Reciprocal Data Transformations and Their Back-Transforms

Abstract

Variable transformations have a long and celebrated history in statistics, one that was rather academically glamorous at least until generalized linear models theory eclipsed their nurturing normal curve theory role. Still, today it continues to be a covered topic in introductory mathematical statistics courses, offering worthwhile pedagogic insights to students about certain aspects of traditional and contemporary statistical theory and methodology. Since its inception in the 1930s, it has been plagued by a paucity of adequate back-transformation formulae for inverse/reciprocal functions. A literature search exposes that, to date, the inequality E(1/X) ≤ 1/(E(X), which often has a sizeable gap captured by the inequality part of its relationship, is the solitary contender for solving this problem. After documenting that inverse data transformations are anything but a rare occurrence, this paper proposes an innovative, elegant back-transformation solution based upon the Kummer confluent hypergeometric function of the first kind. This paper also derives formal back-transformation formulae for the Manly transformation, something apparently never done before. Much related future research remains to be undertaken; this paper furnishes numerous clues about what some of these endeavors need to be.

1. Introduction

Early comprehensive, fruitful statistical advances in normal curve (i.e., Gaussian distribution; e.g., [1]) theory, which benefits from the relative simplicity of its univariate and multivariate mathematical statistics, allowed it to dominate most sectors of statistical analysis methodology for many decades. The advent of its affiliated normal approximation power transformation technique [e.g., Box and Cox [2], who (especially p. 212) present a brief early history of data transformations, tracing these techniques back at least to 1937 (work by Bartlett), and crediting Tukey for considerable contributions about them prior to the publication of their classic Box–Cox paper; others they recognize include Ascombe, Kleczkowski, Moore, and Tidwell; Rojas-Perilla [3] provides an insightful contemporary update to their story] that extended its suitability to many of the hundreds of other univariate random variable (RV) distributions that exist (e.g., [4,5,6]) preserved its prominence until, for example, Nelder and Wedderburn’s formalization and implementation of generalized linear model (GLM; [7]) theory in the early 1970s [8]. Regardless of the data analysis specification error risks affiliated with approximations, recognition of especially normal curve theory’s pedagogic value continues to this day [9]).

Normal curve theory treats continuous interval/ratio measurement scale RVs over a (–∞, ∞) support domain, with Box–Cox [2] power and Manly ([10]; also see [11]) exponential transformations as well as other normal approximations (e.g., [12]) artificially expanding its practical applicability to more limited domains such as the truncated support [0, ∞). Griffith [13], for example, discusses RV transformations together with their accompanying back-transformations, employing fractional calculus to achieve such final results. A serious drawback of this approach is that it applies only to non-negative Box–Cox power transformation exponents. A study [14] using 2010 United States socio-economic/demographic census data, by census tracts (i.e., areal units), for both Dallas County, TX (529 tracts), and the Dallas-Fort Worth-Arlington Metropolitan Statistical Area (DFW MSA; 1324 tracts) containing it, reveals that roughly a third of the 70 (i.e., 35 × 2) selected but commonly utilized attributes measured as either percentages or densities—two time-honored standardization adjustments to geospatial and other aggregate data to minimize size effects—require a negative (i.e., inverse, reciprocal—one having a constant in its numerator and an algebraic expression in its denominator) rather than non-negative power transformation (

Table 1. Attribute counts for which reciprocal Box–Cox power (i.e., inverse polynomial) and Manly exponential transformations maximize the Shapiro–Wilk [16] normality diagnostic statistic ^†.

Geographic Landscape	(Y + δ)−γ		e−βY
	%	Density	%	Density
solely Dallas County	3	14	0	4
both	4	1	7	0
solely Dallas-Fort Worth-Arlington MSA	1	0	3	0

Note: δ denotes a translation/shift parameter, γ denotes a non-negative exponent (a value of zero implies the logarithmic transformation), and β denotes a negative exponential slope coefficient. Comments: no attribute renders a single inverse transformation type across all four specimen attribute variable categories; the LN transformation replaced exponents extremely close to zero (i.e., |γ| ≤ 0.01). ^† Royston [17] devised an algorithm that extends sample size diagnostics from 50 to 2000.

; also see Appendix A). The sizeable proportion of reciprocal transformations reported here testifies to the importance of establishing appropriate back-transformations for this case, too, with a focus on inverse moments rather than the more general inverted distributions (e.g., [15]).

2. Basic Concepts and Methodology

The central issue here concerns the inverse first moment (e.g., [18,19,20]). Although Stephan [21] derives E(l/Y) results for non-negative binomial RVs (i.e., Y = 0 does not exist) in the context of negative exponents, a broad interest in inverse moments barely predates Box and Cox, with the first published mentioning of this phraseology apparently appearing in 1962 (retrieved via a MATHSCINET search on 29 June 2022). Initial attention concentrated on continuous univariate RVs (e.g., [22]) because E(l/Y) does not exist for a discrete univariate RV Y mass function with non-zero mass at Y = 0. Nevertheless, Stephan [21] treats a modified binomial RV, and Kabe [23] devises an expression for truncated binomial and Poisson RV r^th-order inverse moments, with both continuous and discrete research themes being pursued throughout the subsequent decades (e.g., [24,25,26]). Meanwhile, the more recent literature reflects somewhat of a preoccupation with individual RVs (e.g., [27]).

Cressie et al. [28] highlight that the moment generating function of a RV holds information about both its positive and negative integer moments. Unfortunately, as Griffith [13] demonstrates for positive exponent Box–Cox transformations, most empirical transformations involve fractional moments. Regardless, the first relevant proposition is as follows: given certain regularity conditions, an inverse moment can be approximated by its inverse; i.e., E(l/Y) ≈ 1/E(Y). The critical condition is that E(Y) exists and is non-zero. Furthermore, the probability density/mass function support must be positive for E(Y) always to be real. These requirements are the reasons authors devote so much writing about this topic to positive RVs. However, inclusion of a translation (i.e., shift) term δ in a two-parameter transformation allows Y to take on zero, or even negative values, as long as the minimum Y values plus δ is positive. Within the context of maximum likelihood estimation, including a translation parameter δ creates the typical non-regular estimation problem in which the likelihood function becomes unbounded as this parameter approaches −y_min, the minimum RV Y sample value [29] (p. 185). Seber and Wild note that the maximum likelihood estimate of δ is −y_min, exacerbating this situation, and comment that a “satisfactory estimation procedure is needed” [30] (p. 72). An alternative part of the associated complication is that a nonlinear trade-off frequently exists between estimates of the power exponent γ and the translation parameters δ, whereas another is that the range of values for the modified RV depends upon the resulting estimate $\widehat{\mathsf{\delta }}$.

