1. Introduction

This vignette is intended to contain the same validation that is included in the test suite within the cmstatr package, but in a format that is easier for a human to read. The intent is that this vignette will include only those validations that are included in the test suite, but that the test suite may include more tests than are shown in this vignette.

The following packages will be used in this validation. The version of each package used is listed at the end of this vignette.

Throughout this vignette, the testthat package will be used. Expressions such as expect_equal are used to ensure that the two values are equal (within some tolerance). If this expectation is not true, the vignette will fail to build. The tolerance is a relative tolerance: a tolerance of 0.01 means that the two values must be within \(1\%\) of each other. As an example, the following expression checks that the value 10 is equal to 10.1 within a tolerance of 0.01. Such an expectation should be satisfied.

expect_equal(10, 10.1, tolerance = 0.01)

The basis_... functions automatically perform certain diagnostic tests. When those diagnostic tests are not relevant to the validation, the diagnostic tests are overridden by passing the argument override = "all".

3. carbon.fabric Data Set

This data set is example data that is provided with cmstatr. The first few rows of this data are shown below.

head(carbon.fabric)
#>           id test condition batch strength
#> 1 WT-RTD-1-1   WT       RTD     1 137.4438
#> 2 WT-RTD-1-2   WT       RTD     1 139.5395
#> 3 WT-RTD-1-3   WT       RTD     1 150.8900
#> 4 WT-RTD-1-4   WT       RTD     1 141.4474
#> 5 WT-RTD-1-5   WT       RTD     1 141.8203
#> 6 WT-RTD-1-6   WT       RTD     1 151.8821

3.1. Anderson–Darling k-Sample Test

This data was entered into ASAP 2008 [1] and the reported Anderson–Darling k–Sample test statistics were recorded, as were the conclusions.

The value of the test statistic reported by cmstatr and that reported by ASAP 2008 differ by a factor of \(k - 1\), as do the critical values used. As such, the conclusion of the tests are identical. This is described in more detail in the Anderson–Darling k–Sample Vignette.

When the RTD warp-tension data from this data set is entered into ASAP 2008, it reports a test statistic of 0.456 and fails to reject the null hypothesis that the batches are drawn from the same distribution. Adjusting for the different definition of the test statistic, the results given by cmstatr are very similar.

res <- carbon.fabric %>%
  filter(test == "WT") %>%
  filter(condition == "RTD") %>%
  ad_ksample(strength, batch)

expect_equal(res$ad / (res$k - 1), 0.456, tolerance = 0.002)
expect_false(res$reject_same_dist)

res
#> 
#> 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 )

When the ETW warp-tension data from this data set are entered into ASAP 2008, the reported test statistic is 1.604 and it fails to reject the null hypothesis that the batches are drawn from the same distribution. Adjusting for the different definition of the test statistic, cmstatr gives nearly identical results.

res <- carbon.fabric %>%
  filter(test == "WT") %>%
  filter(condition == "ETW") %>%
  ad_ksample(strength, batch)

expect_equal(res$ad / (res$k - 1), 1.604, tolerance = 0.002)
expect_false(res$reject_same_dist)

res
#> 
#> Call:
#> ad_ksample(data = ., x = strength, groups = batch)
#> 
#> N = 18           k = 3            
#> ADK = 3.21       p-value = 0.10208 
#> Conclusion: Samples come from the same distribution ( alpha = 0.025 )

4. Comparison with Examples from CMH-17-1G

4.1 Dataset From Section 8.3.11.1.1

CMH-17-1G [2] provides an example data set and results from ASAP [1] and STAT17 [3]. This example data set is duplicated below:

dat_8_3_11_1_1 <- tribble(
  ~batch, ~strength, ~condition,
  1, 118.3774604, "CTD", 1, 84.9581364, "RTD", 1, 83.7436035, "ETD",
  1, 123.6035612, "CTD", 1, 92.4891822, "RTD", 1, 84.3831677, "ETD",
  1, 115.2238092, "CTD", 1, 96.8212659, "RTD", 1, 94.8030433, "ETD",
  1, 112.6379744, "CTD", 1, 109.030325, "RTD", 1, 94.3931537, "ETD",
  1, 116.5564277, "CTD", 1, 97.8212659, "RTD", 1, 101.702222, "ETD",
  1, 123.1649896, "CTD", 1, 100.921519, "RTD", 1, 86.5372121, "ETD",
  2, 128.5589027, "CTD", 1, 103.699444, "RTD", 1, 92.3772684, "ETD",
  2, 113.1462103, "CTD", 2, 93.7908212, "RTD", 2, 89.2084024, "ETD",
  2, 121.4248107, "CTD", 2, 107.526709, "RTD", 2, 100.686001, "ETD",
  2, 134.3241906, "CTD", 2, 94.5769704, "RTD", 2, 81.0444192, "ETD",
  2, 129.6405117, "CTD", 2, 93.8831373, "RTD", 2, 91.3398070, "ETD",
  2, 117.9818658, "CTD", 2, 98.2296605, "RTD", 2, 93.1441939, "ETD",
  3, 115.4505226, "CTD", 2, 111.346590, "RTD", 2, 85.8204168, "ETD",
  3, 120.0369467, "CTD", 2, 100.817538, "RTD", 3, 94.8966273, "ETD",
  3, 117.1631088, "CTD", 3, 100.382203, "RTD", 3, 95.8068520, "ETD",
  3, 112.9302797, "CTD", 3, 91.5037811, "RTD", 3, 86.7842252, "ETD",
  3, 117.9114501, "CTD", 3, 100.083233, "RTD", 3, 94.4011973, "ETD",
  3, 120.1900159, "CTD", 3, 95.6393615, "RTD", 3, 96.7231171, "ETD",
  3, 110.7295966, "CTD", 3, 109.304779, "RTD", 3, 89.9010384, "ETD",
  3, 100.078562, "RTD", 3, 99.1205847, "RTD", 3, 89.3672306, "ETD",
  1, 106.357525, "ETW", 1, 99.0239966, "ETW2",
  1, 105.898733, "ETW", 1, 103.341238, "ETW2",
  1, 88.4640082, "ETW", 1, 100.302130, "ETW2",
  1, 103.901744, "ETW", 1, 98.4634133, "ETW2",
  1, 80.2058219, "ETW", 1, 92.2647280, "ETW2",
  1, 109.199597, "ETW", 1, 103.487693, "ETW2",
  1, 61.0139431, "ETW", 1, 113.734763, "ETW2",
  2, 99.3207107, "ETW", 2, 108.172659, "ETW2",
  2, 115.861770, "ETW", 2, 108.426732, "ETW2",
  2, 82.6133082, "ETW", 2, 116.260375, "ETW2",
  2, 85.3690411, "ETW", 2, 121.049610, "ETW2",
  2, 115.801622, "ETW", 2, 111.223082, "ETW2",
  2, 44.3217741, "ETW", 2, 104.574843, "ETW2",
  2, 117.328077, "ETW", 2, 103.222552, "ETW2",
  2, 88.6782903, "ETW", 3, 99.3918538, "ETW2",
  3, 107.676986, "ETW", 3, 87.3421658, "ETW2",
  3, 108.960241, "ETW", 3, 102.730741, "ETW2",
  3, 116.122640, "ETW", 3, 96.3694916, "ETW2",
  3, 80.2334815, "ETW", 3, 99.5946088, "ETW2",
  3, 106.145570, "ETW", 3, 97.0712407, "ETW2",
  3, 104.667866, "ETW",
  3, 104.234953, "ETW"
)
dat_8_3_11_1_1
#> # A tibble: 102 x 3
#>    batch strength condition
#>    <dbl>    <dbl> <chr>    
#>  1     1    118.  CTD      
#>  2     1     85.0 RTD      
#>  3     1     83.7 ETD      
#>  4     1    124.  CTD      
#>  5     1     92.5 RTD      
#>  6     1     84.4 ETD      
#>  7     1    115.  CTD      
#>  8     1     96.8 RTD      
#>  9     1     94.8 ETD      
#> 10     1    113.  CTD      
#> # … with 92 more rows

4.1.1 Maximum Normed Residual Test

CMH-17-1G Table 8.3.11.1.1(a) provides results of the MNR test from ASAP for this data set. Batches 2 and 3 of the ETW data is considered here and the results of cmstatr are compared with those published in CMH-17-1G.

