Approaches to the high-throughput synthesis of analogues of dihydroaertiginoic acid

Article abstract of DOI:10.1177/109442810032002

A solution-phase synthesis of dihydroaeruginoic acid (1) was identified for its application towards the high-throughput synthesis of analogues of dihydroaeruginoic acid (1). Development of a solid-phase synthesis of dihydroaeruginoic acid (1) was examined

Full text of DOI:10.1177/109442810032002

ORuRsGseAllN, DIZeAanTI/OBNOAOLTSRTERSAEAPPRICNHGMMEOTDHEORDASTED REGRESSION
To Log or Not to Log : Bootstra p a s
a n Alte rna tive to the Pa ra m e tric
Estim a tion of Mod e ra tion Effe c ts in the
Pre se nc e of Ske we d De p e nd e nt Va ria b le s
CRAIG J. RUSSELL
Unive rsity o f Okla ho m a
MICHELLE A. DEAN
Unive rsity o f No rth Te xa s
When gross deviations from parametric assumptions are observed, conventional
data transformations are often applied with little regard for substantive theoreti-
cal implications. One such transformation involves using the logarithm of posi-
tively skewed dependent variables. Log transformations were shown to severely
decrease estimates of true moderator effects using moderated regression proce-
dures in a Monte Carlo simulation. Estimates of moderator effect sizes were sub-
stantially better estimates of the true latent moderator effect (i.e., larger by a mul-
tiple of 2.6 to 534) when estimated using a simple percentile bootstrap procedure
in the original, positively skewed data. Conclusions with regard to the presence or
absence of a true moderator effect using a simple bootstrap procedure were unaf-
fected by the violation of parametric assumptions in the original, positively
skewed data. In contrast, moderated regression analysis performed on a log-
transformed dependent variable severely increased Type-II error. Implications are
drawn for applied psychological and management research.
At one time or another, almost all investigators in applied psychological and manage-
ment research have been concerned by the assumptions required of common paramet-
ric statistical tests. Investigators typically assume that their samples were drawn from a
single population and rely on the power of the central limit theorem and other paramet-
ric assumptions to draw inferences about latent relationships within that population.
When violations of parametric assumptions are severe, investigators often use some
data transformation designed to minimize the violation. For example, all three empiri-
Authors’Note: We would like to thank Philip Q. Bobko for his comments on an earlier version of this ar-
ticle. Correspondence concerning this article should be addressed to Craig J. Russell, Price College of Busi-
ness, 107 Brooks Drive, University of Oklahoma, Norman, OK 73019-0450; e-mail: cruss@ou.edu.
Organizational Research Methods, Vol. 3 No. 2, April 2000 166-185
© 2000 Sage Publications, Inc.
166
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Russell, Dean / BOOTSTRAPPING MODERATED REGRESSION 167
cal studies reported in a recent special Academy of Management Journal forum on
managerial compensation performed log transformations on compensation data
(Conyon & Peck, 1998; Finkelstein & Boyd, 1998; Sanders & Carpenter, 1998) with
no mention of the purpose or rationale behind these transformations. Presumably, the
log transformations were done to address the presence of heteroskedasticity, that is,
the lack of independence between the mean of Y given X (Y Xi ) and the variance of Y
given X (σ 2y Xi ) that coincides with extreme positive outliers or severe positive skew
(Winer, 1974). Winer’s 1974 text has had a pervasive influence on organizational
research as reflected in the fact that it is the most highly cited publication in the Social
Science Citation Index (Institute for Scientific Information, 1999) between 1957 and
1997 (J. L. Bennett, personal communication, July 12, 1999)—it is difficult to under-
estimate the effect that Winer’s text (and its subsequent updates) has had on organiza-
tional researchers. It could be argued that performing log transformations on posi-
tively skewed dependent variables has become a convention within applied
psychology and management research.
One of the following characteristics is required of studiesusing parametric ordinary
least squares (OLS) procedures to examine linear relationships between variables X
and Y: (a) X and Y are random, bivariate normal or (b) X is fixed and e is normal, where
ei = Yi – biX1 – b0. In the former case, X is random in the sense that investigators do not
specify or control levels of X treatment effects in advance. Instead, X values observed
occur at a frequency dictated by the population probability distribution for X. Common
survey methods employed in research examining voluntary employee turnover (e.g.,
Mobley, Griffeth, Hand, & Meglino, 1979), job satisfaction (Smith, Kendall, & Hulin,
1969), performance prediction (Bray, Campbell, & Grant, 1974; Owens & Schoen-
feldt, 1979), and executive compensation (Finkelstein & Boyd, 1998) provide exam-
ples of random-effects designs. Importantly, when X and Y are distributed bivariate
normal, probabilistic inferences (e.g., conducting hypothesis test of H0: ρ = 0 or esti-
mating confidence intervals can be drawn due to ρ’s presence in the bivariate normal
density function described in Equation 1:
2
2
y − µ
x
y − µ
y
x − µ
x − µ
y
1
x
− 2 ρ
+
(1)
2
σ
σ
σ
σ
y
x
x
y
)
1
f(x,y) =
e 2 (1− ρ
.
2πσxσ y 1− ρ2
If X is not normally distributed, as in the latter case, one may still use the central
limit theorem to assume σ 2y Xi is normally distributed in order to test hypotheses about
ρ.1 In these circumstances, X is often a fixed effect that takes on values occurring in
some known frequency other than what one would have expected if values of X were
drawn at random from the population (e.g., values of X that the investigator selected for
purposes of manipulation). Importantly, regardless of study design, traditional para-
metric procedures cannot be used in conducting hypothesis tests or estimating CIs if
the true probability density function for prediction error (e) is unknown.
