments to pass toggplot2::geom_segment when plottinghorizontal lines connecting levels in groups

label_args

arguments to pass toggplot2::geom_label when plottinglabels

connector_args

arguments to pass toggplot2::geom_point whenplotting the connection between the vertical linesand the horizontal lines connecting levels in groups

Details

Extra options can be included to control aesthetic options. The followingoptions are supported. Any (or all) can be set to a single variablein the data set.

Examples

library(dplyr)carbon.fabric.2 %>%  filter(test == "WT" & condition == "RTD") %>%  nested_data_plot(strength,                   groups = c(batch, panel))# Labels can be filled toocarbon.fabric.2 %>%  filter(test == "WT" & condition == "RTD") %>%  nested_data_plot(strength,                   groups = c(batch, panel),                   fill = batch)

Rank for distribution-free tolerance bound

Description

Calculates the rank order for finding distribution-free tolerancebounds for large samples. This function should only be used forcomputing B-Basis for samples larger than 28 or A-Basis for sampleslarger than 298. This function is used bybasis_nonpara_large_sample().

Usage

nonpara_binomial_rank(n, p, conf)

Arguments

n

the sample size

p

the desired content for the tolerance bound

conf

the confidence level for the desired tolerance bound

Details

This function uses the sum of binomial terms to determine the rankof the ordered statistic that corresponds with the desired tolerancelimit. This approach does not assume any particular distribution. Thisapproach is described by Guenther (1969) and by CMH-17-1G.

The results of this function have been verified against the tables inCMH-17-1G and agreement was found for all sample sizes published inCMH-17-1G for both A- and B-Basis, as well as the sample sizesn+1 andn-1, wheren is the sample size published in CMH-17-1G.

The tables in CMH-17-1G purportedly list the smallest sample sizesfor which a particular rank can be used. That is, for a sample sizeone less than then published in the table, the next lowest rankwould be used. In some cases, the results of this function disagree by arank of one for sample sizes one less than then published in thetable. This indicates a disagreement in that sample size at whichthe rank should change. This is likely due to numericaldifferences in this function and the procedure used to generate the tables.However, the disagreement is limited to sample 6500 for A-Basis; nodiscrepancies have been identified for B-Basis. Since these sample sizes areuncommon for composite materialstesting, and the difference between subsequent order statistics will bevery small for samples this large, this difference will have no practicaleffect on computed tolerance bounds.

Value

The rank corresponding with the desired tolerance bound

References

W. Guenther, “Determination of Sample Size for Distribution-FreeTolerance Limits,” Jan. 1969.Available online:https://www.duo.uio.no/handle/10852/48686

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

See Also

basis_nonpara_large_sample()

Examples

nonpara_binomial_rank(n = 1693, p = 0.99, conf = 0.95)## [1] 11# The above example indicates that for a sample of 1693 observations,# the A-Basis is best approximated as the 11th ordered observation.# In the example below, the same ordered observation would also be used# for a sample of size 1702.nonpara_binomial_rank(n = 1702, p = 0.99, conf = 0.95)## [1] 11

Normalize values to group means

Description

This function computes the mean of each group, then divides eachobservation by its corresponding group mean. This is commonly donewhen pooling data across environments.

Usage

normalize_group_mean(x, group)

Arguments

x

the variable containing the data to normalized

group

the variable containing the groups

Details

Computes the mean for each group, then divides each value by the mean forthe corresponding group.

Value

Returns a vector of normalized values

References

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

Examples

library(dplyr)carbon.fabric.2 %>%filter(test == "WT") %>%  select(condition, strength) %>%  mutate(condition_norm = normalize_group_mean(strength, condition)) %>%  head(10)##    condition strength condition_norm## 1        CTD  142.817      1.0542187## 2        CTD  135.901      1.0031675## 3        CTD  132.511      0.9781438## 4        CTD  135.586      1.0008423## 5        CTD  125.145      0.9237709## 6        CTD  135.203      0.9980151## 7        CTD  128.547      0.9488832## 8        CTD  127.709      0.9426974## 9        CTD  127.074      0.9380101## 10       CTD  126.879      0.9365706

Normalizes strength values to ply thickness

Description

This function takes a vector of strength values and avector of measured thicknesses, and a nominal thicknessand returns the normalized strength.