Within this preceding setting, Hu et al. [31] and Yang et al. [26] propose that, for nonnegative RVs Y, the inverse moment ${\left[\mathsf{\delta }+\mathrm{E}\left(\overline{\mathrm{Y}}\right)\right]}^{-\mathsf{\gamma }}$, where $\overline{\mathrm{Y}}$ is the sample mean, asymptotically approximates $\mathrm{E}\left[{\left(\mathsf{\delta }+\overline{\mathrm{Y}}\right)}^{-\mathsf{\gamma }}\right]$, if RV Y is suitably truncated and satisfies Rosenthal-type inequalities (i.e., specific relationships between moments of order higher than 2 and the variance of partial sums of RVs; [32] (p. 279))—given independent and real centered RVs X_i, i = 1, 2, …, n, for every positive integer n, if E(|X_i|^p) < ∞ for p > 1, where |●| denotes the absolute value of its argument represented by ●, then $\mathrm{E}\left(|{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}{|}^{\mathrm{p}}\right)\le 2^{{\mathrm{p}}^2}\mathrm{MAX}\left\{{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{E}\left({\left|{\mathrm{X}}_{\mathrm{i}}\right|}^{\mathrm{p}}\right),{\left[{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\mathrm{E}\left(\left|{\mathrm{X}}_{\mathrm{i}}\right|\right)\right]}^{\mathrm{p}}\right\}$. Acknowledging that many variants of the adage “a reciprocal moment approximates the reciprocal of that moment” exist, Garcia and Palacios [33] enumerate an additional sufficient condition required for it to be true. More specifically, they address a limit of the form.

$$\underset{\mathrm{n}\to \infty }{\lim }\frac{\mathrm{E}\left[{\left(\mathsf{\delta }+\overline{\mathrm{Y}}\right)}^{-\mathsf{\gamma }}\right]}{{\left[\mathsf{\delta }+\mathrm{E}\left(\overline{\mathrm{Y}}\right)\right]}^{-\mathsf{\gamma }}}=1$$

(1)

This limit holds when non-negative RV Y is expressible, at least asymptotically, as a standard normal RV. However, as Groves and Rothenberg [34] emphasize, the general relationship is given by

$$\mathrm{E}\left[{\left(\mathsf{\delta }+\overline{\mathrm{Y}}\right)}^{-\mathsf{\gamma }}\right] \ge {\left[\mathsf{\delta }+\mathrm{E}\left(\overline{\mathrm{Y}}\right)\right]}^{-\mathsf{\gamma }}$$

(2)

with the gap between the left- (LHS) and right-hand side (RHS) reciprocal polynomials sometimes being very substantial, and the foregoing discussion mostly absorbed by the (near-)equality instance. In addition, this equivalence is adequate only when its transformed distribution exhibits skewness and excess kurtosis of roughly zero (see Appendix A).

2.1. The Manly Back-Transformation for the Negative Exponential Function e^−βY

Conspicuously missing from the entire variable transformation literature is any debate about the inverse Manly transformation and its attendant back-transformation; perhaps surprisingly, the same can be said regarding its positive coefficient version (i.e., e^βY, β > 0; of the 140 empirical attribute variables constituting the database for this paper, six transformations were of this variety). Table 1 suggests that this oversight is problematic. For the inverse case of interest here, the back-transform arithmetic mean, ignoring its seemingly trivial imaginary part involving the Erfc function—the complementary error function defined by $\frac{2}{\sqrt{\mathsf{\pi }}}{{\displaystyle \int }}_{\mathrm{z}}^{\infty }{\mathrm{e}}^{-{\mathrm{t}}^2}\mathrm{dt}$ for argument z—is given by (see Appendix B for its derivation).

$$\frac{1}{2\mathsf{\beta }}\left\{0.577216+\mathrm{LN}\left(\frac{2}{{\mathsf{\sigma }}^2}\right)+\sqrt{\mathsf{\pi }}{\displaystyle \sum }_{\mathrm{k}=1}^{\infty }\frac{{\left(-\frac{{\mathsf{\mu }}^2}{2{\mathsf{\sigma }}^2}\right)}^{\mathrm{k}}}{\mathrm{k}\times \mathsf{\Gamma }\left[\frac{1}{2}+\mathrm{k}\right]}\right\}$$

(3)

where LN denotes natural logarithm, and, respectively, μ and σ, are the mean and the standard deviation of the ideal normal distribution approximated by an inverse Manly transformation. The individual conditional expectations are given by substituting each original transformed value, in turn, for μ in Equation (3).

Table 2. Selected Manly transformed 2010 percentage attribute variables experiencing improvement of their individual correspondences with a Gaussian probability density function.

Dallas County Attribute	Shapiro–Wilk		Raw Data			Back-Transformed Data: Equation (3)
	Y	Y*	ymin †	$\overline{\mathbf{y}}$	ymax	${\mathbf{y}}_{\mathbf{min}}^{\ast }$	$\overline{{\mathit{y}}^{\ast }}$	${\mathbf{y}}_{\mathbf{max}}^{\ast }$
20–29 years of age	0.84	0.97	0	15.94	58.07	1.93	15.85	47.91
30–39 years of age	0.94	0.96	0	15.48	31.41	4.68	15.47	32.22
50–64 years of age	0.94	0.99	0	16.26	56.00	4.11	16.23	42.17

^† the Dallas-Fort Worth International Airport covers an unpopulated census tract, causing the lowest percentage to be zero; respectively, the 2nd-smallest entries are 0, 4.74, and 5.22; Y* denotes the transformed version of RV Y. Note: ${\mathrm{y}}_{\min }^{\ast }$ and ${\mathrm{y}}_{\max }^{\ast }$, respectively, are the (1−3/8)/(n+1/4) and (n−3/8)/(n+1/4) extreme quantile values, n = 529 [35] (p. 176).

tabulates computations for an illustrative application of Equation (3). Following guidelines advocated in Griffith [13], the nearly identical raw and back-transformed arithmetic means imply the presence of little data analysis specification error attributable to employing a normal approximation transformation. Furthermore, for the most part, the reported extremes and their corresponding conditional back-transformed means [based upon the quantiles Blom [35] promotes (see Table 2) imply that these Manly transformations also essentially preserve the ranges of the raw attribute values.

