Statistical model

Astatistical modelis amathematical modelthat embodies a set ofstatistical assumptionsconcerning the generation ofsample data(and similar data from a largerpopulation). A statistical model represents, often in considerably idealized form, thedata-generating process.^[1]When referring specifically toprobabilities,the corresponding term isprobabilistic model.Allstatistical hypothesis testsand allstatistical estimatorsare derived via statistical models. More generally, statistical models are part of the foundation ofstatistical inference.A statistical model is usually specified as a mathematical relationship between one or morerandom variablesand other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman AdèrquotingKenneth Bollen).^[2]

Introduction

Informally, a statistical model can be thought of as astatistical assumption(or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of anyevent.As an example, consider a pair of ordinary six-sideddice.We will study two different statistical assumptions about the dice.

The first statistical assumption is this: for each of the dice, the probability of each face (1, 2, 3, 4, 5, and 6) coming up is⁠1/6⁠.From that assumption, we can calculate the probability of both dice coming up 5: ⁠1/6⁠×⁠1/6⁠ = ⁠1/36⁠. More generally, we can calculate the probability of any event: e.g. (1 and 2) or (3 and 3) or (5 and 6). The alternative statistical assumption is this: for each of the dice, the probability of the face 5 coming up is⁠1/8⁠(because the dice areweighted). From that assumption, we can calculate the probability of both dice coming up 5: ⁠1/8⁠×⁠1/8⁠ = ⁠1/64⁠. We cannot, however, calculate the probability of any other nontrivial event, as the probabilities of the other faces are unknown.

The first statistical assumption constitutes a statistical model: because with the assumption alone, we can calculate the probability of any event. The alternative statistical assumption doesnotconstitute a statistical model: because with the assumption alone, we cannot calculate the probability of every event. In the example above, with the first assumption, calculating the probability of an event is easy. With some other examples, though, the calculation can be difficult, or even impractical (e.g. it might require millions of years of computation). For an assumption to constitute a statistical model, such difficulty is acceptable: doing the calculation does not need to be practicable, just theoretically possible.

Formal definition

In mathematical terms, a statistical model is a pair ( $S,{\mathcal {P}}$ ), where $S$ is the set of possible observations, i.e. thesample space,and ${\mathcal {P}}$ is a set ofprobability distributionson $S$ .^[3]The set ${\mathcal {P}}$ represents all of the models that are considered possible. This set is typically parameterized: ${\mathcal {P}}=\{F_{\theta }:\theta \in \Theta \}$ .The set $\Theta$ defines theparametersof the model. If a parameterization is such that distinct parameter values give rise to distinct distributions, i.e. $F_{\theta _{1}}=F_{\theta _{2}}\Rightarrow \theta _{1}=\theta _{2}$ (in other words, the mapping isinjective), it is said to beidentifiable.^[3]

In some cases, the model can be more complex.

InBayesian statistics,the model is extended by adding a probability distribution over the parameter space $\Theta$ .
A statistical model can sometimes distinguish two sets of probability distributions. The first set ${\mathcal {Q}}=\{F_{\theta }:\theta \in \Theta \}$ is the set of models considered for inference. The second set ${\mathcal {P}}=\{F_{\lambda }:\lambda \in \Lambda \}$ is the set of models that could have generated the data which is much larger than ${\mathcal {Q}}$ .Such statistical models are key in checking that a given procedure isrobust,i.e. that it does not produce catastrophic errors when its assumptions about the data are incorrect.

An example

Suppose that we have a population of children, with the ages of the children distributeduniformly,in the population. The height of a child will bestochasticallyrelated to the age: e.g. when we know that a child is of age 7, this influences the chance of the child being 1.5 meters tall. We could formalize that relationship in alinear regressionmodel, like this: height_i=b₀+b₁age_i+ ε_i,whereb₀is the intercept,b₁is a parameter that age is multiplied by to obtain a prediction of height, ε_iis the error term, andiidentifies 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 admissible for a model of the data—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 dostatistical inference,we would first need to assume some probability distributions for the ε_i.For instance, we might assume that the ε_idistributions arei.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 ( $S,{\mathcal {P}}$ ) as follows. The sample space, $S$ ,of our model comprises the set of all possible pairs (age, height). Each possible value of $\theta$ = (b₀,b₁,σ²) determines a distribution on $S$ ;denote that distribution by $F_{\theta }$ .If $\Theta$ is the set of all possible values of $\theta$ ,then ${\mathcal {P}}=\{F_{\theta }:\theta \in \Theta \}$ .(The parameterization is identifiable, and this is easy to check.)

In this example, the model is determined by (1) specifying $S$ and (2) making some assumptions relevant to ${\mathcal {P}}$ .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 ${\mathcal {P}}$ —as they are required to do.

