Distribution functions for the Lambda prime / non-central Lambda distribution

Density, distribution function, quantile function and random generation for the lambda prime distribution.

Usage

dlambdap(x, df, ncp, log = FALSE)
plambdap(q, df, ncp, lower.tail = TRUE, log.p = FALSE)
qlambdap(p, df, ncp, lower.tail = TRUE, log.p = FALSE)
rlambdap(n, df, ncp)

qlambdap(p, df, ncp, lower.tail = TRUE, log.p = FALSE)

dlambdap(x, df, ncp, log = FALSE)

rlambdap(n, df, ncp)

Arguments

df

the degrees of freedom of the distribution

ncp

the non-centrality parameter (t) of the distribution

lower.tail

logical; if TRUE (default), probabilities are \(P[X <= x]\), otherwise, \(P[X > x]\)

p

vector of probabilities

x, q

vector of quantiles

log, log.p

logical; if TRUE, probabilities / densities are given as logarithms

n

number of observations

Value

dlambdap gives the density, plambdap gives the distribution function (probabilities), qlambdap gives the quantile function, and rlambdap generates a random vector with lambda prime distributed values.

Details

These functions compute LeCoutre's Lambda prime \(\Lambda'\) distribution with df degrees of freedom (denoted df or \(\nu\)) and a non-centrality parameter (denoted ncp or t, and being the observed t-statistic). It is a continuous probability distribution that frequently arises in the sampling distribution of confidence limits for a normal mean and for inferences regarding signal-to-noise or standardized effect sizes. The distribution is generally asymmetric, and its shape adapts based on its parameters. When the non-centrality parameter (t) is zero, or if the degrees of freedom grow large (\(\chi^2_{df} / df \to 0\)), it reduces / converges to the standard normal distribution. Formally: $$\Lambda'_{df}(t) = z + t \sqrt{\chi^2_{df} / df}$$ The non-central t distribution is the non-centrality parameter \(\Lambda\) plus the standard normal z distribution, all divided by the square root of the usual chi-square distribution divided by the degrees of freedom: $$t'_{df}(\Lambda) = (\Lambda + z) / \sqrt{\chi^2_{df} / df}$$ A \(\Lambda'\) distributed random variable can be viewed as a confidence level on a non-central t (with the confidence intervals being computed as percent points of the \(\Lambda'\) distribution).

References

LeCoutre, B. (2007). Another look at confidence intervals for the noncentral t distribution. Journal of Modern Applied Statistical Methods, 6(1), 107–116. https://doi.org/10.22237/jmasm/1177992600

See also

t distribution functions: stats::dt(), stats::pt(), stats::qt(), and stats::rt().

Examples

set.seed(1)
dlambdap(11.1, df = 9, ncp = 10) # 0.1294471
#> [1] 0.1294471
plambdap(11.1, df = 9, ncp = 10) # 0.7134134
#> [1] 0.7134134
qlambdap(0.01, df = 9, ncp = 10) # 4.245347
#> [1] 4.245347
rv <- rlambdap(100, df = 50, ncp = 2)
mean(rv) # 2.077029
#> [1] 2.077029
pv <- plambdap(rv,  df = 50, ncp = 2)
summary(pv)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> 0.01715 0.31579 0.51642 0.52259 0.74839 0.99703 
#     Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#  0.01715 0.31579 0.51642 0.52259 0.74839 0.99703
qv <- qlambdap(pv,  df = 50, ncp = 2)
summary(qv)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#> -0.1677  1.5007  2.0319  2.0770  2.6726  4.7966 
#     Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#  -0.1677  1.5007  2.0319  2.0770  2.6726  4.7966
# absolute difference between the original random vector and
# the quantile vector calculated from the probabilities of the
# original random vector (< 1e-12)
max(abs(qv - rv)) # 0.0000000000002498002
#> [1] 2.498002e-13