As an aside, for a non-reciprocal Manly transformation, the first moment expected value given by Equation (3) simply has a sign change.

2.2. The Box–Cox Back-Transformation for the Inverse Power Function (Y + δ)^−γ

The inverse case of interest here preoccupying applied statisticians and other researchers in their relevant literature writings argues for some form of E(Y*) = 1/E(Y), where variable Y* denotes a Box–Cox inverse transformation. Now this back-transform arithmetic mean, ignoring the imaginary part in the calculation reported by Mathematica 12.3—this outcome seems to be an artifact of the software’s symbolic manipulations (e.g., [36])—is given by (see Appendix B for its derivation).

$$-\mathsf{\delta }+\frac{1}{\sqrt{\mathsf{\pi }}}{2}^{-1-\frac{1}{2\mathsf{\gamma }}}{\mathsf{\sigma }}^{-1/\mathsf{\gamma }}\left\{\left[1+{\left(-1\right)}^{\frac{1}{\mathsf{\gamma }}}\right]\mathsf{\Gamma }\left[\frac{\mathsf{\gamma }-1}{2\mathsf{\gamma }}\right]\times \left[{\displaystyle \sum }_{\mathrm{k}=0}^{\infty }\frac{\sqrt{\mathsf{\pi }} \mathsf{\Gamma }\left[\frac{1}{2\mathsf{\gamma }}+\mathrm{k}\right]}{\mathsf{\Gamma }\left[\frac{1}{2\mathsf{\gamma }}\right] \mathsf{\Gamma }\left[\frac{1}{2}+\mathrm{k}\right]}\frac{{\left(-\frac{{\mathsf{\mu }}^2}{2{\mathsf{\sigma }}^2}\right)}^{\mathrm{k}}}{\mathrm{k}!}\right]+\frac{\sqrt{2}\mathsf{\mu } \left[1-{\left(-1\right)}^{\frac{1}{\mathsf{\gamma }}}\right] \mathsf{\Gamma }\left[1-\frac{1}{2\mathsf{\gamma }}\right]\times \left[{{\displaystyle \sum }}_{\mathrm{k}=0}^{\infty }\frac{\sqrt{\mathsf{\pi }} \mathsf{\Gamma }\left[\frac{1+\mathsf{\gamma }}{2\mathsf{\gamma }}+\mathrm{k}\right]}{2 \mathsf{\Gamma }\left[\frac{1+\mathsf{\gamma }}{2\mathsf{\gamma }}\right] \mathsf{\Gamma }\left[\frac{3}{2}+\mathrm{k}\right]}\frac{{\left(-\frac{{\mathsf{\mu }}^2}{2{\mathsf{\sigma }}^2}\right)}^{\mathrm{k}}}{\mathrm{k}!}\right]}{\mathsf{\sigma }}\right\}$$

(4)

where Γ[•] denotes the standard gamma function with argument •. This expression resembles Equation (3), chiefly because it includes the same type of infinite summations.

Table 3. Selected Box–Cox power transformed 2010 density attribute variables experiencing improvement of their individual correspondences with a Gaussian probability density function.

Attribute	Shapiro–Wilk		Raw Data			$\overline{{\mathit{y}}^{\ast }} \mathbf{Back-Transformed} \mathbf{Mean}{ }^{‡}$
	Y	Y*	ymin †	$\overline{\mathbf{y}}$	ymax	X	via RHS Equation (2)	Equation (4)
Dallas County
occupied housing units	0.715	0.990	0	7.94	62.72	(Y+15.58)−1.97	6.02	7.82
miscellaneous races/ethnicities	0.443	0.997	0	1.38	41.43	(Y+0.03)−0.03	0.66	1.40
20–29 years of age	0.620	0.997	0	3.76	42.07	(Y+0.61)−0.23	2.31	3.86
DFW MSA
occupied housing units ⁑	0.726	0.975	0	5.89	62.72	(Y+10.09)−1.08	4.49	5.89

^† the DFW International Airport covers an unpopulated census tract, creating a zero percentage; ^‡ back-transformed quantities are not guaranteed to be non-negative; ^⁑ included here because it is the sole metroplex density variate for this Box–Cox transformation; Y* denotes the transformed version of RV Y.

tabulates computations for an illustrative application of Equation (4). Again, following guidelines advocated in Griffith [13], the nearly identical raw and back-transformed means imply the presence of little data analysis specification error attributable to employing a normal approximation.

Table 3 results based upon Equation (2) demonstrate the potential superiority of the proposed Box–Cox back-transformation arithmetic mean expression vis à vis contemporary conceptualizations. Evidence supporting Equation (4), beyond that summarized in Appendix B, merits more intensive future scrutiny and research, particularly with regard to the efficacy of ignoring its imaginary part.

3. Applications: More Specimen Empirical Illustrations

Preceding sections present empirical findings for seven of the 49 inverse transformations (see Appendix A) identified for 140 (= 2 × 2 × 35) attribute variables selected from the 2010 US census for either Dallas County or the DFW MSA.

Table 4. Selected Manly transformed 2010% and density attribute variables experiencing improvement in their individual correspondences with a Gaussian probability density function.

