Download The relative importance of size and asymmetry in sexual selection

Behavioral Ecology Vol. 9 No. 6: 546-351
The relative importance of size and
asymmetry in sexual selection
R. ThornhiD' and A. P. M0llerb
•Department of Biology, The University of New Mexico, Albuquerque, NM, 87131-1091, USA, and
b
Laboratoire d'Ecologie, CNRS URA 258, Universite Pierre et Marie Curie, Bat A, 7eme etage,7 quai
St. Bernard, Case 237, F-75252 Paris Cedex 05, France
Developmental stability reflects the ability of individuals to cope with their environment during ontogeny given their genetic
background. An inability to cope with environmental and genetic perturbations is reflected in elevated levels of fluctuating
asymmetry and other measures of developmental instability. Both trait size and symmetry have been implicated as playing an
important role in sexual selection, although their relative importance has never been assessed. We collected information on
the relationship between success in sexual competition and size and asymmetry, respectively, to assess the relative importance
of these two factors in sexual selection. Studies that allowed comparison of the relationships for the same traits' size and
symmetry and success in sexual competition constituted the data, which totaled 73 samples from 33 studies of 29 species. The
average sample-size weighted correlation coefficients between mating success or attractiveness and size and asymmetry, respectively, were used as measures of effect size in a meta-anatysis. Analysis was conducted on samples, studies, and species separately.
We found evidence of an overall larger effect of symmetry at the species level of analysis, but similar effects at the sample or
study levels. The difference in effect size for character size and character symmetry was larger for secondary sexual characters
than for ordinary morphological characters at the level of analysis of samples. The results lend support to the conclusion that
symmetry plays an important general role in sexual selection, especially symmetry of secondary sexual characters. Key words:
buffering capacity, developmental stability, fluctuating asymmetry, mate choice, meta-analysis. [Behav Exol 9:546-551 (1998)]
D
evelopmental stability is defined as the ability of individuals to undergo stable development of their phenotype
under given environmental conditions (review in Mailer and
Swaddle, 1997). An individual's inability to cope with the environment during development is reflected by increased random deviations from perfect symmetry, so-called fluctuating
asymmetry, and an increased frequency of phenodeviants (reviews in Graham ct aL, 1993; Miller and Swaddle, 1997; Palmer and Strobeck, 1986; Parsons, 1990). Both environmental
and genetic perturbations reduce the ability to control developmental processes and hence increase individual measures
of asymmetry (reviewed in Mailer and Swaddle, 1997).
Sexual selection arises from reproductive competition
among individuals for access to individuals of the choosing
sex. Typically, across species, females choose mates, but males
also choose mates when they invest parentally, and relative
parental investment determines the sex that is most sexually
competitive (Darwin, 1871; review in Andersson, 1994). Females may benefit either directly or indirectly from their mate
choice (review in Andersson, 1994), and their choice of symmetrical partners may likewise give rise to either direct or indirect fitness advantages (Gangestad and Thomhill, 1997;
Mailer, 1990, 1993; Thomhill and Gangestad, 1993; Thomhill
and Sauer, 1992; Watson and Thornhill, 1994). For both kinds
of benefits, individual mating success is predicted to decrease
with increasing asymmetry.
Numerous studies of sexual selection in relation to asymmetry have been reported. While some show the predicted
negative relationship between mating success and asymmetry
(e.g., Harvey and Walsh, 1993; Mailer, 1992; Swaddle and
Cuthill, 1994a,b; Thornhill, 1992a,b), others have reported an
Address correspondence to A. P. Moller. E-mail: amoUerChalLsnv.
jussieuJr.
Received 6 October 1997; reviled 20 January 1998; accepted 5 February 1998.
© 1998 International Society for Behavioral Ecology
absence of any effect (e.g., Fiske et al., 1994; Markow and
Ricker, 1992; Ueno, 1994). For example, Andersson and Iwasa
(1996) recently stated that "the role of fluctuating asymmetry
in sexual selection is debated, and its analysis contains many
pitfalls." Similarly, Andersson (1994: 71) stated, in regard to
the relationship between low asymmetry and increased mating
success or sexual attractiveness, that "more studies of a variety
of organisms are needed to clarify how general such patterns
may be in sexual selection." Fields with conflicting results are
commonplace in biology, and because both consistent and
contradictory evidence may follow from perfectly valid studies,
disputes cannot be resolved by simple vote counting without
considering sample size. A fruitful approach is to use metaanahtic techniques to address the generality of a relationship
between two variables, such as sexual selection and asymmetry,
and to attempt to understand the factors responsible for heterogeneity among samples (Cooper and Hedges, 1994; Hedges and Olkin, 1985; Rosenthal, 1991).