Usage

normalize_ply_thickness(strength, measured_thk, nom_thk)

Arguments

strength

the strength to be normalized. Either a vector or a numeric

measured_thk

the measured thickness of the samples. Must be the samelength as strength

nom_thk

the nominal thickness. Must be a single numeric value.

Details

It is often necessary to normalize strength values so that variation inspecimen thickness does not unnecessarily increase variation in strength.See CMH-17-1G, or other references, for information about the cases wherenormalization is appropriate.

Either cured ply thickness or laminate thickness may be used formeasured_thk andnom_thk, as long as the same decisionmade for both values.

The formula applied is:

normalized\,value = test\,value \frac{t_{measured}}{t_{nominal}}

If you need to normalize based on fiber volume fraction (or another method),you will first need to calculate the nominal cured ply thickness (or laminatethickness). Those calculations are outside the scope of this documentation.

Value

The normalized strength values

References

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

Examples

library(dplyr)carbon.fabric.2 %>%select(thickness, strength) %>%  mutate(normalized_strength = normalize_ply_thickness(strength,                                                       thickness,                                                       0.105)) %>%  head(10)##    thickness strength normalized_strength## 1      0.112  142.817            152.3381## 2      0.113  135.901            146.2554## 3      0.113  132.511            142.6071## 4      0.112  135.586            144.6251## 5      0.113  125.145            134.6799## 6      0.113  135.203            145.5042## 7      0.113  128.547            138.3411## 8      0.113  127.709            137.4392## 9      0.113  127.074            136.7558## 10     0.114  126.879            137.7543

Objects exported from other packages

Description

These objects are imported from other packages. Follow the linksbelow to see their documentation.

generics

augment,glance,tidy

See Also

generics::augment()

generics::tidy()

generics::glance()

Separate multiple failure modes into multiple rows

Description

For adata.frame containing a column with (some) multiple failure modes,this function expands thedata.frame by repeating each row with multiplefailure modes so that each row contains only a single failure mode.

Usage

separate_failure_modes(data, failure_mode, sep = "[/, ]+")

Arguments

data

adata.frame

failure_mode

the column indata containing the failure modes

sep

a regular expression with the character(s) separating individualfailure modes. Default"[/, ]+".

Details

When multiple failure modes are reported, they are commonly reported inthe format "LGM/GIT" or "LGM,GIT". This function will separate these multiplefailure modes into multiple rows.

This can be useful when counting the number of coupons exhibited eachfailure mode.

Examples

library(dplyr)data.frame(strength = c(101, 102), fm = c("LGM/GIT", "LGM")) %>%  separate_failure_modes(fm)#### # A tibble: 3 × 2##   strength fm##      <dbl> <chr>## 1      101 LGM## 2      101 GIT## 3      102 LGM

Empirical Survival Function

Description

The empirical survival function (ESF) provides a visualization of adistribution. This is closely related to the empirical cumulativedistribution function (ECDF). The empirical survival function issimply ESF = 1 - ECDF.

Usage

stat_esf(  mapping = NULL,  data = NULL,  geom = "point",  position = "identity",  show.legend = NA,  inherit.aes = TRUE,  n = NULL,  pad = FALSE,  ...)

Arguments

mapping

Set of aesthetic mappings created byaes().

data

The data to be displayed in this layer. This has thesame usage as aggplot2stat function.

geom

The geometric object to use to display the data.

position

Position argument

show.legend

Should this layer be included in the legends?

inherit.aes

IfFALSE, overrides the default aesthetic,rather than combining with them.

n

IfNULL, do not interpolated. Otherwise, thenumber of points to interpolate.

pad

IfTRUE, pad the ESF with additional points⁠(-Inf, 0)⁠ and⁠(0, Inf)⁠.

...

Other arguments to pass on tolayer.

Normal Survival Function

Description

The Normal survival function provides a visualization of adistribution. A normal curve is fit based on the mean and standarddeviation of the data, and the survival function of this normalcurve is plotted. The survival function is simply one minus theCDF.

Usage

stat_normal_surv_func(  mapping = NULL,  data = NULL,  geom = "smooth",  position = "identity",  show.legend = NA,  inherit.aes = TRUE,  n = 100,  pad = FALSE,  ...)

