`cmstatr`

is an R package for analyzing composite material
data for use in the aerospace industry. The statistical methods are
based on those published in CMH-17-1G. This package is intended to
facilitate reproducible statistical analysis of composite materials. In
this tutorial, we’ll explore the basic functionality of
`cmstatr`

.

Before we can actually use the package, we’ll need to load it. We’ll
also load the `dplyr`

package, which we’ll talk about
shortly. There are also a few other packages that we’ll load. These
could all be loaded by loading the `tidyverse`

package
instead.

`cmstatr`

is built with the assumption that the data is in
(so called) tidy
data format. This means that the data is in a data frame and that
each observation (i.e. test result) has its own row and that each
variable has its own column. Included in this package is a sample
composite material data set (this data set is fictional: don’t use it
for anything other than learning this package). The data set
`carbon.fabric.2`

has the expected format. We’ll just show
the first 10 rows of the data for now.

```
carbon.fabric.2 %>%
head(10)
#> test condition batch panel thickness nplies strength modulus failure_mode
#> 1 WT CTD A 1 0.112 14 142.817 9.285 LAT
#> 2 WT CTD A 1 0.113 14 135.901 9.133 LAT
#> 3 WT CTD A 1 0.113 14 132.511 9.253 LAT
#> 4 WT CTD A 2 0.112 14 135.586 9.150 LAB
#> 5 WT CTD A 2 0.113 14 125.145 9.270 LAB
#> 6 WT CTD A 2 0.113 14 135.203 9.189 LGM
#> 7 WT CTD A 2 0.113 14 128.547 9.088 LAB
#> 8 WT CTD B 1 0.113 14 127.709 9.199 LGM
#> 9 WT CTD B 1 0.113 14 127.074 9.058 LGM
#> 10 WT CTD B 1 0.114 14 126.879 9.306 LGM
```

If your data set is not yet in this type of format (note: that the
column names *do not* need to match the column names in the
example), there are many ways to get it into this format. One of the
easier ways of doing so is to use the `tidyr`

package. The
use of this package is outside the scope of this vignette.

Throughout this vignette, we will be using some of the
`tidyverse`

tools for working with data. There are several
ways to work with data in R, but in the opinion of the author of this
vignette, the `tidyverse`

provides the easiest way to do so.
As such, this is the approach used in this vignette. Feel free to use
whichever approach works best for you.

Very often, you’ll want to normalize as-measured strength data to a
nominal cured ply thickness for fiber-dominated properties. Very often,
this will reduce the apparent variance in the data. The
`normalize_ply_thickness`

function can be used to normalize
strength or modulus data to a certain cured ply thickness. This function
takes three arguments: the value to normalize (i.e.. strength or
modulus), the measured thickness and the nominal thickness. In our case,
the nominal cured ply thickness of the material is \(0.0079\). We can then normalize the
warp-tension and fill-compression data as follows:

```
norm_data <- carbon.fabric.2 %>%
filter(test == "WT" | test == "FC") %>%
mutate(strength.norm = normalize_ply_thickness(strength,
thickness / nplies,
0.0079))
norm_data %>%
head(10)
#> test condition batch panel thickness nplies strength modulus failure_mode
#> 1 WT CTD A 1 0.112 14 142.817 9.285 LAT
#> 2 WT CTD A 1 0.113 14 135.901 9.133 LAT
#> 3 WT CTD A 1 0.113 14 132.511 9.253 LAT
#> 4 WT CTD A 2 0.112 14 135.586 9.150 LAB
#> 5 WT CTD A 2 0.113 14 125.145 9.270 LAB
#> 6 WT CTD A 2 0.113 14 135.203 9.189 LGM
#> 7 WT CTD A 2 0.113 14 128.547 9.088 LAB
#> 8 WT CTD B 1 0.113 14 127.709 9.199 LGM
#> 9 WT CTD B 1 0.113 14 127.074 9.058 LGM
#> 10 WT CTD B 1 0.114 14 126.879 9.306 LGM
#> strength.norm
#> 1 144.6248
#> 2 138.8500
#> 3 135.3865
#> 4 137.3023
#> 5 127.8606
#> 6 138.1369
#> 7 131.3364
#> 8 130.4803
#> 9 129.8315
#> 10 130.7794
```

The simplest thing that you will likely do is to calculate a basis
value based of a set of numbers that you consider as unstructured data.
An example of this would be calculating the B-Basis of the
`RTD`

warp tension (`WT`

) data.

There are a number of diagnostic tests that we should run before actually calculating a B-Basis value. We’ll talk about those later, but for now, let’s just get right to checking how the data are distributed and calculating the B-Basis.

