Confidence Intervals for the Variance

Calculates confidence intervals for the variance. Available approachs are the classical one using the ChiSquare distribution, a more robust version proposed by Bonett and the bootstrap options available in the package boot.

VarCI(x, method = c("classic", "bonett", "norm", "basic", "stud", "perc", "bca"),
      conf.level = 0.95, sides = c("two.sided", "left", "right"),
      na.rm = FALSE, R = 999)

Arguments

x: a (non-empty) numeric vector of data values.
method: vector of character strings representing the type of intervals required. The value should be any subset of the values "classic", "bonett", "norm", "basic", "stud", "perc", "bca". See boot.ci.
conf.level: confidence level of the interval.
sides: a character string specifying the side of the confidence interval, must be one of "two.sided" (default), "left" or "right". You can specify just the initial letter. "left" would be analogue to a hypothesis of "greater" in a t.test.
na.rm: logical. Should missing values be removed? Defaults to FALSE.
R: number of bootstrap replicates. Usually this will be a single positive integer. For importance resampling, some resamples may use one set of weights and others use a different set of weights. In this case R would be a vector of integers where each component gives the number of resamples from each of the rows of weights. See boot.

Value

a numeric vector with 3 elements:

var: variance
lwr.ci: lower bound of the confidence interval
upr.ci: upper bound of the confidence interval

Details

The confidence interval for the variance is very sensitive to non-normality in the data. Bonett (2006) has proposed an interval that is nearly exact when the data is normally distributed and provides good performance for moderately non-normal data. See the references for the details.

References

Bonett (2006) Approximate Confidence Interval for Standard Deviation of Nonnormal Distributions, Computational Statistics and Data Analysis, Vol. 50, pp. 775 - 782.
https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/sdconfli.htm (might be outdated)

Author

Andri Signorell <andri@signorell.net>

Examples

VarCI(d.pizza$price, na.rm=TRUE)
#>    var lwr.ci upr.ci 
#>    468    433    508 
VarCI(d.pizza$price, conf.level=0.99, na.rm=TRUE)
#>    var lwr.ci upr.ci 
#>    468    422    521 

x <- c(14.816, 14.863, 14.814, 14.998, 14.965, 14.824, 14.884, 14.838, 14.916,
       15.021, 14.874, 14.856, 14.860, 14.772, 14.980, 14.919)
VarCI(x, conf.level=0.9)
#>     var  lwr.ci  upr.ci 
#> 0.00529 0.00317 0.01092 

# and for the standard deviation
sqrt(VarCI(x, conf.level=0.9))
#>    var lwr.ci upr.ci 
#> 0.0727 0.0563 0.1045 


# from Bonett's paper
# expected results:
# ------------------------------------
#  conf.lvl       sd      lci      uci
# ------------------------------------
#      90.0   0.5168   0.3592   0.9359
#      95.0   0.5168   0.3263   1.0841
#      99.0   0.5168   0.2607   1.5109

p <- c(15.83, 16.01, 16.24, 16.42, 15.33, 15.44, 16.88, 16.31)
sqrt(VarCI(p, method="bonett", conf.level=0.9))
#>    var lwr.ci upr.ci 
#>  0.517  0.359  0.936 
sqrt(VarCI(p, method="bonett"))
#>    var lwr.ci upr.ci 
#>  0.517  0.326  1.084 
sqrt(VarCI(p, method="bonett", conf.level=0.99))
#>    var lwr.ci upr.ci 
#>  0.517  0.261  1.511 

# some bootstrap intervals
VarCI(x, method="norm")
#>     var  lwr.ci  upr.ci 
#> 0.00529 0.00307 0.00825 
VarCI(x, method="perc")
#>     var  lwr.ci  upr.ci 
#> 0.00529 0.00226 0.00792 
VarCI(x, method="bca")
#>     var  lwr.ci  upr.ci 
#> 0.00529 0.00317 0.00886