For Batch 2 of the ETW data, the results match those published in the handbook within a small tolerance. The published test statistic is 2.008.

res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW" & batch == 2) %>%
  maximum_normed_residual(strength, alpha = 0.05)

expect_equal(res$mnr, 2.008, tolerance = 0.001)
expect_equal(res$crit, 2.127, tolerance = 0.001)
expect_equal(res$n_outliers, 0)

res
#> 
#> Call:
#> maximum_normed_residual(data = ., x = strength, alpha = 0.05)
#> 
#> MNR = 2.008274  ( critical value = 2.126645 )
#> 
#> No outliers detected ( alpha = 0.05 )

Similarly, for Batch 3 of the ETW data, the results of cmstatr match the results published in the handbook within a small tolerance. The published test statistic is 2.119

res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW" & batch == 3) %>%
  maximum_normed_residual(strength, alpha = 0.05)

expect_equal(res$mnr, 2.119, tolerance = 0.001)
expect_equal(res$crit, 2.020, tolerance = 0.001)
expect_equal(res$n_outliers, 1)

res
#> 
#> Call:
#> maximum_normed_residual(data = ., x = strength, alpha = 0.05)
#> 
#> MNR = 2.119175  ( critical value = 2.019969 )
#> 
#> Outliers ( alpha = 0.05 ):
#>    Index    Value            
#>        4    80.23348

4.1.2 Anderson–Darling k–Sample Test

For the ETW condition, the ADK test statistic given in [2] is \(ADK = 0.793\) and the test concludes that the samples come from the same distribution. Noting that cmstatr uses the definition of the test statistic given in [4], so the test statistic given by cmstatr differs from that given by ASAP by a factor of \(k - 1\), as described in the Anderson–Darling k–Sample Vignette.

res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW") %>%
  ad_ksample(strength, batch)

expect_equal(res$ad / (res$k - 1), 0.793, tolerance = 0.003)
expect_false(res$reject_same_dist)

res
#> 
#> Call:
#> ad_ksample(data = ., x = strength, groups = batch)
#> 
#> N = 22           k = 3            
#> ADK = 1.59       p-value = 0.59491 
#> Conclusion: Samples come from the same distribution ( alpha = 0.025 )

Similarly, for the ETW2 condition, the test statistic given in [2] is \(ADK = 3.024\) and the test concludes that the samples come from different distributions. This matches cmstatr

res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW2") %>%
  ad_ksample(strength, batch)

expect_equal(res$ad / (res$k - 1), 3.024, tolerance = 0.001)
expect_true(res$reject_same_dist)

res
#> 
#> Call:
#> ad_ksample(data = ., x = strength, groups = batch)
#> 
#> N = 20           k = 3            
#> ADK = 6.05       p-value = 0.0042703 
#> Conclusion: Samples do not come from the same distribution (alpha = 0.025 )

4.1.3 Anderson–Darling Tests for Distribution

CMH-17-1G Section 8.3.11.2.1 contains results from STAT17 for the “observed significance level” from the Anderson–Darling test for various distributions. In this section, the ETW condition from the present data set is used. The published results are given in the following table. The results from cmstatr are below and are very similar to those from STAT17.

Distribution OSL
Normal 0.006051
Lognormal 0.000307
Weibull 0.219
res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW") %>%
  anderson_darling_normal(strength)
expect_equal(res$osl, 0.006051, tolerance = 0.001)
res
#> 
#> Call:
#> anderson_darling_normal(data = ., x = strength)
#> 
#> Distribution:  Normal ( n = 22 ) 
#> Test statistic:  A = 1.052184 
#> OSL (p-value):  0.006051441  (assuming unknown parameters)
#> Conclusion: Sample is not drawn from a Normal distribution ( alpha = 0.05 )
res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW") %>%
  anderson_darling_lognormal(strength)
expect_equal(res$osl, 0.000307, tolerance = 0.001)
res
#> 
#> Call:
#> anderson_darling_lognormal(data = ., x = strength)
#> 
#> Distribution:  Lognormal ( n = 22 ) 
#> Test statistic:  A = 1.568825 
#> OSL (p-value):  0.0003073592  (assuming unknown parameters)
#> Conclusion: Sample is not drawn from a Lognormal distribution ( alpha = 0.05 )
res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW") %>%
  anderson_darling_weibull(strength)
expect_equal(res$osl, 0.0219, tolerance = 0.002)
res
#> 
#> Call:
#> anderson_darling_weibull(data = ., x = strength)
#> 
#> Distribution:  Weibull ( n = 22 ) 
#> Test statistic:  A = 0.8630665 
#> OSL (p-value):  0.02186889  (assuming unknown parameters)
#> Conclusion: Sample is not drawn from a Weibull distribution ( alpha = 0.05 )

4.1.4 Levene’s Test (Between Conditions)

CMH-17-1G Section 8.3.11.1.1 provides results from ASAP for Levene’s test for equality of variance between conditions after the ETW and ETW2 conditions are removed. The handbook shows an F statistic of 0.58, however if this data is entered into ASAP directly, ASAP gives an F statistic of 0.058, which matches the result of cmstatr.

res <- dat_8_3_11_1_1 %>%
  filter(condition != "ETW" & condition != "ETW2") %>%
  levene_test(strength, condition)
expect_equal(res$f, 0.058, tolerance = 0.01)
res
#> 
#> Call:
#> levene_test(data = ., x = strength, groups = condition)
#> 
#> n = 60           k = 3            
#> F = 0.05811631   p-value = 0.943596 
#> Conclusion: Samples have equal variances ( alpha = 0.05 )

4.1.5 Levene’s Test (Between Batches)

CMH-17-1G Section 8.3.11.2.2 provides output from STAT17. The ETW2 condition from the present data set was analyzed by STAT17 and that software reported an F statistic of 0.123 from Levene’s test when comparing the variance of the batches within this condition. The result from cmstatr is similar.

res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW2") %>%
  levene_test(strength, batch)
expect_equal(res$f, 0.123, tolerance = 0.005)
res
#> 
#> Call:
#> levene_test(data = ., x = strength, groups = batch)
#> 
#> n = 20           k = 3            
#> F = 0.1233937    p-value = 0.8847 
#> Conclusion: Samples have equal variances ( alpha = 0.05 )

Similarly, the published value of the F statistic for the CTD condition is \(3.850\). cmstatr produces very similar results.

res <- dat_8_3_11_1_1 %>%
  filter(condition == "CTD") %>%
  levene_test(strength, batch)
expect_equal(res$f, 3.850, tolerance = 0.005)
res
#> 
#> Call:
#> levene_test(data = ., x = strength, groups = batch)
#> 
#> n = 19           k = 3            
#> F = 3.852032     p-value = 0.04309008 
#> Conclusion: Samples have unequal variance ( alpha = 0.05 )

4.1.6 Nonparametric Basis Values

CMH-17-1G Section 8.3.11.2.1 provides STAT17 outputs for the ETW condition of the present data set. The nonparametric Basis values are listed. In this case, the Hanson–Koopmans method is used. The published A-Basis value is 13.0 and the B-Basis is 37.9.

res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW") %>%
  basis_hk_ext(strength, method = "woodward-frawley", p = 0.99, conf = 0.95,
               override = "all")

expect_equal(res$basis, 13.0, tolerance = 0.001)

res
#> 
#> Call:
#> basis_hk_ext(data = ., x = strength, p = 0.99, conf = 0.95, method = "woodward-frawley", 
#>     override = "all")
#> 
#> Distribution:  Nonparametric (Extended Hanson-Koopmans, Woodward-Frawley method)     ( n = 22 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `correct_method_used`,
#>     `sample_size`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> 12.99614
res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW") %>%
  basis_hk_ext(strength, method = "optimum-order", p = 0.90, conf = 0.95,
               override = "all")

expect_equal(res$basis, 37.9, tolerance = 0.001)

res
#> 
#> Call:
#> basis_hk_ext(data = ., x = strength, p = 0.9, conf = 0.95, method = "optimum-order", 
#>     override = "all")
#> 
#> Distribution:  Nonparametric (Extended Hanson-Koopmans, optimum two-order-statistic method)  ( n = 22 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `correct_method_used`,
#>     `sample_size`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 37.88511

4.1.7 Single-Point ANOVA Basis Value

CMH-17-1G Section 8.3.11.2.2 provides output from STAT17 for the ETW2 condition from the present data set. STAT17 reports A- and B-Basis values based on the ANOVA method of 34.6 and 63.2, respectively. The results from cmstatr are similar.

