carbon.fabric
Data Set
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.
library(cmstatr)
library(dplyr)
library(purrr)
library(tidyr)
library(testthat)
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"
.
The following table provides a crossreference between the various functions of the cmstatr
package and the tests shown within this vignette. The sections in this vignette are organized by data set. Not all checks are performed on all data sets.
Function  Tests 

ad_ksample() 
Section 3.1, Section 4.1.2, Section 6.1 
anderson_darling_normal() 
Section 4.1.3, Section 5.1 
anderson_darling_lognormal() 
Section 4.1.3, Section 5.2 
anderson_darling_weibull() 
Section 4.1.3, Section 5.3 
basis_normal() 
Section 5.4 
basis_lognormal() 
Section 5.5 
basis_weibull() 
Section 5.6 
basis_pooled_cv() 
Section 4.2.3, Section 4.2.4, 
basis_pooled_sd() 
Section 4.2.1, Section 4.2.2 
basis_hk_ext() 
Section 4.1.6, Section 5.7, Section 5.8 
basis_nonpara_large_sample() 
Section 5.9 
basis_anova() 
Section 4.1.7 
calc_cv_star() 

cv() 

equiv_change_mean() 
Section 5.11 
equiv_mean_extremum() 
Section 5.10 
hk_ext_z() 
Section 7.3, Section 7.4 
hk_ext_z_j_opt() 
Section 7.5 
k_equiv() 
Section 7.8 
k_factor_normal() 
Section 7.1, Section 7.2 
levene_test() 
Section 4.1.4, Section 4.1.5 
maximum_normed_residual() 
Section 4.1.1 
nonpara_binomial_rank() 
Section 7.6, Section 7.7 
normalize_group_mean() 

normalize_ply_thickness() 

transform_mod_cv_ad() 

transform_mod_cv() 
carbon.fabric
Data SetThis 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 WTRTD11 WT RTD 1 137.4438
#> 2 WTRTD12 WT RTD 1 139.5395
#> 3 WTRTD13 WT RTD 1 150.8900
#> 4 WTRTD14 WT RTD 1 141.4474
#> 5 WTRTD15 WT RTD 1 141.8203
#> 6 WTRTD16 WT RTD 1 151.8821
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 warptension 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.
< carbon.fabric %>%
res 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$reject_same_dist)
res#>
#> Call:
#> ad_ksample(data = ., x = strength, groups = batch)
#>
#> N = 18 k = 3
#> ADK = 0.912 pvalue = 0.95989
#> Conclusion: Samples come from the same distribution ( alpha = 0.025 )
When the ETW warptension data from this data set are entered into ASAP 2008, the reported test statistic is 1.604 and it fails to reject the null hypothesis that the batches are drawn from the same distribution. Adjusting for the different definition of the test statistic, cmstatr
gives nearly identical results.
< carbon.fabric %>%
res filter(test == "WT") %>%
filter(condition == "ETW") %>%
ad_ksample(strength, batch)
expect_equal(res$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 pvalue = 0.10208
#> Conclusion: Samples come from the same distribution ( alpha = 0.025 )
CMH171G [2] provides an example data set and results from ASAP [1] and STAT17 [3]. This example data set is duplicated below:
< tribble(
dat_8_3_11_1_1 ~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 × 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
CMH171G 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 CMH171G.
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.
< dat_8_3_11_1_1 %>%
res 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$n_outliers, 0)
res#>
#> Call:
#> maximum_normed_residual(data = ., x = strength, alpha = 0.05)
#>
#> MNR = 2.008274 ( critical value = 2.126645 )
#>
#> No outliers detected ( alpha = 0.05 )
Similarly, for Batch 3 of the ETW data, the results of cmstatr
match the results published in the handbook within a small tolerance. The published test statistic is 2.119
< dat_8_3_11_1_1 %>%
res filter(condition == "ETW" & batch == 3) %>%
maximum_normed_residual(strength, alpha = 0.05)
expect_equal(res$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
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.
