The K² test could be one of the best tests for assessing
normality, yet its use is limited, likely because it is not commonly included
in standard statistical software, despite being implementable in R through the
moments package. Moreover, its asymptotic approximation has been questioned for
small samples, and no bootstrap version currently exists, even though it is
feasible in R. This simulation study aimed to: (1) verify the linear
independence and nonlinear relationship between √b₁ and b₂; (2) develop an R
script for the K² test in both asymptotic and bootstrap versions; (3) assess
the fit of the bootstrap distribution of the K² statistic to a chi-square
distribution with two degrees of freedom; (4) compare the power of both
implementations against non-normal distributions; and (5) contrast the
bootstrap version of K² with the Shapiro–Wilk test in small samples. A Monte
Carlo simulation with 10,000 replications was conducted, using 16 non-normal
distributions as alternative hypotheses and sample sizes (n) ranging from 20 to
2,000 in increments of 20. Linear independence and a parabolic relationship
between √b₁ and b₂ were confirmed, and the R script was verified to be
functional. The script is available for download as a Word document from a
GitHub repository. The bootstrap distribution of K² converged to a chi-square
distribution for n ≥ 120. The asymptotic version of K² and the Shapiro–Wilk
test showed greater power than the bootstrap version, except for mesokurtic
asymmetric distributions. Bootstrap implementation is recommended in these
cases for n < 120, while the asymptotic version is generally more powerful
and appropriate for n ≥ 20. The developed R script is highly useful for
assessing the normality assumption required by many parametric tests, such as
t-tests and F-tests for comparing means and variances, as well as for
characterising the distribution of a sample of quantitative data; therefore,
its use is recommended for these purposes.
Author(s) Details
José Moral de la
Rubia
School of Psychology, UANL, Mexico.
Please see the book here :- https://doi.org/10.9734/bpi/mcsru/v9/6969
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