What the numbers do not say

Every chapter of this book built a population whose truth was known and asked whether an estimator returned it. That design has a limit that a real study does not share. A simulation knows which trait is causal, which environment is shared, which father is the real one and which competitors are local. A field study knows none of this, and the estimators do not tell it. The recurring lesson of the checking chapters was that the failures that matter most are silent: the fitted model looks the same when it is right and when it is wrong, and the difference lives in a variable that was never measured.

Those silent failures are collected here in one place, because they are what separates an analysis that can be defended from one that merely runs.

The first is the unmeasured cause. A selection gradient separates direct from indirect selection only among the traits in the model, and a condition variable that drives both a trait and fitness hands its selection to the trait without leaving a trace in the residuals. The same variable breaks the breeder’s equation, because the part of the covariance between trait and fitness that runs through the environment is not inherited. No regression diagnostic detects it. The defences are an experiment that moves the trait, measurement of the obvious confounders, or a covariance between fitness and breeding values rather than phenotypes, estimated inside one model rather than in two steps.

The second is a shared environment mistaken for shared genes. Relatives resemble each other for both reasons, and a model with a single source of resemblance reads all of it as additive variance. The data can separate the two only when the design breaks their alignment: half sibs raised in different nests, young moved between mothers, both parents measured so that offspring can be regressed on each separately. Where the design does not break it, no choice of model will.

The third is a quantity estimated in one analysis and used as data in another. Predicted breeding values and behavioural scores averaged over a varying number of tests both carry an error whose size depends on how much information each individual contributed. When that information is linked to fitness, as it is whenever longer-lived animals are measured more often, the second analysis inherits a structure that looks like biology. The fix is always the same in principle: put both questions into one model, so that the uncertainty travels with the estimate.

The fourth is a boundary. Variances cannot be negative, heritabilities cannot exceed one, and estimates of both pile up at the edges when the data are thin. The standard tests assume the parameter lies in the interior of its space; at the edge they are miscalibrated, and the correct reference distribution is a mixture. The directions of weak genetic variance, where constraint lives, are exactly where the estimates of a covariance matrix press against that boundary hardest.

The fifth is a population statistic read as a fixed property. Heritability belongs to one population in one environment, and its denominator moves with every fixed effect in the model. Relatedness, in the sense that Hamilton’s rule needs, is a regression against the population in which competition happens, not a fraction read from a family tree. The benefit and cost in that rule are regression coefficients that change with the frequency of the behaviour. None of these numbers can be carried from one study to another without asking whether the population they describe is the same.

None of this is an argument against the methods. The Price equation is exact in any population, and the chapters showed the Lande-Arnold regression, the breeder’s equation and the animal model recovering the truth whenever their assumptions held. The argument is for writing the assumptions down, for simulating the design before the fieldwork, and for treating each estimate as the output of a model that could have been wrong in a specific, nameable way.

The code in this book is meant to be used that way. Each chapter can be run with the reader’s own sample sizes, correlations and pedigrees in place of the ones chosen here, and the question worth asking of any planned study is the one the chapters asked of every estimator: if the truth were known, would this analysis find it?