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Electronic Supplementary Material
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Indirect selection on female extra-pair reproduction? Comparing the additive
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genetic value of maternal half-sib extra-pair and within-pair offspring
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Jane M. Reid & Rebecca J. Sardell
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Derivation of the equivalence between the difference in breeding value for
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fitness between a female’s EPO and the WPO they replaced and the genetic
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covariance between fitness and male net paternity gain.
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The aim is to estimate the difference in additive genetic breeding value (BV) for fitness
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between a female’s extra-pair offspring (EPO, sired by an extra-pair male) and the
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hypothetical within-pair offspring (WPO, sired by a female’s socially paired mate) that the
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EPO replaced. This difference must be expressed as a genetic covariance in order to be
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estimated directly within an animal model.
17
In general, the expected BV of an offspring equals the average BV of its parents such that:
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E[WEPO] = ½(WF + WEPM)
(Eqn 1)
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E[WWPO] = ½(WF + WWPM)
(Eqn 2)
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where WEPO, WWPO, WF, WEPM and WWPM are the BVs for fitness (W) of EPO, WPO, the female
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and her extra-pair and socially paired males respectively.
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23
1
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The expected difference in BV between a female’s EPO and the WPO it replaced (BV) is
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therefore half the difference in BV between the female’s extra-pair and socially paired
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males:
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BV = E[WEPO] – E[WWPO] = ½(WF + WEPM) - ½(WF + WWPM)
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BV = ½(WEPM – WWPM)
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Averaged over all EPO, the mean difference in BV between EPO and the WPO they replaced
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is therefore:
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E[BV] = ½(EEPO[WEPM] – EEPO[WWPM])
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Verbally, E[BV] is proportional to the BV of extra-pair sires averaged over all EPO minus the
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BV of the socially paired males that were cuckolded by the extra-pair sires, again averaged
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over all EPO (EEPO indicates an expectation over EPO). These quantities can be rewritten as
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averages over sires rather than offspring, such that:
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E[BV] = ½((EM[NEWA]/EM[NE]) - (EM[NCWA]/EM[NC]))
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where NE is the number of EPO a male sired and NC is the number of offspring a male lost
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through cuckoldry and EM[NE] and EM[NC] are these quantities averaged over all males (EM
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indicates an expectation over males). The quantities EM[NEWA] and EM[NCWA] are therefore
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the products of the numbers of EPO sired and offspring lost through cuckoldry and the
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male’s BV for fitness (WA), again averaged over all males.
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However, EM[NE] = EM[Nc] because the mean paternity gain through EPO must equal the
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mean paternity loss through cuckoldry across a population. Hence:
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E[BV] = ½((EM[NEWA] – EM[NCWA])/EM[NE])
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However in general, the covariance between two variables x and y is defined as:
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cov(x,y) = E[(x-µx)(y-µy)]
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where µx and µy are the means of x and y respectively. It follows that:
(Eqn 3)
(Eqn 4)
(Eqn 5)
(Eqn 6)
2
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cov(x,y) = E[xy] - µxµy
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E[xy] = cov(x,y) + µxµy
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E[xy] = cov(x,y) + E[x]E[y]
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Applying Eqn 7 to Eqn 6 gives:
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E[BV] = ½ . (1/EM[NE]) . ((cov(NE,WA) + EM[NE]EM[WA]) – (cov(NC,WA) + EM[NC]EM[WA]))
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where cov(NE,WA) is the covariance between NE and BV for fitness.
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However, in the baseline population the mean BV is zero by definition such that EM[WA]  0.
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Hence:
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E[BV] = ½ . (1/EM[NE]) . (cov(NE,WA) – cov(NC,WA))
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In general by the additive law of covariances:
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cov(NE,WA) – cov(NC,WA) = cov(NE-NC,WA).
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Hence:
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E[BV] = ½(cov(NE-NC,WA))/EM[NE]
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E[BV] = ½(covA(NE-NC,W))/E[NE]
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The mean difference in BV for fitness between EPO and the WPO they replaced can
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therefore be estimated as half the genetic covariance between (NE-NC) and fitness (covA(NE-
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NC,W), where NE is the number of EPO a male sired through extra-pair mating and NC is the
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number of offspring that male lost through cuckoldry), divided by the number of EPO sired
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averaged over males. The quantity NE-NC is a male’s net paternity gain through extra-pair
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reproduction.