res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW2") %>%
  basis_anova(strength, batch, override = "number_of_groups",
              p = 0.99, conf = 0.95)
expect_equal(res$basis, 34.6, tolerance = 0.001)
res
#> 
#> Call:
#> basis_anova(data = ., x = strength, groups = batch, p = 0.99, 
#>     conf = 0.95, override = "number_of_groups")
#> 
#> Distribution:  ANOVA     ( n = 20, r = 3 )
#> The following diagnostic tests were overridden:
#>     `number_of_groups`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> 34.57763
res <- dat_8_3_11_1_1 %>%
  filter(condition == "ETW2") %>%
  basis_anova(strength, batch, override = "number_of_groups")
expect_equal(res$basis, 63.2, tolerance = 0.001)
res
#> 
#> Call:
#> basis_anova(data = ., x = strength, groups = batch, override = "number_of_groups")
#> 
#> Distribution:  ANOVA     ( n = 20, r = 3 )
#> The following diagnostic tests were overridden:
#>     `number_of_groups`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 63.20276

4.2 Dataset From Section 8.3.11.1.2

[2] provides an example data set and results from ASAP [1]. This example data set is duplicated below:

dat_8_3_11_1_2 <- tribble(
  ~batch, ~strength, ~condition,
  1, 79.04517, "CTD", 1, 103.2006, "RTD", 1, 63.22764, "ETW", 1, 54.09806, "ETW2",
  1, 102.6014, "CTD", 1, 105.1034, "RTD", 1, 70.84454, "ETW", 1, 58.87615, "ETW2",
  1, 97.79372, "CTD", 1, 105.1893, "RTD", 1, 66.43223, "ETW", 1, 61.60167, "ETW2",
  1, 92.86423, "CTD", 1, 100.4189, "RTD", 1, 75.37771, "ETW", 1, 60.23973, "ETW2",
  1, 117.218,  "CTD", 2, 85.32319, "RTD", 1, 72.43773, "ETW", 1, 61.4808,  "ETW2",
  1, 108.7168, "CTD", 2, 92.69923, "RTD", 1, 68.43073, "ETW", 1, 64.55832, "ETW2",
  1, 112.2773, "CTD", 2, 98.45242, "RTD", 1, 69.72524, "ETW", 2, 57.76131, "ETW2",
  1, 114.0129, "CTD", 2, 104.1014, "RTD", 2, 66.20343, "ETW", 2, 49.91463, "ETW2",
  2, 106.8452, "CTD", 2, 91.51841, "RTD", 2, 60.51251, "ETW", 2, 61.49271, "ETW2",
  2, 112.3911, "CTD", 2, 101.3746, "RTD", 2, 65.69334, "ETW", 2, 57.7281,  "ETW2",
  2, 115.5658, "CTD", 2, 101.5828, "RTD", 2, 62.73595, "ETW", 2, 62.11653, "ETW2",
  2, 87.40657, "CTD", 2, 99.57384, "RTD", 2, 59.00798, "ETW", 2, 62.69353, "ETW2",
  2, 102.2785, "CTD", 2, 88.84826, "RTD", 2, 62.37761, "ETW", 3, 61.38523, "ETW2",
  2, 110.6073, "CTD", 3, 92.18703, "RTD", 3, 64.3947,  "ETW", 3, 60.39053, "ETW2",
  3, 105.2762, "CTD", 3, 101.8234, "RTD", 3, 72.8491,  "ETW", 3, 59.17616, "ETW2",
  3, 110.8924, "CTD", 3, 97.68909, "RTD", 3, 66.56226, "ETW", 3, 60.17616, "ETW2",
  3, 108.7638, "CTD", 3, 101.5172, "RTD", 3, 66.56779, "ETW", 3, 46.47396, "ETW2",
  3, 110.9833, "CTD", 3, 100.0481, "RTD", 3, 66.00123, "ETW", 3, 51.16616, "ETW2",
  3, 101.3417, "CTD", 3, 102.0544, "RTD", 3, 59.62108, "ETW",
  3, 100.0251, "CTD",                     3, 60.61167, "ETW",
                                          3, 57.65487, "ETW",
                                          3, 66.51241, "ETW",
                                          3, 64.89347, "ETW",
                                          3, 57.73054, "ETW",
                                          3, 68.94086, "ETW",
                                          3, 61.63177, "ETW"
)

4.2.1 Pooled SD A- and B-Basis

CMH-17-1G Table 8.3.11.2(k) provides outputs from ASAP for the data set above. ASAP uses the pooled SD method. ASAP produces the following results, which are quite similar to those produced by cmstatr.

Condition CTD RTD ETW ETW2
B-Basis 93.64 87.30 54.33 47.12
A-Basis 89.19 79.86 46.84 39.69

res <- basis_pooled_sd(dat_8_3_11_1_2, strength, condition,
                         override = "all")

expect_equal(res$basis$value[res$basis$group == "CTD"],
           93.64, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
           87.30, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
           54.33, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW2"],
           47.12, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_sd(data = dat_8_3_11_1_2, x = strength, groups = condition, 
#>     override = "all")
#> 
#> Distribution:  Normal - Pooled Standard Deviation    ( n = 83, r = 4 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `pooled_variance_equal`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> CTD   93.63504 
#> ETW   54.32706 
#> ETW2  47.07669 
#> RTD   87.29555
res <- basis_pooled_sd(dat_8_3_11_1_2, strength, condition,
                       p = 0.99, conf = 0.95,
                       override = "all")
expect_equal(res$basis$value[res$basis$group == "CTD"],
           86.19, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
           79.86, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
           46.84, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW2"],
           39.69, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_sd(data = dat_8_3_11_1_2, x = strength, groups = condition, 
#>     p = 0.99, conf = 0.95, override = "all")
#> 
#> Distribution:  Normal - Pooled Standard Deviation    ( n = 83, r = 4 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `pooled_variance_equal`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> CTD   86.19301 
#> ETW   46.84112 
#> ETW2  39.65214 
#> RTD   79.86205

4.2.2 Pooled SD A- and B-Basis (Mod CV)

After removal of the ETW2 condition, CMH17-STATS reports the pooled A- and B-Basis (mod CV) shown in the following table. cmstatr computes very similar values.

Condition CTD RTD ETW
B-Basis 92.25 85.91 52.97
A-Basis 83.81 77.48 44.47
res <- dat_8_3_11_1_2 %>%
  filter(condition != "ETW2") %>%
  basis_pooled_sd(strength, condition, modcv = TRUE, override = "all")
expect_equal(res$basis$value[res$basis$group == "CTD"],
             92.25, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
             85.91, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
             52.97, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_sd(data = ., x = strength, groups = condition, modcv = TRUE, 
#>     override = "all")
#> 
#> Distribution:  Normal - Pooled Standard Deviation    ( n = 65, r = 3 )
#> Modified CV Approach Used 
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `pooled_variance_equal`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> CTD  92.24927 
#> ETW  52.9649 
#> RTD  85.90454
res <- dat_8_3_11_1_2 %>%
  filter(condition != "ETW2") %>%
  basis_pooled_sd(strength, condition,
                  p = 0.99, conf = 0.95, modcv = TRUE, override = "all")
expect_equal(res$basis$value[res$basis$group == "CTD"],
             83.81, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
             77.48, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
             44.47, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_sd(data = ., x = strength, groups = condition, p = 0.99, 
#>     conf = 0.95, modcv = TRUE, override = "all")
#> 
#> Distribution:  Normal - Pooled Standard Deviation    ( n = 65, r = 3 )
#> Modified CV Approach Used 
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `pooled_variance_equal`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> CTD  83.80981 
#> ETW  44.47038 
#> RTD  77.47589

4.2.3 Pooled CV A- and B-Basis

This data set was input into CMH17-STATS and the Pooled CV method was selected. The results from CMH17-STATS were as follows. cmstatr produces very similar results.

