The ‘compare’ column compares the responses in the two populations. For the respondabilities, *r*(**β**), and evolvabilities, *e*(**β**) and *c*(**β**), the comparison is the ratio of the value in population 1 to that in population 2, when each population is standardized with its own vector or matrix. In other rows, both populations are standardized by the average of the standardization vectors or matrices in the two populations. The row labelled *θ* contains the angles between the response vectors in the two populations, *θ*_{d}.

# Corrigendum

Version of Record online: 26 FEB 2009

DOI: 10.1111/j.1420-9101.2009.01715.x

© 2009 The Authors. Journal Compilation © 2009 European Society For Evolutionary Biology

Additional Information

#### How to Cite

HANSEN, T. F. and HOULE, D. (2009), Corrigendum. Journal of Evolutionary Biology, 22: 913–915. doi: 10.1111/j.1420-9101.2009.01715.x

#### Publication History

- Issue online: 13 MAR 2009
- Version of Record online: 26 FEB 2009

Measuring and comparing evolvability and constraint in multivariate characters

Vol. 21, Issue 5, 1201–1219, Version of Record online: 24 JUL 2008

The paper Hansen & Houle (2008) contains several errors that require correction. The errors are of two kinds. First, an error in the proof of Lemma 1 in Appendix 1, led to incorrect formulas for the mean conditional evolvability, mean respondability, mean autonomy, and mean response difference in the body of the paper. Fortunately, these errors usually have small numerical consequences. Second, Appendix 2 contains numerical errors that make it impossible to repeat calculations from the given information. None of the errors affects the overall conclusions of the paper. We give a concise summary of the important corrections in this corrigendum. A complete version of the corrected material is available online (see Appendix S1 in Supporting Information).

The calculation error in Appendix 1 arose when we omitted a factor 1/2 in the parameters of the beta distribution in the proof of Lemma 1. This led to errors in the formulas for the variances and covariances of quadratic forms given in Lemma 1. As this lemma was used to derive the formulas for the mean conditional evolvability, mean autonomy, mean respondability, and mean response difference, these are all wrong. Additionally, there were typos in the formulas in result 4 and 5. Here we correct the affected results, and the equations in the main text that were derived from them:

**Lemma 1:** If the vector **β** is uniformly distributed on the surface of a unit *k* sphere, **G** and **D** are positive definite matrices of rank *k*, and **GD** =**DG**, then

where *λ* are eigenvalues of **G**, *δ* are eigenvalues of **D**, and the summation is over all *k* eigenvalues.

**Result 3**: If **β** is uniformly distributed on the surface of a unit hypersphere of dimension *k*, and the *k* × *k* matrix **G** is of full rank, then

where *λ*s are the eigenvalues of **G**.

**Result 4:** If **β** is uniformly distributed on the surface of a unit hypersphere of dimension *k*, and the *k* x *k* matrix **M** is of full rank, then

where *λ*s are the eigenvalues of **M**.

**Result 5**: If **β** is uniformly distributed on the surface of a unit hypersphere of dimension *k*, and the *k* x *k* matrix **G** is of full rank, then

where *λ*s are the eigenvalues of **G**.

The correct approximations for mean conditional evolvability, (eqn 5), mean respondability, (p. 1206), mean autonomy, (eqn 6), and mean response difference, (eqn 8), are

where the meaning of the symbols and functions are given in the paper.

The numerical consequences of these changes are less than 10% for examples we have calculated, and in many cases trivially different. The only qualitative improvement is for the mean autonomy () at low dimensions. We also took the opportunity to include an additional correction term in the formula for , yielding a slightly better approximation. Figures 2, 4, and 5 were affected by these errors, plus a minor (and partially compensatory) error in the numerical algorithm used in Figs 2 and 5, but the corrected figures are extremely similar to the originals.

Appendix 2 contained several numerical errors. The selection gradient used to calculate all results in Appendix 2 was **β′** = [0.005/mm, –0.001/mm, 0.01/egg]. The value of this vector was given incorrectly on p. 1217, in the note to Table A3, and in the legend of Table A4. Values of *r*(**β**) in Row 1 of Table A4 were incorrect. Values of *d*(**β**) in row 6 of Table A4 were originally calculated using a **β** of un-standardized length. For consistency, we have recalculated these for a **β** of norm 1. All values except mean unconditional evolvabilities in Table A5 were incorrect due to the errors in the formulae noted above. The corrected Tables are reproduced in their entirety here, with the corrected values shown in bold. Corresponding values in the text should also be changed.

### Acknowledgment

Thanks to J. Stinchcombe for bringing the errors in Appendix 2 to our attention.

### Reference

- 2008. Measuring and comparing evolvability and constraint in multivariate characters. J. Evol. Biol. 21: 1201–1219. &

### Appendices

Statistic | Standardization | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

None* | Mean | Standard deviation | Square root of P | |||||||||

Population | Compare | Population | Compare | Population | Compare | Population | Compare | |||||

1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | |||||

^{}^{}*The units for the responses of each population are a mixture of mm and eggs, and therefore most of these statistics have no clear interpretation. ^{}†The ratio of respondabilities is meaningless on the raw scale. ^{}‡In the columns labelled 1 and 2, this is the angle between **β**and . In the ‘compare’ column, it is the angle between the response vectors in the two populations,*θ*_{d}.^{}§Response difference were calculated from standard length β.
| ||||||||||||

r(β) | 84 | 129 | na† | 0.0125 | 0.0291 | 0.43 | 0.214 | 0.221 | 0.97 | 0.170 | 0.199 | 0.85 |

e(β) | 78 | 111 | 0.70 | 0.0119 | 0.0236 | 0.50 | 0.100 | 0.184 | 0.55 | 0.095 | 0.188 | 0.50 |

c(β) | 29 | 3.8 | 7.67 | 0.0046 | 0.0003 | 15.91 | 0.043 | 0.038 | 1.14 | 0.056 | 0.158 | 0.35 |

a(β) | 0.38 | 0.03 | 0.39 | 0.01 | 0.43 | 0.21 | 0.59 | 0.84 | ||||

θ‡ | 22 | 31 | 16 | 18 | 36 | 15 | 62 | 34 | 51 | 56 | 19 | 78 |

d(β)§ | 54 | 0.010 | 0.17 | 0.23 |

Statistic | Standardization | |||||||
---|---|---|---|---|---|---|---|---|

None* | Mean | Standard deviation | Square root of P | |||||

1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | |

^{}*The units for the responses of each population are a mixture of millimetres and eggs, and these statistics therefore have no clear interpretation.
| ||||||||

40.00 | 63.33 | 0.0051 | 0.0135 | 0.361 | 0.329 | 0.402 | 0.269 | |

50.79 | 83.17 | 0.0070 | 0.0193 | 0.453 | 0.408 | 0.463 | 0.309 | |

13.47 | 4.77 | 0.0016 | 0.0003 | 0.134 | 0.062 | 0.192 | 0.182 | |

0.389 | 0.091 | 0.388 | 0.033 | 0.422 | 0.207 | 0.470 | 0.708 |

### Supporting Information

**Appendix S1** The full, corrected Appendices and corrected figures.

Please note: Wiley-Blackwell are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

Filename | Format | Size | Description |
---|---|---|---|

JEB_1715_sm_AppS1.pdf | 693K | Supporting info item |

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.