(Eqn 7)
(Eqn 8)
(Eqn 9)
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This basic derivation assumes that additive genetic effects on fitness are the same in males
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and females such that the inter-sex genetic correlation rmf  1, and hence that individual
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males sire sons and daughters of equal paternal additive genetic value. It also assumes that
3
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the genetic correlation between a male’s net paternity gain through sons and daughters is
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ca.1. The latter constraint can be verified or relaxed by considering net paternity gain
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through sons and daughters as separate traits. The condition E M[NE] = EM[Nc] must hold
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when NE and NC are evaluated as the numbers of sons and daughters gained through extra-
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pair reproduction and lost through cuckoldry separately (assuming that a female’s offspring
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do not change sex depending on their paternity). The mean differences in BV between a
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female’s EP sons and the WP sons they replaced, and between a female’s EP daughters and
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the WP daughters they replaced, can therefore be estimated as:
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E[BVS] = ½(covA(NES-NCS,W)/E[NES])
(Eqn 10)
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E[BVD] = ½(covA(NED-NCD,W)/E[NED])
(Eqn 11)
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where the subscripts S and D indicate values through sons and daughters respectively.
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Distribution of kinship
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Due to the depth, completeness and high inter-connectedness of the song sparrow
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pedigree, some degree of non-zero kinship (k) was detectable in the vast majority of all
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pairwise comparisons among all individuals retained in the pedigrees used for the current
100
analyses. For example, across the 2432 individuals in the pedigree used for the analysis of
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offspring survival to recruitment, only 1.6% of all pairwise k values were zero. Figure S1
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shows the full distribution of k and the same distribution without the zero values. These
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distributions are not overwhelmingly skewed or zero-inflated and mean kinship was
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relatively high compared to many other wild population pedigrees [e.g. 1-2]. There was
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therefore substantial power to estimate additive genetic variances (VA) and covariances
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(covA). Indeed, a heritability of only h2prob  0.07 was detectable. This power stemmed
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primarily from comparisons among second-order, third-order and more distant relatives:
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only 0.10% of all pairwise comparisons (and 0.11% of pairwise comparisons for which k > 0)
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comprised full-sibs. To further quantify power we estimated the ‘reliability’ (r2) as the ratio
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of the variance in posterior modal BV across all individuals to the posterior modal V A; r2  1
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when BVs are accurately estimated, and r2  h2 when estimated BVs entirely reflect
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individual phenotype [23]. Reliability r2 = 0.75 for offspring recruitment.
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Figure S1. Distributions of kinship (k) across (a) all pairwise comparisons among all 2432
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individuals in the pedigree and (b) all pairwise comparisons where k > 0. The mean, median,
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minimum, maximum and inter-quartile range (IQR) were 0.065, 0.061, 0.000, 0.471 and
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0.043-0.079 across all pairwise comparisons and 0.066, 0.062, 0.0001, 0.471 and 0.045-
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0.080 excluding zeros. Bar width is 0.005.
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Recruitment analyses
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Unlike the more typical use of generalized linear (mixed) models, the primary aim of an
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animal model is not necessarily to explain or account for the maximum possible proportion
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of variation in the trait of interest through known fixed effects. Indeed, the degree to which
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any fixed effects should be included in animal models is debatable [3]. This is because the
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total phenotypic variance is of interest; a trait’s heritability (h2) is estimated as the ratio of
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additive genetic variance (VA) to the total variance that remains after accounting for fixed
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effects. Estimated heritabilities will therefore increase as more fixed effects are fitted (and
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hence as a greater proportion of remaining variation reflects additive genetic effects rather
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than any environmental source [3]). Such high heritabilities are likely to be misleading in
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the context of understanding evolution. However, failure to fit appropriate fixed effects
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that describe major environmental variation can bias comparisons of estimated breeding
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values among groups of animals of interest [4,5]. There is therefore no straightforward
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‘correct’ answer to which fixed effects should be fitted. The most judicious approach may
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be to fit effects that describe major environmental differences between groups of
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individuals within which selection can or is likely to act, but to avoid fitting further fixed
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effects that describe more detailed aspects of underlying ecology or environmental
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variation.
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Given these considerations, our basic animal model structure for offspring recruitment
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included the following fixed effects:
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1) an offspring’s natal year, modelled as 16 levels across the 16 studied cohorts,
because mean recruitment varies among song sparrow cohorts [6]
7
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2) a linear regression of recruitment on an offspring’s inbreeding coefficient (f, hence
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estimating inbreeding depression), because mean recruitment shows inbreeding
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depression in song sparrows and failing to control for inbreeding depression can
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inflate estimates of VA [7,8]
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3) an offspring’s sex, modelled as two levels, because mean recruitment differs
between male and female song sparrows [6]
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4) an offspring’s extra-pair status, modelled as two levels (EPO and WPO), to minimise
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the degree to which any environmental variation associated with paternity status
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could confound the estimated genetic effects of interest [4,5]
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5) a status by sex interaction, because relative recruitment differs between male and
female EPO versus WPO in mixed-paternity song sparrow broods [9].