Recently, Mailer and Thornhill (1998), in a meta-analysis
of sexual selection in relation to asymmetry in 146 samples
from 65 studies of 42 species of all known studies (insects,
fish, birds and mammals), found a moderately negative, statistically significant relationship between asymmetry and mating success or attractiveness to the opposite sex (average, sample-size weighted Pearson's product-moment correlation r =
—.42 for studies and r = —.34 for species). Moderator variables that explained heterogeneity among studies and species
included greater effects for males than for females, when a
secondary sexual character rather than an ordinary trait was
studied, in experimental as opposed to observational studies,
and for traits not involved with mobility as compared to traits
affecting mobility.
The aim of the present study was to test whether the magnitude of the effect of asymmetry on sexual selection was similar to that of character size per se, which is known from numerous studies to be related to sexual selection (see review in
Andersson, 1994). Large males often experience an advantage
Thornhill and Mellcr • Size and asymmetry in sexual selection
in terms of sexual selection because of their success in malemale competition, or because of superior ability to acquire
resources preferred by females (Andersson, 1994). Obviously,
we could only include studies in which information on both
size and asymmetry was available. The test consisted of pairwise comparisons of the magnitude of a standardized effect
size for asymmetry and size of the same characters in the large
sample of studies composing the meta-analysis mentioned
above. A second objective of this study was to test whether the
difference in effect size for character size and symmetry differed between secondary sexual characters and ordinary morphological traits. It has been suggested that an evolutionary
history of directional selection gives rise to increased levels of
asymmetry (Meller and Pomiankowski, 1993), and some empirical evidence suggests that asymmetry is indeed larger in
secondary sexual characters than in ordinary morphological
characters (e.g., Meller and Hdglund, 1991). Therefore,
choosy individuals or contestants in intrasexual competition
should be better able to distinguish between the asymmetry
of target individuals when inspecting secondary sexual characters rather than ordinary morphological traits.
MATERIALS AND METHODS
The data set used for the present study is from Mailer and
Thornhill (1998), which was based on (1) a comprehensive
search of the literature and of the Internet for information
on the relationship between asymmetry and sexual selection,
and (2) correspondence between the authors and behavioral
ecologists conducting research on this relationship. Because
we investigated the relative effect size for symmetry versus
character size for the same trait, the only studies included
from Mailer and Thornhill (1998) were those that had one
or more estimates that allowed calculation of an effect size for
the relationship between asymmetry and character size, and
mating success or a mate preference, respectively. We were
able to obtain unpublished correlation coefficients for character size for a few additional studies that were also included
in this study. In all cases of studies used in the present study,
data for size and asymmetry are based on the same trait and
involve the same individuals and, thus, sample size (see Mellcr
and Thornhill, 1998, for further details). All the data are presented in Table 1.
Meta-analysis consists of obtaining an estimate of the magnitude of a general effect of interest from each of a number
of studies, calculating average effects, and elucidating moderator variables that account for heterogeneity among studies.
We calculated effect sizes as the Pearson product-moment correlation coefficients, following the procedures outlined in Rosenthal (1991), Hedges and Olkin (1985), and Cooper and
Hedges (1994). Effect sizes were reported in the original references in forms other than r (t, F, x* statistics, Mann-Whitney
U, Spearman or Kendall rank-order correlations), which were
then converted using formulas in Rosenthal (1991). We calculated effect sizes for samples, studies, and spedes by using
a sample-size weighted mean of the relevant samples in the
study (Rosenthal, 1991). A sample was defined as a single test
for an effect The difference in effect size between symmetry
and character size per se was tested by first converting the
effect size for asymmetry into an effect for symmetry to obtain
similar signs for the two measures of effect size by changing
the sign of the correlation coefficient for the asymmetry effect
size. Hence we decided to work on effect sizes with similar
signs for size and symmetry to make the results more intelligible. For example, an effect of size and asymmetry in the
same direction, but with a stronger effect for the first variable,
should give rise to a consistent positive difference in effect
size.