< dat_8_3_11_1_1 %>%
res 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 pvalue = 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
< dat_8_3_11_1_1 %>%
res 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 pvalue = 0.0042703
#> Conclusion: Samples do not come from the same distribution (alpha = 0.025 )
CMH171G 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 
< dat_8_3_11_1_1 %>%
res filter(condition == "ETW") %>%
anderson_darling_normal(strength)
expect_equal(res$osl, 0.006051, tolerance = 0.001)
res#>
#> Call:
#> anderson_darling_normal(data = ., x = strength)
#>
#> Distribution: Normal ( n = 22 )
#> Test statistic: A = 1.052184
#> OSL (pvalue): 0.006051441 (assuming unknown parameters)
#> Conclusion: Sample is not drawn from a Normal distribution ( alpha = 0.05 )
< dat_8_3_11_1_1 %>%
res filter(condition == "ETW") %>%
anderson_darling_lognormal(strength)
expect_equal(res$osl, 0.000307, tolerance = 0.001)
res#>
#> Call:
#> anderson_darling_lognormal(data = ., x = strength)
#>
#> Distribution: Lognormal ( n = 22 )
#> Test statistic: A = 1.568825
#> OSL (pvalue): 0.0003073592 (assuming unknown parameters)
#> Conclusion: Sample is not drawn from a Lognormal distribution ( alpha = 0.05 )
< dat_8_3_11_1_1 %>%
res filter(condition == "ETW") %>%
anderson_darling_weibull(strength)
expect_equal(res$osl, 0.0219, tolerance = 0.002)
res#>
#> Call:
#> anderson_darling_weibull(data = ., x = strength)
#>
#> Distribution: Weibull ( n = 22 )
#> Test statistic: A = 0.8630665
#> OSL (pvalue): 0.02186889 (assuming unknown parameters)
#> Conclusion: Sample is not drawn from a Weibull distribution ( alpha = 0.05 )
CMH171G Section 8.3.11.1.1 provides results from ASAP for Levene’s test for equality of variance between conditions after the ETW and ETW2 conditions are removed. The handbook shows an F statistic of 0.58, however if this data is entered into ASAP directly, ASAP gives an F statistic of 0.058, which matches the result of cmstatr
.
< dat_8_3_11_1_1 %>%
res filter(condition != "ETW" & condition != "ETW2") %>%
levene_test(strength, condition)
expect_equal(res$f, 0.058, tolerance = 0.01)
res#>
#> Call:
#> levene_test(data = ., x = strength, groups = condition)
#>
#> n = 60 k = 3
#> F = 0.05811631 pvalue = 0.943596
#> Conclusion: Samples have equal variances ( alpha = 0.05 )
CMH171G 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.
< dat_8_3_11_1_1 %>%
res 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 pvalue = 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.
< dat_8_3_11_1_1 %>%
res filter(condition == "CTD") %>%
levene_test(strength, batch)
expect_equal(res$f, 3.850, tolerance = 0.005)
res#>
#> Call:
#> levene_test(data = ., x = strength, groups = batch)
#>
#> n = 19 k = 3
#> F = 3.852032 pvalue = 0.04309008
#> Conclusion: Samples have unequal variance ( alpha = 0.05 )
CMH171G Section 8.3.11.2.1 provides STAT17 outputs for the ETW condition of the present data set. The nonparametric Basis values are listed. In this case, the Hanson–Koopmans method is used. The published ABasis value is 13.0 and the BBasis is 37.9.
< dat_8_3_11_1_1 %>%
res filter(condition == "ETW") %>%
basis_hk_ext(strength, method = "woodwardfrawley", p = 0.99, conf = 0.95,
override = "all")
expect_equal(res$basis, 13.0, tolerance = 0.001)
res#>
#> Call:
#> basis_hk_ext(data = ., x = strength, p = 0.99, conf = 0.95, method = "woodwardfrawley",
#> override = "all")
#>
#> Distribution: Nonparametric (Extended HansonKoopmans, WoodwardFrawley method) ( n = 22 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `correct_method_used`,
#> `sample_size`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> 12.99614
< dat_8_3_11_1_1 %>%
res filter(condition == "ETW") %>%
basis_hk_ext(strength, method = "optimumorder", p = 0.90, conf = 0.95,
override = "all")
expect_equal(res$basis, 37.9, tolerance = 0.001)
res#>
#> Call:
#> basis_hk_ext(data = ., x = strength, p = 0.9, conf = 0.95, method = "optimumorder",
#> override = "all")
#>
#> Distribution: Nonparametric (Extended HansonKoopmans, optimum twoorderstatistic method) ( n = 22 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `correct_method_used`,
#> `sample_size`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 37.88511
CMH171G Section 8.3.11.2.2 provides output from STAT17 for the ETW2 condition from the present data set. STAT17 reports A and BBasis values based on the ANOVA method of 34.6 and 63.2, respectively. The results from cmstatr
are similar.