Attribute	Shapiro–Wilk			%				Density
	Y→Y*			Dallas County		DFW MSA		Dallas County
	DC%	DFW%	Density	$\overline{\mathbf{y}}$	Equation (3)	$\overline{\mathbf{y}}$	Equation (3)	$\overline{\mathbf{y}}$	Equation (3)
owner occupied			0.96→0.98					3.24	3.24
20–29 years of age	0.84→0.97	0.83→0.99		Table 2		14.70	14.62
30–39 years of age	0.94→0.96	0.95→0.98		Table 2		14.93	14.92
40–49 years of age		0.958→0.962	0.83→0.99			14.58	14.58	2.73	2.72
50–64 years of age	0.94→0.99	0.97→0.997	0.95→0.99	Table 2		16.91	16.91	2.98	2.98
65+ years of age			0.95→0.99					1.62	1.62
manufacturing employment	0.97→0.995	0.97→0.998		9.54	9.54	10.92	10.92
wholesale employment		0.96→0.99				3.57	3.57
retail employment	0.92→0.99	0.96→0.995		10.87	10.86	11.37	11.37
professional employment	0.95→0.995	0.95→0.997		13.32	13.32	12.18	12.20
education employment		0.992→0.997				17.70	17.71
miscellaneous employment	0.90→0.996	0.92→0.997		5.45	5.45	5.09	5.09

Note: DC denotes Dallas County; no DFW MSA attribute variable densities underwent this transformation; gray denotes alternative or no required attribute variate transformation; bold italic font denotes extreme alignment improvements; Y* denotes the transformed version of RV Y.

compilation uncovers a strong tendency for Manly and Box–Cox inverse transformations to be competitive in situations for which the exponent γ is relatively large in absolute value (i.e., |γ| > 2); for example, the percentage of retail employment, whose respective goodness-of-fit error sums of squares (ESSs) are 5.48 and 5.94 [with an accompanying total sum of squares (TSS) of 525.8] yields an exponent of −8.44, well below the lower limit of −2 in Tukey’s [37] transformation ladder of reasonable powers (ranging from −2 to 2).

Table 4 furnishes numerical outcomes extremely supportive of this aforementioned contention. All back-transformed arithmetic means are nearly identical to their raw data counterparts, implying the presence of little data analysis specification error attributable to employing a normal approximation transformation. This type of conclusion almost always is the expectation when the mean percentage is roughly 50; in the suite of cases investigated here, percentages range from roughly 3% to 18%, which are substantially less than 50%. One reason these consequences may appear so good is that the worst raw data Shapiro–Wilk (S-W) statistic is 0.83, which is low but not excessively low; one raw data diagnostic statistic is 0.992, which is significantly less than one, but reflects considerable symmetry (i.e., its companion skewness measure is 0.31, which improves to 0.01 with the Manly transformation), and a distributional form approaching a bell-shaped curve.

Figure 1 portrays the two extreme specimens, with regard to their S-W normality diagnostic statistics, appearing in Table 4. The transformed plots are inversely related to their affiliated raw data plots, by construction. Although both raw data diagnostic statistics are significantly less than one, these graphics disclose noticeably better alignment for the 0.83→0.99, and questionably better alignment for the 0.992→0.997, increase in S-W cases. Regardless, in both instances, Equation (3) furnishes an excellent back-transformation as judged by a comparison of the raw and back-transformed data arithmetic means.

Table 5. Selected Box–Cox power transformed 2010% and density attribute variables experiencing individual Gaussian probability density function correspondence improvement.

Attribute	Shapiro–Wilk			%				Density
	Y→Y*			Dallas County		DFW MSA		Dallas County
	DC%	DFW%	Density	$\overline{\mathbf{y}}$	Equation (4)	$\overline{\mathbf{y}}$	Equation (4)	$\overline{\mathbf{y}}$	Equation (4)
some college			0.82→0.99					2.29	2.31
associate degree	0.97→0.99			5.31	5.31
no public assistance			0.73→0.99					7.68	7.64
occupied housing units			0.72→0.99					Table 3
vacant housing units	0.89→0.996	0.88→0.999	0.55→0.996	9.26	9.29	8.27	8.27	1.01	1.09
under 20 years of age			0.71→0.996					6.01	6.01
20–29 years of age			0.62→0.997					Table 3
30–39 years of age			0.70→0.996					3.42	3.48
65+ years of age	0.91→0.998			9.33	9.33
retail employment			0.70→0.99					1.10	1.10
transportation employment			0.67→0.99					0.51	0.51
financial employment	0.95→0.996		0.68→0.99	9.22	9.23			0.94	0.96
professional employment			0.63→0.99					1.46	1.49
education employment			0.68→0.99					1.57	1.56
arts employment	0.94→0.997	0.93→0.997	0.63→0.997	9.11	9.12	8.34	8.35	1.05	1.05
miscellaneous employment			0.57→0.996					0.57	0.57
public admin. Employees	0.63→0.97	0.78→0.98		2.75	2.73	3.09	3.08
Hispanic		0.85→0.98				27.60	28.30
miscellaneous race/ethnicity	0.66→0.99	0.72→0.997	0.44→0.997	6.68	6.82	7.33	7.39	Table 3

Note: DC denotes Dallas County; only the DFW MSA occupied housing units attribute variable densities underwent this transformation (see Table 3); gray denotes alternative or no required attribute variate transformation; bold italic font denotes extreme alignment improvements; Y* denotes the transformed version of RV Y.

also furnishes extremely supportive numerical outcomes. Although not as similar as the Manly pairings, all Box–Cox back-transformed arithmetic means are nearly identical to their raw data counterparts, again implying the presence of little data analysis specification error attributable to employing a normal approximation transformation. In addition, Table 5 compilation reveals a strong tendency for Box–Cox logarithmic and inverse transformations to be competitive in situations for which the exponent γ lies in the interval [0, −0.1]. For example, the Dallas County associate degree percentage variable has goodness-of-fit ESSs of 0.9700 for the logarithmic, and 0.9622 for the Box–Cox negative power ($\widehat{\mathsf{\gamma }}\approx -0.43$), transformations (TSS = 523.8); however, $\widehat{\mathsf{\gamma }}$ is not sufficiently close enough to zero to justify replacing this latter with this former transformation.

