Statistical model

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A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of some sample data and similar data from a larger population. A statistical model represents, often in considerably idealized form, the data-generating process.

The assumptions embodied by a statistical model describe a set of probability distributions, some of which are assumed to adequately approximate the distribution from which a particular data set is sampled. The probability distributions inherent in statistical models are what distinguishes statistical models from other, non-statistical, mathematical models.

A statistical model is usually specified by mathematical equations that relate one or more random variables and possibly other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen).^[1]

All statistical hypothesis tests and all statistical estimators are derived from statistical models. More generally, statistical models are part of the foundation of statistical inference.

Formal definition

In mathematical terms, a statistical model is usually thought of as a pair (${\displaystyle S,{\mathcal {P}}}$), where ${\displaystyle S}$ is the set of possible observations, i.e. the sample space, and ${\displaystyle {\mathcal {P}}}$ is a set of probability distributions on ${\displaystyle S}$.^[2]

The intuition behind this definition is as follows. It is assumed that there is a "true" probability distribution induced by the process that generates the observed data. We choose ${\displaystyle {\mathcal {P}}}$ to represent a set (of distributions) which contains a distribution that adequately approximates the true distribution. Note that we do not require that ${\displaystyle {\mathcal {P}}}$ contains the true distribution, and in practice that is rarely the case. Indeed, as Burnham & Anderson state, "A model is a simplification or approximation of reality and hence will not reflect all of reality"^[3]—whence the saying "all models are wrong".

The set ${\displaystyle {\mathcal {P}}}$ is almost always parameterized: ${\displaystyle {\mathcal {P}}=\{P_{\theta }:\theta \in \Theta \}}$. The set ${\displaystyle \Theta }$ defines the parameters of the model. A parameterization is generally required to have distinct parameter values give rise to distinct distributions, i.e. ${\displaystyle P_{\theta _{1}}=P_{\theta _{2}}\Rightarrow \theta _{1}=\theta _{2}}$ must hold (in other words, it must be injective). A parameterization that meets the requirement is said to be identifiable.^[2]

An example

Suppose that we have a population of school children, with the ages of the children distributed uniformly, in the population. The height of a child will be stochastically related to the age: e.g. when we know that a child is of age 7, this influences the chance of the child being 5 feet tall. We could formalize that relationship in a linear regression model, like this: height_i = b₀ + b₁age_i + ε_i, where b₀ is the intercept, b₁ is a parameter that age is multiplied by in obtaining a prediction of height, ε_i is the error term, and i identifies the child. This implies that height is predicted by age, with some error.

An admissible model must be consistent with all the data points. Thus, a straight line (height_i = b₀ + b₁age_i) cannot be the equation for a model of the data. The line cannot be the equation for a model, unless it exactly fits all the data points—i.e. all the data points lie perfectly on the line. The error term, ε_i, must be included in the equation, so that the model is consistent with all the data points.

To do statistical inference, we would first need to assume some probability distributions for the ε_i. For instance, we might assume that the ε_i distributions are i.i.d. Gaussian, with zero mean. In this instance, the model would have 3 parameters: b₀, b₁, and the variance of the Gaussian distribution.

We can formally specify the model in the form (${\displaystyle S,{\mathcal {P}}}$) as follows. The sample space, ${\displaystyle S}$, of our model comprises the set of all possible pairs (age, height). Each possible value of ${\displaystyle \theta }$ = (b₀, b₁, σ²) determines a distribution on ${\displaystyle S}$; denote that distribution by ${\displaystyle P_{\theta }}$. If ${\displaystyle \Theta }$ is the set of all possible values of ${\displaystyle \theta }$, then ${\displaystyle {\mathcal {P}}=\{P_{\theta }:\theta \in \Theta \}}$. (The parameterization is identifiable, and this is easy to check.)

In this example, the model is determined by (1) specifying ${\displaystyle S}$ and (2) making some assumptions relevant to ${\displaystyle {\mathcal {P}}}$. There are two assumptions: that height can be approximated by a linear function of age; that errors in the approximation are distributed as i.i.d. Gaussian. The assumptions are sufficient to specify ${\displaystyle {\mathcal {P}}}$—as they are required to do.