As noted above, one common violation of parametric assumptions occurs when the
variance of Y given X (σ 2y Xi ) is a function of the conditional mean (Y Xi ). Efforts
examining severely positively skewed Y distributions routinely occur in applied psy-
chological research, particularly in compensation research. Skewed compensation
distributions are caused by a number of factors, including the increasing span of pay
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168 ORGANIZATIONAL RESEARCH METHODS
ranges as the pay range midpoint increases (England & Pierson, 1990) and the extreme
levels of executive compensation typically reported in U.S. corporations.2 Both fac-
tors result in a lack of independence betweenY Xi and σ 2y Xi , violating the homoske-
dasticity assumption (Winer, 1974).
Winer (1974, pp. 398-401) described a number of transformations that correct or at
least lessen violations of some parametric assumptions. Log transformations of vari-
ables demonstrating highly positive or negative skew yield a more bell-shaped fre-
quency distribution, whereY Xi and σe are relatively uncorrelated. Winer (1974,
p. 400) noted that log transforms are particularly effective in stabilizing the condi-
tional variance of Y given X when the independence of error terms is violated due to
σ 2y = k 2 Xi2 or when Y has a great deal of positive skew (Olds, Mattson, & Oldeh,
i
1956).3
The usual effect of such transforms is to lessen the prediction error for values of Y
occurring at the extreme tail of the positively skewed dependent variable, conse-
quently increasingrx2y in additive models used to predict log Y. The SYSTAT6.0 for Win-
dows: Statistics (SPSS, 1996, pp. 252-257) manual described one such example in
which gross domestic product (GDP) per capita (X) was used to predict military spend-
ing (Y) in a sample of 57 countries. In this example, rx2y goes from .417 to .734 in the
presence of log transformation.
Importantly, the resultant model using the transformed data is Ymil$
=
+β log
1
X
+log
10β 0
e , which does not technically adhere to OLS characteristics (e.g.,
10 GDP 10
unbiased, minimum-variance parameter estimates). This model is perfectly service-
able if prediction is the investigator’s main concern—inferences about the accuracy of
prediction can be drawn from rxy. Note that probabilistic inferences cannot be drawn
$
for rxy, b0, b1, or Y unless one assumes that the log10e term in Ymil$
=
+β log
1
X
+log
e
10
10β 0
is normally distributed. We are aware of no research stream
10 GDP
(theory based or otherwise) that holds that log10 of e is normal.4 Regardless, the model
must have some theoretical meaning if explanation is the investigator’s main concern.
For example, it is unclear what theory or policy implications should be drawn from
finding that the log of salary is differentially related to organizational tenure for men
and women.5 The authors are unaware of any studies examining interactive models
providing a theoretical rationale justifying nonlinear (monotonic or nonmonotonic)
transformations in applied psychological or management research (although concepts
like the diminishing marginal utility of money may provide such a rationale in the
future). Enhanced statistical elegance achieved via nonlinear transformations has not
been accompanied by theory-based rationale justifying its use.
Nonlinear transformations can cause more uncertainty in interpreting tests of mod-
eration than they resolve. Investigators generally need to estimate sample sizes
required for replications and extensions of past research. Investigators examining pre-
viously reported data on the GDP–military spending relationships will solve
(1− rx2y )
rxy = tα / 2
N − 2
for N, estimating that they need a sample size, when rx2y = .417, that is approximately
four times as large as the sample required when rx2y = .734 at α = .05. Again, absent
theoretical rationale, arguments can be mounted for either estimate.
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Russell, Dean / BOOTSTRAPPING MODERATED REGRESSION 169
Importantly, nonlinear (monotonic and nonmonotonic) transformations of original
data create a number of problems for OLS applications used to detect moderator
effects. Busemeyer and Jones (1983) demonstrated monotonic transformations could
$
be found that cause Y values generated from a truly additive model (e.g.,Y = b0 + b1X1
$
+ b2X2) to yield support for a multiplicative model (e.g.,Y = b0 + b1X1 + b2X2 + b3X1X2)
and vice versa. Hence, reports of significant and nonsignificant interaction effects
after having performed log transformation on Y remain open to alternative interpreta-
tion (cf. Henderson & Frederickson, 1996; Sanders & Carpenter, 1998).
In sum, investigators often face circumstances in which data are clearly not bivari-
ate normal, e is not normal, and/or heteroskedasticity is present. Nonlinear transfor-
mations generate unknown levels of distortion in the many estimates of moderator
effects required to test theories in management and applied psychology (Busemeyer &
Jones, 1983; Russell & Bobko, 1992). Investigators’continued use of nonlinear trans-
forms to test moderator effects (e.g., Henderson & Frederickson, 1996; Kuhn &
Sweetman, 1998; Sanders & Carpenter, 1998) will result in literatures characterized
by mixed findings containing frequent Type-I and -II errors. This will be especially
true when other investigators do not use nonlinear transformations in studying the
same phenomena (e.g., Gomez-Mejia, Tosi, & Hinkin, 1987). Severe consequences
for theory development will result.
The bootstrap is a relatively new method of empirically estimating characteristics
of population distributions from sample data (Efron, 1979) that holds remarkable
implications for these applied research issues. Unfortunately, Mooney and Duval
(1993) noted that “the bootstrap is . . . foreign to most social scientists schooled in the
traditional parametric approach to inference” (p. 27). This study briefly reviews the
bootstrap literature and reports the results of a Monte Carlo simulation demonstrating
how log transformations can yield spuriously low estimates of moderator effect sizes
(i.e., ∆R2). Finally, a bootstrap approach to detect interaction effects when authors
would otherwise employ log transformations and traditional OLS techniques is pre-
sented, and implications for applied psychological and management research are
offered.
Bootstra p Estim a tion Proc e d ure s
Bootstrapping holds promise as a statistical estimation technique yielding precise
estimates of population distributions from sample data. Bootstrapping estimates the
population distribution of a statistic (e.g., rxy) by iteratively resampling cases from a set
of observed data. Basically, B bootstrap samples of size N are taken with replacement
from the original sample of size N and saved to a file. An investigation using B = 1,000
bootstrap samples of size N is able to approximate the actual sampling distribution that
would have been obtained if multiple independent samples of size N were drawn from
the population (Efron & Tibshirani, 1993).
