transform_mod_cv: Transforms data according to the modified CV rule

Description

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

Two versions of this transformation are implemented. The first version, transform_mod_cv(), transforms the data in a single group (with no other structure) according to the modified CV rules.

The second version, transform_mod_cv_ad(), transforms data that is structured according to both condition and batch, as is commonly done for the Anderson--Darling k-Sample and Anderson-Darling tests when pooling across environments.

Usage

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

Value

A vector of transformed data

Arguments

x: a vector of data to transform
condition: a vector indicating the condition to which each observation belongs
batch: a vector indicating the batch to which each observation belongs

Details

The modified CV transformation takes the general form:

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

Where $S_i^*$ is the modified standard deviation (mod CV times mean) for the $ith$ group; $S_i$ is the standard deviation for the $ith$ group, $\bar{x_i}$ is the group mean and $x_{ij}$ is the observation.

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

transform_mod_cv_ad() takes a vector containing the observations plus a vector containing the corresponding conditions and a vector containing the batches. This function first calculates the modified CV value from the data from each condition (independently). Then, within each condition, the transformation above is applied to produce the transformed data $x'$. This transformed data is further transformed using the following equation.

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

Where:

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

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

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

Examples

Run this code

# Transform data according to the modified CV transformation
# and report the original and modified CV for each condition

library(dplyr)
carbon.fabric %>%
filter(test == "FT") %>%
  group_by(condition) %>%
  mutate(trans_strength = transform_mod_cv(strength)) %>%
  head(10)

## # A tibble: 10 x 6
## # Groups:   condition [1]
##    id         test  condition batch strength trans_strength
##                              
##  1 FT-RTD-1-1 FT    RTD           1     126.           126.
##  2 FT-RTD-1-2 FT    RTD           1     139.           141.
##  3 FT-RTD-1-3 FT    RTD           1     116.           115.
##  4 FT-RTD-1-4 FT    RTD           1     132.           133.
##  5 FT-RTD-1-5 FT    RTD           1     129.           129.
##  6 FT-RTD-1-6 FT    RTD           1     130.           130.
##  7 FT-RTD-2-1 FT    RTD           2     131.           131.
##  8 FT-RTD-2-2 FT    RTD           2     124.           124.
##  9 FT-RTD-2-3 FT    RTD           2     125.           125.
## 10 FT-RTD-2-4 FT    RTD           2     120.           119.

# The CV of this transformed data can be computed to verify
# that the resulting CV follows the rules for modified CV

carbon.fabric %>%
  filter(test == "FT") %>%
  group_by(condition) %>%
  mutate(trans_strength = transform_mod_cv(strength)) %>%
  summarize(cv = sd(strength) / mean(strength),
            mod_cv = sd(trans_strength) / mean(trans_strength))

## # A tibble: 3 x 3
##   condition     cv mod_cv
##           
## 1 CTD       0.0423 0.0612
## 2 ETW       0.0369 0.0600
## 3 RTD       0.0621 0.0711