We’ll use an Anderson–Darling test to check if the data are normally
distributed. The `cmstatr`

package provides the function
`anderson_darling_normal`

and related functions for other
distributions. We can run an Anderson–Darling test for normality on the
warp tension RTD data as follows. We’ll perform this test on the
normalized strength.

```
norm_data %>%
filter(test == "WT" & condition == "RTD") %>%
anderson_darling_normal(strength.norm)
#>
#> Call:
#> anderson_darling_normal(data = ., x = strength.norm)
#>
#> Distribution: Normal ( n = 28 )
#> Test statistic: A = 0.3805995
#> OSL (p-value): 0.3132051 (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Normal distribution ( alpha = 0.05 )
```

Now that we know that this data follows a normal distribution (since
the observed significance level (OSL) of the Anderson–Darling test is
greater than \(0.05\)), we can proceed
to calculate a basis value based based on the assumption of normally
distributed data. The `cmstatr`

package provides the function
`basis_normal`

as well as related functions for other
distributions. By default, the B-Basis value is calculated, but other
population proportions and confidence bounds can be specified (for
example, specify `p = 0.99, conf = 0.99`

for A-Basis).

```
norm_data %>%
filter(test == "WT" & condition == "RTD") %>%
basis_normal(strength.norm)
#> `outliers_within_batch` not run because parameter `batch` not specified
#> `between_batch_variability` not run because parameter `batch` not specified
#>
#> Call:
#> basis_normal(data = ., x = strength.norm)
#>
#> Distribution: Normal ( n = 28 )
#> B-Basis: ( p = 0.9 , conf = 0.95 )
#> 129.9583
```

We see that the calculated B-Basis is \(129.96\). We also see two messages issued
by the `cmstatr`

package. These messages relate to the
automated diagnostic tests performed by the basis calculation functions.
In this case we see messages that two of the diagnostic tests were not
performed because we didn’t specify the batch of each observation. The
batch is not required for calculating single-point basis values, but it
is required for performing batch-to-batch variability and within-batch
outlier diagnostic tests.

The `basis_normal`

function performs the following
diagnostic tests by default:

- Within batch outliers using
`maximum_normed_residual()`

- Between batch variability using
`ad_ksample()`

- Outliers using
`maximum_normed_residual()`

- Normality of data using
`anderson_darling_normal()`

There are two ways that we can deal with the two messages that we
see. We can pass in a column that specifies the batch for each
observation, or we can override those two diagnostic tests so that
`cmstatr`

doesn’t run them.

To override the two diagnostic tests, we set the argument
`override`

to a list of the names of the diagnostic tests
that we want to skip. The names of the diagnostic tests that were not
run are shown between back-ticks (`) in the message. Our call to
`basis_normal()`

would be updated as follows:

```
norm_data %>%
filter(test == "WT" & condition == "RTD") %>%
basis_normal(strength.norm,
override = c("outliers_within_batch",
"between_batch_variability"))
#>
#> Call:
#> basis_normal(data = ., x = strength.norm, override = c("outliers_within_batch",
#> "between_batch_variability"))
#>
#> Distribution: Normal ( n = 28 )
#> The following diagnostic tests were overridden:
#> `outliers_within_batch`,
#> `between_batch_variability`
#> B-Basis: ( p = 0.9 , conf = 0.95 )
#> 129.9583
```

Obviously, you should be cautious about overriding the diagnostic tests. There are certainly times when it is appropriate to do so, but sound engineering judgment is required.

The better approach would be to specify the batch. This can be done as follows:

```
norm_data %>%
filter(test == "WT" & condition == "RTD") %>%
basis_normal(strength.norm, batch)
#> Warning: `between_batch_variability` failed: Anderson-Darling k-Sample test
#> indicates that batches are drawn from different distributions
#>
#> Call:
#> basis_normal(data = ., x = strength.norm, batch = batch)
#>
#> Distribution: Normal ( n = 28 )
#> The following diagnostic tests failed:
#> `between_batch_variability`
#> B-Basis: ( p = 0.9 , conf = 0.95 )
#> 129.9583
```

Now that batch is specified, we see that one of the diagnostic tests actually fails: the Anderson–Darling k-Sample test shows that the batches are not drawn from the same (unspecified) distribution. We can run this diagnostic test directly to investigate further:

```
norm_data %>%
filter(test == "WT" & condition == "RTD") %>%
ad_ksample(strength.norm, batch)
#>
#> Call:
#> ad_ksample(data = ., x = strength.norm, groups = batch)
#>
#> N = 28 k = 3
#> ADK = 6.65 p-value = 0.0025892
#> Conclusion: Samples do not come from the same distribution (alpha = 0.025 )
```

For the Anderson–Darling k-Sample test, \(\alpha=0.025\) is normally used. In this case the p-value is \(p=0.0026\), so it is no where near \(\alpha\) (note the number of decimal places).

