Generalizing Contextual Analysis

Abstract

Okasha, in Evolution and the Levels of Selection, convincingly argues that two rival statistical decompositions of covariance, namely contextual analysis and the neighbour approach, are better causal decompositions than the hierarchical Price approach. However, he claims that this result cannot be generalized in the special case of soft selection and argues that the Price approach represents in this case a better option. He provides several arguments to substantiate this claim. In this paper, I demonstrate that these arguments are flawed and argue that neither the Price equation nor the contextual and neighbour partitionings sensu Okasha are adequate causal decompositions in cases of soft selection. The Price partitioning is generally unable to detect cross-level by-products and this naturally also applies to soft selection. Both contextual and neighbour partitionings violate the fundamental principle of determinism that the same cause always produces the same effect. I argue that a fourth partitioning widely used in the contemporary social sciences, under the generic term of ‘hierarchical linear model’ and related to contextual analysis understood broadly, addresses the shortcomings of the three other partitionings and thus represents a better causal decomposition. I then defend this model against the argument that because it predicts that there is some organismal selection in some specific cases of segregation distortion then it should be rejected. I show that cases of segregation distortion that intuitively seem to contradict the conclusion drawn from the hierarchical linear model are in fact cases of multilevel selection 2 while the assessment of the different partitionings are restricted to multilevel selection 1.

Appendix

Okasha (2005, 718–719) provides a definition of $\beta_{4}$ in terms of collective characters and particle characters by demonstrating that there is the simple relation following relation between $\beta_{2}$ and $\beta_{4}$. Given that neighbourhood character is defined as $X_{jk} = \frac{{nZ_{k} - z_{jk} }}{n - 1}$, we can deduct that

$$\beta_{4} = \frac{n - 1}{n}\beta_{2}$$

where $n$ is the number of particles in a collective.

Although Okasha does not provide a demonstration of it, it is also useful to express $\beta_{3}$ in terms of collective and particle characters. In fact, this will allows us to highlight the difference between the direct effect of particle character on particle fitness, controlling for collective character and the direct effect of particle character on particle fitness, controlling for neighbourhood character. This also highlights the straightforward mathematical link between contextual and neighbour partitionings. This can be done as follows. Assuming $e_{jk}$ is nil we have:

$$w_{jk} = \beta_{3} z_{jk} + \beta_{4} X_{k} = \beta_{3} z_{jk} + \beta_{2} \frac{{nZ_{k} - z_{jk} }}{n }$$

This expression can be rearranged as follows:

$$w_{jk} = \left( {\beta_{3} - \frac{{\beta_{2} }}{n }} \right)z_{jk} + \beta_{2} Z_{k}$$

Because both the contextual and Okasha’s version of the neighbourhood regression models are models for particle fitness, we know that:

$$w_{jk} = \beta_{1} z_{jk} + \beta_{2} Z_{k} = \beta_{3} z_{jk} + \beta_{4} X_{k}$$

And thus it follows that:

$$\beta_{1} z_{jk} + \beta_{2} Z_{k} = \left( {\beta_{3} - \frac{{\beta_{2} }}{n }} \right)z_{jk} + \beta_{2} Z_{k}$$

This implies that:

$$\beta_{1} = \beta_{3} - \frac{{\beta_{2} }}{n }$$

and therefore that:

$$\beta_{3} = \beta_{1} + \frac{{\beta_{2} }}{n }$$

Recall that Okasha defines $\beta_{3}$ as the partial regression coefficient of fitness on particle character, controlling for neighbourhood character. We can now express it verbally in terms of particle and collective characters. Following the definitions of $\beta_{1}$, $\beta_{2}$ and $n$ provided in the main text, $\beta_{3}$ is the sum of the partial regression coefficient of particle fitness on particle character, controlling for collective character and the partial regression coefficient of fitness on collective character, controlling for particle character, divided by the number of particles in the collective.