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In practice, estimates of the latter two effects did not differ significantly from zero and
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estimates of VA and h2 remained quantitatively similar whether or not they were included in
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the animal model (Table S1).
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environmental variables that could be hypothesised to influence offspring recruitment in
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general or in relation to individual or brood paternity status were not fitted. Fixed rather
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than random year effects were modelled because selection on recruitment (survival to age
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one year) is likely to act primarily within years. However, VA and h2 were still significantly
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greater than zero when random rather than fixed year effects were modelled (posterior
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mode for h2latent: 0.08; 95%CI: 0.03-0.15).
Further fixed effects describing specific ecological or
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The 2196 offspring were produced by 217 different mothers and 215 different genetic
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fathers and reared by 215 different social fathers. Overall, 138 (64%) mothers and 131
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(61%) genetic fathers that contributed any offspring contributed ≥1 WPO and ≥1 EPO, while
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70 (32%) mothers and 71 (33%) genetic fathers contributed ≥1 WPO but no EPO and 9 (4%)
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mothers and 13 (6%) genetic fathers contributed ≥1 EPO but no WPO. Furthermore, ca.50%
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of mothers and fathers that contributed ≥1 WP daughter or son also contributed ≥1 EPO of
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the same sex. Therefore, across the population, there was substantial congruence in the
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identities of parents of WPO and EPO.
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Random effects of an individual’s natal brood, natal territory, mother identity and/or
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social father identity were initially fitted to estimate variances due to consistent effects of
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brood, location and parents providing care and minimise the degree to which such effects
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could potentially confound estimates of VA [10]. However estimates of these variances
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were close to zero and estimates of fixed effects, VA and h2 remained quantitatively similar
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whether or not they were included in the animal model (Table S2). The lack of detectable
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variance due to brood, territory and provisioning parents may be because the focal trait
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(survival from ca.6 days post-hatch to recruitment at age one year) primarily covers a period
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when offspring are no longer associated with their natal territory, parents or brood mates.
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Effects of natal and rearing environment may therefore be expected to be small relative to
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environmental conditions experienced during the summer, autumn, winter and spring after
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fledging.
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Table S1. Posterior modes (and 95% credible intervals) for additive genetic variance, latent-
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and probability-scale heritabilities and inbreeding depression in survival to recruitment
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estimated across 2196 known-sex song sparrow offspring, and effects of sex and extra-pair
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status. The full model and models that did not include non-significant fixed effects are
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presented. h2prob was estimated assuming µR = 0.19. Small discrepancies among estimates
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of the same parameter from different model runs are expected due to Monte Carlo error.
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additive genetic
latent-scale
probability-scale
inbreeding
sex
extra-pair
extra-pair
variance
heritability
heritability
depression
(male vs
status
status by sex
(VA)
(h2latent)
(h2prob)
(f)
female)
(EPO vs WPO)
interaction
0.61
0.13
0.07
-9.2
0.37
-0.24
0.11
(0.21 – 1.35)
(0.05 – 0.24)
(0.03 – 0.14)
(-14.4 – -5.9)
(0.09 – 0.72)
(-0.67 – 0.23)
(-0.36 – 0.85)
0.60
0.14
0.07
-9.9
0.50
-0.16
(0.18 – 1.25)
(0.05 – 0.23)
(0.02 – 0.14)
(-14.1 – -5.6)
(0.14 – 0.70)
(-0.43 – 0.19)
0.62
0.13
0.07
-9.9
0.40
(0.22 – 1.30)
(0.05 – 0.23)
(0.03 – 0.13)
(-14.1 – -5.7)
(0.16 – 0.73)
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Table S2. Posterior modes (and 95% credible intervals) for additive genetic variance, latent-
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scale heritabilities and inbreeding depression in survival to recruitment estimated across
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2196 known-sex song sparrow offspring, and effects of sex and extra-pair status.
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Components of variance due to brood, natal territory, mother identity and social father
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identity were additionally estimated. Variance components are bounded to zero, and small
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discrepancies among estimates of the same parameter from different model runs are
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expected due to Monte Carlo error.
