Measuring selection when you have too many traits that are too correlated

Author blog: Pro­fess­or John Stinch­combe explains why mul­ti­col­lin­ear­ity is not the end of the road for meas­ur­ing selec­tion on bio­lo­gic­al traits.

Nat­ur­al selec­tion is the engine of adapt­ive evol­u­tion­ary change, and it’s safe to say that evol­u­tion­ary bio­lo­gists since Dar­win have devoted enorm­ous effort to under­stand­ing it. How do we meas­ure it? How strong is it? Does it act the same way on males and females, on life his­tory and mor­pho­logy? How spa­tially or tem­por­ally vari­able is it?  The strength, con­sist­ency, and mode of nat­ur­al selec­tion has pro­found implic­a­tions for many fun­da­ment­al evol­u­tion­ary ques­tions. Empir­ic­al pro­gress requires a reli­able way to meas­ure selec­tion act­ing on the indi­vidu­al traits we care about.

The Lande-Arnold Revolution

The Lande-Arnold approach gave us a straight­for­ward way to meas­ure nat­ur­al selec­tion, and revo­lu­tion­ized our under­stand­ing of it. By per­form­ing mul­tiple regres­sions of rel­at­ive fit­ness on mul­tiple traits, empir­i­cists had a means to estim­ate the strength and dir­ec­tion of selec­tion (Fig 1). Its appeal is multi-faceted. Imple­ment­a­tion is via mul­tiple regres­sion, eas­ily with­in the wheel­house of almost all bio­lo­gists. The estim­ates obtained– par­tial regres­sion coef­fi­cients or selec­tion gradi­ents– have dir­ect con­nec­tions to quant­it­at­ive genet­ic the­ory for pre­dict­ing evol­u­tion­ary responses. Selec­tion gradi­ents also dis­tin­guish dir­ect selec­tion on the trait of interest from selec­tion on cor­rel­ated traits included in the stat­ist­ic­al mod­el. With these tools in hand, evol­u­tion­ary bio­lo­gists set about meas­ur­ing nat­ur­al selec­tion in numer­ous con­texts: the wild, the green­house and growth cham­ber, and exper­i­ment­al meso­cosms. A vast lit­er­at­ure has developed around the Lande-Arnold approach, includ­ing stat­ist­ic­al meth­ods and issues of exper­i­ment­al design, typ­ic­al strengths of selec­tion, ways to com­pare selec­tion amongst dif­fer­ent spe­cies and traits, and many oth­er issues.

A stub­born problem

Since the advent of the Lande-Arnold approach, how­ever, one issue has remained espe­cially stub­born: how to make sense of selec­tion on prin­cip­al com­pon­ent (PC) scores rather than on tra­di­tion­al traits. PC scores have two main advant­ages: first, they reduce the dimen­sion­al­ity of the ana­lys­is. One might be able to sum­mar­ize many to dozens of tra­di­tion­al traits in a few PC axes that describe most of the vari­ation (e.g., Fig. 2 has two pan­els rather than the N pan­els in Fig. 1). In fact, Lande and Arnold per­formed PCA on the Bum­pus data­set for this reas­on. Second, PC scores are uncor­rel­ated with each oth­er. This fea­ture becomes advant­age­ous when the ori­gin­al traits are so highly cor­rel­ated that a regres­sion of rel­at­ive fit­ness on all traits encoun­ters mul­ti­col­lin­ear­ity. Under mul­ti­col­lin­ear­ity, traits are so highly cor­rel­ated that it becomes impossible to dis­tin­guish their sep­ar­ate influ­ences on rel­at­ive fit­ness. Using PC scores is a bit like hav­ing your cake and eat­ing it too: inform­a­tion from all the traits is present in PC scores, but knotty issues of mul­ti­col­lin­ear­ity and dimen­sion­al­ity appear to go away.