Samples within studies and multiple studies of single spedes
may lack statistical independence. Traditionally, this problem
has been solved by investigating the predictions at the level
of samples, studies, and spedes. If qualitatively similar conclusions are reached at all three levels, this implies that the level
of analysis is relatively unimportant We are unaware of ways
of performing meta-analyses in a phylogenetic framework, although such a solution eventually may become possible.
Weighted mean effect sizes were calculated using Fisher's
transformation of r to Z, (Sokal and Rohlf, 1996) and calculation of a mean weighted Zn where
mean weighted Z,a
1 uyZySw^
where u>j is the weight factor for analysis unit j , with Wj equaling Nj- 3 in the present case (Rosenthal, 1991). The Z\was
converted to ? and the significance level of whether r differed
significantly from zero was subsequently determined, following Hedges and Olkin (1985), using mean weighted utj = (Nj
— S)fl(N
— 3), where Nia the sum of the sample sizes in
the analysis, and Nj — 3 is summed over all j from 1 to A,
which is the number of analysis units (e.g., * = 29 for spedes
analysis). Confidence intervals on these r values were calculated using a modification of Hedges and Olkin (1985). The
null hypothesis was that the effect size was zero, and lack of
overlap of the 95% confidence intervals with zero would reject
the null hypothesis. Furthermore, we evaluated the null hypothesis by calculating a standard normal deviate, which has
a critical value of 1.96 for statistical significance at the 5%
level. Although the effects of symmetry and size in relation to
sexual selection were based on the same individuals and thus
were dependent, the overall correlation between the two effects was near zero (see Results). Thus, we calculated a confidence interval on the difference between symmetry and size
mean weighted Rvalues as the difference in Z,± 1.96/(N —
3k), where 1.96 was the two-tailed critical value of the standard
normal distribution; N and k are as described above.
Because secondary sexual characters have been suggested
to have larger asymmetry than ordinary morphological characters due to a recent evolutionary history of directional selection, we investigated whether the difference in effect size
between character size and character symmetry was larger for
secondary sexual characters than for ordinary morphological
traits. The null hypothesis was that the difference would be
zero, and lack of overlap of the 95% confidence interval (C3)
with zero would reject the null hypothesis. The characters
were classified as being secondary sex traits according to the
original source references; the classification is shown in Table 1.
RESULTS
Statistical independence of effect sixes for symmetry and
Unweighted Pearson's correlation coefficients for the correlation between symmetry and a measure of sexual selection
and size and a measure of sexual selection were small and
insignificant at all three levels of our analysis (samples, n =
73, r « .025, p = .833; studies, n - 33, r = -.007, p = .971;
spedes, n = 29, r = .150, p = .441).
Analyses haurd on fmpi»« as units of analysis
We analyzed the effect size for sexual selection and symmetry
and size, respectively, by comparing effect sizes at the levels
of samples, studies and spedes. For the 73 samples, the
weighted mean effect size for symmetry was r •= .191 (95% CI
0.158, 0.223); the weighted mean effect size for character size
was r = .175 (95% (3 0.143, 0.2O9). Both effects were signif-