Condition CTD RTD ETW ETW2
B-Basis 90.89 85.37 56.79 50.55
A-Basis 81.62 76.67 50.98 45.40
res <- basis_pooled_cv(dat_8_3_11_1_2, strength, condition, override = "all")
expect_equal(res$basis$value[res$basis$group == "CTD"],
             90.89, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
             85.37, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
             56.79, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW2"],
             50.55, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_cv(data = dat_8_3_11_1_2, x = strength, groups = condition, 
#>     override = "all")
#> 
#> Distribution:  Normal - Pooled CV    ( n = 83, r = 4 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `normalized_variance_equal`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> CTD   90.88018 
#> ETW   56.78337 
#> ETW2  50.54406 
#> RTD   85.36756
res <- basis_pooled_cv(dat_8_3_11_1_2, strength, condition,
                       p = 0.99, conf = 0.95, override = "all")
expect_equal(res$basis$value[res$basis$group == "CTD"],
             81.62, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
             76.67, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
             50.98, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW2"],
             45.40, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_cv(data = dat_8_3_11_1_2, x = strength, groups = condition, 
#>     p = 0.99, conf = 0.95, override = "all")
#> 
#> Distribution:  Normal - Pooled CV    ( n = 83, r = 4 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `normalized_variance_equal`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> CTD   81.60931 
#> ETW   50.97801 
#> ETW2  45.39158 
#> RTD   76.66214

4.2.4 Pooled CV A- and B-Basis (Mod CV)

This data set was input into CMH17-STATS and the Pooled CV method was selected with the modified CV transform. Additionally, the ETW2 condition was removed. The results from CMH17-STATS were as follows. cmstatr produces very similar results.

Condition CTD RTD ETW
B-Basis 90.31 84.83 56.43
A-Basis 80.57 75.69 50.33
res <- dat_8_3_11_1_2 %>%
  filter(condition != "ETW2") %>%
  basis_pooled_cv(strength, condition, modcv = TRUE, override = "all")
expect_equal(res$basis$value[res$basis$group == "CTD"],
             90.31, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
             84.83, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
             56.43, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_cv(data = ., x = strength, groups = condition, modcv = TRUE, 
#>     override = "all")
#> 
#> Distribution:  Normal - Pooled CV    ( n = 65, r = 3 )
#> Modified CV Approach Used 
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `normalized_variance_equal`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> CTD  90.30646 
#> ETW  56.42786 
#> RTD  84.82748
res <- dat_8_3_11_1_2 %>%
  filter(condition != "ETW2") %>%
  basis_pooled_cv(strength, condition, modcv = TRUE,
                  p = 0.99, conf = 0.95, override = "all")
expect_equal(res$basis$value[res$basis$group == "CTD"],
             80.57, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "RTD"],
             75.69, tolerance = 0.001)
expect_equal(res$basis$value[res$basis$group == "ETW"],
             50.33, tolerance = 0.001)
res
#> 
#> Call:
#> basis_pooled_cv(data = ., x = strength, groups = condition, p = 0.99, 
#>     conf = 0.95, modcv = TRUE, override = "all")
#> 
#> Distribution:  Normal - Pooled CV    ( n = 65, r = 3 )
#> Modified CV Approach Used 
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_group_variability`,
#>     `outliers_within_group`,
#>     `pooled_data_normal`,
#>     `normalized_variance_equal`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> CTD  80.56529 
#> ETW  50.32422 
#> RTD  75.6817

5. Other Data Sets

This section contains various small data sets. In most cases, these data sets were generated randomly for the purpose of comparing cmstatr to other software.

5.1 Anderson–Darling Test (Normal)

The following data set was randomly generated. When this is entered into STAT17 [3], that software gives the value \(OSL = 0.465\), which matches the result of cmstatr within a small margin.

dat <- data.frame(
  strength = c(
    137.4438, 139.5395, 150.89, 141.4474, 141.8203, 151.8821, 143.9245,
    132.9732, 136.6419, 138.1723, 148.7668, 143.283, 143.5429,
    141.7023, 137.4732, 152.338, 144.1589, 128.5218
  )
)
res <- anderson_darling_normal(dat, strength)

expect_equal(res$osl, 0.465, tolerance = 0.001)

res
#> 
#> Call:
#> anderson_darling_normal(data = dat, x = strength)
#> 
#> Distribution:  Normal ( n = 18 ) 
#> Test statistic:  A = 0.2849726 
#> OSL (p-value):  0.4648132  (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Normal distribution ( alpha = 0.05 )

5.2 Anderson–Darling Test (Lognormal)

The following data set was randomly generated. When this is entered into STAT17 [3], that software gives the value \(OSL = 0.480\), which matches the result of cmstatr within a small margin.

dat <- data.frame(
  strength = c(
    137.4438, 139.5395, 150.89, 141.4474, 141.8203, 151.8821, 143.9245,
    132.9732, 136.6419, 138.1723, 148.7668, 143.283, 143.5429,
    141.7023, 137.4732, 152.338, 144.1589, 128.5218
  )
)

res <- anderson_darling_lognormal(dat, strength)

expect_equal(res$osl, 0.480, tolerance = 0.001)

res
#> 
#> Call:
#> anderson_darling_lognormal(data = dat, x = strength)
#> 
#> Distribution:  Lognormal ( n = 18 ) 
#> Test statistic:  A = 0.2774652 
#> OSL (p-value):  0.4798148  (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Lognormal distribution ( alpha = 0.05 )

5.3 Anderson–Darling Test (Weibull)

The following data set was randomly generated. When this is entered into STAT17 [3], that software gives the value \(OSL = 0.179\), which matches the result of cmstatr within a small margin.

dat <- data.frame(
  strength = c(
    137.4438, 139.5395, 150.89, 141.4474, 141.8203, 151.8821, 143.9245,
    132.9732, 136.6419, 138.1723, 148.7668, 143.283, 143.5429,
    141.7023, 137.4732, 152.338, 144.1589, 128.5218
  )
)

res <- anderson_darling_weibull(dat, strength)

expect_equal(res$osl, 0.179, tolerance = 0.002)

res
#> 
#> Call:
#> anderson_darling_weibull(data = dat, x = strength)
#> 
#> Distribution:  Weibull ( n = 18 ) 
#> Test statistic:  A = 0.5113909 
#> OSL (p-value):  0.1787882  (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Weibull distribution ( alpha = 0.05 )

5.4 Normal A- and B-Basis

The following data was input into STAT17 and the A- and B-Basis values were computed assuming normally distributed data. The results were 120.336 and 129.287, respectively. cmstatr reports very similar values.

dat <- c(
  137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
  132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
  137.4732, 152.3380, 144.1589, 128.5218
)

res <- basis_normal(x = dat, p = 0.99, conf = 0.95, override = "all")
expect_equal(res$basis, 120.336, tolerance = 0.0005)
res
#> 
#> Call:
#> basis_normal(x = dat, p = 0.99, conf = 0.95, override = "all")
#> 
#> Distribution:  Normal    ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `anderson_darling_normal`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> 120.3549
res <- basis_normal(x = dat, p = 0.9, conf = 0.95, override = "all")
expect_equal(res$basis, 129.287, tolerance = 0.0005)
res
#> 
#> Call:
#> basis_normal(x = dat, p = 0.9, conf = 0.95, override = "all")
#> 
#> Distribution:  Normal    ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `anderson_darling_normal`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 129.2898

5.5 Lognormal A- and B-Basis

The following data was input into STAT17 and the A- and B-Basis values were computed assuming distributed according to a lognormal distribution. The results were 121.710 and 129.664, respectively. cmstatr reports very similar values.

dat <- c(
  137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
  132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
  137.4732, 152.3380, 144.1589, 128.5218
)

res <- basis_lognormal(x = dat, p = 0.99, conf = 0.95, override = "all")
expect_equal(res$basis, 121.710, tolerance = 0.0005)
res
#> 
#> Call:
#> basis_lognormal(x = dat, p = 0.99, conf = 0.95, override = "all")
#> 
#> Distribution:  Lognormal     ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `anderson_darling_lognormal`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> 121.7265
res <- basis_lognormal(x = dat, p = 0.9, conf = 0.95, override = "all")
expect_equal(res$basis, 129.664, tolerance = 0.0005)
res
#> 
#> Call:
#> basis_lognormal(x = dat, p = 0.9, conf = 0.95, override = "all")
#> 
#> Distribution:  Lognormal     ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `anderson_darling_lognormal`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 129.667

5.6 Weibull A- and B-Basis

The following data was input into STAT17 and the A- and B-Basis values were computed assuming data following the Weibull distribution. The results were 109.150 and 125.441, respectively. cmstatr reports very similar values.