We can plot the distribution of this data and make a judgment call about whether to continue.

```
norm_data %>%
filter(test == "WT" & condition == "RTD") %>%
group_by(batch) %>%
ggplot(aes(x = strength.norm, color = batch)) +
stat_normal_surv_func() +
stat_esf() +
ggtitle("Distribution of Data For Each Batch")
```

We can also run the other diagnostic test by themselves. These are described in more detail in the following sections.

In this section, we’ll use the fill-compression data from the
`carbon.fabric.2`

data set.

After checking that there are a sufficient number of conditions,
batches and specimens and that the failure modes are consistent, we
would normally check if there are outliers within each batch and
condition. The maximum normed residual test can be used for this. The
`cmstatr`

package provides the function
`maximum_normed_residual`

to do this. First, we’ll group the
data by condition and batch, then run the test on each group. The
`maximum_normed_residual`

function returns an object that
contains a number of values. We’ll create a `data.frame`

that
contains those values.

In order to do this, we need to use the `nest`

function
from the `tidyr`

package. This is explained in detail here.
Basically, `nest`

allows a column of `list`

s or a
column of `data.frame`

s to be added to a
`data.frame`

. Once nested, we can use the `glance`

method to unpack the values returned by
`maximum_normed_residual`

into a one-row
`data.frame`

, and then use `unnest`

to flatten
this into a single `data.frame`

.

```
norm_data %>%
filter(test == "FC") %>%
group_by(condition, batch) %>%
nest() %>%
mutate(mnr = map(data,
~maximum_normed_residual(data = .x, x = strength.norm)),
tidied = map(mnr, glance)) %>%
select(-c(mnr, data)) %>% # remove unneeded columns
unnest(tidied)
#> # A tibble: 15 × 6
#> # Groups: condition, batch [15]
#> condition batch mnr alpha crit n_outliers
#> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 CTD A 1.48 0.05 1.89 0
#> 2 CTD B 1.72 0.05 1.89 0
#> 3 CTD C 1.52 0.05 1.89 0
#> 4 RTD A 1.34 0.05 1.89 0
#> 5 RTD B 1.47 0.05 1.89 0
#> 6 RTD C 1.52 0.05 1.89 0
#> 7 ETD A 1.66 0.05 1.89 0
#> 8 ETD B 1.53 0.05 1.89 0
#> 9 ETD C 1.40 0.05 1.89 0
#> 10 ETW A 1.45 0.05 1.89 0
#> 11 ETW B 1.83 0.05 1.89 0
#> 12 ETW C 1.76 0.05 1.89 0
#> 13 ETW2 A 1.85 0.05 1.89 0
#> 14 ETW2 B 1.54 0.05 1.89 0
#> 15 ETW2 C 1.38 0.05 2.02 0
```

None of the groups have outliers, so we can continue.

Next, we will use the Anderson–Darling k-Sample test to check that
each batch comes from the same distribution within each condition. We
can use the `ad_ksample`

function from `cmstatr`

to do so. Once again, we’ll use `nest`

/`unnest`

and `glance`

to do so.

```
norm_data %>%
filter(test == "FC") %>%
group_by(condition) %>%
nest() %>%
mutate(adk = map(data, ~ad_ksample(data = .x,
x = strength.norm,
groups = batch)),
tidied = map(adk, glance)) %>%
select(-c(data, adk)) %>% # remove unneeded columns
unnest(tidied)
#> # A tibble: 5 × 8
#> # Groups: condition [5]
#> condition alpha n k sigma ad p reject_same_dist
#> <chr> <dbl> <int> <int> <dbl> <dbl> <dbl> <lgl>
#> 1 CTD 0.025 18 3 0.944 1.76 0.505 FALSE
#> 2 RTD 0.025 18 3 0.944 1.03 0.918 FALSE
#> 3 ETD 0.025 18 3 0.944 0.683 0.997 FALSE
#> 4 ETW 0.025 18 3 0.944 0.93 0.954 FALSE
#> 5 ETW2 0.025 19 3 0.951 1.74 0.513 FALSE
```

For all conditions, the Anderson–Darling k-Sample test fails to reject the hypothesis that each batch comes from the same (unspecified) distribution. We can thus proceed to pooling the data.

Just as we did when checking for outlier within each condition and each batch, we can pool all the batches (within each condition) and check for outliers within each condition.

```
norm_data %>%
filter(test == "FC") %>%
group_by(condition) %>%
nest() %>%
mutate(mnr = map(data, ~maximum_normed_residual(data = .x,
x = strength.norm)),
tidied = map(mnr, glance)) %>%
select(-c(mnr, data)) %>% # remove unneeded columns
unnest(tidied)
#> # A tibble: 5 × 5
#> # Groups: condition [5]
#> condition mnr alpha crit n_outliers
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 CTD 2.38 0.05 2.65 0
#> 2 RTD 2.06 0.05 2.65 0
#> 3 ETD 2.05 0.05 2.65 0
#> 4 ETW 2.34 0.05 2.65 0
#> 5 ETW2 2.07 0.05 2.68 0
```

We find no outliers, so we can continue.