There are many advantages to using the bootstrap technique. First, it is not
restricted by the normality assumptions of parametric tests. The percentile bootstrap-
ping method (Efron & Tibshirani, 1993, chapter 13) generates information about the
latent population distribution, permitting estimation of CIs directly from the boot-
strapped sampling distribution (e.g., if B = 1,000 bootstrap samples are taken, the boot-
strap correlations rb representing the 25th and 975th largest values constitute the lower
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170 ORGANIZATIONAL RESEARCH METHODS
and upper points of a 95% CI). The graphical interpretation of rb frequency distribu-
tions also yields insight into characteristics of the latent population distribution
(Efron & Tibshirani, 1993). When the sample is drawn from a population with a single
value of ρ, the central limit theorem dictates that the rb frequency distribution will rap-
idly approximate the normal distribution as B and N increase. A multimodal rb fre-
quency distribution would suggest that the sample was drawn from multiple popula-
tions, each with its own value of ρ. Second, information about the form of the original
sample is retained with no loss of distributional information. Rasmussen (1987) noted
that loss of information does occur when nonparametric techniques convert data to
ranks. Lunneborg (1985) described bootstrapping as falling between parametric and
nonparametric procedures for making probabilistic inferences.
Rasmussen (1987) presented the following simple example of a bootstrap proce-
dure. Suppose a researcher wants to test the null hypothesis that ρxy = 0 between
first-year grade point averages (GPA) and Graduate Record Exam (GRE) scores using
data obtained from 10 graduate students (H0: ρGPA, GRE = 0). First, an initial bootstrap
sample (B1) is randomly drawn with replacement from these 10 observations, yielding
the possibility of some observations being represented more than once in the bootstrap
sample, whereas other observations may not be included. A single bootstrap sample
may include the following cases: 5, 2, 8, 6, 2, 7, 9, 6, 1, and 2. Note that due to random
sampling with replacement, Case 2 was included more than once, whereas Case 3 was
not included in this first bootstrap sample (B1). The 10 cases may result in a correlation
of, say, .59. This procedure is repeated a large number of times (e.g., B = 1,000), and
each rb is saved to a separate file. Second, the bootstrap correlations (rb) are rank
ordered with the 25th and 975th correlations representing 95% CI end points. Finally,
the null hypothesis H0: ρGPA, GRE = 0 is tested by determining whether zero falls within
the CI (Rasmussen, 1987).
Studies examining similarities in results obtained from bootstrap and normal the-
ory approaches when parametric assumptions are met test the bootstrap’s ability to
estimate true latent population distributions (e.g., Diaconis & Efron, 1983; Efron,
1985, 1986; Lunneborg, 1985). These studies resulted in bootstrap statistics (e.g., esti-
mates of CIs) that were extremely close to those generated from parametric
approaches. Bickel and Freedman (1981; Freedman, 1981) demonstrated that the
bootstrap was asymptotically valid for many statistics with known population prob-
ability distributions (e.g., t and OLS regression statistics). However, the procedure is
perhaps of most value in drawing inferences about statistics with unknown popula-
tion probability distributions (e.g., medians or mixed samples drawn from multiple
populations).
Some issues remain unresolved in using bootstrapping to conduct hypothesis test-
ing, most revolving around the relative accuracy of parametric versus bootstrap proce-
dures in estimating probability intervals at the extreme tails of known (i.e., normal)
distributions. However, the simple percentile bootstrap method of estimating CIs
described above provides “good theoretical coverage properties as well as reasonable
stability in practice” (Efron & Tibshirani, 1993, p. 169). “Good theoretical coverage”
refers to CIs that (a) accurately estimate the probability of the population parameter
falling within the CI and (b) divide coverage error equally across the two tails.6
Empirical comparisons of bootstrap and traditional OLS regression procedures’
abilities to detect moderator effects when the dependent variable is positively skewed
are presented below.
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Russell, Dean / BOOTSTRAPPING MODERATED REGRESSION 171
Monte Ca rlo Sim ula tion
De sig n
In typical random-effects designs, investigators do not know how independent vari-
ables and prediction error are distributed. In fixed-effects designs, investigators typi-
cally control or specify independent variable levels, although the dependent Y distribu-
tion will be a function of the independent variable(s) and prediction error (e)
distributions. Classical measurement theory presumes Yi =YTi + e, whereYTi is the true
latent value of Y for person i. WhenYTi is a function of some X (e.g., Y = b0 + b1X1 + e or
Y = b0 + b1X1 + b2X2 + b3X1X2 + e), X1, X2, or e must be nonnormal in order for the
observed Yi to be nonnormally distributed. Consequently, to simulate the kinds of data
that investigators might encounter in either random- or fixed-effects designs, data
were generated in nine Monte Carlo simulations in which independent variables X1
and X2 and prediction error (e) systematically varied across normal, uniform, and chi-
square distributions. Normal distributions were selected to simulate multivariate nor-
mal conditions in random-effects designs. Uniform distributions were selected to
simulate fixed-effects experimental designs. Chi-square distributions for X and e
simulated positively skewed Y distributions such as those found in compensation
research.
Sa m p le
Simulation data were generated for combinations of X1, X2, and e distributions
using the SYSTAT9 (SPSS, 1996) computer package. Five thousand samples of N =
113 paired X1, X2 observations were drawn at random from all possible combinations
of normal, uniform, and chi-square population distributions X1, X2, and e (Guzzo, Jette,
& Katzell [1985] reported a mean N = 113 across studies in a meta-analysis of
compensation-based intervention programs). Results are only reported for conditions
in which X1 and X2 were drawn from identical population distributions, although the
results when X1 and X2 were drawn from different population distributions were con-
sistent with those reported below.7 Note that 5,000 samples of N = 113 were drawn for
every combination of X, Y, and e distributions described below as per Mooney’s (1997)
suggestions for conducting Monte Carlo simulations, resulting in nine sets of 5,000
samples of N = 113. All aspects of the Monte Carlo simulation were replicated using
5,000 samples of N = 226 and N = 56 (i.e., using samples twice and one half as large as
N = 113). Identical patterns of results emerged and are available from the first author
on request.