Behavioral Ecology Vol. 9 No. 6
548
Table 1
Species, effect site (Pearson product-moment correlation coefficients) for the relationship between rharartrr size and symmetry, respectively,
and a measure of sexual selection, tatmplr size, and type of character for different trwHn
Specie*
Insects
Cotnagrion fnuila
Drvsophila mojavtnsis
D. mojavtnsis
D. nigrospiracula
D. nigrospimcula
D. pstudoobscura
D. pstudoobscura
D. pstudoobscura
D. pstudoobscura
D. pstudoobscura
D. simulant
D. simulans
D. simulans
D. simulans
D. simulans
D. simulans
D. simulans
Eyprtpocnemts plorons
Eypvtpocnevus plorans
Ischnura denticolHs
I. denticolHs
I. dmticolHs
Musca donustica
M. domtstica
M. domtstica
M. domtstica
M. domtstica
M. domtstica
At domtstica
M. domtstica
MjrmeUotettix maculatus
Myrmtltotittix maculata
Panorpa japonica
P. japonica
P. japonica
P. japonica
P. japonica
Piatjcjpha caUgata
PolybUpharis opaca
PotybUpharix opaca
Scatophaga sttrcoraria
Sepsis cynipsta
SienurtUa mtlanura
S. mttanura
S. mttanura
S. mtlanura
S. mttanura
S. mtlanura
S. mdanura
S. mtlanura
Fish
Copadichromis thinos
Cjprmodon ptcostnsis
Cjprinodon ptcostnsis
Birds
Anas platyrftynchos
Anas platyrhjnchos
GalUnago media
G. media
C media
Hirundo rustics
H. rustica
H. rustica
H. rustica
Passer domtsticus
Pavo cristatus
r (size)
r (lymmetry)
n
Character
Reference
.144
-.146
.783
.042
.085
.215
.113
.107
.121
.533
-.262
-.029
-344
.186
.894
-.733
492
44
44
.028
.041
.029
-.020
.008
.094
-.009
-.095
.033
.183
.041
-.038
-.042
.270
.145
.218
.084
-.078
-.021
-.180
.376
-.172
.096
.105
.041
.174
.045
-.037
-.046
-.114
-.008
-.012
-.048
.2550
.0046
-.1243
.0228
-.0175
.1710
.0174
.4293
.1130
.0380
-.1758
-.1575
-.3951
-.0476
-.0988
-.0842
-.0360
-.0100
-.0500
.2972
.4443
.4245
.0799
.1502
.1192
.6428
.3072
.4422
.3885
3423
.630
.8600
.0980
.4624
.6800
.8390
.8400
-.0301
3200
.8540
3966
.2430
.2210
.1191
.1267
.1740
.3582
3843
3895
.6209
100
100
100
100
100
216
230
95
95
95
100
100
50
50
50
100
100
50
30
30
22
121
25
21
25
55
103
84
87
106
26
33
71
38
69
38
34
26
Wing
Bristles
Wing
Bristles
Bristles
Wing
Arista! branch
Wing
Bristles
Sex comb*
Aristal branch
Wing
Wing
Bristles
Bristles
Bristles
Sex comb1
Tibia
Tibia
Hindwing
Forewing
Tibia
Tibia
Tibia
Tibia
Tibia
Wing
Wing
Wing
Wing
Stridulator*
Stridulator*
Wing
Wing
Wing
Wing
Pheromone*
Tibia color*
Wing
Wing
Wing
Tibia
Elytra
Elytra
Elytra
Elytra
Antenna*
Antenna*
Antenna*
Antenna*
Harvey and Walsh (1993)
Markow and Ricker (1992)
Markow and Ricker (1992)
Polak (1997)
Polak (1997)
Markow and Ricker (1992)
Markow and Ricker (1992)
Markow and Ricker (1992)
Markow et al. (1996)
Markow et al. (1996)
Markow and Ricker (1992)
Markow and Ricker (1992)
Markow and Ricker (1992)
Markow and Ricker (1992)
Markow and Ricker (1992)
Markow et al. (19%)
Markow et al. (1996)
Castro et al. (manuscript)
Castro et al. (manuscript)
Cordoba-Aguilar (1995)
C6rdob*Aguilar (1995)
Cordoba-Aguilar (1995)
MeUer (1996)
Meller (1996)
MeUer (1996)
Mailer (1996)
MeUer (1996)
Meller (1996)
MeUer (1996)
MeUer (1996)
MeUer (1998)
MeUer (1998)
Thornhill (1992b)
Thornhill (1992a)
Thornhill (1992a)
Thorahul (1992b)
Thornhill (1992b)
Jennions (1996)
Poulsen et al. (1994)
Poulsen et aL (1994)
Liggett et al. (1993)
Allen and Simmons (19%)
MeUer and Zamora-Munoz
MeUeT and Zamora-Munoi
MeUer and Zamora-Munoz
MeUer and Zamora-Munoz
MeUer and Zamora-Munoz
MeUer and Zamora-Munoz
MeUer and Zamora-Munoz
MeUer and Zamora-Munoz
-.046
.125
.167
.0520
.4446
-5280
41
153
52
Bower*
Skeleton
Skeleton
Taylor et aL (manuscript)
Kodric-Brown (1998)
