Function designed to help examine distributions. It also includes an option for assessing multivariate normality using the (squared) Mahalanobis distance. A generic function, some methods, and constructor (as.testDistribution) and function to check class (is.testDistribution) also are provided.

Note that for the use argument, several options are possible. By default it is “complete.obs”, which uses only cases with complete data on all variables. Another option is “pairwise.complete.obs”, which uses all available data for each variable indivdiually to estimate the means and variances, and all pairwise complete observation pairs for each covariance. Because the same cases are not used for all estimates, it is possible to obtain a covariance matrix that is not positive definite (e.g., correlations > +1 or < -1).

Finally, the last option is “fiml”, which uses full information maximum likelihood estimates of the means and covariance matrix. Depending on the number of cases, missing data patterns, and variables, this may be quite slow and computationally demanding.

The robust argument determines whether to use robust estimates or not when calculating densities, etc. By default it is FALSE, but if TRUE and a univariate or multivariate normal distribution is tested, then robust estimates of the means and covariance matrix (a variance if univariate) will be used based on covMcd from the robustbase package.

## Usage

testDistribution(x, ...)

as.testDistribution(x)

is.testDistribution(x)

# S3 method for default
testDistribution(
x,
distr = c("normal", "beta", "chisq", "f", "gamma", "geometric", "nbinom", "poisson",
"uniform", "mvnormal"),
na.rm = TRUE,
starts,
extremevalues = c("no", "theoretical", "empirical"),
ev.perc = 0.001,
use = c("complete.obs", "pairwise.complete.obs", "fiml"),
robust = FALSE,
...
)

## Arguments

x

The data as a single variable or vector to check the distribution unless the distribution is “mvnormal” in which case it should be a data frame or data table.

...

Additional arguments. If these include mu and sigma and the distribution is multivariate normal, then it will use the passed values instead of calculating the mean and covariances of the data.

distr

A character string indicating the distribution to be tested. Currently one of: “normal”, “beta”, “chisq” (chi-squared), “f”, “gamma”, “geometric”, “nbinom” (negative binomial), “poisson”, “uniform”, or “mvnormal” for multivariate normal where Mahalanobis distances are calculated and compared against a Chi-squared distribution with degrees of freedom equal to the number of variables.

na.rm

A logical value whether to omit missing values. Defaults to TRUE.

starts

A named list of the starting values. Not required for all distributions. Passed on to fitdistr which fits the maximum likelihood estimates of the distribution parameters.

extremevalues

A character vector whether to indicate extreme values. Should be “no” to do nothing, “empirical” to show extreme values based on the observed data percentiles, or “theoretical” to show extreme values based on percentiles of the theoretical distribution.

ev.perc

Percentile to use for extreme values. For example if .01, then the lowest 1 percent and highest 1 percent will be labelled extreme values. Defaults to the lowest and highest 0.1 percent.

use

A character vector indicating how the moments (means and covariance matrix) should be estimated in the presence of missing data when distr = mvnormal. The default is to use complete observations, but full information maximum likelihood based on functions in lavaan is also available. See details.

robust

A logical whether to use robust estimation or not. Currently only applies to normally distributed data (univariate or multivariate). Also, when robust = TRUE, only complete observations are used (i.e., use = "complete.obs"). See details.

## Value

A logical whether or not an object is of class

testDistribution or an object of the same class.

A list with information about the distribution (parameter estimates, name, log likelihood (useful for comparing the fit of different distributions to the data), and a dataset with the sorted data and theoretical quantiles.

SEMSummary

## Examples


## example data
set.seed(1234)
d <- data.table::data.table(
Ynorm = rnorm(200),
Ybeta = rbeta(200, 1, 4),
Ychisq = rchisq(200, 8),
Yf = rf(200, 5, 10),
Ygamma = rgamma(200, 2, 2),
Ynbinom = rnbinom(200, mu = 4, size = 9),
Ypois = rpois(200, 4))