dat <- c(
  137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
  132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
  137.4732, 152.3380, 144.1589, 128.5218
)

res <- basis_weibull(x = dat, p = 0.99, conf = 0.95, override = "all")
expect_equal(res$basis, 109.150, tolerance = 0.005)
res
#> 
#> Call:
#> basis_weibull(x = dat, p = 0.99, conf = 0.95, override = "all")
#> 
#> Distribution:  Weibull   ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `anderson_darling_weibull`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> 109.651
res <- basis_weibull(x = dat, p = 0.9, conf = 0.95, override = "all")
expect_equal(res$basis, 125.441, tolerance = 0.005)
res
#> 
#> Call:
#> basis_weibull(x = dat, p = 0.9, conf = 0.95, override = "all")
#> 
#> Distribution:  Weibull   ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `anderson_darling_weibull`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 125.7338

5.7 Extended Hanson–Koopmans A- and B-Basis

The following data was input into STAT17 and the A- and B-Basis values were computed using the nonparametric (small sample) method. The results were 99.651 and 124.156, respectively. cmstatr reports very similar values.

dat <- c(
  137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
  132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
  137.4732, 152.3380, 144.1589, 128.5218
)

res <- basis_hk_ext(x = dat, p = 0.99, conf = 0.95,
                    method = "woodward-frawley", override = "all")
expect_equal(res$basis, 99.651, tolerance = 0.005)
res
#> 
#> Call:
#> basis_hk_ext(x = dat, p = 0.99, conf = 0.95, method = "woodward-frawley", 
#>     override = "all")
#> 
#> Distribution:  Nonparametric (Extended Hanson-Koopmans, Woodward-Frawley method)     ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `correct_method_used`,
#>     `sample_size`
#> A-Basis:   ( p = 0.99 , conf = 0.95 )
#> 99.65098
res <- basis_hk_ext(x = dat, p = 0.9, conf = 0.95,
                    method = "optimum-order", override = "all")
expect_equal(res$basis, 124.156, tolerance = 0.005)
res
#> 
#> Call:
#> basis_hk_ext(x = dat, p = 0.9, conf = 0.95, method = "optimum-order", 
#>     override = "all")
#> 
#> Distribution:  Nonparametric (Extended Hanson-Koopmans, optimum two-order-statistic method)  ( n = 18 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `correct_method_used`,
#>     `sample_size`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 124.1615

5.8 Extended Hanson–Koopmans B-Basis

The following random numbers were generated.

dat <- c(
  139.6734, 143.0032, 130.4757, 144.8327, 138.7818, 136.7693, 148.636,
  131.0095, 131.4933, 142.8856, 158.0198, 145.2271, 137.5991, 139.8298,
  140.8557, 137.6148, 131.3614, 152.7795, 145.8792, 152.9207, 160.0989,
  145.1920, 128.6383, 141.5992, 122.5297, 159.8209, 151.6720, 159.0156
)

All of the numbers above were input into STAT17 and the reported B-Basis value using the Optimum Order nonparametric method was 122.36798. This result matches the results of cmstatr within a small margin.

res <- basis_hk_ext(x = dat, p = 0.9, conf = 0.95,
                    method = "optimum-order", override = "all")
expect_equal(res$basis, 122.36798, tolerance = 0.001)
res
#> 
#> Call:
#> basis_hk_ext(x = dat, p = 0.9, conf = 0.95, method = "optimum-order", 
#>     override = "all")
#> 
#> Distribution:  Nonparametric (Extended Hanson-Koopmans, optimum two-order-statistic method)  ( n = 28 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `correct_method_used`,
#>     `sample_size`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 122.3638

The last two observations from the above data set were discarded, leaving 26 observations. This smaller data set was input into STAT17 and that software calculated a B-Basis value of 121.57073 using the Optimum Order nonparametric method. cmstatr reports a very similar number.

res <- basis_hk_ext(x = head(dat, 26), p = 0.9, conf = 0.95,
                    method = "optimum-order", override = "all")
expect_equal(res$basis, 121.57073, tolerance = 0.001)
res
#> 
#> Call:
#> basis_hk_ext(x = head(dat, 26), p = 0.9, conf = 0.95, method = "optimum-order", 
#>     override = "all")
#> 
#> Distribution:  Nonparametric (Extended Hanson-Koopmans, optimum two-order-statistic method)  ( n = 26 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `correct_method_used`,
#>     `sample_size`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 121.5655

The same data set was further reduced such that only the first 22 observations were included. This smaller data set was input into STAT17 and that software calculated a B-Basis value of 128.82397 using the Optimum Order nonparametric method. cmstatr reports a very similar number.

res <- basis_hk_ext(x = head(dat, 22), p = 0.9, conf = 0.95,
                    method = "optimum-order", override = "all")
expect_equal(res$basis, 128.82397, tolerance = 0.001)
res
#> 
#> Call:
#> basis_hk_ext(x = head(dat, 22), p = 0.9, conf = 0.95, method = "optimum-order", 
#>     override = "all")
#> 
#> Distribution:  Nonparametric (Extended Hanson-Koopmans, optimum two-order-statistic method)  ( n = 22 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `correct_method_used`,
#>     `sample_size`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 128.8224

5.9 Large Sample Nonparametric B-Basis

The following data was input into STAT17 and the B-Basis value was computed using the nonparametric (large sample) method. The results was 122.738297. cmstatr reports very similar values.

dat <- c(
  137.3603, 135.6665, 136.6914, 154.7919, 159.2037, 137.3277, 128.821,
  138.6304, 138.9004, 147.4598, 148.6622, 144.4948, 131.0851, 149.0203,
  131.8232, 146.4471, 123.8124, 126.3105, 140.7609, 134.4875, 128.7508,
  117.1854, 129.3088, 141.6789, 138.4073, 136.0295, 128.4164, 141.7733,
  134.455,  122.7383, 136.9171, 136.9232, 138.8402, 152.8294, 135.0633,
  121.052,  131.035,  138.3248, 131.1379, 147.3771, 130.0681, 132.7467,
  137.1444, 141.662,  146.9363, 160.7448, 138.5511, 129.1628, 140.2939,
  144.8167, 156.5918, 132.0099, 129.3551, 136.6066, 134.5095, 128.2081,
  144.0896, 141.8029, 130.0149, 140.8813, 137.7864
)

res <- basis_nonpara_large_sample(x = dat, p = 0.9, conf = 0.95,
                                  override = "all")
expect_equal(res$basis, 122.738297, tolerance = 0.005)
res
#> 
#> Call:
#> basis_nonpara_large_sample(x = dat, p = 0.9, conf = 0.95, override = "all")
#> 
#> Distribution:  Nonparametric (large sample)  ( n = 61 )
#> The following diagnostic tests were overridden:
#>     `outliers_within_batch`,
#>     `between_batch_variability`,
#>     `outliers`,
#>     `sample_size`
#> B-Basis:   ( p = 0.9 , conf = 0.95 )
#> 122.7383

5.10 Acceptance Limits Based on Mean and Extremum

Results from cmstatr’s equiv_mean_extremum function were compared with results from HYTEQ. The summary statistics for the qualification data were set as mean = 141.310 and sd=6.415. For a value of alpha=0.05 and n = 9, HYTEQ reported thresholds of 123.725 and 137.197 for minimum individual and mean, respectively. cmstatr produces very similar results.

res <- equiv_mean_extremum(alpha = 0.05, mean_qual = 141.310, sd_qual = 6.415,
                           n_sample = 9)
expect_equal(res$threshold_min_indiv, 123.725, tolerance = 0.001)
expect_equal(res$threshold_mean, 137.197, tolerance = 0.001)
res
#> 
#> Call:
#> equiv_mean_extremum(mean_qual = 141.31, sd_qual = 6.415, n_sample = 9, 
#>     alpha = 0.05)
#> 
#> For alpha = 0.05 and n = 9 
#> ( k1 = 2.741054 and k2 = 0.6410852 )
#>                   Min Individual    Sample Mean    
#>      Thresholds:     123.7261         137.1974

Using the same parameters, but using the modified CV method, HYTEQ produces thresholds of 117.024 and 135.630 for minimum individual and mean, respectively. cmstatr produces very similar results.

res <- equiv_mean_extremum(alpha = 0.05, mean_qual = 141.310, sd_qual = 6.415,
                           n_sample = 9, modcv = TRUE)
expect_equal(res$threshold_min_indiv, 117.024, tolerance = 0.001)
expect_equal(res$threshold_mean, 135.630, tolerance = 0.001)
res
#> 
#> Call:
#> equiv_mean_extremum(mean_qual = 141.31, sd_qual = 6.415, n_sample = 9, 
#>     alpha = 0.05, modcv = TRUE)
#> 
#> Modified CV used: CV* = 0.06269832 ( CV = 0.04539665 )
#> 
#> For alpha = 0.05 and n = 9 
#> ( k1 = 2.741054 and k2 = 0.6410852 )
#>                   Min Individual    Sample Mean    
#>      Thresholds:     117.0245          135.63

5.11 Acceptance Based on Change in Mean

Results from cmstatr’s equiv_change_mean function were compared with results from HYTEQ. The following parameters were used. A value of alpha = 0.05 was selected.