When X1 and X2 observations were drawn at random from a normal population dis-
tribution, µ and σ were set at µ = 3 and σ = 1. Variables X1 and X2 within each data set
were then rounded to the nearest integer (yielding values ranging from 1 to 5, i.e.,
5-point Likert scales) in order to simulate measurement circumstances commonly
encountered in applied psychological and management research. Uniform X1 and X2
data sets were drawn from a population containing integer values between 1 and 5,
inclusive. Additional X1 and X2 data sets were drawn from chi-square distributions
with three degrees of freedom. These steps resulted in nine Monte Carlo data sets when
the three possible X distributions (normal, uniform, chi-square) were combined with
the three possible e distributions (normal, uniform, chi-square).
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172 ORGANIZATIONAL RESEARCH METHODS
Three dependent variables were generated within each data set to reflect large,
medium, and small effect sizes. Equations 2, 3, and 4 were used to generate values for
Y1, Y2, and Y3 within each of the nine data sets:
Y1 = .75X1X2 + .25e
Y2 = .50X1X2 + .50e
Y3 = .25X1X2 + .75e
(2)
(3)
(4)
Under the e-equals-normal condition, prediction error e was drawn from a normal
population with a mean and standard deviation set equal to the mean and standard de-
viation of the X1X2 product term with which it was paired. Under the e = uniform condi-
tion, e was randomly drawn from a uniform population distribution ranging from 1 to
20. Under the e = chi-square condition, e was randomly drawn from a chi-square popu-
lation distribution with 9 degrees of freedom (where 9 is the mean population value for
all X1X2 product terms regardless of sample X1 and X2 distribution characteristics).
Hence, three dependent variables, Y1, Y2, and Y3, reflecting large, medium, and small
moderator effect sizes were available to be examined within each of the nine data sets.
Ana lyse s
All tests of interaction effects used moderated regression analysis (Bobko, 1995;
Darlington, 1968; Saunders, 1955, 1956). The F test of H0: ∆R2 = 0, where ∆R2 =
2
2
$
$
Rmultiplicative − Radditive for the equationsY = b0 + b1X1 + b2X2 + b3X1X2 andY = b0 + b1X1
+ b2X2, respectively, constitutes the test of an interaction effect when X1 and X2 are
interval scale measures. The strategy and organizational theory literatures commonly
refer to this as the Chow test (Chow, 1960).
To provide a point of reference, samples of N = 50,000 for each combination of X1
and X2 distribution were generated separately for purposes of estimating E(∆R2) when
e = 0. When X1 and X2 were normal, uniform, and chi-square, E(∆R2) = .057, .077, and
.256, respectively. These values should be considered asymptotes or what would occur
under circumstances of perfect, error-free prediction. The addition of prediction error
will slowly decrease E(∆R2), for example, if when X1 and X2 are distributed as
chi-square the true prediction model is Y = .1X1X2 + .9e, then clearly E(∆R2) ≠ .256.
Regardless, it should be noted that these are expected values of ∆R2 and actual values
observed might be larger or smaller when Y does or does not include prediction error
(e.g., Russell & Bobko [1992] observed∆R2 to be greater than E[∆R2] for some subjects).
Re sults
Table 1a reports results of moderated regression analyses performed on the three
effect sizes (Y1, Y2, and Y3) in the nine different combinations of X and e distributions
(i.e., normal, uniform, and chi-square X1 and X2 distributions paired with normal, uni-
form, and chi-square e distributions). Moderator effect sizes are captured by the
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Russell, Dean / BOOTSTRAPPING MODERATED REGRESSION 173
median ∆R2 column, containing the 2,500th largest value of ∆R2 obtained from the
5,000 samples of N = 113. Although F statistics testing H0: ∆R2 = 0 can be derived for
median ∆R2 values, only the ones derived for normally distributed prediction error
meet parametric assumptions and are interpretable (i.e., statistics reported in bold in
Table 1a). Regardless, the 2.5 and 97.5 percentile values of ∆R2 were identified within
the set of 5,000 Monte Carlo N = 113 samples.8 As the expected value of the F statistic
testing H0: ∆R2 = 0 is F = 1.0, one would reject H0 using logic that underlies simple per-
centile bootstrap applications when the F statistic
∆R2 / (4 − 3)
(1− Rm2ultiplicative ) / (113 − 4)
(i.e., F1, 109
=
)
for the moderator effect cutting off the lower 2.5% of the 5,000 ∆R2 is greater than 1
(i.e., when F = 1.0 falls outside of the 95% ∆R2 CI). Median values of ∆R2 reported in
Table 1a for which the lower 2.5 percentile values generated F greater than 1.0 are indi-
cated in italic.
Interestingly, profiles of ∆R2 for large, medium, and small effect sizes for interpret-
able equations in Table 1a (i.e., those meeting OLS assumptions) are .047, .024, .006;
.067, .041, .009; and .221, .191, .087 for X1 and X2 distributions drawn from normal,
uniform, and chi-square populations of X1 and X2, respectively. Not surprisingly,
smaller values of ∆R2 are observed as the effect size decreases across Y1, Y2, and Y3. The
pattern of effect sizes across normal, uniform, and chi-square distributions is consis-
tent with McClelland and Judd’s (1993) demonstration that multiplicative effect sizes
are maximized in designs using extreme values of X1 and X2. Normally distributed X1
and X2 will have the fewest extreme X1X2 observations due to low probabilities in the
extreme tails of the normal distribution. Uniform and chi-square distributions for X1
and X2 will have increasingly more frequent extreme observations in the tails of an
X1X2 distribution, respectively.