Figure 2 portrays the two extreme specimens, with regard to their S-W normality diagnostic statistics, appearing in Table 5. As before, the transformed plots are inversely related to their affiliated raw data plots, which is by construction. Although both raw data diagnostic statistics are significantly less than one, these graphics disclose noticeably better alignment for the 0.44→0.997, and modestly better alignment for the 0.97→0.99, increase in S-W cases. Regardless, in both instances, Equation (4) furnishes an excellent back-transformation as judged by a comparison of the raw and back-transformed data arithmetic means.

In summary, the back-transformations proposed in this paper perform extremely well across a wide range of arbitrarily selected variates. The Manly negative exponential back-transformation seems to accomplish its goal better than the Box–Cox negative power back-transformation. Nonetheless, both appear to be superior to the Equation (2) proposition frequently endorsed, studied, and presumably applied in the literature. The average absolute error for the 49 specimen variables is roughly 1%, with a maximum of slightly less than 7%. Figure 3 portrays features of these errors, which overwhelming ratify Equations (3) and (4); see Appendix Figure A1 for a more comprehensive visualization.

4. Discussion

Normal curve theory no longer enjoys the statistical methodology dominance it held prior to the advent of GLM theory and practice. Yet, a perusal of introductory mathematical statistics textbooks divulges that teaching about variable transformations is customary. This is an excellent place in a curriculum to treat normal RV back-transformations. After all, as Lesch and Jeske [9] (p. 277) point out, “Although the modern computing environment [coupled with mathematical statistics advances] has obviously alleviated the necessity of [a normal] approximation, it is still both historically relevant and quite insightful from an instructional perspective.” In keeping with this contention, the assessment presented in this paper urges future research pursuits addressing normal back-transformations for inverse RVs. Evidence provided in it contends that the Manly transformation, coupled with its accompanying back-transformation, exhibits considerable promise, especially for large negative power exponent values; the Manly transformation appears to preserve the Tukey power exponents ladder and augment its two ends, replacing these exponents when they become too extreme, a notion consistent with both parsimony and the use of an ESS criterion to help decide upon a particular transformation (i.e., Manly or Box–Cox power).

Given the preceding materials, at this time, the five ensuing themes of this section merit more thorough discussion to complete this paper.

4.1. The Inverse Back-Transformation Conceptualization

To date, reliable general inverse back-transformations continue to be a tool eluding applied statisticians, even after the emergence of a sizeable literature seeking these instruments. Conceivably, Equation (2) represents the prevailing best case scenario; unfortunately, Table 3 documents that this option can supply poor results. Furthermore, Manly [10] formulated an additional transformation that has been, and is, all but ignored in practice. One appealing advantage of his construction is that it substitutes for more extreme Box–Cox power exponents whose data calculations generate massively large or minutely small numerical values. An important contribution here is the derivation of the back-transformation for Manly’s invention.

GLM theory furnishes another crucial modern-day component to understanding data transformations and their back-transformations. Initially, the only option was to work with normal curve theory. Today, side-by-side analyses completed with it and the appropriate GLM technique allow a detailed examination of how well a transformation-based normal curve theory approach works. This type of insight can become indispensable in large or massive data settings. GLM estimation often requires an iteratively reweighted least squares routine, which essentially involves repetition of calculus-guided estimation, whereas a normal approximation might allow a linear regression substitution, dramatically reducing daunting computational demands and burdens to solve a problem.

Table 6. Comparative GLM results for certain specimen attribute variables.

Table 2 and Table 3 Attribute			$\overline{\mathit{y}}$	via RHS Equation (2)	Equations (3) and (4)	GLM
						Beta- Binomial	Gamma-Poisson
per cent	Dallas County	20–29 years of age	15.94		15.85	15.60
		30–39 years of age	15.48		15.47	15.43
		50–64 years of age	16.26		16.23	15.90
density		occupied housing units	7.94	6.02	7.82		7.94
		miscellaneous races/ethnicities	1.38	0.66	1.40		1.37
		20–29 years of age	3.76	2.31	3.86		3.75
	DFW	occupied housing units	5.89	4.49	5.89		5.88

summarizes illustrative GLM estimation output for the variates appearing in Table 2 and Table 3.

Georeferenced data tend to be extraordinarily overdispersed. Accordingly, Table 6 tabulates calculations that utilized beta-binomial parametric mixture regression, and gamma-Poisson parametric (i.e., negative binomial) mixture, rather than Poisson, regression to accommodate for any excess variation. The reported GLM estimates further corroborate the validity of Equations (3) and (4).

4.2. Some Mathematics Underlying the Inverse Back-Transformations

Griffith [13] derives positive Box–Cox power exponent back-transformation formulae using fractional calculus (with a detailed appendix overview of this topic; e.g., [38]). These derivations encompass complicated, sophisticated sums having arguments written as powers of and ratios containing μ and σ combined with gamma functions.

Not surprisingly, then, Equations (3) and (4) build upon similar complex arithmetic operations. The Kummer confluent hypergeometric function, a degenerate mathematical construct introduced in the early 1800s [39], has two of its three regular singular points merge into an irregular singularity (hence, the term confluent in its description), and is the solution to the following differential equation:

$$\mathrm{z}\frac{{\mathrm{dw}}^2}{{\mathrm{d}}^2\mathrm{z}}+\left(\mathrm{b}-\mathrm{z}\right)\frac{\mathrm{dw}}{\mathrm{dz}}-\mathrm{aw}=0$$

where d denotes derivative, with a regular singular point at z = 0 and an irregular singular point at z = ∞. This solution is reminiscent of the fractional calculus form, containing the general summation

$${\displaystyle \sum }_{\mathrm{n}=0}^{\infty }\frac{{\left(a\right)}_n{\mathrm{z}}^{\mathrm{n}}}{{\left(b\right)}_n\mathrm{n}!}$$

where the Prochhammer symbol ${\left(\mathrm{x}\right)}_{\mathrm{n}}$ denoting the rising factorial x(x−1) (x−2)…(x − n+1. Its integral representation is given by (after Abramowitz and Stegun [40] (p. 505)),

$$\frac{\mathsf{\Gamma }\left(1/2\right)}{\mathsf{\Gamma }\left(1/2-\mathrm{a}\right)\mathsf{\Gamma }\left(\mathrm{a}\right)}{{\displaystyle \int }}_0^1{\mathrm{e}}^{-\frac{{\mathsf{\mu }}^2\mathrm{t}}{2{\sigma }^2}}{\mathrm{t}}^{\mathrm{a}-1}{\left(1-\mathrm{t}\right)}^{-\mathrm{a}-1/2}\mathrm{dt}$$

whereas, after taking the first partial derivative of its numerator with respect to a and then setting a to 0 for Equation (3), the numerator becomes

$${{\displaystyle \int }}_0^1\frac{{\mathrm{e}}^{-\frac{{\mathsf{\mu }}^2\mathrm{t}}{2{\mathsf{\sigma }}^2}}}{\mathrm{t} \sqrt{1-\mathrm{t}}}\mathrm{dt}$$

