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A multivariate analysis of genetic constraints to life history evolution in a wild population of red deer.

Walling CA, Morrissey MB, Foerster K, Clutton-Brock TH, Pemberton JM, Kruuk LE - Genetics (2014)

Bottom Line: We use factor analytic modeling of the genetic variance-covariance matrix ( G: ) to reduce the dimensionality of the problem and take a multivariate approach to estimating genetic constraints.We found limited support for genetic constraint through genetic covariances between traits, both within sex and between sexes.We discuss these results with respect to other recent findings and to the problems of estimating these parameters for natural populations.

View Article: PubMed Central - PubMed

Affiliation: Institute of Evolutionary Biology, School of Biological Sciences, University of Edinburgh, Edinburgh, EH9 3JT, United Kingdom craig.walling@ed.ac.uk.

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The simulated distribution of estimates of θ and e(β) for females. (A) θ1_f; (B) θ2_f; (C) θ3_f; (D) θ4_f; (E) e(βf); (F) e(βf)nc; (G) Re_f produced by carrying through the errors in the estimation of Gf and βf. Values <0 cannot exist except for θ4_f, and thus the distributions are presented to aid in interpretation of whether the simulated distributions are distinct from zero, i.e., have a normal distribution that is not highly concentrated near (ramped up against) zero. Dashed lines show the position of the “best estimate”, i.e., the estimate when using the maximum-likelihood estimate of the parameters of Gf and βf; this is the value given in Table 2 and Table 3. For E and F, solid lines show the position of the average evolvability over random selection gradients (ēf); see Materials and Methods for details.
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fig2: The simulated distribution of estimates of θ and e(β) for females. (A) θ1_f; (B) θ2_f; (C) θ3_f; (D) θ4_f; (E) e(βf); (F) e(βf)nc; (G) Re_f produced by carrying through the errors in the estimation of Gf and βf. Values <0 cannot exist except for θ4_f, and thus the distributions are presented to aid in interpretation of whether the simulated distributions are distinct from zero, i.e., have a normal distribution that is not highly concentrated near (ramped up against) zero. Dashed lines show the position of the “best estimate”, i.e., the estimate when using the maximum-likelihood estimate of the parameters of Gf and βf; this is the value given in Table 2 and Table 3. For E and F, solid lines show the position of the average evolvability over random selection gradients (ēf); see Materials and Methods for details.

Mentions: Estimates of β and G have associated error and thus so do values calculated from them [e.g., , θ, and e(β)]. Errors in these estimates were approximated using an MC simulation algorithm (see also Morrissey et al. 2012a). Briefly, we drew 100,000 multivariate random normal (MVN) values of S, P, and G, using the maximum-likelihood estimates of these parameters (from ASReml) as the mean and the variance covariance matrices of these parameter estimates as the variance (again these are given in ASReml). These 100,000 values were then combined as appropriate in Equations 5, 4, 6, 7, and 9 to produce 100,000 estimates of β, , θ, e(β), and Re. The 95% credible interval (CI) around these values was then calculated using the quantile function in R and used as an estimate of the 95% credible interval around each parameter estimate. It should be noted that this method assumes the sampling errors in the estimates of variances and covariances are multivariate normal. For angles θ1, θ2, θ3, θ5_bs, and θ6_bs (which are all defined as angles between two vectors) and for all values of evolvability, estimates cannot be negative and thus interpreting a lack of overlap of the 95% CI with zero as indicative of the value differing from zero is not valid. As such, statistical hypothesis tests have limited meaning and we therefore assessed statistical support for substantially nonzero values by examining the distribution of MC samples (see Figure 2, Figure 3, and Figure 4). In practice, this involves visual inspection of the distributions of estimates and, when the distribution is concentrated close to zero (i.e., is associated with left truncation and strong right skew), drawing conclusions equivalent to those associated with failure to reject a hypothesis.


A multivariate analysis of genetic constraints to life history evolution in a wild population of red deer.

Walling CA, Morrissey MB, Foerster K, Clutton-Brock TH, Pemberton JM, Kruuk LE - Genetics (2014)

The simulated distribution of estimates of θ and e(β) for females. (A) θ1_f; (B) θ2_f; (C) θ3_f; (D) θ4_f; (E) e(βf); (F) e(βf)nc; (G) Re_f produced by carrying through the errors in the estimation of Gf and βf. Values <0 cannot exist except for θ4_f, and thus the distributions are presented to aid in interpretation of whether the simulated distributions are distinct from zero, i.e., have a normal distribution that is not highly concentrated near (ramped up against) zero. Dashed lines show the position of the “best estimate”, i.e., the estimate when using the maximum-likelihood estimate of the parameters of Gf and βf; this is the value given in Table 2 and Table 3. For E and F, solid lines show the position of the average evolvability over random selection gradients (ēf); see Materials and Methods for details.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4256783&req=5

fig2: The simulated distribution of estimates of θ and e(β) for females. (A) θ1_f; (B) θ2_f; (C) θ3_f; (D) θ4_f; (E) e(βf); (F) e(βf)nc; (G) Re_f produced by carrying through the errors in the estimation of Gf and βf. Values <0 cannot exist except for θ4_f, and thus the distributions are presented to aid in interpretation of whether the simulated distributions are distinct from zero, i.e., have a normal distribution that is not highly concentrated near (ramped up against) zero. Dashed lines show the position of the “best estimate”, i.e., the estimate when using the maximum-likelihood estimate of the parameters of Gf and βf; this is the value given in Table 2 and Table 3. For E and F, solid lines show the position of the average evolvability over random selection gradients (ēf); see Materials and Methods for details.
Mentions: Estimates of β and G have associated error and thus so do values calculated from them [e.g., , θ, and e(β)]. Errors in these estimates were approximated using an MC simulation algorithm (see also Morrissey et al. 2012a). Briefly, we drew 100,000 multivariate random normal (MVN) values of S, P, and G, using the maximum-likelihood estimates of these parameters (from ASReml) as the mean and the variance covariance matrices of these parameter estimates as the variance (again these are given in ASReml). These 100,000 values were then combined as appropriate in Equations 5, 4, 6, 7, and 9 to produce 100,000 estimates of β, , θ, e(β), and Re. The 95% credible interval (CI) around these values was then calculated using the quantile function in R and used as an estimate of the 95% credible interval around each parameter estimate. It should be noted that this method assumes the sampling errors in the estimates of variances and covariances are multivariate normal. For angles θ1, θ2, θ3, θ5_bs, and θ6_bs (which are all defined as angles between two vectors) and for all values of evolvability, estimates cannot be negative and thus interpreting a lack of overlap of the 95% CI with zero as indicative of the value differing from zero is not valid. As such, statistical hypothesis tests have limited meaning and we therefore assessed statistical support for substantially nonzero values by examining the distribution of MC samples (see Figure 2, Figure 3, and Figure 4). In practice, this involves visual inspection of the distributions of estimates and, when the distribution is concentrated close to zero (i.e., is associated with left truncation and strong right skew), drawing conclusions equivalent to those associated with failure to reject a hypothesis.

Bottom Line: We use factor analytic modeling of the genetic variance-covariance matrix ( G: ) to reduce the dimensionality of the problem and take a multivariate approach to estimating genetic constraints.We found limited support for genetic constraint through genetic covariances between traits, both within sex and between sexes.We discuss these results with respect to other recent findings and to the problems of estimating these parameters for natural populations.

View Article: PubMed Central - PubMed

Affiliation: Institute of Evolutionary Biology, School of Biological Sciences, University of Edinburgh, Edinburgh, EH9 3JT, United Kingdom craig.walling@ed.ac.uk.

Show MeSH