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additive genetic
additional variance
latent-scale
inbreeding
sex
extra-pair
extra-pair
variance
component
heritability
depression
(male vs
status
status by sex
(h2latent)
(f)
female)
(EPO vs WPO)
interaction
(VA)
0.63
brood:
0.13
-9.9
0.38
-0.24
0.09
(0.21 – 1.45)
0.006
(0.05 – 0.24)
(-15.4 – -5.8)
(0.07 – 0.73)
(-0.73 – 0.21)
(-0.36 – 0.84)
(<0.0001 – 0.63)
0.64
territory:
0.14
-9.6
0.34
-0.30
0.15
(0.24 – 1.48)
0.001
(0.05 – 0.25)
(-15.2 – -5.5)
(0.08 – 0.72)
(-0.73 – 0.19)
(-0.31 – 0.88)
(<0.0001 – 0.19)
0.61
mother:
0.13
-9.2
0.33
-0.22
0.17
(0.16 – 1.31)
0.002
(0.05 – 0.23)
(-14.8 – -5.6)
(0.08 – 0.72)
(-0.70 – 0.24)
(-0.40 – 0.82)
(<0.0001 – 0.29)
0.64
father:
0.13
-9.7
0.39
-0.31
0.14
(0.15 – 1.28)
0.003
(0.04 – 0.23)
(-14.3 – -5.2)
(0.09 – 0.72)
(-0.74 – 0.21)
(-0.41 – 0.84)
(<0.0001 – 0.43)
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Net paternity gain analyses
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Male net paternity gain (NE-NC) varied from 11 to -11, 6 to -6 and 5 to -5 through all
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offspring, sons and daughters respectively. Median net paternity gain was zero in all three
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cases. The distributions were approximately symmetrical but not normal due to excess
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zeros (figure S2). These zeros comprised males that gained and lost zero paternity through
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EPR, and males that sired the same number of EPO as they lost through cuckoldry.
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However, analyses assumed Gaussian error distributions as the most appropriate
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distribution possible.
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Five immigrant males, totalling 18 male-years, were excluded from analyses because
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f is undefined for immigrants (as opposed to their offspring). However net paternity gain
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was zero in 12 of 18 male-years and the summed total net paternity gain was -4 across the
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remaining 6 male-years. Excluding phenotypic data from the immigrant males therefore
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caused very minimal violation of the assumption that E[N E] = E[NC]. However fixed year
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effects were included in models to account for the extremely slight among-year variation in
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mean NE-NC caused by excluding immigrant males.
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The 293 observed males were reared by 142 different mothers and 144 different
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social fathers. Estimates of VA, h2pat and inbreeding depression in male net paternity gain
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remained similar when random effects of mother and social father identities were modelled
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(table S3). Exploratory analyses showed that net paternity gain did not vary with male age.
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Covariances between net paternity gain and recruitment due to mother and social father
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identities were close to zero.
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12
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Figure S2. Distributions of net paternity gain across (a) all offspring (NE-NC), (b) sons (NES-
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NCS) and (c) daughters (NED-NCD) calculated across all adult male song sparrows alive in each
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year during 1993-2008.
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Table S3.
Posterior modes (and 95% credible intervals) for variance components,
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heritabilities and inbreeding depression in male net paternity gain estimated across 293
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individual males totalling 738 male-years. Variance components are bounded to zero and
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small discrepancies among estimates of the same parameter from different model runs are
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expected due to Monte Carlo error.
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additive genetic
permanent
additional
residual
heritability
inbreeding
variance
individual
variance
variance
(h2pat)
depression
(VA)
variance
component
(VR)
(f)
(VPI)
0.23
0.001
mother: 0.001
4.22
0.06
-2.51
(<0.001 – 0.51)
(<0.001 – 0.23)
(<0.001 – 0.31)
(3.78 – 4.75)
(<0.001 – 0.11)
(-5.78 – 1.04)
0.21
0.001
father: 0.001
4.30
0.05
-2.26
(<0.001 – 0.53)
(<0.001 – 0.23)
(<0.001 – 0.21)
(3.79 – 4.73)
(<0.001 – 0.11)
(-5.65 – 1.30)
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Model priors & specifications
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Parameter expanded priors on variance components used normally distributed working
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parameter priors with mean zero and variance 1000 and inverse-Wishart distributed
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location effect priors with degree of belief and limit variance of one (forming a scaled non-
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central F-distribution [11,12]).
278
genetic
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alpha.V=diag(2)*1000)). Such priors are relatively uninformative and facilitate mixing when
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variance components are close to zero [12]. However, estimated variance components
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remained similar when models were rerun with non-parameter expanded inverse Wishart
282
priors across reasonable variation in prior parameter values (including combinations of
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diagonal and off-diagonal values for V of 0.1-1 and -0.5–0.5 and values for nu of 0.002-0.1).