Parameter Qualification Sample
Mean 9.24 9.02
SD 0.162 0.15785
n 28 9

HYTEQ gives an acceptance range of 9.115 to 9.365. cmstatr produces similar results.

res <- 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)
expect_equal(res$threshold, c(9.115, 9.365), tolerance = 0.001)
res
#> 
#> 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)
#> 
#> For alpha = 0.05 
#>                   Qualification        Sample      
#>           Number        28               9         
#>             Mean       9.24             9.02       
#>               SD      0.162           0.15785      
#>           Result               FAIL               
#>    Passing Range       9.114712 to 9.365288

After selecting the modified CV method, HYTEQ gives an acceptance range of 8.857 to 9.623. cmstatr produces similar results.

res <- 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)
expect_equal(res$threshold, c(8.857, 9.623), tolerance = 0.001)
res
#> 
#> 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

6. Comparison with Literature

In this section, results from cmstatr are compared with values published in literature.

6.1 Anderson–Darling K–Sample Test

[4] provides example data that compares measurements obtained in four labs. Their paper gives values of the ADK test statistic as well as p-values.

The data in [4] is as follows:

dat_ss1987 <- data.frame(
  smoothness = c(
    38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0,
    39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8,
    34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0,
    34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8
  ),
  lab = c(rep("A", 8), rep("B", 8), rep("C", 8), rep("D", 8))
)
dat_ss1987
#>    smoothness lab
#> 1        38.7   A
#> 2        41.5   A
#> 3        43.8   A
#> 4        44.5   A
#> 5        45.5   A
#> 6        46.0   A
#> 7        47.7   A
#> 8        58.0   A
#> 9        39.2   B
#> 10       39.3   B
#> 11       39.7   B
#> 12       41.4   B
#> 13       41.8   B
#> 14       42.9   B
#> 15       43.3   B
#> 16       45.8   B
#> 17       34.0   C
#> 18       35.0   C
#> 19       39.0   C
#> 20       40.0   C
#> 21       43.0   C
#> 22       43.0   C
#> 23       44.0   C
#> 24       45.0   C
#> 25       34.0   D
#> 26       34.8   D
#> 27       34.8   D
#> 28       35.4   D
#> 29       37.2   D
#> 30       37.8   D
#> 31       41.2   D
#> 32       42.8   D

[4] lists the corresponding test statistics \(A_{akN}^2 = 8.3926\) and \(\sigma_N = 1.2038\) with the p-value \(p = 0.0022\). These match the result of cmstatr within a small margin.

res <- ad_ksample(dat_ss1987, smoothness, lab)

expect_equal(res$ad, 8.3926, tolerance = 0.001)
expect_equal(res$sigma, 1.2038, tolerance = 0.001)
expect_equal(res$p, 0.00226, tolerance = 0.01)

res
#> 
#> Call:
#> ad_ksample(data = dat_ss1987, x = smoothness, groups = lab)
#> 
#> N = 32           k = 4            
#> ADK = 8.39       p-value = 0.002255 
#> Conclusion: Samples do not come from the same distribution (alpha = 0.025 )

7. Comparison with Published Factors

Various factors, such as tolerance limit factors, are published in various publications. This section compares those published factors with those computed by cmstatr.

7.1 Normal kB Factors

B-Basis tolerance limit factors assuming a normal distribution are published in CMH-17-1G. Those factors are reproduced below and are compared with the results of cmstatr. The published factors and those computed by cmstatr are quite similar.

tribble(
  ~n, ~kB_published,
  2, 20.581, 36, 1.725, 70, 1.582, 104, 1.522,
  3, 6.157, 37, 1.718, 71, 1.579, 105, 1.521,
  4, 4.163, 38, 1.711, 72, 1.577, 106, 1.519,
  5, 3.408, 39, 1.704, 73, 1.575, 107, 1.518,
  6, 3.007, 40, 1.698, 74, 1.572, 108, 1.517,
  7, 2.756, 41, 1.692, 75, 1.570, 109, 1.516,
  8, 2.583, 42, 1.686, 76, 1.568, 110, 1.515,
  9, 2.454, 43, 1.680, 77, 1.566, 111, 1.513,
  10, 2.355, 44, 1.675, 78, 1.564, 112, 1.512,
  11, 2.276, 45, 1.669, 79, 1.562, 113, 1.511,
  12, 2.211, 46, 1.664, 80, 1.560, 114, 1.510,
  13, 2.156, 47, 1.660, 81, 1.558, 115, 1.509,
  14, 2.109, 48, 1.655, 82, 1.556, 116, 1.508,
  15, 2.069, 49, 1.650, 83, 1.554, 117, 1.507,
  16, 2.034, 50, 1.646, 84, 1.552, 118, 1.506,
  17, 2.002, 51, 1.642, 85, 1.551, 119, 1.505,
  18, 1.974, 52, 1.638, 86, 1.549, 120, 1.504,
  19, 1.949, 53, 1.634, 87, 1.547, 121, 1.503,
  20, 1.927, 54, 1.630, 88, 1.545, 122, 1.502,
  21, 1.906, 55, 1.626, 89, 1.544, 123, 1.501,
  22, 1.887, 56, 1.623, 90, 1.542, 124, 1.500,
  23, 1.870, 57, 1.619, 91, 1.540, 125, 1.499,
  24, 1.854, 58, 1.616, 92, 1.539, 126, 1.498,
  25, 1.839, 59, 1.613, 93, 1.537, 127, 1.497,
  26, 1.825, 60, 1.609, 94, 1.536, 128, 1.496,
  27, 1.812, 61, 1.606, 95, 1.534, 129, 1.495,
  28, 1.800, 62, 1.603, 96, 1.533, 130, 1.494,
  29, 1.789, 63, 1.600, 97, 1.531, 131, 1.493,
  30, 1.778, 64, 1.597, 98, 1.530, 132, 1.492,
  31, 1.768, 65, 1.595, 99, 1.529, 133, 1.492,
  32, 1.758, 66, 1.592, 100, 1.527, 134, 1.491,
  33, 1.749, 67, 1.589, 101, 1.526, 135, 1.490,
  34, 1.741, 68, 1.587, 102, 1.525, 136, 1.489,
  35, 1.733, 69, 1.584, 103, 1.523, 137, 1.488
) %>%
  arrange(n) %>%
  mutate(kB_cmstatr = k_factor_normal(n, p = 0.9, conf = 0.95)) %>%
  rowwise() %>%
  mutate(diff = expect_equal(kB_published, kB_cmstatr, tolerance = 0.001)) %>%
  select(-c(diff))
#> # A tibble: 136 x 3
#> # Rowwise: 
#>        n kB_published kB_cmstatr
#>    <dbl>        <dbl>      <dbl>
#>  1     2        20.6       20.6 
#>  2     3         6.16       6.16
#>  3     4         4.16       4.16
#>  4     5         3.41       3.41
#>  5     6         3.01       3.01
#>  6     7         2.76       2.76
#>  7     8         2.58       2.58
#>  8     9         2.45       2.45
#>  9    10         2.36       2.35
#> 10    11         2.28       2.28
#> # … with 126 more rows

7.2 Normal kA Factors

A-Basis tolerance limit factors assuming a normal distribution are published in CMH-17-1G. Those factors are reproduced below and are compared with the results of cmstatr. The published factors and those computed by cmstatr are quite similar.

