knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) # If any of the required packages are unavailable, # don't re-run the code required <- c("dplyr", "ggplot2", "tidyr", "cmstatr", "purrr") if (!all(unlist(lapply(required, function(pkg) { requireNamespace(pkg, quietly = TRUE)} )))) { knitr::opts_chunk$set(eval = FALSE) }

`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.

library(cmstatr) library(dplyr) library(ggplot2) library(tidyr) library(purrr)

`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)

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)

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)

# Verify that the AD test always provides the same conclusion # If this assertion fails, the Vignette needs to be re-written if (0.05 >= (norm_data %>% filter(test == "WT" & condition == "RTD") %>% anderson_darling_normal(strength.norm))$osl) { stop("Unexpected vale for Anderson-Darling test") }

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)

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"))

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)

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)

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)

if ((norm_data %>% filter(test == "FC") %>% group_by(condition, batch) %>% summarise( n_outliers = maximum_normed_residual(x = strength.norm)$n_outliers ) %>% ungroup() %>% summarise(n_outliers = sum(n_outliers)))[[1]] != 0) { stop("Unexpected number of outliers") }

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)

if (!all(!(norm_data %>% filter(test == "FC") %>% group_by(condition) %>% summarise(different_dist = ad_ksample(x = strength.norm, groups = batch)$reject_same_dist ))$different_dist)) { stop("Unexpected ADK result") }

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)

if ((norm_data %>% filter(test == "FC") %>% group_by(condition) %>% summarise( n_outliers = maximum_normed_residual(x = strength.norm)$n_outliers ) %>% ungroup() %>% summarise(n_outliers = sum(n_outliers)))[[1]] != 0) { stop("Unexpected number of outliers") }

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)

if (!(norm_data %>% filter(test == "FC") %>% levene_test(strength.norm, condition))$reject_equal_variance) { stop("Unexpected result from Levene's test") }

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)

if ((norm_data %>% filter(test == "FC") %>% mutate( strength_norm_group = normalize_group_mean(strength.norm, condition)) %>% levene_test(strength_norm_group, condition))$reject_equal_variance) { stop("Unexpected value from Levene's test") }

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)

if ((norm_data %>% filter(test == "FC") %>% mutate( strength_norm_group = normalize_group_mean(strength.norm, condition)) %>% anderson_darling_normal(strength_norm_group))$osl <= 0.05) { stop("Unexpected value from AD test") }

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)

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)

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)

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`

.

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