Coefficient of quartile variation (\(CQV\)) is a measure of relative dispersion that is based on interquartile range (IQR). Since cqv is unitless, it is useful for comparison of variables with different units. It is also a measure of homogeneity (Altunkaynak & Gamgam, 2018; Bonett, 2006).
The population coefficient of quartile variation is:
\[
CQV = \biggl(\frac{Q_3-Q_1}{Q_3+Q_1}\biggr)\times100
\] where \(q_3\) and \(q_1\) are the population third quartile (i.e., \(75^{th}\) percentile) and first quartile (i.e., \(25^{th}\) percentile), respectively. Almost always, we analyze data from samples but want to generalize it as the population’s parameter (Albatineh, Kibria, Wilcox, & Zogheib, 2014). Its sample’s estimate is given as:
\[
cqv = \biggl(\frac{q_3-q_1}{q_3+q_1}\biggr)\times100
\] There are different methods for the calculation of confidence intervals (CI) for CQV. All of them are fruitful and have particular use cases. For sake of versatility, we cover almost all of these methods in cvcqv package. Here, we explain them along with some examples:
Bonett (Bonett, 2006) introduced the following confidence interval for CQV:
\[
\exp\{\ln{(D/S)c\ \pm\ Z_{1-\alpha/2}\sqrt{v} }\}
\] where \(c = n/(n-1)\) is a centering adjustment which helps to equalize the tail error probabilities (Altunkaynak & Gamgam, 2018; Bonett, 2006). \(D = \hat{Q3}-\hat{Q1}\) and \(S = \hat{Q3}+\hat{Q1}\) are the sample \(25^{th}\) and \(75^{th}\) percentiles, respectively; \(Z_{1-\alpha/2}\) is the \(1-\alpha/2\) quantile of the standard normal distribution. Computation of \(v\) which is \(Var\{\ln{(D/S)}\}\) is long and a bit complicated, but has been implemented for cqv function:
x <- c(
0.2, 0.5, 1.1, 1.4, 1.8, 2.3, 2.5, 2.7, 3.5, 4.4,
4.6, 5.4, 5.4, 5.7, 5.8, 5.9, 6.0, 6.6, 7.1, 7.9
)
cqv_versatile(
x,
na.rm = TRUE,
digits = 3,
method = "bonett"
)## $method
## [1] "cqv with Bonett 95% CI"
##
## $statistics
## est lower upper
## 45.625 24.785 77.329Thanks to package boot (Canty & Ripley, 2017) we can obtain bootstrap CI around \(cqv\):
## $method
## [1] "cqv with adjusted bootstrap percentile (BCa) 95% CI"
##
## $statistics
## est lower upper
## 45.625 21.527 79.698In conclusion, we can observe CIs calculated by all available methods:
## $method
## [1] "All methods"
##
## $statistics
## est lower upper
## bonett 45.625 24.785 77.329
## norm 45.625 19.410 70.462
## basic 45.625 19.398 73.303
## percent 45.625 17.947 71.852
## bca 45.625 25.532 83.405
## description
## bonett cqv with Bonett 95% CI
## norm cqv with normal approximation 95% CI
## basic cqv with basic bootstrap 95% CI
## percent cqv with bootstrap percentile 95% CI
## bca cqv with adjusted bootstrap percentile (BCa) 95% CIAlbatineh, A. N., Kibria, B. M., Wilcox, M. L., & Zogheib, B. (2014). Confidence interval estimation for the population coefficient of variation using ranked set sampling: A simulation study. Journal of Applied Statistics, 41(4), 733–751. https://doi.org/10.1080/02664763.2013.847405
Altunkaynak, B., & Gamgam, H. (2018). Bootstrap confidence intervals for the coefficient of quartile variation. Communications in Statistics: Simulation and Computation, 0(0), 1–9. https://doi.org/10.1080/03610918.2018.1435800
Bonett, D. G. (2006). Confidence interval for a coefficient of quartile variation. Computational Statistics and Data Analysis, 50(11), 2953–2957. https://doi.org/10.1016/j.csda.2005.05.007
Canty, A., & Ripley, B. (2017). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-20.