Its final solution has the imaginary term Erfc[μ/σ√2]πi, whose contribution to Equation (3) appears to be rather trivial (e.g.,

Table 7. Complex solutions to Equations (3) and (4) for specimen attribute variables.

Table 2 and Table 3 Attribute			Real	Imaginary	Transformed σ
per cent	Dallas County	20–29 years of age	15.8513	−0.0531i	0.1468
		30–39 years of age	15.4698	−0.0000i	0.0816
		50–64 years of age	16.2338	−0.0000i	0.1042
density		occupied housing units	7.8183	0.4051i	0.0009
		miscellaneous races/ethnicities	1.4017	−0.0000i	0.0367
		20–29 years of age	3.8601	0.0065i	0.1321
	DFW	occupied housing units	5.8898	0.0772i	0.0152

NOTE: i denotes the imaginary number iota

; the magnitude of the complex number essentially is its real part), and thus has been discarded here.

Meanwhile, Equation (4) embraces two specific Kummer confluent hypergeometric functions, the first with a = 1/(2γ) and b = 1/2, and the second with a = (1+γ)/(2γ) and b = 3/2, each pair of which substitutes into

$$\frac{\mathsf{\Gamma }\left(\mathrm{b}\right)}{\mathsf{\Gamma }\left(\mathrm{b}-\mathrm{a}\right)\mathsf{\Gamma }\left(\mathrm{a}\right)}{{\displaystyle \int }}_0^1{\mathrm{e}}^{-\frac{{\mathsf{\mu }}^2\mathrm{t}}{2{\sigma }^2}}{\mathrm{t}}^{\mathrm{a}-1}{\left(1-\mathrm{t}\right)}^{\mathrm{b}-\mathrm{a}-1}\mathrm{dt}$$

Together, these mathematical functions are the source of the imaginary part for Equation (4), which accordingly is twofold: (−σ)^1/γ and −(−σ)^1/γ−1. These two terms are not totally ignorable, jointly or separately, although their final composite imaginary part seems to be. This particular conjecture warrants future scrutiny and research.

Wolfram Mathematica 12.3, for example, implements the Kummer confluent hypergeometric function for both symbolic and numerical manipulations (see https://reference.wolfram.com/language/ref/Hypergeometric1F1.html for its operationalization in Mathematica 12.3 (accessed on 6 July 2022)). Support for this latter maneuver comprises arithmetical evaluation to arbitrary numerical precision. Furthermore, this function’s executable capabilities include automatic cycling through lists of values, such as those comprising a transformed dataset in need of back-transforming. Its principal shortcoming is that it can encounter under- and over-flow calculation warnings and failures, as the next section shows.

4.3. The Specimen Empirical Example

A principal objective of the specimen data examined in this paper is to exemplify the relatively large number of times applied statisticians can encounter the necessity for adopting inverse transformations during normal curve theory exercises with their own data. The literature seems to lack any narratives about Manly back-transformations in general, let alone explanations directing their use for inverse (i.e., negative exponential) transformation cases. This paper not only fills that knowledge gap, but it also furnishes more definitive and rigorous Box–Cox inverse back-transformations. The benchmark here is a comparison of raw data and back-transformed arithmetic means (see [13]). However, Fisher’s [41] probability integral transform together with Angus’s [42] quantile function theorems, which may be stated as follows, enable one of its extensions to an entire dataset: for data values constituting any attribute variable transformable to a formal RV (e.g., the normal), this transformation is exact if the underlying distribution is the true one, and approximate in large samples if the distribution was fitted to these data. This theory is the foundation sustaining the extreme back-transformed values reported in Table 2, which build upon Blom’s [35] uniform-based systematic sample spanning a probability density function support.

Table 8. Summary statistics calculated with complete dataset back-transformations.

Table 2 and Table 3 Attribute		$\overline{\mathit{y}}$		$\tilde{\mathit{y}}$		sy		Skewness		Excess Kurtosis
		Ob-Served	A	Ob-Served	A	Ob-Served	A	ob-Served	A	Ob-Served	A
			S		S		S		S		S
per cent	Dallas County
	20–29 years of age	15.94	17.36	14.25	15.85	7.83	7.97	2.00	1.04	6.11	1.20
			15.77		14.63		7.09		1.43		4.23
	30–39 years of age	15.48	15.77	14.81	15.47	4.37	4.34	0.90	0.42	2.04	0.30
			15.47		15.18		4.25		0.40		0.27
	50–64 years of age	16.26	16.83	15.48	16.23	5.40	5.50	1.32	0.76	5.79	1.20
			16.24		15.73		5.18		0.64		0.80
den-sity	occupied housing units ‡	7.94	8.68	6.20	7.60	7.37	5.02	2.96	0.61	12.23	−0.55
			7.76		6.06		7.36		4.56		36.02
	miscellaneous races/ethnicities †	1.38	2.82	0.62	1.34	2.52	4.90	8.77	5.87	123.72	51.12
			1.38		0.65		2.41		5.87		51.13
	20–29 years of age ⁑	3.76	7.34	2.38	3.77	4.74	10.08	3.68	3.05	18.17	10.71
			3.78		2.33		5.24		5.85		53.87
	DFW MSA
	occupied housing units §	5.89	7.77	4.88	6.00	6.01	6.15	3.22	1.29	16.57	1.26
			5.86		4.49		6.20		5.04		53.05