If X1, X2, or e are highly positively skewed, as they are when drawn from χ 2df =3 popu-
lations, Y will demonstrate some skewness. Investigators following Winer’s (1974)
convention would perform a log transform on Y in hope of permitting probabilistic
inferences that are possible when parametric assumptions are met. Table 1b reports
moderated regression results when Y1, Y2, and Y3 were subjected to a log10 transforma-
tion for the five X1, X2, and e combinations involving skewed chi-square distributions
(when X1, X2, or e are positively skewed, Y will be positively skewed). Moderated
regression effect sizes for the nontransformed Y1, Y2, and Y3 (Table 1a) are 2.7 to 15
times larger than effect sizes observed for log-transformed Y1, Y2, and Y3 (Table 1b).
Perhaps most interestingly, effect sizes for the one data set that meets parametric
assumptions (X1 and X2 distributed as chi-square, e distributed normally) go from ∆R2 =
.221 to ∆R2 = .032 for Y1 and log10Y1, respectively; from ∆R2 = .191 to ∆R2 = .041 for Y2
and log10Y2, respectively; and from ∆R2 = .087 to ∆R2 = .025 for Y3 and log10Y3, respec-
tively. Hence, moderated regression effect sizes are 3.5 to 6.9 times larger and more
likely to correctly detect the true latent population moderator effect when estimated
from the nontransformed data, although investigators following convention would
have log-transformed Y1, Y2, and Y3 before conducting the analyses. The stronger the
moderator effect, the larger the difference between effect sizes derived from nontrans-
formed versus log-transformed Ys.
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174 ORGANIZATIONAL RESEARCH METHODS
Table 1a
Monte Carlo Moderated Regression Analyses for Different X
and e Distributions: 5,000 Random Samples of N = 113
(Y1 = .75X1X2 + .25e, Y2 = .5X1X2 + .5e, and Y3 = .25X1X2 + .75e)
Median (Rm2ultiplicative − Ra2dditive = ∆R2)
X1 and X2 Distribution
Characteristic
e Distribution
Characteristic
Rm2ultiplicative
Ra2dditive
DR2
Normal: µ = 3, σ = 1, rounded
Normal: e = 9, σ = 5a
to nearest integer
Y1
Y2
Y3
.943
.786
.298
.918
.771
.288
.047
.024
.006
Uniform random number
between 1 and 18
Y1
Y2
Y3
.919
.726
.293
.893
.709
.281
.047
.025
.007
Chi-square: df = 9b
Y1
Y2
Y3
.943
.732
.383
.918
.712
.373
.047
.029
.008
Uniform random number:
distribution from 1 to 5,
rounded to nearest integer
Normal: e = 9, σ = 5a
Y1
Y2
Y3
.956
.803
.360
.920
.778
.347
.067
.041
.009
Uniform random number
between 1 and 20
Y1
Y2
Y3
.960
.648
.431
.924
.616
.420
.068
.041
.009
Chi-square: df = 9b
Y1
Y2
Y3
.969
.713
.496
.933
.680
.484
.070
.046
.012
Chi-square: df = 3, rounded
to nearest integer
Normal: e = 9, σ = 5a
Y1
Y2
Y3
.991
.910
.561
.873
.798
.477
.221
.191
.087
Uniform random number
between 1 and 18
Y1
Y2
Y3
.992
.924
.658
.874
.815
.589
.220
.190
.086
Chi-square: df = 9b
Y1
Y2
Y3
.992
.939
.647
.874
.827
.561
.221
.197
.104
Note. N = 113 is the average N across k = 330 effect sizes reported in a meta-analysis of Guzzo,
Jette, and Katzell (1985). Data presented in bold met parametric assumptions and are interpret-
able.Results presented in italics had the 2.5 percentile value of F for ∆R2 greater than 1, hence H0:
F = 1.0 did not fall in the 95% CI.
a. M and SD for all normally distributed error terms were set to be equal to the M and SD for the
product X1X2.
b. The expected value of the chi-square distribution is equal to its df. Hence, with df = 9, the ex-
pected midpoint of the error distribution is equal to the mean of the X1X2 product term.
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Russell, Dean / BOOTSTRAPPING MODERATED REGRESSION 175
Table 1b
Moderated Regression Results for Log10(Y1) = .75X1X2 + .25e,
Log10(Y2) = .5X1X2 + .5e, and Log10(Y3) = .25X1X2 + .75e
Median (Rm2ultiplicative − Ra2dditive = ∆R2)
X1 and X2 Distribution
Characteristic
e Distribution
Characteristic
Rm2ultiplicative
Ra2dditive
DR2
Normal
Chi-square: df = 9a
Log10Y1
.876
.749
.343
.873
.743
.334
.006c
.007
.005
Log10Y2
Log10Y3
Uniform random number
Chi-square: df = 3
Chi-square: df = 9a
Log10Y1
.948
.724
.440
.944
.714
.431
.008
.010
.008
Log10Y2
Log10Y3
Normal: e = 9, SD = 5b
Log10Y1
.874
.748
.501
.856
.720
.475
.032
.041
.025
Log10Y2
Log10Y3
Uniform random number
between 1 and 20
Log10Y1
.862
.716
.498
.843
.688
.473
.032
.040
.025
Log10Y2
Log10Y3
Chi-square: df = 9a
Log10Y1
.829
.805
.618
.808
.772
.585
.035
.051
.039
Log10Y2
Log10Y3
a. The expected value of the chi-square distribution is equal to its df. Hence, with df = 9, the ex-
pected midpoint of the error distribution is equal to the mean of the X1X2 product term.
b. M and SD for all normally distributed error terms were set to be equal to the M and SD for the
product X1X2.
c. The F statistic for the 2.5 percentile value of ∆R2 was greater than 1.