While meas­ur­ing selec­tion on PC scores seems use­ful, it induces new draw­backs. Many bio­lo­gists have an intu­ition about traits like date of first flower­ing, branch num­ber, and growth rate; how­ever, relat­ing a lin­ear com­bin­a­tion of all three of these traits to an organism’s nat­ur­al his­tory or eco­logy is much harder. In addi­tion, one of the major advances spawned by the Lande-Arnold revolu­tion was to com­pare selec­tion across stud­ies: is selec­tion on date of first repro­duc­tion sim­il­ar between annu­als and per­en­ni­als?  When traits are on a com­mon scale, these com­par­is­ons are pos­sible. Com­par­ing selec­tion on PC1 scores across stud­ies is much harder, espe­cially giv­en dif­fer­ences in what traits enter the ana­lys­is and how vari­able they are. For these and oth­er reas­ons, meas­ur­ing selec­tion on PC scores has been roundly criticized.

An altern­at­ive approach

In our new Com­ment & Opin­ion piece, pub­lished in Evol­u­tion Let­ters, we sug­gest an altern­at­ive that allows invest­ig­at­ors to meas­ure selec­tion on PC scores and quant­it­at­ively inter­pret them in light of the ori­gin­al traits. The approach seems well suited to deal­ing with mul­ti­col­lin­ear­ity or dimen­sion­al­ity prob­lems. Our approach is to quant­it­at­ively com­bine inform­a­tion about how PC scores relate to rel­at­ive fit­ness with inform­a­tion about how PC scores relate to the ori­gin­al traits, yield­ing an estim­ate of how traits are asso­ci­ated with fit­ness. Prac­tic­ally, this involves some lin­ear algebra—multiplying a mat­rix of eigen­vectors from the ori­gin­al PCA by a vec­tor of selec­tion estim­ates obtained using PC scores as traits—to obtain estim­ates of selec­tion, but on the ori­gin­al traits (Fig 3). We applied this approach to our own data and to lit­er­at­ure examples to illus­trate its effectiveness.

Our approach requires one to use a sub­set of PC axes and PC scores as traits: using them all returns the same answers as tra­di­tion­al regres­sion. Using a sub­set entails decisions about how many PC axes are required to cap­ture vari­ation in the traits without rein­tro­du­cing mul­ti­col­lin­ear­ity. We sug­gest omit­ting the trail­ing PC axes describ­ing rel­at­ively little vari­ation (or mainly sampling vari­ation), which stat­ist­ic­al the­ory shows are respons­ible for mul­ti­col­lin­ear­ity, if it exists. It is import­ant to note that decision-mak­ing like this is endem­ic to selec­tion ana­lys­is: invest­ig­at­ors must decide how many cor­rel­ated traits to meas­ure and include, or how many PC axes to include, espe­cially when one recog­nizes how many fea­tures of an organ­ism we could con­ceiv­ably measure.

Gen­er­al lessons

We see three gen­er­al les­sons. First, meas­ur­ing selec­tion on PC scores does not have to be aban­doned due to inter­pret­a­tion chal­lenges. For cases when PC axes have many com­pon­ent traits, with dif­fer­ent signs and mag­nitudes to their load­ings, work­ing in terms of the ori­gin­al bio­lo­gic­al traits is much sim­pler and more intu­it­ive. Second, it is tempt­ing to inter­pret selec­tion on PC axes as selec­tion on the trait that loads most heav­ily on those axes. The lit­er­at­ure examples we reviewed reveal that this isn’t always true: those same traits also load on the remain­ing PC axes, which them­selves can be under selec­tion. Pro­ject­ing selec­tion estim­ates for all PC scores back into terms of the ori­gin­al traits gives an over­all pic­ture of selec­tion. Finally, and per­haps most import­ant, our approach shows how mul­ti­col­lin­ear­ity and high-dimen­sion­al data do not need to be a stop­ping point for selec­tion ana­lys­is. In these situ­ations, invest­ig­at­ors are forced to do some­thing: either drop traits, change hypo­theses, or forgo estim­at­ing selec­tion gradi­ents. We sug­gest that in the case of mul­ti­col­lin­ear­ity, or high-dimen­sion­al data­sets like expres­sion, volat­iles, or meta­bol­ites, meas­ur­ing selec­tion on PC scores and then pro­ject­ing the estim­ates back into the terms of the ori­gin­al traits is a prom­ising way forward.

John Stinch­combe is Pro­fess­or of Eco­logy & Evol­u­tion­ary Bio­logy at the Uni­ver­sity of Toronto. The ori­gin­al paper is freely avail­able to read and down­load from Evol­u­tion Let­ters.