General remarks

A statistical model is a special class ofmathematical 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 arestochastic.In the above example with children's heights, ε is a stochastic variable; without that stochastic variable, the model would be deterministic. Statistical models are often used even when the data-generating process being modeled is deterministic. For instance,coin tossingis, in principle, a deterministic process; yet it is commonly modeled as stochastic (via aBernoulli process). Choosing an appropriate statistical model to represent a given data-generating process is sometimes extremely difficult, and may require knowledge of both the process and relevant statistical analyses. Relatedly, the statisticianSir David Coxhas said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".^[4]

There are three purposes for a statistical model, according to Konishi & Kitagawa:^[5]

Predictions
Extraction of information
Description of stochastic structures

Those three purposes are essentially the same as the three purposes indicated by Friendly & Meyer: prediction, estimation, description.^[6]

Dimension of a model

Suppose that we have a statistical model ( $S,{\mathcal {P}}$ ) with ${\mathcal {P}}=\{F_{\theta }:\theta \in \Theta \}$ .In notation, we write that $\Theta \subseteq \mathbb {R} ^{k}$ where $k$ is a positive integer ( $\mathbb {R}$ denotes thereal numbers;other sets can be used, in principle). Here, $k$ is called thedimensionof the model. The model is said to beparametricif $\Theta$ has finite dimension.^{[citation needed]}As an example, if we assume that data arise from a univariateGaussian distribution,then we are assuming that

{\mathcal {P}}=\left\{F_{\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): this leads to the same statistical model as was used in the example with children's heights. 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 the set of all possible lines has dimension 2, even though geometrically, a line has dimension 1.)

Although formally $\theta \in \Theta$ is a single parameter that has dimension $k$ ,it is sometimes regarded as comprising $k$ separate parameters. For example, with the univariate Gaussian distribution, $\theta$ is formally a single parameter with dimension 2, but it is often regarded as comprising 2 separate parameters—the mean and the standard deviation. A statistical model isnonparametricif the parameter set $\Theta$ is infinite dimensional. A statistical model issemiparametricif it has both finite-dimensional and infinite-dimensional parameters. Formally, if $k$ is the dimension of $\Theta$ and $n$ is the number of samples, both semiparametric and nonparametric models have $k\rightarrow \infty$ as $n\rightarrow \infty$ .If $k/n\rightarrow 0$ as $n\rightarrow \infty$ ,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 Coxhas said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies".^[7]

Nested models

Two statistical models arenestedif 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 = b 0 + b 1 x + b 2 x 2 + ε, ε ~ 𝒩(0, σ 2)

has, nested within it, the linear model

y = b 0 + b 1 x + ε, ε ~ 𝒩(0, σ 2)

—we constrain the parameter $b 2$ 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 an example where they have the same dimension, the set of positive-mean Gaussian distributions is nested within the set of all Gaussian distributions; they both have dimension 2.

Comparing models

Comparing statistical models is fundamental for much ofstatistical inference.Konishi & Kitagawa (2008,p. 75) 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." Common criteria for comparing models include the following:R²,Bayes factor,Akaike information criterion,and thelikelihood-ratio testtogether with its generalization, therelative likelihood.

Another way of comparing two statistical models is through the notion ofdeficiencyintroduced byLucien Le Cam.^[8]

Notes

^Cox 2006,p. 178
^Adèr 2008,p.280
^^a ^bMcCullagh 2002
^Cox 2006,p. 197
^Konishi & Kitagawa 2008,§1.1
^Friendly & Meyer 2016,§11.6
^Cox 2006,p. 2
^Le Cam, Lucien (1964)."Sufficiency and Approximate Sufficiency".Annals of Mathematical Statistics.35(4).Institute of Mathematical Statistics:1429.doi:10.1214/aoms/1177700372.

References

Adèr, H. J.(2008), "Modelling", in Adèr, H. J.;Mellenbergh, G. J.(eds.),Advising on Research Methods: A consultant's companion,Huizen, The Netherlands: Johannes van Kessel Publishing, pp. 271–304.
Burnham, K. P.; Anderson, D. R. (2002),Model Selection and Multimodel Inference(2nd ed.), Springer-Verlag.
Cox, D. R.(2006),Principles of Statistical Inference,Cambridge University Press.
Friendly, M.;Meyer, D. (2016),Discrete Data Analysis with R,Chapman & Hall.
Konishi, S.; Kitagawa, G. (2008),Information Criteria and Statistical Modeling,Springer.
McCullagh, P.(2002),"What is a statistical model?"(PDF),Annals of Statistics,30(5): 1225–1310,doi:10.1214/aos/1035844977.