In sum, moderated regression effect sizes derived from a Monte Carlo simulation of
5,000 N = 113 samples drawn from normal, uniform, and chi-square e and X distribu-
tions are 2.7 to 15 times more likely to detect true latent moderator effects (i.e., reject
H0: ∆R2 = 0) when the dependent variable has not been subjected to a log transforma-
tion. The final portion of this study demonstrates how primary researchers would
apply a simple bootstrap procedure in analyzing data obtained from a single sample
and confirming implications of the Monte Carlo results (i.e., that inferences drawn
from bootstrap-generated CIs about moderator effects are expected to exhibit less
Type-II error).
Bootstra p De m onstra tion
Sa m p le s
As a rule, researchers generally face circumstances in which they have data gath-
ered from a single sample, not 5,000 samples. Hence, to simulate what individual
researchers typically encounter, nine samples of N = 113 paired X1, X2 observations
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176 ORGANIZATIONAL RESEARCH METHODS
were created at random from normal, uniform, and chi-square population distributions
using the SYSTAT9 computer package. When X1 and X2 observations were drawn at
random from a normal population distribution, µ and σ were set at µ = 3 and σ = 1. As
in the Monte Carlo simulation and consistent with measurement circumstances com-
monly encountered in applied psychological and management research, X1 and X2 data
sets were rounded to the nearest integer, yielding values from 1 to 5. Uniform X1 and X2
data sets were drawn from a population containing integer values between 1 and 5,
inclusive, yielding X1 = 3.012, σx1 = 1.438, and X2 = 2.889, σx2 = 1.394, respectively.
Finally, X1 and X2 data sets were drawn from chi-square distributions with 3 degrees of
freedom, yielding X1 = 2.986, σx1 = 2.344, and X2 = 3.008, σx2 = 2.660, respectively.
Three dependent variables were generated within each sample using Equations 2, 3,
and 4 described in the Monte Carlo simulation above. Error terms (e) were drawn from
the same populations as described in the Monte Carlo simulation above, with their
means and standard deviations set equal to the X1X2 product term means and standard
deviations.
Ana lyse s
Tests of interaction effects using moderated regression analysis were performed
using dependent variables Y1, Y2, Y3, LogY1, LogY2, and LogY3 in each of the nine sam-
ples. In addition, B = 1,000 bootstrap estimates of ∆R2 were derived for all dependent
variables in each of the nine samples using the percentile bootstrap method described
above.9
Re sults
Table 2a reports results of moderated regression analyses performed on the nine
samples of N = 113 containing different combinations of X1, X2, and e distributions
(i.e., normal, uniform, and chi-square distributions of X1 and X2 paired with normal,
uniform, and chi-square e distributions). Although F statistics are reported for mod-
eration effects in all nine combinations, only the three derived for normally distributed
prediction error meet parametric assumptions and are interpretable (i.e., statistics
reported in bold in Table 2a). Moderator effect sizes are captured by the ∆R2 column
(the F statistic tests H0: ∆R2 = 0) (Bobko, 1995; Darlington, 1968).
∆R2 for Y1 in the three interpretable equations in Table 2a are .049, .068, and .216 for
X1 and X2 distributions drawn from normal, uniform, and chi-square populations,
respectively. This profile of effect sizes is again consistent with the observation that
normal X1 and X2 will have the fewest extreme X1X2 observations due to low probabili-
ties in the extreme tails of the normal distribution and results reported in the Monte
Carlo study reported above.
Figure 1 demonstrates when X1, X2, or e were highly positively skewed, as they are
when drawn from χ 2df =3 populations, Y exhibited some positive skewness. Investiga-
tors following convention would perform a log transform on Y hoping to permit the
probabilistic inferences that are possible when parametric assumptions are met. Table
2b reports moderated regression results when Y was subjected to a log10 transformation
for the five X1, X2, and e combinations with skewed chi-square distributions (i.e.,
skewed Y distributions appear only when X1, X2, or e distributions were positively
skewed). Consistent with the Monte Carlo findings reported above, moderated regres-
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Russell, Dean / BOOTSTRAPPING MODERATED REGRESSION 177
sion effect sizes for the original nontransformed data were two to seven times larger
than effect sizes observed for log-transformed data. Effect sizes for the one data set
that met parametric assumptions (X1 and X2 distributed as chi-square, e distributed nor-
mally) went from ∆R2 = .216 to ∆R2 = .030 when Y wassubjected to log transformation.
Hence, moderated regression effect size was 7.2 times larger when it was (correctly)
estimated from nontransformed data.
However, CIs around ∆R2 can be derived via bootstrapping procedures regardless
of how X1, X2, e, or Y are distributed. Table 3 reports bootstrap estimates of the 2.5 per-
centile values of the moderated regression effect size ∆R2 taken from B = 1,000 boot-
strap samples of size N = 113 for the five situations in which Y is positively skewed,
that is, those that are subject to log transformation using the current methodological
convention. Interestingly, median effect sizes across 1,000 bootstrap samples were
between 2.6 and 534 times larger than ∆R2 effect sizes resulting from analyses con-
ducted after Y was log transformed (see Table 2b). This suggests that to be equally
likely to be detected, moderator effect sizes when Y is skewed and log transformed
must be 2.6 to 534 times as large as those observed under conditions when Y is not log
transformed and ∆R2 is estimated from the median bootstrap ∆R2 value. Alternatively,
other things being equal, the sample size needed to correctly reject H0: ∆R2 = 0 would
need to be 6.76 to 285,156 times as large when Y is skewed and log transformed in
these samples. Investigators using OLS moderated regression and log-transformed
dependent variables would be much more likely to fail to detect true interaction effects
(Type-II error).