Kodric-Brown (1998)
.150
.440
-.244
-.107
.174
.650
-.162
.242
.914
.1400
-5000
-.1888
-.2290
.2732
.4450
.9268
.1500
.1300
.6500
3342
36
36
Tarsus
Tarsus
Wing
Color*
Omland (manuscript)
Omland (manuscript)
Fiske et aL (1994)
FiskeetaL (1994)
Fiske et al. (1994)
MeUer (1992)
Meller (1993)
MeUer (1994)
MeUer (1994)
Kimball (1995)
Hasegawa (1995)
354
316
.163
368
291
100
100
100
36
100
100
100
22
21
22
61
33
367
390
14
16
Tarsus
Tail*
Tail*
TailWing
Badge*
Train*
(1997)
(1997)
(1997)
(1997)
(1997)
(1997)
(1997)
(1997)
ThornhiH and Mailer • Size and asymmetry in sexual selection
549
Table 1, continued
Species
Ptilonorhynchus violacrus
Taeniopjgia guttata
Tachjdntta bicoior
Tttrao utrix
Tttrao Utrix
Vidua macroura
Mammals
Homo sapiens
Homo sapiens
Ourcbia oxtribi
r (symmetry)
n
Character
Reference
.532
.320
.360
.120
.100
.195
.587
.7489
-.0200
.1500
.0700
-.2500
33
Bower*
Plumage*
Tail
Tarsus
Tail feather*
Tail*
Borgia (1985)
Swaddle and Cuthul (1994a)
Dunn ct aL (1994)
Rintamaki et aL (1997)
Rintamakd et al. (1997)
Savalli (1998)
.420
-.072
.080
.2994
.0754
.3676
61
500
42
Skeleton
Thornhill et al. (1995)
Manning et al. (1996)
Arcese (1994)
r(size)
10
21
42
46
7
Breast*
Horn*
' Secondary sexual character.
icanuy larger than zero (symmetry, statistic = 15.70; size, statistic •= 14.45, p < .001). The effect sizes for symmetry and
size were not significantly different, as the 95% (3 of the difference includes zero (-0.026, 0.042).
After we divided all samples into those considered to be
secondary sexual characters and other characters, we calculated the mean weighted effect size for character size and symmetry. For morphological traits not classified as secondary sexual characters (n = 48), weighted mean effect size for character size was r = .184 (95% CI 0.152,0.210) and for character
symmetry r = .174 (95% C3 0.132, 0.200). These means are
significantly greater than zero (size, statistic = 12.95; symmetry, statistic = 12.19; all p < .001), and they are not significantly different (95% Q of difference, -0.029, 0.51). For
morphological characters classified as secondary sexual characters (n •= 25), weighted mean effect size differed significantly and the effect for character symmetry was larger than
the effect size for character size: size, r = .153, 95% d 0.088,
0.216; symmetry, r = .241, 95% a 0.174, 0.299; 95% Q of
difference, 0.022, 0.154). Both means exceed zero (size, statistics ™ 6.48; symmetry, statistic = 10.184; all p < .001).
Analyses based on studies as units of analysis
When analyzed on the basis of the 33 studies, weighted mean
effect size for character symmetry was r = .198 (95% CJ 0.152,
0.245); weighted mean effect size for character size was r =
.154 (95% a 0.108, 0.201). The means are significantly greater than zero (size, statistic = 8.99; symmetry, statistic «• 1138;
all p < .001), and they are not significantly different (95% CI
of difference, -0.003, 0.093).
Analyses based on species as units of analysis
Analysis at the level of the 29 species revealed a weighted
mean effect size for character symmetry of r = .219 (95% Q
0.169, 0.260); the weighted mean effect size for character size
was r = .140 (95% Q 0.086, 0.191). The means are significantly greater than zero (size, statistic = 7.29; symmetry, statistic »= 11.61; all p < -001), and the means are significantly
different (95% a of difference, 0.030, 0.136).
In conclusion, the effect size for symmetry was significantly
larger than that for character size in analysis across species,
but not in the analyses of samples or studies.
