Statistical models and estimation

A breeder records phenotypes and wants a defensible estimate of average performance. A statistical model turns that question into assumptions and unknown quantities that can be estimated from records.

From observations to an estimate

For a first model, let phenotype \(y_i\) of animal \(i\) among \(n\) observed animals satisfy

\[ y_i=\mu+\varepsilon_i, \qquad \varepsilon_i\sim N(0,\sigma^2). \]

The parameter \(\mu\) is the fixed but unknown population mean and \(\sigma\) is the assumed residual standard deviation. The normal, independent, common- variance residual model is an assumption, not a biological fact. Family, management, or environmental structure can make it inadequate.

An estimator is a rule applied to possible datasets, while an estimate is the number it returns for this dataset. Thus the sample mean \(\bar y\) is an estimator of \(\mu\), and \(\hat\mu\) is its realised estimate. A prediction instead concerns an unobserved outcome, such as a future offspring phenotype. Estimating a mean and predicting a new record are related but different tasks.

Code
d <- readRDS("../../demo-data/main/demo_data.rds")
y <- d$pheno$y_cont_qtl_snp[1:5]
mean(y)
[1] -1.518102

The result is about \(-1.518\). It estimates the mean under this simple model; it is neither the true mean of every possible animal nor a prediction that all five animals should have the same phenotype. The data are simulated project data, not measurements from an empirical trial.

Likelihood holds data fixed and compares parameters

A probability model asks which data could arise if \(\mu\) and \(\sigma\) were known. A likelihood uses the same formula after data are observed, holding \(\mathbf y\) fixed and comparing candidate parameter values:

\[ L(\mu;\mathbf y)=\prod_{i=1}^{n} \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{-\frac{(y_i-\mu)^2}{2\sigma^2}\right\}. \]

Likelihood is not a probability distribution over \(\mu\): in frequentist estimation, the parameter is fixed and unknown. Its maximiser is the maximum-likelihood estimate. Software commonly uses log-likelihoods because they turn products into sums.

Code
mu_grid <- seq(-4, 2, length.out = 401)
log_likelihood <- vapply(
  mu_grid,
  function(mu) sum(dnorm(y, mean = mu, sd = 3, log = TRUE)),
  numeric(1)
)
mu_grid[which.max(log_likelihood)]
[1] -1.525

The grid gives \(-1.525\), close to the exact sample mean because grid points are 0.015 apart. With known \(\sigma\), maximising this normal likelihood gives \(\hat\mu=\bar y\). Here \(\sigma=3\) is supplied to focus on estimating \(\mu\).

The same estimate minimises squared-error loss,

\[Q(\mu)=\sum_{i=1}^{n}(y_i-\mu)^2.\]

That agreement depends on the chosen normal model and loss. Absolute-error loss targets a median, and another probability model can yield a different estimator.

Identifiability limits what records can separate

An identifiable parameter has a unique implication for the distribution of observed data. If two parameter values always give the same fitted values, the data cannot distinguish them.

For \(\mathbf y=\mathbf X\boldsymbol\beta+\boldsymbol\varepsilon\), suppose two columns of the \(n\times p\) marker matrix \(\mathbf X\) are exact copies. Increasing one coefficient and decreasing the other by the same amount leaves \(\mathbf X\boldsymbol\beta\) unchanged. Their separate coefficients are not identified, although their combined fitted contribution can be. Near-copy markers create collinearity: effects may be identifiable but unstable. A marker coefficient is therefore not proof of a unique causal locus.

Identifiability also does not establish out-of-sample prediction quality. A well-estimated training-population parameter can still predict poorly in a new generation or environment. The next page adds fixed management effects and correlated animal effects before genomic relationship matrices and REML.

Exercises

  1. A rule returns 2.1 for one dataset while a model parameter is \(\theta\). Which is the estimator and which is the estimate?

The estimator is the rule mapping data to a value. The value 2.1 is its estimate for this dataset; \(\theta\) is the fixed but unknown parameter.

  1. After observing data, what varies in a likelihood comparison?

The observations are held fixed and candidate parameter values vary. Likelihood ranks how well those values explain the same observed data.

  1. Can two exact-copy marker columns have separately identified regression coefficients from phenotypes alone?

No. Opposite coefficient changes can leave every fitted value unchanged, so an extra constraint or representation is needed to separate the effects.