Disc ussion
This study demonstrated a fundamental problem in the detection of latent modera-
tion effects when log transforms are used to correct positively skewed dependent vari-
ables. Specifically, the increased probability of Type-II error was demonstrated in a
Monte Carlo simulation generating 5,000 samples from known population distribu-
tions and in a subsequent bootstrap analysis of individual simulated samples. Results
suggested severe decrements in the statistical power required to test moderation
regression effects (i.e., H0: ∆R2 = 0) that result from log transformations. These decre-
ments occurred when parametric assumptions were in fact met (i.e., the data reported
in bold in Tables 1a and 2a) and when parametric assumptions were not met. Graphi-
cally, log transformations change the Y distribution shape, effectively decreasing Y
variance by reducing the degree to which extremely positive Y values deviate from the
mean. If these extreme Y values were created by an interaction between one or more
positively skewed independent variables (e.g., when X1 and X2 are distributed as chi-
square), log transformations of Y effectively disguise the extreme values of Y that
should result from the product of extreme X1 and X2 values as less extreme values,
effectively yielding a logY variable that exhibits less variance than the original raw Y
observations. Although Type-I error is always possible (cf. Aguinis & Pierce, 1998), it
is clear that log transformations of positively skewed dependent variables greatly
enhance the probability of Type-II error.
(text continues on p. 182)
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Table 2a
Moderated Regression Analyses for Different X and e Distributions: N = 113 (Y1 = .75X1X2 + .25e, Y2 = .5X1X2 + .5e, and Y3 = .25X1X2 + .75e)
X1 and X2 Distribution Characteristic
e Distribution Characteristic
Normal: e = 8.789, SD = 4.996a
R2multiplicative
R2additive
DR2
F1, 109
p Value
Normal: µ = 3, σ = 1, rounded to nearest integer
Y1
Y2
Y3
.799
.587
.331
.750
.561
.322
.049
.026
.009
5.616
2.910
0.990
< .01
> .05
> .05
Uniform random number between 1 and 20
Y1
Y2
Y3
.880
.601
.291
.821
.580
.282
.059
.021
.009
6.834
2.357
0.990
< .01
> .05
> .05
Chi-square: df = 9b
Y1
Y2
Y3
.932
.599
.333
.887
.567
.329
.045
.032
.004
5.136
3.603
0.438
< .05
> .05
> .05
Uniform random number: distribution from 1 to 5,
rounded to nearest integer
Normal: e = 8.702, SD = 6.029a
Y1
Y2
Y3
.825
.633
.436
.757
.590
.425
.068
.043
.011
7.953
4.898
1.110
< .01
< .05
> .05
Uniform random number between 1 and 20
Y1
Y2
Y3
.831
.599
.395
.787
.580
.383
.044
.019
.012
4.561
2.111
1.324
< .05
> .05
> .05
Chi-square: df = 9b
Y1
Y2
Y3
.813
.701
.521
.792
.651
.479
.071
.050
.042
9.269
5.737
4.779
< .01
< .05
< .05

Chi-square: df = 3, rounded to nearest integer
Normal: e = 8.982, SD = 6.163a
Y1
Y2
Y3
.846
.620
.461
.630
.460
.355
.216
.160
.106
30.030
12.976
12.924
< .001
< .001
< .001
Uniform random number between 1 and 20
Y1
Y2
Y3
.906
.782
.506
.688
.642
.446
.214
.140
.060
29.677
17.744
6.957
< .01
< .01
< .01
Chi-square: df = 9b
Y1
Y2
Y3
.949
.682
.466
.799
.581
.426
.150
.101
.040
19.235
12.246
4.542
< .01
< .01
< .05
Note.Only statistics appearing in bold are interpretable under parametric assumptions and F statistics test H0:∆R2 = 0.N = 113 is the average N across k = 330 effect sizes re-
ported in a meta-analysis of Guzzo, Jette, and Katzell (1985).
a. M and SD for all normally distributed error terms were set to be equal to the M and SD for the product X1X2.
b.The expected value of the chi-square distribution is equal to its df.Hence, with df = 9, the expected midpoint of the error distribution is equal to the mean of the X1X2 product
term.

Figure 1: Frequency Distribution for X1X2 and Y (X1 and X2 are chi-square distributed and Y = .75X1X2 + .25e)
Note. The variable e is drawn from random normal or random uniform distributions with e and σe set equal to X1X2 and σX1X2, respectively.

Table 2b
Moderated Regression Results for Log10(Y1) = .75X1X2 + .25e, Log10(Y2) = .5X1X2 + .5e, and Log10(Y3) = .25X1X2 + .75e
X1 and X2 Distribution Characteristic
Normal
e Distribution Characteristic
Chi-square: df = 9a
R2multiplicative
R2additive
DR2
F1, 109
p Value
Log10Y1
Log10Y2
.757
.511
.409
.736
.500
.402
.021
.011
.007
2.338
1.100
0.768
> .05
> .05
> .05
Log10Y3
Uniform random number
Chi-square: df = 3
Chi-square: df = 9a
Log10Y1
Log10Y2
.880
.567
.399
.877
.566
.399
.003
.001
.000
0.328
0.109
0.087
> .05
> .05
> .05
Log10Y3
Normal: e = 8.808, SD = 6.029b
Log10Y1
Log10Y2
Log10Y3
.553
.399
.278
.523
.385
.269
.030
.014
.009
3.371
1.548
0.990
> .05
> .05
> .05
Uniform random number between 1 and 20
Log10Y1
Log10Y2
.611
.456
.342
.550
.425
.331
.061
.031
.011
7.081
3.487
1.100
< .05
> .05
> .05
Log10Y3
Chi-square: df = 9a
Log10Y1
.678
.444
.367
.588
.399
.348
.090
.045
.019
10.780
5.136
2.111
< .01
< .05
> .05
Log10Y2
Log10Y3
a.The expected value of the chi-square distribution is equal to its df.Hence, with df = 9, the expected midpoint of the error distribution is equal to the mean of the X1X2 product
term.
b. M and SD for all normally distributed error terms were set to be equal to the M and SD for the product X1X2.

