Validating niche-construction theory through path analysis

Abstract

Under the conventional view of evolution, species over time come to exhibit those characteristics that best enable them to survive and reproduce in their preexisting environments. Niche construction provides a second evolutionary route to establishing the adaptive fit, or match, between organism and environment, viewing such matches as dynamical products of a two-way process involving organisms both responding to problems posed by environments as well as setting themselves new problems by changing their environments through further niche construction. Not surprising, the analysis of niche construction is complicated. For example, variables of interest might contain measurement error, or some variables might not be observable. In other cases, variables might not be datable and have to be measured at the same date. A time-series generalization of path analysis, which itself can be viewed as a version of simultaneous-equation analysis, offers a means of highlighting causal relationships in complex systems of niche construction by graphically representing a hypothesis of causality between variables and, in some instances, providing an estimated weight that a hypothesized causal variable has on another variable. Path analysis forces researchers to specify how variables relate to one another and encourages development of clear and logical theories concerning the processes that influence a particular outcome. As we show through a case study—the coevolution of cattle husbandry and the tolerance for milk consumption—path analysis can also call attention to potential areas of weakness and ambiguity in data sets and how they are used in constructing inferences.

Appendix

Freedman (2004) is an excellent source for the warnings and difficulties of trying to infer causality from regression analysis. Sensible theory and interpretation must always be used, even if the “fit” is good and all the usual specification tests are passed. The theory in the LP example is quite convincing about the qualitative directions of the causal arrows in the path diagram, but suppose we want to estimate the quantitative size of those paths of causality and conduct quantitative statistical inference. How would we do it, what problems are likely to arise, and how can we fix them?

The main text of the paper focuses on how to statistically test for two-way feedback effects that distinguish NCT from SET using the expository device of linear time-series analysis. The text illustrated this problem using the example of lactase persistence (LP), which was modeled using a linear time-series dynamical system with several environmental variables and a one-locus, two-allele system. In some sense, the LP example is not a good choice to illustrate the approach because the causal mechanisms behind the spread (and absence) of LP are quite well understood (e.g., Ingram et al. 2009).

Population-genetics modeling, however, inherently generates nonlinear dynamical systems. Here, we construct some simple models of NCT versus SET, including a simple nonlinear dynamic population-genetics model where the fitness functions depend not only on allelic frequencies but also on environmental stock variables (e.g., stock of animals capable of producing dairy products), which may have been augmented by activities in the past. Examples of mathematical models of niche construction are Laland et al. (1999), Creanza et al. (2012), and Han and Hui (2014).

We begin by writing down a discrete time-simple dynamic model of interactions between organisms and environment by adapting Laland et al.’s (2009) Eqs. (1a, 1b) and (2a, 2b) and executing a “small noise” expansion to get a linear time-series model. We then use the resulting linear model to discuss some statistical-identification issues in distinguishing NCT from SET.

We have as a mathematical metaphor for SET, where all feedback is from E to O and not vice versa,

$$ {O}_{t+1}=f\left({O}_t,{E}_t,\varepsilon {e}_{1,t+1}\right),\ {E}_{t+1}=g\left({E}_t,\varepsilon {e}_{2,t+1}\right) $$

(A1)

Differentiating (A1) with respect to ε and evaluating it at ε = 0, we get

$$ d{O}_{t+1}/d\varepsilon ={f}_Od{O}_t/d\varepsilon +{f}_Ed{E}_t/d\varepsilon +{f}_e{e}_{1,t+1},\ d{E}_{t+1}/d\varepsilon ={g}_Ed{E}_t/d\varepsilon +{g}_e{e}_{2,t+1} $$

(A2)

Here, partial derivatives are denoted by subscripts, partial derivatives are evaluated at ε = 0, and O _t , E _t denote the state of the organism and the state of the environment at date t. Equation (A2) is a linear time-series model for the perturbations.