DISCUSSION
Only recendy has fluctuating asymmetry been implied to play
a role in sexual selection, with the first study appearing in
1985 (Borgia, 1985). As is the case for all novel approaches
and developments in a field, a subsequent surge of interest
generates both supportive and contradictory evidence (Hunt,
1997). This is not surprising for evolutionary biology, given
that it is a science not generally based on consistent principles. Selection and other evolutionary agents differ between
species and in time and space within species. Often in reviews,
hypotheses are evaluated by vote-counting procedures without
any consideration of sample sizes. Studies will be statistically
significant depending on the magnitude of an effect and the
sample size. If the power of a statistical test is low, for example,
due to small sample size or large within-group variance, we
cannot necessarily accept the null hypothesis of no relationship. This is the situation in which meta-analytic approaches
become useful because they can help quantitatively assess the
magnitude of effects after taking sample size and potential
moderator variables into account (Cooper and Hedges, 1994;
Hedges and Olkin, 1985; Rosenthal, 1991). We have used this
approach to assess the relative importance of size and symmetry in sexual selection. Sample size varied from 7 to 500
across the studies analyzed in this paper (Table 1).
The general role of asymmetry in sexual selection has been
disputed because several studies have shown no effects, whereas others have demonstrated highly significant effects. In the
present study, using studies that include symmetry and size of
the same trait in relation to success in sexual competition, we
demonstrated that both size and symmetry are significantly
and positively related to sexual selection. Our analysis was conducted at three levels. Analysis using all samples for which
both size and symmetry effects were available and analysis using means for studies revealed no significant difference between the effect size of symmetry and sexual selection and
size and sexual selection. The difference at the species level
of analysis, however, was significant, with symmetry having the
larger effect size. The sample size for studies (n •= 33) and
species (n = 29) were moderate. More research in which investigators include analysis of both size and symmetry in relation to mating success or attractiveness to the opposite sex
is needed to more accurately measure the relative importance
of size and symmetry in sexual selection.
The near-zero correlation between the influences of symmetry versus size on sexual competition indicates that size and
symmetry play independent or, as indicated by the analysis at
the level of species, weakly similar, roles in the sexual selection
systems of different species. The extent to which this may be
due to size and asymmetry signaling different kinds of benefits
needs careful theoretical and empirical consideration.
A study of asymmetry and sexual selection based on a much
larger sample of studies revealed an average effect size of 0.34
for species (Moller and Thomhill, 1998), which ia somewhat
higher than the value found in the present study (0.219).
550
Hence, the sample used in the present study (because it contained only studies with information on effects of both character size and symmetry) may have been somewhat biased toward a weak effect Obviously, a sample with a weak effect of
symmetry would be conservative in the sense that it would
make any difference between the effect size of size and asymmetry more difficult to detect.
We found a significantly larger difference in effect size between size and symmetry of characters for secondary sexual
traits as compared to ordinary morphological traits at the level
of sample analysis. Studies and species were not compared
because some studies were of only one type of character. It
has been suggested that secondary sexual characters demonstrate larger degrees of asymmetry because of a recent evolutionary history of directional selection (Mailer and Hoglund, 1991; Mpller and Pomiankowski, 1993). A larger degree
of asymmetry in secondary sexual characters as compared to
ordinary morphological traits should be more easily perceived
by females and hence give rise to a larger effect size. This
prediction was supported by our analyses.
Recently, sexual selection was reviewed extensively by Andersson {1994). w n o found considerable evidence for its presence in a wide variety of organisms (see Andersson, 1994: Table 6A). Such compilations of positive evidence are of great
value, but summaries of negative and positive studies are even
more valuable. The relative importance of sexual selection
can be assessed only from quantitative approaches such as
meta-analysis. Meta-analysis estimates the overall effect of a
relationship in a number of studies and furthermore allows
quantitative assessment of the importance of moderator variables presumed to influence the magnitude of a particular
relationship. This was the approach used in the present study,
which suggests differences in the magnitude of effect sizes for
the relative importance of size and symmetry in sexual selection. Although size has been implicated to play an important
role in sexual selection (review in Andersson, 1994), our study
confirms this. The effect of size in sexual selection, however,
is no greater and is perhaps smaller than that of symmetry.
In conclusion, the relative importance of character size and
symmetry was evaluated meta-anah/tically for all available studies, and the calculations demonstrated significant relationships for both size and symmetry, but symmetry appears to
have a slightly greater effect, especially in studies of secondary
sexual traits.
J. P. Camacho, R. Kimball, J. T. Manning, K. Omland, and J. Taylor
kindly provided unpublished information. The research of A.P.M. was
supported by a grant from the Danish Natural Science Research
Council. S. Gangettad provided statistical advice.
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