^‡ the back-transformed imaginary part ranges in magnitude from; 1.7 × 10⁻⁵ to 16.8; the three smallest and thirteen largest value back-transforms failed correct computations. ^† the back-transformed imaginary part ranges in magnitude from 1.7 × 10⁻¹³ to 2.8 × 10⁻¹⁶. ^⁑ the back-transformed imaginary part ranges in magnitude from 4.5 × 10⁻¹¹ to 75.4; the smallest and four largest value back-transforms failed correct computations. ^§ the back-transformed imaginary part ranges in magnitude from 1.0 × 10⁻⁵ to 25.8; the 34 smallest and 11 largest value back-transforms failed correct computations. Note: A denotes analytical, calculated with Equations (3) or (4); S denotes simulation based upon 10,000 replications.

continues inspections initiated with Table 2 and Table 3; the left-hand amount in each column is the observed quantity, whereas the right-hand stack is the analytical algebraic Equations (3) and (4) quantity on the top, and the simulated parametric resampling quantity on the bottom. Numerical failures of Mathematica 12.3 in the outermost tails of normal probability density functions (see Table 8 notes) prompted verification by simulation. Of note is that Box–Cox transformations creating small means and variances may suffer from numerical distortions during their back-transformations, requiring this type of remedial intervention The protocol for this paper was to draw a systematic sample of values based upon the Blom [35] calculated CDF percentages, namely (r_i − 3/8)/(n + 1 + 4). This strategy failed for occupied housing units and Dallas County 20–29 years of age densities, because they involve extreme cumulative percentages that are excessive outliers in the normal distribution tails. Its replacement strategy was to draw 10,000 random samples of size n (= 529 or 1324) from a posited ideal normal probability distribution, rejecting negative values (<0.38% of the selections for one, and none for another Dallas County attribute variable; <0.04% for the DFW MSA variate), sort them in ascending order, and then compute a back-transformation using Mathematica 12.3 for each of the n summary averages. This procedural switch causes differences between certain Table 3 and Table 8 entries. One outcome is a modest number of negative values (e.g., smallest) and non-monotonicity in the very largest (e.g., misrepresentations attributable to underflow calculations), miscalculations not certified by the simulation exercises. In addition, because they are conditional means, this complication motivating a trimming (i.e., similar to data Winsorizing) of these inadmissible values is in keeping with back-transformed values shrinking toward their mean.

Table 8 highlights possible back-transformation confusion between the mean and the median, with reference to a data analysis specification error appraisal criterion, because the ideal transformed RVs are flawlessly Gaussian, and hence these two quantities are the same. Figure 3a portrays a near-perfect matching that this table convincingly contradicts, both with analytical and with replication simulation displays. Rather, it endorses the Equation (3) Manly back-transformation, while raising serious questions about any general improvements Equation (4) might offer Box–Cox back-transformations vis à vis the RHS of Equation (2); this deficiency may be an artifact of simply ignoring the imaginary part of the complex number solutions generated by the Kummer confluent hypergeometric function. In other words, the Box–Cox inverse back-transformation comparisons here signify a potential for its use to introduce moderate-to-severe specification error into a data analysis. In general, Table 8 standard error tabulations are consistent with shrinkage conjectures, whereas, more or less, skewness and kurtosis tabulations are consistent with smoothing expectations. In a nutshell, Table 8 results imply a need for considerable comparative future research.

4.4. Alternative Transformations

The Box–Cox power and Manly exponential data transformations are not unique; Yeo–Johnson [12] transformations, for example, do not complete the set of possibilities, either. History reveals that alternatives exist for especially proportions and percentages, two of the most popular being the logit and the arcsine, this latter being the target of some derision (e.g., [43]).

The logit transform is given by the natural logarithm LN[p/(1 − p)], where 0 < p < 1 is an empirical probability, equivalent to a percentage (when multiplied by 100). It maps probability values in the interval (0, 1) {\displaystyle (0,1)} (0, 1) to real numbers in the range (−∞, +∞) {\displaystyle (−\infty, +\infty)} (–∞, ∞), paralleling the real number support for the normal probability density function. One constraining weakness of this conceptualization is that p ≠ 0, 1. Therefore, its slightly more general form may be written as LN[(p + Δ)/(1 − p + 2Δ)], Δ > 0, which allows 0 ≤ p ≤ 1; it also may be written as LN{k(p + Δ)/[k(1 − p + 2Δ)]}, where k = 100 is usual (i.e., the values become percentages), and k = 1 in the preceding empirical probabilities example. Its back-transformation is 1/(1 + e^−x). Meanwhile, the inverse for this function is LN[(1 − p)/p], with a back-transformation of e^−x/(1 + e^−x). In other words, the notion of an inverse transformation is inconsequential in this context, because estimation is either for p or for (1 − p). Furthermore, it directly relates to binomial regression (see Table 6).

Table 9. Selected specimen attribute summary statistics for the logit back-transformation.

Table 2 Attribute	Raw Data	Manly Transformation	Logistic Transformation
	S-W	S-W	S-W	Δ	Back-Transformation E(Y)
20–29 years of age	0.83346	0.97257	0.93988	3	14.90
30–39 years of age	0.93624	0.96403	0.95726	11	15.22
50–64 years of age	0.93457	0.99154	0.97782	6	15.79

Note: S-W denotes the Shapiro–Wilk normality diagnostic statistic.

documents that this variable transformation is not uniformly better than those studied in this paper (e.g., its S-W falls between the raw and the Manly transformed outcomes). In addition, indications from evidence conveyed in Table 6 are that it may well be inferior to its comparable precedingly reflected upon beta-binomial operationalization or Equation (3) output.

4.5. Alternative RV Specifications

Not only do alternative transformations exist, but alternative RV specifications also exist. Perhaps the logarithm is the one deserving the most consideration and contemplation when it competes with an inverse Box–Cox transformation with a power exponent within the interval (0, −0.10); Vélez et al. [44] establish a more precise case-specific lower bound via confidence intervals (CIs) for $\widehat{\mathsf{\lambda }}$, accompanied by the standard criterion based upon whether or not zero falls within a CI. Its back-transformation is well-known to be ${\mathrm{e}}^{\mathsf{\mu }+\frac{{\mathsf{\sigma }}^2}{2}}$; fortunately, analytical formulae exist for all of its entries in Table 9. The other competition previously mentioned is between the Manly negative exponential and the Box–Cox power exponent of −γ < −2 transformations; the Box–Cox option in this latter case automatically should revert to its Manly competitor on the basis of numerical—for example underflow—difficulties alone.