A “mathematical metaphor” for NCT is

$$ {O}_{t+1}=f\left({O}_t,{E}_t,\varepsilon {e}_{1,t+1}\right),\ {E}_{t+1}=g\left({O}_t,{E}_t,\varepsilon {e}_{2,t+1}\right) $$

(A3)

with derivative equation

$$ d{O}_{t+1}/d\varepsilon ={f}_Od{O}_t/d\varepsilon +{f}_Ed{E}_t/d\varepsilon +{f}_e{e}_{1,t+1},\ d{E}_{t+1}/d\varepsilon ={g}_Od{O}_t/d\varepsilon +{g}_Ed{E}_t/d\varepsilon +{g}_e{e}_{2,t+1} $$

(A4)

Here, there is feedback from organisms to environment because of niche construction. If we had data on proxies for the perturbations, {dO _s /dε, dE _s /dε, s = 1, 2, …}, we could potentially design a statistical test of (A2) against (A1) by estimating the slope coefficient, g _O , and testing the null hypothesis that g _O is zero against the alternative that it is not zero.

If we put the state of the organism to be the fraction of organisms—humans in this case—having the LP allele at date t, i.e., O _t  = LP _t , and define the environmental-state variable, E _t  = M _t , then the linear system (1a) in the text is the same as equation (A4), provided the partial derivatives are constant in time. If the regression condition is E{e _it + 1|LP _t , M _t } = 0, i = 1, 2 for all dates t, then we can run simple ordinary least squares (OLS) to test the null hypothesis H ₀ : α ₁₂ = 0. Rejection of the null is consistent with NCT being true. Discussion is in the main text. If we replace the scalar E _t with the vector E _t  = (M _t , N _t ), then Eq. (A4) would correspond to Eq. (1b) in the main text, provided the partial derivatives in (A4) are constant in time. The main text discusses the relevant null hypothesis in this case.

When might the partial derivatives be constant in time? One example is when the perturbations occur around a deterministic steady state of the system. What should one do if the underlying system is too nonlinear for linear approximations to be of workable quality? In this case, we would recommend a nonlinear testing approach (e.g., Bahadori and Liu 2013; Diks and Panchenko 2006; Lee et al. 1993), as explained in the main text.

A potentially useful analogy to testing NCT versus SET is testing endogenous growth theory (EGT) against conventional exogenous growth theory (CGT) in economics (Jones 1995). EGT argues that increases in human research-and-development (R&D), much like increases in niche construction, create a permanent effect on growth, whereas CGT would argue that the effect of increased R&D is transient and wears off over time. Jones argued that statistical methods for testing for “unit roots” could be used to test for EGT effects if one has an explicit measure for the amount of R&D conducted at each date. His methods shifted the debate between advocates of EGT and CGT when he found little evidence for the persistent effect on productivity of the economy that was predicted by EGT.

By analogy, if we had an explicit measure for the amount of niche construction being done at each date, we could adapt Jones’s methods to test NCT versus SET. Unfortunately, finding data on quantitative measures of niche construction at different time periods is difficult. Thus, if we use this route, we must have a proxy for the actual amount of niche construction. It might be difficult to detect niche-construction effects even if they are there. As an example, consider the following modification of Eqs. (A3) and (A4):

$$ \begin{array}{l}L{P}_{t+1}=f\left(L{P}_t,{E}_t,\varepsilon {e}_{1,t+1}\right),\ \\ {}{E}_{t+1}={\mu}_E+{\rho}_E{E}_t+{\beta}_{LP}L{P}_t+{\beta}_{-LP}\left(1-L{P}_t\right)+{e}_{2,t+1}\\ {}{\beta}_{LP}>{\beta}_{-LP}\ge 0\end{array} $$

(A5)

The idea here is that humans with and without the LP allele can indulge in environmental modification—for example, by the husbandry of animals with potential dairy-production capability for whatever reasons, some of which may be totally independent of consumption of dairy products. In (A5), we assume β _LP  > β _− LP  ≥ 0 to capture the idea that humans with the LP allele should be more interested in adding and caring for animals with dairy-product potential than humans not carrying the allele. However, humans without the LP allele may have β _LP  > β _− LP  ≥ 0. Suppose we cannot observe the actual amount of dairy husbandry but have observations on LP _t . We can look at parameter values in (A5) that are consistent with feedback from LP _t to E _t + 1 but where this feedback is hard to detect statistically. Note that we have not required the function f(.) in (A5) to be linear. Nonlinearity in f(.) allows one to treat threshold phenomena—for example, LP being an advantageous allele only when the population of dairy animals reaches a threshold level.