Often it is desirable to pool data across several environments. There are two methods for doing so: “pooled standard deviation” and “pooled CV” (CV is an abbreviation for Coefficient of Variation)

First, we will check for equality of variance among the conditions.
We will do so using Levene’s test. The `cmstatr`

package
provides the function `levene_test`

to do so.

```
norm_data %>%
filter(test == "FC") %>%
levene_test(strength.norm, condition)
#>
#> Call:
#> levene_test(data = ., x = strength.norm, groups = condition)
#>
#> n = 91 k = 5
#> F = 5.260731 p-value = 0.0007727083
#> Conclusion: Samples have unequal variance ( alpha = 0.05 )
```

The result from Levene’s test indicates that the variance for each condition is not equal. This indicates that the data cannot be pooled using the “pooled standard deviation” method.

We can check if the data can be pooled using the “pooled CV” method.
We’ll start by normalizing the data from each group to the group’s mean.
The `cmstatr`

package provides the function
`normalize_group_mean`

for this purpose.

```
norm_data %>%
filter(test == "FC") %>%
mutate(
strength_norm_group = normalize_group_mean(strength.norm, condition)) %>%
levene_test(strength_norm_group, condition)
#>
#> Call:
#> levene_test(data = ., x = strength_norm_group, groups = condition)
#>
#> n = 91 k = 5
#> F = 1.839645 p-value = 0.1285863
#> Conclusion: Samples have equal variances ( alpha = 0.05 )
```

The Levene’s test thus shows the variances of the pooled data are equal. We can move on to performing an Anderson–Darling test for normality on the pooled data.

```
norm_data %>%
filter(test == "FC") %>%
mutate(
strength_norm_group = normalize_group_mean(strength.norm, condition)) %>%
anderson_darling_normal(strength_norm_group)
#>
#> Call:
#> anderson_darling_normal(data = ., x = strength_norm_group)
#>
#> Distribution: Normal ( n = 91 )
#> Test statistic: A = 0.3619689
#> OSL (p-value): 0.3812268 (assuming unknown parameters)
#> Conclusion: Sample is drawn from a Normal distribution ( alpha = 0.05 )
```

The Anderson–Darling test indicates that the pooled data is drawn from a normal distribution, so we can continue with calculating basis values using the “pooled CV” method.

```
norm_data %>%
filter(test == "FC") %>%
basis_pooled_cv(strength.norm, condition, batch)
#>
#> Call:
#> basis_pooled_cv(data = ., x = strength.norm, groups = condition,
#> batch = batch)
#>
#> Distribution: Normal - Pooled CV ( n = 91, r = 5 )
#> B-Basis: ( p = 0.9 , conf = 0.95 )
#> CTD 85.09241
#> ETD 66.55109
#> ETW 51.43626
#> ETW2 45.81318
#> RTD 78.2274
```

The conditions listed in the output above are in alphabetical order. This probably isn’t what you want. Instead, you probably want the conditions listed in a certain order. This can be done by ordering the data first as demonstrated below. You’re probably just do this one in at the start of your analysis.

```
norm_data %>%
mutate(condition = ordered(condition,
c("CTD", "RTD", "ETD", "ETW", "ETW2"))) %>%
filter(test == "FC") %>%
basis_pooled_cv(strength.norm, condition, batch)
#>
#> Call:
#> basis_pooled_cv(data = ., x = strength.norm, groups = condition,
#> batch = batch)
#>
#> Distribution: Normal - Pooled CV ( n = 91, r = 5 )
#> B-Basis: ( p = 0.9 , conf = 0.95 )
#> CTD 85.09241
#> RTD 78.2274
#> ETD 66.55109
#> ETW 51.43626
#> ETW2 45.81318
```

Eventually, once you’ve finished calculating all your basis values,
you’ll probably want to set specification requirements or evaluate
site/process equivalency. `cmstatr`

has functionality to do
both.

Let’s say that you want to develop specification limits for fill compression that you’re going to put in your material specification. You can do this as follows:

```
carbon.fabric.2 %>%
filter(test == "FC" & condition == "RTD") %>%
equiv_mean_extremum(strength, n_sample = 5, alpha = 0.01)
#>
#> Call:
#> equiv_mean_extremum(df_qual = ., data_qual = strength, n_sample = 5,
#> alpha = 0.01)
#>
#> For alpha = 0.01 and n = 5
#> ( k1 = 3.071482 and k2 = 1.142506 )
#> Min Individual Sample Mean
#> Thresholds: 69.89842 82.16867
```

If you’re determining equivalency limits for modulus, a different
approach is generally used so that bilateral limits are set.
`cmstatr`

can do this as well, using the function
`equiv_change_mean`

.