## testing and graphing
testDistribution(d$Ybeta, "beta", starts = list(shape1 = 1, shape2 = 4)) #>$Data
#>                 X           Y OriginalOrder   isEV    YDeviates
#>             <num>       <num>         <int> <fctr>        <num>
#>   1: 0.0009548787 0.002899638            20     No  0.001944759
#>   2: 0.0026222465 0.006542140           122     No  0.003919893
#>   3: 0.0041999262 0.007116305           124     No  0.002916379
#>   4: 0.0057324381 0.007488806           119     No  0.001756368
#>   5: 0.0072360495 0.008381499             7     No  0.001145450
#>  ---
#> 196: 0.5896978140 0.658553356           192     No  0.068855542
#> 197: 0.6125406443 0.680790719           171     No  0.068250075
#> 198: 0.6411187285 0.689096550           175     No  0.047977822
#> 199: 0.6804840872 0.694824897           181     No  0.014340810
#> 200: 0.7510216428 0.697959189           198     No -0.053062454
#>
#> $Distribution #>$Distribution$d #> function (x, shape1, shape2, ncp = 0, log = FALSE) #> { #> if (missing(ncp)) #> .Call(C_dbeta, x, shape1, shape2, log) #> else .Call(C_dnbeta, x, shape1, shape2, ncp, log) #> } #> <bytecode: 0x55d7a0185c18> #> <environment: namespace:stats> #> #>$Distribution$q #> function (p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE) #> { #> if (missing(ncp)) #> .Call(C_qbeta, p, shape1, shape2, lower.tail, log.p) #> else .Call(C_qnbeta, p, shape1, shape2, ncp, lower.tail, #> log.p) #> } #> <bytecode: 0x55d7a0185128> #> <environment: namespace:stats> #> #>$Distribution$Name #> [1] "Beta" #> #>$Distribution$fit #> shape1 shape2 #> 1.09047254 4.43181099 #> (0.09708525) (0.46533971) #> #>$Distribution$LL #> 'log Lik.' 131.529 (df=2) #> #> #>$EVLimits
#> [1] -Inf  Inf
#>
#> $NOK #> [1] 200 #> #>$distr
#> [1] "beta"
#>
#> $na.rm #> [1] TRUE #> #>$extremevalues
#> [1] "no"
#>
#> $ev.perc #> [1] 0.001 #> #>$use
#> [1] "complete.obs"
#>
#> $robust #> [1] FALSE #> #> attr(,"class") #> [1] "testDistribution" testDistribution(d$Ychisq, "chisq", starts = list(df = 8))
#> $Data #> X Y OriginalOrder isEV YDeviates #> <num> <num> <int> <fctr> <num> #> 1: 1.020806 1.119129 75 No 0.09832363 #> 2: 1.409747 1.386369 54 No -0.02337731 #> 3: 1.647133 1.476380 78 No -0.17075244 #> 4: 1.829535 1.785484 149 No -0.04405142 #> 5: 1.981898 1.948253 56 No -0.03364471 #> --- #> 196: 17.443427 19.520193 34 No 2.07676616 #> 197: 18.145695 19.541868 154 No 1.39617310 #> 198: 19.071980 20.585244 55 No 1.51326441 #> 199: 20.451772 20.984934 170 No 0.53316198 #> 200: 23.332634 24.064860 12 No 0.73222569 #> #>$Distribution
#> $Distribution$d
#> function (x, df, ncp = 0, log = FALSE)
#> {
#>     if (missing(ncp))
#>         .Call(C_dchisq, x, df, log)
#>     else .Call(C_dnchisq, x, df, ncp, log)
#> }
#> <bytecode: 0x55d79f897f08>
#> <environment: namespace:stats>
#>
#> $Distribution$q
#> function (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
#> {
#>     if (missing(ncp))
#>         .Call(C_qchisq, p, df, lower.tail, log.p)
#>     else .Call(C_qnchisq, p, df, ncp, lower.tail, log.p)
#> }
#> <bytecode: 0x55d79f8974c0>
#> <environment: namespace:stats>
#>
#> $Distribution$Name
#> [1] "Chi-squared"
#>
#> $Distribution$fit
#>       df
#>   7.7424489
#>  (0.2605917)
#>
#> $Distribution$LL
#> 'log Lik.' -544.7285 (df=1)
#>
#>
#> $EVLimits #> [1] -Inf Inf #> #>$NOK
#> [1] 200
#>
#> $distr #> [1] "chisq" #> #>$na.rm
#> [1] TRUE
#>
#> $extremevalues #> [1] "no" #> #>$ev.perc
#> [1] 0.001
#>
#> $use #> [1] "complete.obs" #> #>$robust
#> [1] FALSE
#>
#> attr(,"class")
#> [1] "testDistribution"