Even though (A5) is a simple, essentially trivial dynamic model, it directs our attention to two potential problems in statistically detecting NCT effects, even though the effects are present. The first is where the difference, β _LP  − β _− LP , is small. A regression of an observable measure of E _t + 1 on E _t would deliver estimates of the constant term, μ _E  + β _− LP , the slope effect, β _LP  − β _− LP , and the persistence parameter, ρ _E . If β _LP  − β _− LP is close to zero, one would conclude there is no NCT effect, even though it is present. If the persistence parameter, ρ _E  > 0, is close to one, the NCT effect on the ultimate state of the environment could be very large, even though it would be extremely difficult to detect.

The second problem is the case where β _− LP  = 0 and β _LP  > 0 is small. It is clear from the discussion above that it would be empirically difficult to detect the NCT effect, β _LP  > 0. However, the NCT effect could be very important if ρ _E  > 0 is very close to one. Indeed, if μ _E  = 0, so that the environment would decay to nothing if there were no niche construction, a very small β _LP  > 0, coupled with ρ _E  > 0 very close to one, would lead to a large ultimate effect on the environment, E, even if only a small fraction of the population has the LP allele.

A crude approach to detecting NCT effects in the two cases above might be to take the current state of the environment as a proxy for niche-construction activity and examine how fast a surprise increase in the environment—perhaps a result of some kind of observable positive shock to the environmental-state variable—wears off on the measure of organism state once we have a measure of the state of the organisms. In other words, we could plot the empirical impulse-response function (Jones 1995) of the system to a positive shock to the state of the environment. A rapid decay rate would be consistent with the absence of NCT effects, and a much slower decay rate would be consistent with the presence of NCT effects. Admittedly, this would be a crude, indirect way of dealing with the problems exposed in the two cases above, but perhaps it is better than nothing.

Here is another example where NCT effects might be difficult to detect when present. Suppose niche construction is not “free” to an organism, meaning it might take energy, time, or some other resource away from the organism that it could have used for something else that would increase fitness. A simple model of this effect is

$$ {O}_{t+1}=f\left({O}_t,{E}_t,{A}_t,\varepsilon {e}_{1,t+1}\right),\ {E}_{t+1}=g\left({E}_t,{A}_t,\varepsilon {e}_{2,t+1}\right) $$

(A6)

where an increase in A _t increases g(.) but where f(.) is increasing then decreasing in A _t . This assumption on f(.) reflects diminishing returns and increasing marginal costs to the organism in terms of its reproductive success for the next period.

Suppose there is selection for organisms to solve the problem

$$ { \max}_{A_t}{\displaystyle \int \Big\{}f\left({O}_t,{E}_t,{A}_t,\varepsilon {e}_{1,t+1}\right)\Big\}d{F}_{e_1} $$

(A7)

This maximization problem implies that A _t is a function of (O _t , E _t ), call it A(O _t , E _t ). If A(O _t , E _t ) is a function of E _t alone, and if we do not observe A _t , then clearly we will not detect NCT-type feedback into g(.) by trying to relate E _t + 1 to E _t by any method. When might this happen? A sufficient condition is

$$ {\partial}^2f/\partial A\partial O=0 $$

(A8)

