On the use of relative brain size 1

11 1. There is a long standing interest in the effect of relative brain size on other life history variables in a 12 comparative context. Historically, residuals have been used to calculate these effects, but more recently 13 it has been recognised that regression on residuals is not good practice. Instead, absolute brain size 14 and body size are included in a multiple regression, with the idea that this controls for allometry. 15 2. I use a simple simulation to illustrate how a case with brain size as response variable differs from a case 16 with relative brain size as predictor variable. I use the simulated data to test which modelling approach 17 can estimate the underlying causal effects for each case. 18 3. The results show that a multiple regression models with both body size and another variable as predictor 19 variable and brain size as response variable work well. However, if relative brain size is a predictor 20 variable, a multiple regression fails to correctly estimate the effects. 21 4. I propose the use of structural equation models to simultaneously estimate relative brain size and its 22 effect on the third variable and discuss other potential methods. 23

will present the underlying assumed causal structure presented in Figure 1. To avoid discussion about 48 directionality I have simply named the third variable z. In the first case both body size and z cause brain size 49 (see Figure 1a). To decide how to best estimate the direct effect of z on brain size, one needs to make sure 50 that all back-door paths are closed. In other words, all arrows that points towards z need to be considered. 51 But since there are no such arrows in this case, there is no need to include additional variables. However, 52 body size explains most of the variation in brain size. Therefore it is still a good idea to include this variable 53 to get a more precise estimate of the effect of z on brain size. Or in other words on relative brain size, since 54 allometry is now accounted for. 55 The second case is causally more opaque (see Figure 1b). In general the question of interest is how cognitive 56 ability influences the third variable z. But since this cannot be measured reliably across species (therefore 57 denoted by a U for unobserved), relative brain size is used as a proxy. This is where it becomes tricky. The 58 assumption is that it is the extra bit of brain that is caused by the need for cognitive ability. And it is 59 therefore relative brain size that needs to be included as proxy. The variable we initially measured is absolute 60 brain size, and is a collider in this DAG. A collider is a variable that is caused by two or more other variables.

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When including a collider in the analysis it opens up a back-door path, in our case through body size. This 62 is problematic, because the estimated effect of brain size on z now also contains some of the effect of body 63 size on z. Therefore the estimated effect of body size on z will be biased. There is no way to correct for this 64 using the variables in the DAG in a multiple regression. To include both relative brain size and body size one 65 needs a system that contains regressions with brain size as response (of body size) and as predictor (of the 66 third variable).

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A structural equation model is such a system. It contains regressions for each variable and allows brain size 68 to be response and predictor variable simultaneously (Bowen and Guo 2011). When fitted using a Bayesian 69 approach, information flows in both directions, since the likelihood is computed for the whole system at each

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Case I: relative brain size as response variable 90 In the first case absolute brain size is caused by body size and z. The interest of the study is to what extent 91 z causes additional increase in brain size. In other words, to what extend z correlates with relative brain size.

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I simulated 20 data sets with 100 species with the following structure: 93 body size ∼ normal(0, 1) z ∼ normal(0, 1) brain size ∼ normal(µ brain , 1) µ brain = 1 * body size + 1 * z I analysed the resulting data with a frequentist linear model and with the Bayesian equivalent. Then I plotted Bayesian), they actually tested the effect of absolute brain size, since including body size only accounts 129 for allometry in the response variable. The simulations in this paper showed that including body size as 130 additional predictor to control for allometric scaling of brain size works well if relative brain size is the 131 response variable, but not if relative brain size is a predictor variable. Perhaps counter-intuitively, the effect 132 of brain size was still estimated correctly by all models. It was the body size effect that was biased in the linear models. The use of such models would lead to the incorrect inference that there was no effect of body 134 size at all.

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One way to create some intuition about what is going on is that absolute brain size (which was the actual 136 predictor variable included in the linear models) contains information about both body size and relative brain 137 size. In a sense this variable controls for the body size effect already, leaving the direct effect of body size 138 zero. In a case where body size itself does not have an effect on z, it does not actually need to be included at 139 all. Using absolute brain size would be fine. The variation in brain size due to allometric scaling would just 140 create noise. Using absolute brain size would of course lead to a less precise estimate of the effect of brain 141 size, so using relative brain size from a structural equation model would still be preferable. the difference between the observed and predicted brain size from one model as predictor for a second model.

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A potentially even more powerful approach was put forward by Smaers and Vinicius (2009) and involves 146 reconstructing the ancestral states of both body size and brain size. Recently this approach was used to 147 test brain evolution in mammals and it was shown that relatively large brains can be achieved by highly for two reasons. First, the largest non-cognitive effects were due to a shift in the slope of the body to brain 152 relation. Such a shift would be less likely to affect results when considering a lower order taxonomic group 153 (e.g., only primates) and could be partially accounted for by including multiple slopes (e.g., one per family).

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Second, running a Bayesian multiregime OU modeling approach is not straightforward and becomes really 155 difficult when many covariates are included. The best compromise is to first study the allometric patterns in 156 the taxon of interest and then decide if residual brain size can be used as proxy for cognitive ability.

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For this paper I drew all variables from normal distributions with zero mean and standard deviation one. I 158 furthermore simulated z as a function of relative brain size, rather than simulating both z and brain size 159 as a function of the unobserved cognitive ability (as depicted in Figure 1b). I chose to do this to illustrate 165 span of birds. They recognised the critique on using residuals, but still used this approach, because they 167 wanted to remove the allometric effects from brain size and not from innovation propensity. This was a valid 168 argument since the original critique assumed a case where x 1 and x 2 are both causing y (Freckleton 2002).

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When x 1 and x 2 are correlated, the use of a residual of y ∼ x 1 creates a bias. In the case of life history and 170 innovativeness the causal structure is different and this particular problem does not arise. However, in an 171 empirical study measurement error, missing data and phylogenetic covariance still need to be accounted for.

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When using residuals, uncertainty does not propagate because models are run sequentially. The aim of this paper was to show how a case where relative brain size is a response variable differs from 181 a case where it is a predictor variable. From a simple simulation it becomes clear that one cannot include 182 absolute brain size and body size as covariates if relative brain size is theorised to predict the third variable.

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Multiple other approaches can be used instead and the design of the model should be based on the assumed 184 causal structure of the system. I provide a simple structural equation model, which works for the simulated 185 data. Future studies should make sure all back-door paths are closed for their DAG and potentially include 186 additional components to control for measurement error, missing data and phylogeny. The use of these 187 models is not limited to relative brain size, but can be used for any comparative study in which multiple 188 causal paths are of interest.

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Acknowledgements. I would like to thank all anonymous reviewers for valuable input. Their contribution 190 helped to clarify the manuscript and gain new insights. com/simeonqs/On_the_use_of_relative_brain_size.