3  Terminology: variables, parameters, and the rest

It’s worth pinning down the vocabulary, because these words get used loosely and the confusion causes real mistakes. The clean distinction is between the things that vary within a single run of your model and the things that are fixed across that run.

A variable is a quantity that changes across observations, over time, or across space. In a population model, the population size at each time step is a variable; so are the covariates you measured (temperature, rainfall, site). Variables split further into a few roles worth naming:

• Response (or dependent) variables: what you’re trying to explain or predict.
• Predictor (or independent, explanatory, covariate) variables: what you use to do the explaining.
• State variables: in a dynamic model, the quantities that describe the system at a moment in time and get updated at each step (e.g. abundance, biomass). The distinction between a state variable and a parameter is the one people most often blur: the state evolves as the model runs, the parameters do not.
• Rate variables: the instantaneous rate of change of a state variable, typically a derivative (dN/dt) or a flux term, and usually itself a function of state variables and parameters. In the Lotka-Volterra predator-prey model, the predation rate aNP is a rate variable built from the state variables N and P and the parameter a.

A parameter is usually a quantity that is fixed for a given run and governs how the variables relate: the knobs of the model. In logistic growth, the growth rate r and carrying capacity K are parameters; the population size is a state variable. Some parameters are set by you from theory or prior knowledge; others are what you estimate from data (this is exactly where the identifiability and confounding issues of the previous section bite).

Now parameters can also vary, and then it’s debatable whether they should remain named “parameters” or become “variables”. For example, r could vary over time to represent changes in population growth rate.

A hyperparameter is a parameter of a distribution over other parameters, and it shows up in two related but distinct settings:

• In hierarchical (multilevel) models: when parameters are themselves treated as draws from a higher-level distribution, that higher-level distribution has its own parameters, and those are the hyperparameters. If each site has its own growth rate rᵢ, and you model those site-level rates as coming from a common distribution with mean μ and spread σ, then μ and σ are hyperparameters. This is the machinery behind the “students in schools in districts / quadrats in sites in regions” nesting from earlier.
• In machine learning: the term is used more loosely for settings you choose rather than fit, such as a learning rate, the number of layers, or a regularisation strength. Same idea (a quantity one level up that governs the fitting), different flavour.

Two more words you’ll meet constantly. A constant is a fixed quantity you never estimate and never vary (a known conversion factor, π). And data are the observed values you condition on: the realised measurements of your variables, as opposed to the model quantities you infer. Keeping “the thing I measured” (data), “the thing that evolves” (state variable), and “the knob I’m estimating” (parameter) firmly separated in your own head will save you from a surprising share of modelling errors.

In statistical modelling we also talk about latent variables. These are variables we haven’t observed, but are estimated by the model.