Table 10. Equation (4) and Box–Cox LN specimen attribute back-transformation comparisons.

Attribute	$\overline{\mathbf{y}}$		$\tilde{\mathbf{y}}$		sy		Skewness		Excess Kurtosis
	Equation (4)	LN	Equation (4)	LN	Equation (4)	LN	Equation (4)	LN	Equation (4)	LN
	Actual		Actual		Actual		Actual		Actual
Dallas County
% arts employment (γ = −0.03)	9.13	9.12	8.27	8.25	5.16	5.18	1.14	1.15	2.37	2.46
	9.11		8.15		5.14		0.99		1.14
% miscellaneous race/ethnicity (γ = −0.08)	7.19	6.63	4.03	4.12	11.28	8.38	6.04	5.63	63.10	87.66
	6.68		3.96		8.06		3.37		16.74
miscellaneous race/ethnicity density (γ = −0.03)	1.40	1.37	0.65	0.66	2.69	2.51	6.68	11.23	73.59	523.76
	1.38		0.62		2.52		8.77		123.72
arts employment density (γ = −0.04)	1.06	1.05	0.59	0.59	1.56	1.52	5.20	6.88	47.39	145.41
	1.05		0.62		1.41		3.67		20.17
miscellaneous employment density (γ = −0.08)	0.57	0.57	0.36	0.36	0.72	0.68	4.47	4.67	36.94	55.09
	0.57		0.37		0.76		5.95		61.19
DFW MSA
% financial employment (γ = −0.01)	8.95	8.93	8.24	8.22	4.96	4.96	0.93	0.95	1.45	1.63
	8.92		8.16		4.95		0.92		1.52
% vacant houses (γ = −0.05)	8.26	8.27	7.14	7.15	5.16	5.16	1.68	1.70	4.94	5.57
	8.27		7.27		5.18		1.76		5.77
% miscellaneous race/ethnicity (γ = −0.06)	7.38	7.32	4.94	5.00	8.29	7.87	4.30	4.28	32.60	44.58
	7.33		4.82		7.54		2.86		12.39
% Hispanic (γ = −0.09)	28.29	27.88	20.12	20.51	28.50	25.82	4.05	3.46	29.45	26.91
	27.64		19.24		22.07		1.22		0.62

contents corroborate the exchangeability of the Box–Cox logarithmic and Equation (4) results when a negative power exponent is close to 0; Griffith [13] accentuates this point for its mirror positive γ interval (0, 0.10). Both back-transformations furnish competitive and reasonably accurate mean, median, and variance estimates. In contrast, because of smoothing effects induced by a transformation and its subsequent back-transformation, skewness and kurtosis frequently undergo the kinds of alterations that materialized in Table 10. One valuable insight and takeaway from this extended discussion is that parsimony is a useful concurrent criterion when selecting a data transformation, a contention alluded to by the Tukey ladder of powers. The newly stated analytical back-transformation solution provided by Equations (3) and (4) forge this as well as other new comprehensions about variable transformations.

5. Final Remarks

In conclusion, a cadre of statistical methodologist have been and are obsessed with trying to compel inverse/reciprocal/negative back-transformations to adhere to the functional form E(1/X) ≈ 1/E(X). However, disappointing sequels to their efforts often follow the application of this specific answer prototype, to which certain Table 3 and Table 6 entries attest. Nonetheless, determining such a solution is very important in general because many empirical attribute variables appear to require a transformation containing a negative exponent in order to improve, for example, their frequency distribution alignment with a bell-shaped curve, or stabilize their variance. One of the most important contributions of this paper is the pair of Equations (3) and (4), which furnish a solution defying the quest to exploit the relationship E(1/X) ≈ 1/E(X). Its accompanying critical implication is that the Kummer confluent hypergeometric function of the first kind supplies the necessary formula to excogitate an appropriate, accurate reciprocal function back-transformation solution.

In keeping with Freedman and Modarres [45], among others, Equation (3) needs a collection of algebraic formulae for the median, the variance, skewness, and kurtosis, replicating what presently is available for the logarithmic back-transformation, for example, to complement it. In addition, it needs a numerically sound implementation that avoids the normal tail computational adulteration issues currently encountered with Mathematica 12.3, and most likely other symbolic algebra software packages (e.g., Maplesoft; https://www.maplesoft.com/products/maple/features/symbolicnumericmath.aspx, accessed on 6 July 2022). One implication emerging here is that perseverance with the applicable algebraic manipulations should be productive; after all, this is the approach that rendered Equations (3) and (4).

Equation (4), a second novel contribution of this paper, needs considerable refinement that effectively and definitively handles its imaginary part. The real-world attribute variables explored in this paper repeatedly exhibited monotonically decreasing covarying magnitudes of their real and imaginary parts. Table 8 notes communicate that some of these amounts are not necessarily trivial in size. This pernicious Equation (4) property needs to be resolved. Nevertheless, the real number part of its output (a la Table 5, Table 6, Table 9 and Table 10) tends to match both designated observed data statistics and measures generated by competing back-transformations. The attendant chief implication here derives from the simulation experiments précised in this paper, namely both the imaginary part of the numbers, and the corrupted tail calculations by Mathematica 12.3, appear to be vestiges of symbolic manipulation rules (e.g., [36]) combined with machine and software precision and other computational inadequacies. Consequently, a refinement of Equation (4) should be void of complex numbers. This situation is reminiscent of, and encouraged by, Cardan’s formulas versus trigonometric solutions for determining the three roots of cubic equations.

Finally, the ultimate advancement spawned by this paper is completion of the back-transformation conceptualization devised by Griffith [13], extending his positive power exponents composition to embrace negative power exponents. The primary implication stemming from this particular provision is that a unified back-transformation theory is draftable now.