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Coefficient of Variation

Source: R/coefVar.R
coefVar.Rd

Calculates the coefficient of variation and its confidence limits using various methods.

Usage

coefVar(x, ...)

# Default S3 method
coefVar(x, weights = NULL, unbiased = FALSE, na.rm = FALSE, ...)

# S3 method for class 'lm'
coefVar(x, unbiased = FALSE, na.rm = FALSE, ...)

# S3 method for class 'aov'
coefVar(x, unbiased = FALSE, na.rm = FALSE, ...)

coefVarCI(
  x,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("nct", "vangel", "mckay", "naive"),
  weights = NULL,
  unbiased = FALSE,
  na.rm = FALSE,
  ...
)

Arguments

x

a non-empty numeric vector of data values, or a fitted model for the lm/aov methods

...

further arguments

weights

a numeric vector of weights the same length as x giving the weights to use for elements of x

unbiased

logical; whether to apply a bias correction. See Details. Defaults to FALSE.

na.rm

logical. Should missing values be removed? Defaults to FALSE, in which case missing values are an error.

conf.level

confidence level of the interval. If set to NA (the default), only the point estimate is returned.

sides

character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals().

method

character string specifying the confidence interval method: "nct" (default), "vangel", "mckay", or "naive". Partial matching is supported. See Details.

Value

an unnamed numeric scalar containing the coefficient of variation for coefVar(). If recycling in coefVarCI() yields a single case, it returns a named numeric vector with elements:

est

point estimate of the coefficient of variation.

lci

lower confidence interval bound.

uci

upper confidence interval bound.

Otherwise, coefVarCI() returns a numeric matrix with one row per case and the columns est, lci, and uci.

Details

In order for the coefficient of variation to be an unbiased estimate of the true population value, the coefficient of variation is corrected as: $$ CV_{korr} = CV \cdot \left( 1 - \frac{1}{4\cdot(n-1)} + \frac{1}{n} \cdot CV^2 + \frac{1}{2 \cdot (n-1)^2} \right) $$

For determining the confidence intervals for the coefficient of variation a number of methods have been proposed. coefVarCI() currently supports four different methods. The details for the methods are given in the specific references.

The "naive" method is based on dividing the standard confidence limit for the standard deviation by the sample mean.

McKay's approximation is asymptotically exact as n goes to infinity. McKay recommends this approximation only if the coefficient of variation is less than 0.33. Note that if the coefficient of variation is greater than 0.33, either the normality of the data is suspect or the probability of negative values in the data is non-negligible. In this case, McKay's approximation may not be valid. Also, it is generally recommended that the sample size should be at least 10 before using McKay's approximation.

Vangel's modified McKay method is more accurate than the McKay method in most cases, particularly for small samples. According to Vangel, the unmodified McKay is only more accurate when both the coefficient of variation and alpha are large. However, if the coefficient of variation is large, then this implies either that the data contains negative values or the data does not follow a normal distribution. In this case, neither the McKay or the modified McKay should be used. In general, the Vangel's modified McKay method is recommended over the McKay method. It generally provides good approximations as long as the data is approximately normal and the coefficient of variation is less than 0.33.

See also: https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/coefvacl.htm

nct uses the noncentral t-distribution to calculate the confidence intervals. See Smithson (2003).

sides names the side on which the finite bound lies: "left" yields \([lci, \infty)\), "right" yields \((-\infty, uci]\).

**Note:** Analytic (precision) weights are not supported. For likelihood-based weighted variance estimation, see stats::cov.wt().

Note

Parts of the code contributed by Michael Smithson.

References

McKay, A. T. (1932). Distribution of the coefficient of variation and the extended t distribution, Journal of the Royal Statistical Society, 95, 695–698.

Johnson, B. L., Welch, B. L. (1940). Applications of the non-central t-distribution. Biometrika, 31, 362–389.

Mark Vangel (1996) Confidence Intervals for a Normal Coefficient of Variation, American Statistician, Vol. 15, No. 1, pp. 21-26.

Kelley, K. (2007). Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach. Behavior Research Methods, 39 (4), 755-766

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1-24

Smithson, M.J. (2003) Confidence Intervals, Quantitative Applications in the Social Sciences Series, No. 140. Thousand Oaks, CA: Sage. pp. 39-41

Steve Verrill (2003) Confidence Bounds for Normal and Lognormal Distribution Coefficients of Variation, Research Paper 609, USDA Forest Products Laboratory, Madison, Wisconsin.

Verrill, S. and Johnson, R.A. (2007) Confidence Bounds and Hypothesis Tests for Normal Distribution Coefficients of Variation, Communications in Statistics Theory and Methods, Volume 36, No. 12, pp 2187-2206.

See also

meanX(), sdX(), (both supporting weights)

Other dispersion: iqrX(), madX(), meanAD(), meanSE(), rangeX(), varX()

Examples


set.seed(15)
x <- runif(100)
coefVar(x)
#> [1] 0.5092566
coefVarCI(x)
#>       est       lci       uci 
#> 0.5092566 0.4351644 0.6151409 

# Coefficient of variation for a linear model
r.lm <- lm(Fertility ~ ., swiss)
coefVar(r.lm)
#> [1] 0.1021544

# conf.level is recycled, so several levels can be requested at once.
# Reproduces the NIST reference table for the ZARR13 dataset (see the
# URL above) from a sample rebuilt to its published mean and sd, so the
# example needs no network access.
set.seed(1)
zarr <- 9.261460 + 0.022789 * as.vector(scale(rnorm(195)))

round(coefVarCI(zarr, method = "vangel", sides = "two.sided",
                conf.level = c(.5, .8, .9, .95, .99, .999)), 5)
#>          est     lci     uci
#> [1,] 0.00246 0.00238 0.00255
#> [2,] 0.00246 0.00231 0.00263
#> [3,] 0.00246 0.00227 0.00269
#> [4,] 0.00246 0.00224 0.00273
#> [5,] 0.00246 0.00217 0.00283
#> [6,] 0.00246 0.00210 0.00294

# ---------------------------------------------------------
#  Confidence    Coefficient          Lower          Upper
#   Value (%)   of Variation          Limit          Limit
# ---------------------------------------------------------
#        50.0        0.00246        0.00238        0.00255
#        80.0        0.00246        0.00231        0.00263
#        90.0        0.00246        0.00227        0.00269
#        95.0        0.00246        0.00224        0.00273
#        99.0        0.00246        0.00217        0.00283
#        99.9        0.00246        0.00210        0.00294