< dat_8_3_11_1_1 %>%
res filter(condition == "ETW2") %>%
basis_anova(strength, batch, override = "number_of_groups",
p = 0.99, conf = 0.95)
expect_equal(res$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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> 34.57763
< dat_8_3_11_1_1 %>%
res 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`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 63.20276
[2] provides an example data set and results from ASAP [1]. This example data set is duplicated below:
< tribble(
dat_8_3_11_1_2 ~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"
)
CMH171G 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 

BBasis  93.64  87.30  54.33  47.12 
ABasis  89.19  79.86  46.84  39.69 
< basis_pooled_sd(dat_8_3_11_1_2, strength, condition,
res 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`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> CTD 93.63504
#> ETW 54.32706
#> ETW2 47.07669
#> RTD 87.29555
< basis_pooled_sd(dat_8_3_11_1_2, strength, condition,
res 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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> CTD 86.19301
#> ETW 46.84112
#> ETW2 39.65214
#> RTD 79.86205
After removal of the ETW2 condition, CMH17STATS reports the pooled A and BBasis (mod CV) shown in the following table. cmstatr
computes very similar values.
Condition  CTD  RTD  ETW 

BBasis  92.25  85.91  52.97 
ABasis  83.81  77.48  44.47 
< dat_8_3_11_1_2 %>%
res 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`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> CTD 92.24927
#> ETW 52.9649
#> RTD 85.90454
< dat_8_3_11_1_2 %>%
res 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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> CTD 83.80981
#> ETW 44.47038
#> RTD 77.47589
This data set was input into CMH17STATS and the Pooled CV method was selected. The results from CMH17STATS were as follows. cmstatr
produces very similar results.
Condition  CTD  RTD  ETW  ETW2 

BBasis  90.89  85.37  56.79  50.55 
ABasis  81.62  76.67  50.98  45.40 
< basis_pooled_cv(dat_8_3_11_1_2, strength, condition, override = "all")
res 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`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> CTD 90.88018
#> ETW 56.78337
#> ETW2 50.54406
#> RTD 85.36756
< basis_pooled_cv(dat_8_3_11_1_2, strength, condition,
res 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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> CTD 81.60931
#> ETW 50.97801
#> ETW2 45.39158
#> RTD 76.66214
This data set was input into CMH17STATS and the Pooled CV method was selected with the modified CV transform. Additionally, the ETW2 condition was removed. The results from CMH17STATS were as follows. cmstatr
produces very similar results.
Condition  CTD  RTD  ETW 

BBasis  90.31  84.83  56.43 
ABasis  80.57  75.69  50.33 
< dat_8_3_11_1_2 %>%
res 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`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> CTD 90.30646
#> ETW 56.42786
#> RTD 84.82748
< dat_8_3_11_1_2 %>%
res 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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> CTD 80.56529
#> ETW 50.32422
#> RTD 75.6817
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.
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.
< data.frame(
dat 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
)
)< anderson_darling_normal(dat, strength)
res
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 (pvalue): 0.4648132 (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Normal distribution ( alpha = 0.05 )
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.
< data.frame(
dat 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
)
)
< anderson_darling_lognormal(dat, strength)
res
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 (pvalue): 0.4798148 (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Lognormal distribution ( alpha = 0.05 )
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.
< data.frame(
dat 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
)
)
< anderson_darling_weibull(dat, strength)
res
expect_equal(res$osl, 0.179, tolerance = 0.002)
res#>
#> Call:
#> anderson_darling_weibull(data = dat, x = strength)
#>
#> Distribution: Weibull ( n = 18 )
#> Test statistic: A = 0.5113909
#> OSL (pvalue): 0.1787882 (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Weibull distribution ( alpha = 0.05 )
The following data was input into STAT17 and the A and BBasis values were computed assuming normally distributed data. The results were 120.336 and 129.287, respectively. cmstatr
reports very similar values.