182 ORGANIZATIONAL RESEARCH METHODS
Table 3
Bootstrap 95% Confidence Intervals of Moderated Regression Analyses:
B = 1,000, N = 113, and Simulation Data Derived From
Y1 = .75X1X2 + .25e, Y2 = .5X1X2 + .5e, and Y3 = .25X1X2 + .75e
X1 and X2 Distribution
Characteristic
e Distribution
Characteristic
R2multiplicative
R2additive
DR2
Normal
Chi-square: df = 9a
Y1
Y2
Y3
.833
.602
.433
.663
.482
.343
.170c
.120c
.090
Uniform random number
Chi-square: df = 3
Chi-square: df = 9a
Y1
Y2
Y3
.917
.689
.450
.700
.529
.340
.217c
.160c
.110c
Normal: e = 8.808,
SD = 6.029b
Y1
Y2
Y3
.817
.555
.311
.607
.422
.178
.210c
.133c
.133c
Uniform random
number between 1 and 20
Y1
.862
.499
.311
.666
.349
.201
.196c
.150c
.111c
Y2
Y3
Chi-square: df = 9a
Y1
Y2
Y3
.890
.609
.522
.656
.448
.402
.234c
.161c
.120c
Note. N = 113 is the average N across k = 330 effect sizes reported in a meta-analysis of Guzzo,
Jette, and Katzell (1985).
a. The expected value of the chi-square distribution is equal to its df. Hence, with df = 9, the ex-
pected midpoint of the error distribution is equal to the mean of the X1X2 product term.
b. M and SD for all normally distributed error terms were set to be equal to the M and SD for the
product X1X2.
c. The F statistic for the 2.5 percentile value of ∆R2 was greater than 1.
Fortunately, results also indicated that bootstrapping procedures provide a viable
alternative to traditional, parametric statistical procedures for detecting moderator
effects regardless of how X1, X2, and e are distributed. In fact, in situations in which
convention dictates that Y should be subjected to log transformation, log transforma-
tions caused extremely severe decrements in statistical power for parametric OLS pro-
cedures relative to bootstrap procedures. Data simulated here are commonly found in
compensation research, where parametric procedures are commonly used after Y has
been subjected to a routine log transformation (e.g., Henderson & Frederickson, 1996;
Sanders & Carpenter, 1998).
Of course, log transformations could be justified on some theoretical basis. The
authors are unaware of any theoretical rationale put forth by compensation theory or
any other area of applied psychological or management research to justify such a trans-
formation in the presence of a multiplicative model. Furthermore, the authors have
never seen any discussion of the theoretical underpinnings of latent models that result
β
+β log
1
X
10
0
$
from such a transformation, such asY = 10
(SPSS, 1996). As a result, any
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Russell, Dean / BOOTSTRAPPING MODERATED REGRESSION 183
gains in statistical elegance and predictive power (i.e., for additive models) stemming
from log transformations are not currently matched by gains in theoretical insight.
Null results for tests of moderation in studies employing log transformations are
expected to frequently reflect Type-II errors when a true latent moderation process is
present.
In sum, when hypothesized models involve interaction effects, applied psychologi-
cal and management research would benefit from a routine application of the bootstrap
procedures. Although they do not replace common parametric procedures, bootstrap
applications are appropriate when parametric assumptions are not viable (e.g., when
heteroskedasticity is present due to a positively skewed dependent variable). Nonlin-
ear monotonic transformations may achieve necessary statistical conditions for para-
metric inferences in OLS applications to additive models (Busemeyer & Jones, 1983;
Winer, 1974). The results indicated that nonlinear monotonic transformations erode
investigators’ capacity to assess theory-based predictions of moderation effects (e.g.,
estimates of moderation effect ∆R2). Importantly, bootstrapping provides an alterna-
tive method of assessing theory-based inferences of moderation effects from data that
cannot be assessed with comparable statistical power by conventional procedures.
Note s
1. Note that if X is not normally distributed and Y is a linear function of X, Y will also likely
not be normally distributed.
2. Bergman, Scarpello, and Hills (1998) and Milkovich and Newman (1996) noted how pay
ranges are generally a constant or increasing percentage of the range midpoint. Hence, as the pay
range midpoint (and mean) increases, the variation in observed salaries around the midpoint in-
creases.
3. Of course, weighted least square (WLS) procedures would also resolve the heteroskedas-
ticity problem. However, it would do so by migrating what was a nonlinear transformation
paired with ordinary least squares into the internal optimal weighting procedures that are char-
acteristic of WLS.
4. We thank an anonymous reviewer for bringing this to our attention.
5. We thank an anonymous reviewer for this example.
6. See Efron and Tibshirani (1993), chapter 14 (pp. 178-201) for a discussion of alternatives
to the simple bootstrap. Specifically, the bias-corrected and accelerated (BCa) and the approxi-
mate bootstrap confidence (ABC) interval methods are marginally more complex techniques
that overcome most shortcomings associated with the simple bootstrap.
7. The first author will provide these results on request.
8. ∆R2 for the 2.5 and 97.5 percentile values are available from the first author on request.
9. See Efron and Tibshirani (1993), chapter 14 (pp. 178-201) for more elaborate bootstrap pro-
cedures exhibiting certainstatistical elegance that might yield morerobust confidenceintervals.
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Craig J. Russell is the J.C. Penney Chair of Business Leadership at the Price School of Business, University
of Oklahoma and holds a joint appointment as a professor of psychology. He received his Ph.D. in business
administration from the University of Iowa. He pursues research in management and leadership develop-
ment, personnel selection, and research methods.
Michelle A. Dean is an assistant professor in the Department of Management, College of Business Admin-
istration, University of North Texas. She received her Ph.D. in business administration from Louisiana State
University. Her research focuses on performance prediction in personnel selection systems, with recent em-
phasis on biodata criterion and construct validity.
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