Examples of functions f(O, E, A, e) that satisfy (A8) are

$$ \begin{array}{l}f\left(O,E,A,e\right)={f}_1\left(O,E,e\right){f}_2\left(A,E,e\right)\\ {}f\left(O,E,A,e\right)={f}_1\left(O,E,e\right)+{f}_2\left(A,E,e\right)\end{array} $$

(A9)

When might NCT effects be detected in an empirical study but in reality are not actually there, i.e., a false positive occurs? Consider the dynamics

$$ {O}_{t+1}=f\left({O}_t,{E}_t,{Z}_t,\varepsilon {e}_{1,t+1}\right),\ {E}_{t+1}=g\left({E}_t,{Z}_t,\varepsilon {e}_{2,t+1}\right) $$

(A10)

where Z _t is an unobserved state variable at period t. If the dynamics were linear with constant coefficients, we could write them in the form of a bi-variate auto-regression:

$$ \begin{array}{l}{E}_{t+1}={\alpha}_{10}+{\alpha}_{11}{E}_t+{\alpha}_{12}{O}_t+{\alpha}_{13}{Z}_t+{e}_{1,t+1},\ t=1,2,\dots, T\\ {}{O}_{t+1}={\alpha}_{20}+{\alpha}_{21}{E}_t+{\alpha}_{22}{O}_t+{\alpha}_{23}{Z}_t+{e}_{2,t+1},\ t=1,2,\dots, T\end{array} $$

(A11)

We could use the omitted variable formula (Greene 2003, Section 8.2.1) to indicate that typically there will be bias in the estimate of α ₁₂, even when the true value is zero. Because it is difficult to exclude omitted variables, Z _t , unless one has strong theory, omitted variables are likely to be present and cause false positives of NCT effects even when they are not really there. Using theory to argue for the presence (or absence) of omitted variables may help sharpen theoretical discussions in adducing arguments in support for (or against) NCT.

Estimating selection strength

We have been asked on occasion to compare our approach to others, such as that of Itan et al. (2009). We can do that by considering their Eq. (3):

$$ \begin{array}{l}p\hbox{'}=\left[\left(1+s\right){p}^2+\left(1+s\right)pq\right]/\left[1+s\left({p}^2+pq\right)\right]\\ {}N\hbox{'}=N\left(1+s\left({p}^2+2pq\right)\right)\end{array} $$

(A12)

where their p,p’ and N,N’ are, in our notation,

$$ p\equiv {p}_t=L{P}_t,p\hbox{'}\equiv {p}_{t+1}=L{P}_{t+1},q\equiv 1-p,N\equiv {N}_t,N\hbox{'}\equiv {N}_{t+1} $$

(A13)

The Itan et al. (2009) population-genetics system (A12) is a one-locus, two-allele diploid model, where fitnesses are specified as w ₁₁ = w ₁₂ = 1 + s for homozygotes and heterozygotes with the LP allele and w ₂₂ = 1 for homozygotes without the LP allele, and where s > 0 denotes the relative fitness of those individuals with the LP allele. Populations containing individuals with the LP allele are assumed to grow faster.

The analogy to (A12) in our model is

$$ \begin{array}{l}L{P}_{t+1}={a}_{30}+{b}_{32}{C}_t+{a}_{31}{M}_t+{a}_{33}L{P}_t+{a}_{34}F{M}_t+{a}_{35}FM{P}_t+{e_3}_{,t+1}\\ {}{N}_{t+1}={a}_{20}+{b}_{22}{C}_t+{a}_{21}{M}_t+{a}_{22}{N}_t+{e}_{2,t+1}\end{array} $$

(A14)

which can be viewed as a linearization of the Itan et al. (2009) model, where their selection coefficient, “s,” is not constant but rather a function:

$$ {s}_{t+1}\equiv S\left({C}_t,{M}_t,L{P}_t,F{M}_t,FM{P}_t,{e}_{t+1}\right) $$

(A15)

If we inserted (A15) in place of s in Itan et al.’s Eq. (A12) and linearized them around p _t  = 0, we would obtain a system much like (A14) except the coefficients would be time dependent.