< c(
dat 137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
137.4732, 152.3380, 144.1589, 128.5218
)
< basis_normal(x = dat, p = 0.99, conf = 0.95, override = "all")
res expect_equal(res$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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> 120.3549
< basis_normal(x = dat, p = 0.9, conf = 0.95, override = "all")
res expect_equal(res$basis, 129.287, tolerance = 0.0005)
res#>
#> Call:
#> basis_normal(x = dat, p = 0.9, conf = 0.95, override = "all")
#>
#> Distribution: Normal ( n = 18 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `anderson_darling_normal`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 129.2898
The following data was input into STAT17 and the A and BBasis values were computed assuming distributed according to a lognormal distribution. The results were 121.710 and 129.664, respectively. cmstatr
reports very similar values.
< c(
dat 137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
137.4732, 152.3380, 144.1589, 128.5218
)
< basis_lognormal(x = dat, p = 0.99, conf = 0.95, override = "all")
res expect_equal(res$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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> 121.7265
< basis_lognormal(x = dat, p = 0.9, conf = 0.95, override = "all")
res expect_equal(res$basis, 129.664, tolerance = 0.0005)
res#>
#> Call:
#> basis_lognormal(x = dat, p = 0.9, conf = 0.95, override = "all")
#>
#> Distribution: Lognormal ( n = 18 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `anderson_darling_lognormal`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 129.667
The following data was input into STAT17 and the A and BBasis values were computed assuming data following the Weibull distribution. The results were 109.150 and 125.441, respectively. cmstatr
reports very similar values.
< c(
dat 137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
137.4732, 152.3380, 144.1589, 128.5218
)
< basis_weibull(x = dat, p = 0.99, conf = 0.95, override = "all")
res expect_equal(res$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`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> 109.651
< basis_weibull(x = dat, p = 0.9, conf = 0.95, override = "all")
res expect_equal(res$basis, 125.441, tolerance = 0.005)
res#>
#> Call:
#> basis_weibull(x = dat, p = 0.9, conf = 0.95, override = "all")
#>
#> Distribution: Weibull ( n = 18 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `anderson_darling_weibull`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 125.7338
The following data was input into STAT17 and the A and BBasis values were computed using the nonparametric (small sample) method. The results were 99.651 and 124.156, respectively. cmstatr
reports very similar values.
< c(
dat 137.4438, 139.5395, 150.8900, 141.4474, 141.8203, 151.8821, 143.9245,
132.9732, 136.6419, 138.1723, 148.7668, 143.2830, 143.5429, 141.7023,
137.4732, 152.3380, 144.1589, 128.5218
)
< basis_hk_ext(x = dat, p = 0.99, conf = 0.95,
res method = "woodwardfrawley", override = "all")
expect_equal(res$basis, 99.651, tolerance = 0.005)
res#>
#> Call:
#> basis_hk_ext(x = dat, p = 0.99, conf = 0.95, method = "woodwardfrawley",
#> override = "all")
#>
#> Distribution: Nonparametric (Extended HansonKoopmans, WoodwardFrawley method) ( n = 18 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `correct_method_used`,
#> `sample_size`
#> ABasis: ( p = 0.99 , conf = 0.95 )
#> 99.65098
< basis_hk_ext(x = dat, p = 0.9, conf = 0.95,
res method = "optimumorder", override = "all")
expect_equal(res$basis, 124.156, tolerance = 0.005)
res#>
#> Call:
#> basis_hk_ext(x = dat, p = 0.9, conf = 0.95, method = "optimumorder",
#> override = "all")
#>
#> Distribution: Nonparametric (Extended HansonKoopmans, optimum twoorderstatistic method) ( n = 18 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `correct_method_used`,
#> `sample_size`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 124.1615
The following random numbers were generated.
< c(
dat 139.6734, 143.0032, 130.4757, 144.8327, 138.7818, 136.7693, 148.636,
131.0095, 131.4933, 142.8856, 158.0198, 145.2271, 137.5991, 139.8298,
140.8557, 137.6148, 131.3614, 152.7795, 145.8792, 152.9207, 160.0989,
145.1920, 128.6383, 141.5992, 122.5297, 159.8209, 151.6720, 159.0156
)
All of the numbers above were input into STAT17 and the reported BBasis value using the Optimum Order nonparametric method was 122.36798. This result matches the results of cmstatr
within a small margin.
