Why estimate linear and full (linear, quadratic, and correlational) selection coefficients separately?

"We then fitted a linear regression including all three life-history traits to estimate the vector of linear selection gradients, β, for each sex (Lande and Arnold 1983). A quadratic regression model incorporating all linear, quadratic, and cross-product terms was then used to estimate the matrix of nonlinear selection gradients, γ, for each sex. "

This is from the paper Evidence for strong intralocus sexual conflict in the Indian meal moth, Ploida interpunctella. I am not sure why they estimate these gradients separetly rather than taking them just from the one full model - any suggestions? Is this normal or have they done it for a reason?

For some similar work see these two papers:

• Gosden et al which seems to use only the linear gradients from one model with sex as an effect.

• Stearns et al which uses linear, quadratic and correlational estimated from one multiple regressions model per sex (similar to the main paper I cite first but not using a linear terms only model for the linear coefficient).

I have mailed the corresponding author on the quoted paper to ask why, I will let you know what he says if he replies

1 Answer

This is directly following the advice of Lande & Arnold (1983), saying:

Linear multiple regression can be used first to estimate the forces of directional selection, $\beta$, and their standard errors. Then a quadratic multiple regression (16) or (Al), can be used to estimate the forces of stabilizing selection, $\gamma$, with their standard errors. The regression (16) provides the best quadratic approximation to the selective surface (although valid estimates of $\beta$ can be obtained only from a purely linear regression, or by using the orthogonal regression (Al)).

The reason is that the estimates of direction selection is influenced by the higher order terms in the full model that includes quadratic and correlational selection, and the linear model therefore has the best estimates of the change in mean value over a generation of selection ($s = \bar{z}_{after} - \bar{z}_{before}$). However, the full model is the better representation of the fitness surface. There are also other methods to approximate the fitness surface.

For the record, I used the same approach in my master's thesis many years ago.

• Thanks - what did you do your masters thesis on and who with? (I did mine on multivariate nonlinear selection in a secondary sexual trait hence I should have known the answer to the above, it's just been too long!) Talk in chat? Dec 9 '14 at 16:11
• Actually, after rereading the Lande Arnold paper, this will be problematic only "If the character distribution before selection displays multivariate skewness (non- zero third moments), the linear and quadratic terms are correlated and estimates of β, depend on whether or not the quadratic terms are included in the regression." Thus, if I understand correctly, you can use orthogonal polynomials using poly(x,2) in R or run a model with only the linear and a second model with both the linear and the quadratic term. Is that correct? Aug 20 '18 at 19:43
• The third moment in the sentence means the skewness. Does that mean the same thing as covariance = 0? Aug 20 '18 at 19:45