## for chi-square distribution, extreme values only on
## the right tail
testDistribution(d$Ychisq, "chisq", starts = list(df = 8), extremevalues = "empirical", ev.perc = .1) #>$Data
#>              X         Y OriginalOrder   isEV   YDeviates
#>          <num>     <num>         <int> <fctr>       <num>
#>   1:  1.020806  1.119129            75     No  0.09832363
#>   2:  1.409747  1.386369            54     No -0.02337731
#>   3:  1.647133  1.476380            78     No -0.17075244
#>   4:  1.829535  1.785484           149     No -0.04405142
#>   5:  1.981898  1.948253            56     No -0.03364471
#>  ---
#> 196: 17.443427 19.520193            34    Yes  2.07676616
#> 197: 18.145695 19.541868           154    Yes  1.39617310
#> 198: 19.071980 20.585244            55    Yes  1.51326441
#> 199: 20.451772 20.984934           170    Yes  0.53316198
#> 200: 23.332634 24.064860            12    Yes  0.73222569
#>
#> $Distribution #>$Distribution$d #> function (x, df, ncp = 0, log = FALSE) #> { #> if (missing(ncp)) #> .Call(C_dchisq, x, df, log) #> else .Call(C_dnchisq, x, df, ncp, log) #> } #> <bytecode: 0x55d79f897f08> #> <environment: namespace:stats> #> #>$Distribution$q #> function (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE) #> { #> if (missing(ncp)) #> .Call(C_qchisq, p, df, lower.tail, log.p) #> else .Call(C_qnchisq, p, df, ncp, lower.tail, log.p) #> } #> <bytecode: 0x55d79f8974c0> #> <environment: namespace:stats> #> #>$Distribution$Name #> [1] "Chi-squared" #> #>$Distribution$fit #> df #> 7.7424489 #> (0.2605917) #> #>$Distribution$LL #> 'log Lik.' -544.7285 (df=1) #> #> #>$EVLimits
#>               90%
#>     -Inf 13.60881
#>
#> $NOK #> [1] 200 #> #>$distr
#> [1] "chisq"
#>
#> $na.rm #> [1] TRUE #> #>$extremevalues
#> [1] "empirical"
#>
#> $ev.perc #> [1] 0.1 #> #>$use
#> [1] "complete.obs"
#>
#> $robust #> [1] FALSE #> #> attr(,"class") #> [1] "testDistribution" testDistribution(d$Ychisq, "chisq", starts = list(df = 8),
extremevalues = "theoretical", ev.perc = .1)
#> $Data #> X Y OriginalOrder isEV YDeviates #> <num> <num> <int> <fctr> <num> #> 1: 1.020806 1.119129 75 No 0.09832363 #> 2: 1.409747 1.386369 54 No -0.02337731 #> 3: 1.647133 1.476380 78 No -0.17075244 #> 4: 1.829535 1.785484 149 No -0.04405142 #> 5: 1.981898 1.948253 56 No -0.03364471 #> --- #> 196: 17.443427 19.520193 34 Yes 2.07676616 #> 197: 18.145695 19.541868 154 Yes 1.39617310 #> 198: 19.071980 20.585244 55 Yes 1.51326441 #> 199: 20.451772 20.984934 170 Yes 0.53316198 #> 200: 23.332634 24.064860 12 Yes 0.73222569 #> #>$Distribution
#> $Distribution$d
#> function (x, df, ncp = 0, log = FALSE)
#> {
#>     if (missing(ncp))
#>         .Call(C_dchisq, x, df, log)
#>     else .Call(C_dnchisq, x, df, ncp, log)
#> }
#> <bytecode: 0x55d79f897f08>
#> <environment: namespace:stats>
#>
#> $Distribution$q
#> function (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
#> {
#>     if (missing(ncp))
#>         .Call(C_qchisq, p, df, lower.tail, log.p)
#>     else .Call(C_qnchisq, p, df, ncp, lower.tail, log.p)
#> }
#> <bytecode: 0x55d79f8974c0>
#> <environment: namespace:stats>
#>
#> $Distribution$Name
#> [1] "Chi-squared"
#>
#> $Distribution$fit
#>       df
#>   7.7424489
#>  (0.2605917)
#>
#> $Distribution$LL
#> 'log Lik.' -544.7285 (df=1)
#>
#>
#> $EVLimits #> [1] 0.00000 13.01761 #> #>$NOK
#> [1] 200
#>
#> $distr #> [1] "chisq" #> #>$na.rm
#> [1] TRUE
#>
#> $extremevalues #> [1] "theoretical" #> #>$ev.perc
#> [1] 0.1
#>
#> $use #> [1] "complete.obs" #> #>$robust
#> [1] FALSE
#>
#> attr(,"class")
#> [1] "testDistribution"

if (FALSE) {

testDistribution(d$Yf, "uniform") testDistribution(d$Ypois, "geometric")

testDistribution(d$Yf, "f", starts = list(df1 = 5, df2 = 10)) testDistribution(d$Ygamma, "gamma")
testDistribution(d$Ynbinom, "poisson") testDistribution(d$Ynbinom, "nbinom")
testDistribution(d$Ypois, "poisson") ## compare log likelihood of two different distributions testDistribution(d$Ygamma, "normal")$Distribution$LL
testDistribution(d$Ygamma, "gamma")$Distribution$LL testDistribution(d$Ynorm, "normal")
testDistribution(c(d$Ynorm, 10, 1000), "normal", extremevalues = "theoretical") testDistribution(c(d$Ynorm, 10, 1000), "normal",
extremevalues = "theoretical", robust = TRUE)

testDistribution(mtcars, "mvnormal")

## for multivariate normal mahalanobis distance
## which follows a chi-square distribution, extreme values only on
## the right tail
testDistribution(mtcars, "mvnormal", extremevalues = "empirical",
ev.perc = .1)
testDistribution(mtcars, "mvnormal", extremevalues = "theoretical",
ev.perc = .1)

rm(d) ## cleanup
}