Backgrounds: One of the best statistics for measuring variability in a qualitative variable is Kvalseth's Standard Deviation from the Mode (SDM ). It is capitalized to denote its population value (parameter) and written in lowercase to represent its sample value (statistic). The author applies the delta method to derive an asymptotic standard error and to obtain a Wald-type confidence interval. Although the calculation of sdm statistic is not complex, this measure is seldom used.
Aims: To disseminate and deepen the knowledge of the sdm statistic and to facilitate its use.
Objectives: 1) To determine its sampling distribution under the
normality hypothesis, 2) to compare the asymptotic and bootstrap standard
errors, and 3) to develop an R script for point and interval (asymptotic and
bootstrap) estimation of this measure.
Study Design: This is a methodological study using bootstrap
simulation.
Methodology: For the first objective, samples of 30 different
sizes (30, 50 to 950 in increments of 50, 1000 to 5000 in increments of 500,
and 10,000) were drawn from five binomial distributions and two multinomial
distributions. Using bootstrap, 210 sampling distributions of sdm statistics were generated. Symmetry was tested
with the D'Agostino test, mesokurtosis with the Anscombe-Glynn test, and
normality with the Shapiro-Francia and Kolmogorov-Smirnov-Lilliefors tests. For
the second objective, the Snedecor-Cochran F-test for variance equivalence was
used. The developed script was applied to a multinomial variable representing
migration motives.
Results: The hypothesis that the asymptotic sampling distribution
is normal was upheld. The asymptotic and bootstrap standard errors were very
similar, with an average absolute difference of 0.0020 and a maximum absolute
difference of 0.0199. Both standard errors showed convergence, as the
correlation between their absolute difference and sample size was less than or
equal to -0.8 in the seven bootstrap sampling distributions of the sdm statistic (across 7 types of distribution,
with 30 paired data corresponding to 30 sample sizes).
Conclusion: For samples of 1000 and above, the sampling
distribution of sdm statistic is normal, and its asymptotic error
is adequate. However, for smaller samples, it is better to use bootstrap to
compute the standard error and the bias-corrected and accelerated percentile
method to obtain the confidence interval. The developed R script allows
performing all these calculations, in addition to representing the sample
through a frequency table and two graphs.
Author
(s) Details
José Moral de la
Rubia
School of Psychology, Universidad Autónoma de Nuevo León, Monterrey,
Mexico.
Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v8/2812
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