General remarks

A statistical model is a special class of mathematical model. What distinguishes a statistical model from other mathematical models is that a statistical model is non-deterministic. Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values, but instead have probability distributions; i.e. some of the variables are stochastic. In the example above, ε is a stochastic variable; without that variable, the model would be deterministic.

Statistical models are often used even when the physical process being modeled is deterministic. For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process).

There are three purposes for a statistical model, according to Konishi & Kitagawa.^[4]

• Predictions
• Extraction of information
• Description of stochastic structures

Dimension of a model

Suppose that we have a statistical model (${\displaystyle S,{\mathcal {P}}}$) with ${\displaystyle {\mathcal {P}}=\{P_{\theta }:\theta \in \Theta \}}$. The model is said to be parametric if ${\displaystyle \Theta }$ has a finite dimension. In notation, we write that ${\displaystyle \Theta \subseteq \mathbb {R} ^{k}}$ where k is a positive integer (${\displaystyle \mathbb {R} }$ denotes the real numbers; other sets can be used, in principle). Here, k is called the dimension of the model.

As an example, if we assume that data arise from a univariate Gaussian distribution, then we are assuming that

${\displaystyle {\mathcal {P}}=\left\{P_{\mu ,\sigma }(x)\equiv {\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right):\mu \in \mathbb {R} ,\sigma >0\right\}}$.

In this example, the dimension, k, equals 2.

As another example, suppose that the data consists of points (x, y) that we assume are distributed according to a straight line with i.i.d. Gaussian residuals (with zero mean). Then the dimension of the statistical model is 3: the intercept of the line, the slope of the line, and the variance of the distribution of the residuals. (Note that in geometry, a straight line has dimension 1.)

Although formally ${\displaystyle \theta \in \Theta }$ is a single parameter that has dimension k, it is sometimes regarded as comprising k separate parameters. For example, with the univariate Gaussian distribution, ${\displaystyle \theta }$ is a single parameter with dimension 2, but it is sometimes regarded as comprising 2 separate parameters—the mean and the standard deviation.

A statistical model is nonparametric if the parameter set ${\displaystyle \Theta }$ is infinite dimensional. A statistical model is semiparametric if it has both finite-dimensional and infinite-dimensional parameters. Formally, if k is the dimension of ${\displaystyle \Theta }$ and n is the number of samples, both semiparametric and nonparametric models have ${\displaystyle k\rightarrow \infty }$ as ${\displaystyle n\rightarrow \infty }$. If ${\displaystyle k/n\rightarrow 0}$ as ${\displaystyle n\rightarrow \infty }$, then the model is semiparametric; otherwise, the model is nonparametric.

Parametric models are by far the most commonly used statistical models. Regarding semiparametric and nonparametric models, Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies".^[5]

Nested models

Two statistical models are nested if the first model can be transformed into the second model by imposing constraints on the parameters of the first model. As an example, the set of all Gaussian distributions has, nested within it, the set of zero-mean Gaussian distributions: we constrain the mean in the set of all Gaussian distributions to get the zero-mean distributions. As a second example, the quadratic model

y = b₀ + b₁x + b₂x² + ε,    ε ~ 𝒩(0, σ²)

has, nested within it, the linear model

y = b₀ + b₁x + ε,    ε ~ 𝒩(0, σ²)

—we constrain the parameter b₂ to equal 0.

In both those examples, the first model has a higher dimension than the second model (for the first example, the zero-mean model has dimension 1). Such is often, but not always, the case. As a different example, the set of positive-mean Gaussian distributions, which has dimension 2, is nested within the set of all Gaussian distributions.

Comparing models

It is assumed that there is a "true" probability distribution underlying the observed data, induced by the process that generated the data. The main goal of model selection is to make statements about which elements of ${\displaystyle {\mathcal {P}}}$ are most likely to adequately approximate the true distribution.

Models can be compared to each other by exploratory data analysis or confirmatory data analysis. In exploratory analysis, a variety of models are formulated and an assessment is performed of how well each one describes the data. In confirmatory analysis, a previously formulated model or models are compared to the data. Common criteria for comparing models include R², Bayes factor, and the likelihood-ratio test together with its generalization relative likelihood.

Konishi & Kitagawa state: "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling. They are typically formulated as comparisons of several statistical models."^[6] Relatedly, Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".^[7]