< basis_hk_ext(x = dat, p = 0.9, conf = 0.95,
res method = "optimumorder", override = "all")
expect_equal(res$basis, 122.36798, tolerance = 0.001)
res#>
#> Call:
#> basis_hk_ext(x = dat, p = 0.9, conf = 0.95, method = "optimumorder",
#> override = "all")
#>
#> Distribution: Nonparametric (Extended HansonKoopmans, optimum twoorderstatistic method) ( n = 28 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `correct_method_used`,
#> `sample_size`
#> BBasis: ( 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 BBasis value of 121.57073 using the Optimum Order nonparametric method. cmstatr
reports a very similar number.
< basis_hk_ext(x = head(dat, 26), p = 0.9, conf = 0.95,
res method = "optimumorder", override = "all")
expect_equal(res$basis, 121.57073, tolerance = 0.001)
res#>
#> Call:
#> basis_hk_ext(x = head(dat, 26), p = 0.9, conf = 0.95, method = "optimumorder",
#> override = "all")
#>
#> Distribution: Nonparametric (Extended HansonKoopmans, optimum twoorderstatistic method) ( n = 26 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `correct_method_used`,
#> `sample_size`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 121.5655
The same data set was further reduced such that only the first 22 observations were included. This smaller data set was input into STAT17 and that software calculated a BBasis value of 128.82397 using the Optimum Order nonparametric method. cmstatr
reports a very similar number.
< basis_hk_ext(x = head(dat, 22), p = 0.9, conf = 0.95,
res method = "optimumorder", override = "all")
expect_equal(res$basis, 128.82397, tolerance = 0.001)
res#>
#> Call:
#> basis_hk_ext(x = head(dat, 22), p = 0.9, conf = 0.95, method = "optimumorder",
#> override = "all")
#>
#> Distribution: Nonparametric (Extended HansonKoopmans, optimum twoorderstatistic method) ( n = 22 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `correct_method_used`,
#> `sample_size`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 128.8224
The following data was input into STAT17 and the BBasis value was computed using the nonparametric (large sample) method. The results was 122.738297. cmstatr
reports very similar values.
< c(
dat 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
)
< basis_nonpara_large_sample(x = dat, p = 0.9, conf = 0.95,
res override = "all")
expect_equal(res$basis, 122.738297, tolerance = 0.005)
res#>
#> Call:
#> basis_nonpara_large_sample(x = dat, p = 0.9, conf = 0.95, override = "all")
#>
#> Distribution: Nonparametric (large sample) ( n = 61 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`,
#> `outliers`,
#> `sample_size`
#> BBasis: ( p = 0.9 , conf = 0.95 )
#> 122.7383
Results from cmstatr
’s equiv_mean_extremum
function were compared with results from HYTEQ. The summary statistics for the qualification data were set as mean = 141.310
and sd=6.415
. For a value of alpha=0.05
and n = 9
, HYTEQ reported thresholds of 123.725 and 137.197 for minimum individual and mean, respectively. cmstatr
produces very similar results.
< equiv_mean_extremum(alpha = 0.05, mean_qual = 141.310, sd_qual = 6.415,
res n_sample = 9)
expect_equal(res$threshold_min_indiv, 123.725, tolerance = 0.001)
expect_equal(res$threshold_mean, 137.197, tolerance = 0.001)
res#>
#> Call:
#> equiv_mean_extremum(mean_qual = 141.31, sd_qual = 6.415, n_sample = 9,
#> alpha = 0.05)
#>
#> For alpha = 0.05 and n = 9
#> ( k1 = 2.741054 and k2 = 0.6410852 )
#> Min Individual Sample Mean
#> Thresholds: 123.7261 137.1974
Using the same parameters, but using the modified CV method, HYTEQ produces thresholds of 117.024 and 135.630 for minimum individual and mean, respectively. cmstatr
produces very similar results.
< equiv_mean_extremum(alpha = 0.05, mean_qual = 141.310, sd_qual = 6.415,
res n_sample = 9, modcv = TRUE)
expect_equal(res$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
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.
< equiv_change_mean(alpha = 0.05, n_sample = 9, mean_sample = 9.02,
res sd_sample = 0.15785, n_qual = 28, mean_qual = 9.24,
sd_qual = 0.162)
expect_equal(res$threshold, c(9.115, 9.365), tolerance = 0.001)
res#>
#> Call:
#> equiv_change_mean(n_qual = 28, mean_qual = 9.24, sd_qual = 0.162,
#> n_sample = 9, mean_sample = 9.02, sd_sample = 0.15785, alpha = 0.05)
#>
#> For alpha = 0.05
#> Qualification Sample
#> Number 28 9
#> Mean 9.24 9.02
#> SD 0.162 0.15785
#> Result FAIL
#> Passing Range 9.114712 to 9.365288
After selecting the modified CV method, HYTEQ gives an acceptance range of 8.857 to 9.623. cmstatr
produces similar results.