tribble(
  ~n, ~kA_published,
  2, 37.094, 36, 2.983, 70, 2.765, 104, 2.676,
  3, 10.553, 37, 2.972, 71, 2.762, 105, 2.674,
  4, 7.042, 38, 2.961, 72, 2.758, 106, 2.672,
  5, 5.741, 39, 2.951, 73, 2.755, 107, 2.671,
  6, 5.062, 40, 2.941, 74, 2.751, 108, 2.669,
  7, 4.642, 41, 2.932, 75, 2.748, 109, 2.667,
  8, 4.354, 42, 2.923, 76, 2.745, 110, 2.665,
  9, 4.143, 43, 2.914, 77, 2.742, 111, 2.663,
  10, 3.981, 44, 2.906, 78, 2.739, 112, 2.662,
  11, 3.852, 45, 2.898, 79, 2.736, 113, 2.660,
  12, 3.747, 46, 2.890, 80, 2.733, 114, 2.658,
  13, 3.659, 47, 2.883, 81, 2.730, 115, 2.657,
  14, 3.585, 48, 2.876, 82, 2.727, 116, 2.655,
  15, 3.520, 49, 2.869, 83, 2.724, 117, 2.654,
  16, 3.464, 50, 2.862, 84, 2.721, 118, 2.652,
  17, 3.414, 51, 2.856, 85, 2.719, 119, 2.651,
  18, 3.370, 52, 2.850, 86, 2.716, 120, 2.649,
  19, 3.331, 53, 2.844, 87, 2.714, 121, 2.648,
  20, 3.295, 54, 2.838, 88, 2.711, 122, 2.646,
  21, 3.263, 55, 2.833, 89, 2.709, 123, 2.645,
  22, 3.233, 56, 2.827, 90, 2.706, 124, 2.643,
  23, 3.206, 57, 2.822, 91, 2.704, 125, 2.642,
  24, 3.181, 58, 2.817, 92, 2.701, 126, 2.640,
  25, 3.158, 59, 2.812, 93, 2.699, 127, 2.639,
  26, 3.136, 60, 2.807, 94, 2.697, 128, 2.638,
  27, 3.116, 61, 2.802, 95, 2.695, 129, 2.636,
  28, 3.098, 62, 2.798, 96, 2.692, 130, 2.635,
  29, 3.080, 63, 2.793, 97, 2.690, 131, 2.634,
  30, 3.064, 64, 2.789, 98, 2.688, 132, 2.632,
  31, 3.048, 65, 2.785, 99, 2.686, 133, 2.631,
  32, 3.034, 66, 2.781, 100, 2.684, 134, 2.630,
  33, 3.020, 67, 2.777, 101, 2.682, 135, 2.628,
  34, 3.007, 68, 2.773, 102, 2.680, 136, 2.627,
  35, 2.995, 69, 2.769, 103, 2.678, 137, 2.626
) %>%
  arrange(n) %>%
  mutate(kA_cmstatr = k_factor_normal(n, p = 0.99, conf = 0.95)) %>%
  rowwise() %>%
  mutate(diff = expect_equal(kA_published, kA_cmstatr, tolerance = 0.001)) %>%
  select(-c(diff))
#> # A tibble: 136 x 3
#> # Rowwise: 
#>        n kA_published kA_cmstatr
#>    <dbl>        <dbl>      <dbl>
#>  1     2        37.1       37.1 
#>  2     3        10.6       10.6 
#>  3     4         7.04       7.04
#>  4     5         5.74       5.74
#>  5     6         5.06       5.06
#>  6     7         4.64       4.64
#>  7     8         4.35       4.35
#>  8     9         4.14       4.14
#>  9    10         3.98       3.98
#> 10    11         3.85       3.85
#> # … with 126 more rows

7.3 Nonparametric B-Basis Extended Hanson–Koopmans

Vangel [5] provides extensive tables of \(z\) for the case where \(i=1\) and \(j\) is the median observation. This section checks the results of cmstatr’s function against those tables. Only the odd values of \(n\) are checked so that the median is a single observation. The unit tests for the cmstatr package include checks of a variety of values of \(p\) and confidence, but only the factors for B-Basis are checked here.

tribble(
  ~n, ~z,
  3,  28.820048,
  5,  6.1981307,
  7,  3.4780112,
  9,  2.5168762,
  11, 2.0312134,
  13, 1.7377374,
  15, 1.5403989,
  17, 1.3979806,
  19, 1.2899172,
  21, 1.2048089,
  23, 1.1358259,
  25, 1.0786237,
  27, 1.0303046,
) %>%
  rowwise() %>%
  mutate(
    z_calc = hk_ext_z(n, 1, ceiling(n / 2), p = 0.90, conf = 0.95)
  ) %>%
  mutate(diff = expect_equal(z, z_calc, tolerance = 0.0001)) %>% 
  select(-c(diff))
#> # A tibble: 13 x 3
#> # Rowwise: 
#>        n     z z_calc
#>    <dbl> <dbl>  <dbl>
#>  1     3 28.8   28.8 
#>  2     5  6.20   6.20
#>  3     7  3.48   3.48
#>  4     9  2.52   2.52
#>  5    11  2.03   2.03
#>  6    13  1.74   1.74
#>  7    15  1.54   1.54
#>  8    17  1.40   1.40
#>  9    19  1.29   1.29
#> 10    21  1.20   1.20
#> 11    23  1.14   1.14
#> 12    25  1.08   1.08
#> 13    27  1.03   1.03

7.4 Nonparametric A-Basis Extended Hanson–Koopmans

CMH-17-1G provides Table 8.5.15, which contains factors for calculating A-Basis values using the Extended Hanson–Koopmans nonparametric method. That table is reproduced in part here and the factors are compared with those computed by cmstatr. More extensive checks are performed in the unit test of the cmstatr package. The factors computed by cmstatr are very similar to those published in CMH-17-1G.

tribble(
  ~n, ~k,
  2, 80.0038,
  4, 9.49579,
  6, 5.57681,
  8, 4.25011,
  10, 3.57267,
  12, 3.1554,
  14, 2.86924,
  16, 2.65889,
  18, 2.4966,
  20, 2.36683,
  25, 2.131,
  30, 1.96975,
  35, 1.85088,
  40, 1.75868,
  45, 1.68449,
  50, 1.62313,
  60, 1.5267,
  70, 1.45352,
  80, 1.39549,
  90, 1.34796,
  100, 1.30806,
  120, 1.24425,
  140, 1.19491,
  160, 1.15519,
  180, 1.12226,
  200, 1.09434,
  225, 1.06471,
  250, 1.03952,
  275, 1.01773
) %>%
  rowwise() %>%
  mutate(z_calc = hk_ext_z(n, 1, n, 0.99, 0.95)) %>%
  mutate(diff = expect_lt(abs(k - z_calc), 0.0001)) %>% 
  select(-c(diff))
#> # A tibble: 29 x 3
#> # Rowwise: 
#>        n     k z_calc
#>    <dbl> <dbl>  <dbl>
#>  1     2 80.0   80.0 
#>  2     4  9.50   9.50
#>  3     6  5.58   5.58
#>  4     8  4.25   4.25
#>  5    10  3.57   3.57
#>  6    12  3.16   3.16
#>  7    14  2.87   2.87
#>  8    16  2.66   2.66
#>  9    18  2.50   2.50
#> 10    20  2.37   2.37
#> # … with 19 more rows

7.5 Factors for Small Sample Nonparametric B-Basis

CMH-17-1G Table 8.5.14 provides ranks orders and factors for computing nonparametric B-Basis values. This table is reproduced below and compared with the results of cmstatr. The results are similar. In some cases, the rank order (\(r\) in CMH-17-1G or \(j\) in cmstatr) and the the factor (\(k\)) are different. These differences are discussed in detail in the vignette Extended Hanson-Koopmans.