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These analyses indicate that our data are informative for estimating the main variance
285
components of interest, and specifically VA and covA(NE-NC,W).
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≥2005000 iterations, burn-in ≥5000 and thinning interval ≥2000 except that bivariate
287
analysis of sex-specific recruitment required 10010000 iterations with burn-in and thinning
288
interval 10000 to ensure autocorrelation of <0.05.
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recruitment was repeated on a dataset with sex randomised across offspring. The posterior
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distribution for rmf was similar to that estimated across the real data. Estimates of VA and h2
291
in net paternity gain and recruitment, and the genetic covariance, also remained similar
292
when analyses were fitted using maximum likelihood in DMU.
covariance
of
Bivariate model priors were similar but specified prior
zero
(such
that
V=diag(2),
nu=2,
alpha.mu=c(0,0),
Final analyses used
Bivariate analysis of sex-specific
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295
296
15
297
Immigrants
298
Estimates of BVs can be biased if a population comprises multiple genetic groups that are
299
not explicitly modelled [5]. In song sparrows, the parents of EPO and WPO are largely
300
congruent and are therefore unlikely to represent different genetic groups (see
301
‘Recruitment analyses’). However bias could arise if the occasional immigrants to Mandarte
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(1.1year-1 on average) comprise a different genetic group from the natives that form the
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baseline (founder) pedigree generation. In such cases, each individual’s estimated BV
304
should be corrected for the relative contribution of each genetic group to its ancestry [13].
305
Group effects can be estimated as the mean BV across the hypothetical (‘phantom’) parents
306
of the founders of each putative genetic group [13], in our case immigrants versus native
307
founders. However, the distributions of BVs for offspring recruitment (survival from 6 days
308
post-hatch to age one year) were very similar across phantom parents of native founders
309
and subsequent immigrants; the 95%CI for each group substantially overlapped the
310
posterior mean for the other (natives: posterior mean -0.002, 95%CI -0.22-0.23; immigrants:
311
posterior mean 0.005, 95%CI -0.26-0.25). The difference in mean BV between the two
312
groups did not differ from zero (posterior mode: -0.008; 95%CI: -0.35-0.35). There was
313
therefore no evidence that the native founders and immigrants constituted different genetic
314
groups with respect to offspring recruitment.
315
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321
Dominance genetic variance
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Estimates of VA can be biased if dominance genetic effects and dominance genetic variance
323
(VD) are not explicitly modelled [14-16]. Furthermore, estimating individual dominance
324
genetic effects could in theory allow direct test of the hypothesis that females produce EPO
325
of higher non-additive genetic value than the WPO they replaced (one ‘compatible genes’
326
hypothesis explaining female EPR). However, power to estimate dominance genetic effects
327
and hence VD across the song sparrow dataset is currently low. Indeed, VD has not yet been
328
robustly estimated in any wild vertebrate population. This is because information to
329
estimate dominance genetic effects and VD stems primarily from phenotypic comparisons
330
among full-sibs [11], and full-sibs comprise a small proportion of all available phenotypic
331
comparisons in the song sparrow dataset (see ‘Distribution of kinship’).
332
occurrence of EPR reduces the number of full-sibs (and increases the number of half-sibs).
333
There is therefore little power to estimate dominance genetic effects with sufficient
334
precision to test the hypothesis that females produce EPO of higher dominance genetic
335
value. However, this same data structure means that estimates of VA and covA are primarily
336
informed by comparisons among half-sibs and more distant relatives.
337
resemblance among such relatives does not generally include a component due to
338
dominance (except for some specific and rare types of relatives such as double-first cousins,
339
[11]). Our estimates of VA derived from models that do not include dominance genetic
340
effects or VD are therefore unlikely to be substantially biased. Indeed, preliminary analyses
341
suggest that estimating VD within the animal model for offspring recruitment scarcely alters
342
the estimate of VA (notwithstanding that estimates of VD are imprecise). Rather, in general,
343
unestimated VD is more likely to be estimated as permanent or common environment
344
variance [10,16]. By modelling a fixed regression on individual coefficient of inbreeding (f)
Indeed, the
Phenotypic
17
345
we minimise the degree to which dominance genetic effects (as opposed to V D) can bias
346
estimates of VA [16], and account for any resemblance among relatives that is due to
347
correlated inbreeding depression in populations with variance in family size [17].
348
349
350
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