< equiv_change_mean(alpha = 0.05, n_sample = 9, mean_sample = 9.02,
res sd_sample = 0.15785, n_qual = 28, mean_qual = 9.24,
sd_qual = 0.162, modcv = TRUE)
expect_equal(res$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
In this section, results from cmstatr
are compared with values published in literature.
[4] provides example data that compares measurements obtained in four labs. Their paper gives values of the ADK test statistic as well as pvalues.
The data in [4] is as follows:
< data.frame(
dat_ss1987 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 pvalue \(p = 0.0022\). These match the result of cmstatr
within a small margin.
< ad_ksample(dat_ss1987, smoothness, lab)
res
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 pvalue = 0.002255
#> Conclusion: Samples do not come from the same distribution (alpha = 0.025 )
Various factors, such as tolerance limit factors, are published in various publications. This section compares those published factors with those computed by cmstatr
.
BBasis tolerance limit factors assuming a normal distribution are published in CMH171G. 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 × 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
ABasis tolerance limit factors assuming a normal distribution are published in CMH171G. 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 × 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
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 BBasis 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 × 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
CMH171G provides Table 8.5.15, which contains factors for calculating ABasis 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 CMH171G.
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 × 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
CMH171G Table 8.5.14 provides ranks orders and factors for computing nonparametric BBasis 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 CMH171G or \(j\) in cmstatr
) and the the factor (\(k\)) are different. These differences are discussed in detail in the vignette Extended HansonKoopmans.
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 × 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
CMH171G Table 8.5.12 provides factors for computing BBasis 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 × 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
CMH171G Table 8.5.13 provides factors for computing BBasis 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 × 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
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 × 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
This copy of this vignette was build on the following system.
sessionInfo()
#> R version 4.1.1 (20210810)
#> Platform: x86_64pclinuxgnu (64bit)
#> Running under: Ubuntu 20.04.3 LTS
#>
#> Matrix products: default
#> BLAS: /usr/lib/x86_64linuxgnu/blas/libblas.so.3.9.0
#> LAPACK: /usr/lib/x86_64linuxgnu/lapack/liblapack.so.3.9.0
#>
#> locale:
#> [1] LC_CTYPE=en_CA.UTF8 LC_NUMERIC=C
#> [3] LC_TIME=en_CA.UTF8 LC_COLLATE=C
#> [5] LC_MONETARY=en_CA.UTF8 LC_MESSAGES=en_CA.UTF8
#> [7] LC_PAPER=en_CA.UTF8 LC_NAME=C
#> [9] LC_ADDRESS=C LC_TELEPHONE=C
#> [11] LC_MEASUREMENT=en_CA.UTF8 LC_IDENTIFICATION=C
#>
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#>
#> other attached packages:
#> [1] testthat_3.0.4 purrr_0.3.4 cmstatr_0.9.1 tidyr_1.1.4 ggplot2_3.3.5
#> [6] dplyr_1.0.7
#>
#> loaded via a namespace (and not attached):
#> [1] highr_0.9 pillar_1.6.3 bslib_0.3.0 compiler_4.1.1
#> [5] jquerylib_0.1.4 kSamples_1.29 tools_4.1.1 pkgload_1.2.2
#> [9] digest_0.6.28 viridisLite_0.4.0 jsonlite_1.7.2 evaluate_0.14
#> [13] lifecycle_1.0.1 tibble_3.1.4 gtable_0.3.0 pkgconfig_2.0.3
#> [17] rlang_0.4.11.9001 cli_3.0.1 DBI_1.1.1 rstudioapi_0.13
#> [21] yaml_2.2.1 xfun_0.26 fastmap_1.1.0 withr_2.4.2
#> [25] stringr_1.4.0 knitr_1.35 desc_1.4.0 SuppDists_1.19.5
#> [29] generics_0.1.0 vctrs_0.3.8 sass_0.4.0 rprojroot_2.0.2
#> [33] grid_4.1.1 tidyselect_1.1.1 glue_1.4.2 R6_2.5.1
#> [37] fansi_0.5.0 rmarkdown_2.11 waldo_0.3.1 farver_2.1.0
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