tribble(
  ~n, ~r, ~k,
  2, 2, 35.177,
  3, 3, 7.859,
  4, 4, 4.505,
  5, 4, 4.101,
  6, 5, 3.064,
  7, 5, 2.858,
  8, 6, 2.382,
  9, 6, 2.253,
  10, 6, 2.137,
  11, 7, 1.897,
  12, 7, 1.814,
  13, 7, 1.738,
  14, 8, 1.599,
  15, 8, 1.540,
  16, 8, 1.485,
  17, 8, 1.434,
  18, 9, 1.354,
  19, 9, 1.311,
  20, 10, 1.253,
  21, 10, 1.218,
  22, 10, 1.184,
  23, 11, 1.143,
  24, 11, 1.114,
  25, 11, 1.087,
  26, 11, 1.060,
  27, 11, 1.035,
  28, 12, 1.010
) %>% 
  rowwise() %>%
  mutate(r_calc = hk_ext_z_j_opt(n, 0.90, 0.95)$j) %>%
  mutate(k_calc = hk_ext_z_j_opt(n, 0.90, 0.95)$z)
#> # A tibble: 27 x 5
#> # Rowwise: 
#>        n     r     k r_calc k_calc
#>    <dbl> <dbl> <dbl>  <int>  <dbl>
#>  1     2     2 35.2       2  35.2 
#>  2     3     3  7.86      3   7.86
#>  3     4     4  4.50      4   4.51
#>  4     5     4  4.10      4   4.10
#>  5     6     5  3.06      5   3.06
#>  6     7     5  2.86      5   2.86
#>  7     8     6  2.38      6   2.38
#>  8     9     6  2.25      6   2.25
#>  9    10     6  2.14      6   2.14
#> 10    11     7  1.90      7   1.90
#> # … with 17 more rows

7.6 Nonparametric B-Basis Binomial Rank

CMH-17-1G Table 8.5.12 provides factors for computing B-Basis values using the nonparametric binomial rank method. Part of that table is reproduced below and compared with the results of cmstatr. The results of cmstatr are similar to the published values. A more complete comparison is performed in the units tests of the cmstatr package.

tribble(
  ~n, ~rb,
  29, 1,
  46, 2,
  61, 3,
  76, 4,
  89, 5,
  103, 6,
  116, 7,
  129, 8,
  142, 9,
  154, 10,
  167, 11,
  179, 12,
  191, 13,
  203, 14
) %>%
  rowwise() %>%
  mutate(r_calc = nonpara_binomial_rank(n, 0.9, 0.95)) %>%
  mutate(test = expect_equal(rb, r_calc)) %>%
  select(-c(test))
#> # A tibble: 14 x 3
#> # Rowwise: 
#>        n    rb r_calc
#>    <dbl> <dbl>  <dbl>
#>  1    29     1      1
#>  2    46     2      2
#>  3    61     3      3
#>  4    76     4      4
#>  5    89     5      5
#>  6   103     6      6
#>  7   116     7      7
#>  8   129     8      8
#>  9   142     9      9
#> 10   154    10     10
#> 11   167    11     11
#> 12   179    12     12
#> 13   191    13     13
#> 14   203    14     14

7.7 Nonparametric A-Basis Binomial Rank

CMH-17-1G Table 8.5.13 provides factors for computing B-Basis values using the nonparametric binomial rank method. Part of that table is reproduced below and compared with the results of cmstatr. The results of cmstatr are similar to the published values. A more complete comparison is performed in the units tests of the cmstatr package.

tribble(
  ~n, ~ra,
  299, 1,
  473, 2,
  628, 3,
  773, 4,
  913, 5
) %>%
  rowwise() %>%
  mutate(r_calc = nonpara_binomial_rank(n, 0.99, 0.95)) %>%
  mutate(test = expect_equal(ra, r_calc)) %>%
  select(-c(test))
#> # A tibble: 5 x 3
#> # Rowwise: 
#>       n    ra r_calc
#>   <dbl> <dbl>  <dbl>
#> 1   299     1      1
#> 2   473     2      2
#> 3   628     3      3
#> 4   773     4      4
#> 5   913     5      5

7.8 Factors for Equivalency

Vangel’s 2002 paper provides factors for calculating limits for sample mean and sample extremum for various values of \(\alpha\) and sample size (\(n\)). A subset of those factors are reproduced below and compared with results from cmstatr. The results are very similar for values of \(\alpha\) and \(n\) that are common for composite materials.

read.csv(system.file("extdata", "k1.vangel.csv",
                     package = "cmstatr")) %>%
  gather(n, k1, X2:X10) %>%
  mutate(n = as.numeric(substring(n, 2))) %>%
  inner_join(
    read.csv(system.file("extdata", "k2.vangel.csv",
                         package = "cmstatr")) %>%
      gather(n, k2, X2:X10) %>%
      mutate(n = as.numeric(substring(n, 2))),
    by = c("n" = "n", "alpha" = "alpha")
  ) %>%
  filter(n >= 5 & (alpha == 0.01 | alpha == 0.05)) %>% 
  group_by(n, alpha) %>%
  nest() %>%
  mutate(equiv = map2(alpha, n, ~k_equiv(.x, .y))) %>%
  mutate(k1_calc = map(equiv, function(e) e[1]),
         k2_calc = map(equiv, function(e) e[2])) %>%
  select(-c(equiv)) %>% 
  unnest(cols = c(data, k1_calc, k2_calc)) %>% 
  mutate(check = expect_equal(k1, k1_calc, tolerance = 0.0001)) %>%
  select(-c(check)) %>% 
  mutate(check = expect_equal(k2, k2_calc, tolerance = 0.0001)) %>%
  select(-c(check))
#> # A tibble: 12 x 6
#> # Groups:   alpha, n [12]
#>    alpha     n    k1    k2 k1_calc k2_calc
#>    <dbl> <dbl> <dbl> <dbl>   <dbl>   <dbl>
#>  1  0.05     5  2.53 0.852    2.53   0.853
#>  2  0.01     5  3.07 1.14     3.07   1.14 
#>  3  0.05     6  2.60 0.781    2.60   0.781
#>  4  0.01     6  3.13 1.04     3.13   1.04 
#>  5  0.05     7  2.65 0.725    2.65   0.725
#>  6  0.01     7  3.18 0.968    3.18   0.968
#>  7  0.05     8  2.7  0.679    2.70   0.679
#>  8  0.01     8  3.22 0.906    3.22   0.906
#>  9  0.05     9  2.74 0.641    2.74   0.641
#> 10  0.01     9  3.25 0.854    3.25   0.855
#> 11  0.05    10  2.78 0.609    2.78   0.609
#> 12  0.01    10  3.28 0.811    3.28   0.811

8. Session Info

This copy of this vignette was build on the following system.

sessionInfo()
#> R version 4.1.0 (2021-05-18)
#> Platform: x86_64-apple-darwin17.0 (64-bit)
#> Running under: macOS Catalina 10.15.7
#> 
#> Matrix products: default
#> BLAS:   /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib
#> 
#> locale:
#> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] testthat_3.0.4 tidyr_1.1.3    purrr_0.3.4    dplyr_1.0.7    cmstatr_0.9.0 
#> 
#> loaded via a namespace (and not attached):
#>  [1] compiler_4.1.0    pillar_1.6.1      tools_4.1.0       kSamples_1.2-9   
#>  [5] pkgload_1.2.1     digest_0.6.27     evaluate_0.14     memoise_2.0.0    
#>  [9] lifecycle_1.0.0   tibble_3.1.2      gtable_0.3.0      pkgconfig_2.0.3  
#> [13] rlang_0.4.11      cli_3.0.0         yaml_2.2.1        pkgdown_1.6.1    
#> [17] xfun_0.24         fastmap_1.1.0     withr_2.4.2       stringr_1.4.0    
#> [21] knitr_1.33        SuppDists_1.1-9.5 generics_0.1.0    desc_1.3.0       
#> [25] fs_1.5.0          vctrs_0.3.8       systemfonts_1.0.2 rprojroot_2.0.2  
#> [29] grid_4.1.0        tidyselect_1.1.1  glue_1.4.2        R6_2.5.0         
#> [33] textshaping_0.3.5 fansi_0.5.0       rmarkdown_2.9     waldo_0.2.5      
#> [37] ggplot2_3.3.5     magrittr_2.0.1    scales_1.1.1      htmltools_0.5.1.1
#> [41] ellipsis_0.3.2    MASS_7.3-54       colorspace_2.0-2  ragg_1.1.3       
#> [45] utf8_1.2.1        stringi_1.6.2     munsell_0.5.0     cachem_1.0.5     
#> [49] crayon_1.4.1

9. References

[1] K. S. Raju and J. S. Tomblin, “AGATE Statistical Analysis Program,” Wichita State University, ASAP-2008, 2008.

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

[3] Materials Sciences Corporation, “CMH-17 Statistical Analysis for B-Basis and A-Basis Values,” Materials Sciences Corporation, Horsham, PA, STAT-17 Rev 5, Jan. 2008.

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

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