Arguments

mapping

Set of aesthetic mappings created byaes().

data

The data to be displayed in this layer. This has thesame usage as aggplot2stat function.

geom

The geometric object to use to display the data.

position

Position argument

show.legend

Should this layer be included in the legends?

inherit.aes

IfFALSE, overrides the default aesthetic,rather than combining with them.

n

IfNULL, do not interpolated. Otherwise, thenumber of points to interpolate.

pad

IfTRUE, pad the ESF with additional points⁠(-Inf, 0)⁠ and⁠(0, Inf)⁠.

...

Other arguments to pass on tolayer.

Transforms data according to the modified CV rule

Description

Transforms data according to the modified coefficient of variation (CV)rule. This is used to add additional variance to datasets withunexpectedly low variance, which is sometimes encountered duringtesting of new materials over short periods of time.

Two versions of this transformation are implemented. The first version,transform_mod_cv(), transforms the data in a single group (withno other structure) according to the modified CV rules.

The secondversion,transform_mod_cv_ad(), transforms data that is structuredaccording to both condition and batch, as is commonly done forthe Anderson–Darling k-Sample and Anderson-Darling tests when poolingacross environments.

Usage

transform_mod_cv_ad(x, condition, batch)transform_mod_cv(x)

Arguments

x

a vector of data to transform

condition

a vector indicating the condition to which eachobservation belongs

batch

a vector indicating the batch to which each observationbelongs

Details

The modified CV transformation takes the general form:

\frac{S_i^*}{S_i} (x_{ij} - \bar{x_i}) + \bar{x_i}

WhereS_i^* is the modified standard deviation(mod CV times mean) fortheith group;S_i is the standard deviationfor theith group,\bar{x_i} isthe group mean andx_{ij} is the observation.

transform_mod_cv() takes a vectorcontaining the observations and transforms the data.The equation above is used, and all observationsare considered to be from the same group.

transform_mod_cv_ad() takes a vector containing the observationsplus a vector containing the corresponding conditions and a vectorcontaining the batches. This function first calculates the modifiedCV value from the data from each condition (independently). Then,within each condition, the transformationabove is applied to produce the transformed datax'.This transformed data is further transformed using the followingequation.

x_{ij}'' = C (x'_{ij} - \bar{x_i}) + \bar{x_i}

Where:

C = \sqrt{\frac{SSE^*}{SSE'}}

SSE^* = (n-1) (CV^* \bar{x})^2 - \sum(n_i(\bar{x_i}-\bar{x})^2)

SSE' = \sum(x'_{ij} - \bar{x_i})^2

Value

A vector of transformed data

See Also

calc_cv_star()

cv()

Examples

# Transform data according to the modified CV transformation# and report the original and modified CV for each conditionlibrary(dplyr)carbon.fabric %>%filter(test == "FT") %>%  group_by(condition) %>%  mutate(trans_strength = transform_mod_cv(strength)) %>%  head(10)## # A tibble: 10 x 6## # Groups:   condition [1]##    id         test  condition batch strength trans_strength##    <chr>      <chr> <chr>     <int>    <dbl>          <dbl>##  1 FT-RTD-1-1 FT    RTD           1     126.           126.##  2 FT-RTD-1-2 FT    RTD           1     139.           141.##  3 FT-RTD-1-3 FT    RTD           1     116.           115.##  4 FT-RTD-1-4 FT    RTD           1     132.           133.##  5 FT-RTD-1-5 FT    RTD           1     129.           129.##  6 FT-RTD-1-6 FT    RTD           1     130.           130.##  7 FT-RTD-2-1 FT    RTD           2     131.           131.##  8 FT-RTD-2-2 FT    RTD           2     124.           124.##  9 FT-RTD-2-3 FT    RTD           2     125.           125.## 10 FT-RTD-2-4 FT    RTD           2     120.           119.# The CV of this transformed data can be computed to verify# that the resulting CV follows the rules for modified CVcarbon.fabric %>%  filter(test == "FT") %>%  group_by(condition) %>%  mutate(trans_strength = transform_mod_cv(strength)) %>%  summarize(cv = sd(strength) / mean(strength),            mod_cv = sd(trans_strength) / mean(trans_strength))## # A tibble: 3 x 3##   condition     cv mod_cv##   <chr>      <dbl>  <dbl>## 1 CTD       0.0423 0.0612## 2 ETW       0.0369 0.0600## 3 RTD       0.0621 0.0711
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