Statistical model

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Statistical model

A statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions. If the model has only one equation it is called a single-equation model, whereas if it has more than one equation, it is known as a multiple-equation model.

In mathematical terms, a statistical model is frequently thought of as a pair where is the set of possible observations and the set of possible probability distributions on .

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Statistical inference

Inferential statistics or statistical induction comprises the use of statistics to make inferences concerning some unknown aspect of a population....
enables us to make statements about which element(s) of this set are likely to be the true one.

Three notions are sufficient to describe all statistical models.

We choose a statistical unit
Statistical unit

In different statistical disciplines, the statistical unit is the source of a random variable. In epidemiology, the usual term is sampling unit, whereas experimental unit is often used in experimental design....
, such as a person, to observe directly. Multiple observations of the same unit over time is called longitudinal research
Longitudinal study

A longitudinal study is a correlational research study that involves repeated observations of the same items over long periods of time — often many decades....
. Observations of multiple statistical attributes is a common way of studying relationships among the attributes of a single unit.
Our interest may be in a statistical population
Statistical population

In statistics, a statistical population is a Set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population....
(or set) of similar units rather than in any individual unit. Survey sampling
Survey sampling

In statistics, survey sampling involves selecting a sample from a finite population. It is an important part of planning statistical research and design of experiments....
offers an example of this type of modeling.
Our interest may focus on a statistical assembly
Statistical assembly

In statistics, for example in statistical quality control, a statistical assembly is a collection of parts or components which makes up a statistical unit....
where we examine functional subunits of the statistical unit
Statistical unit

In different statistical disciplines, the statistical unit is the source of a random variable. In epidemiology, the usual term is sampling unit, whereas experimental unit is often used in experimental design....
. For example, Physiology
Physiology

Physiology is the study of the mechanical, physical, and biochemical functions of living organisms. Physiology has traditionally been divided between plant physiology and animal and all living things physiology but the principles of physiology are universal, no matter what particular organism is being studied....
modeling probes the organs which compose the unit. A common model for this type of research is the stimulus-response model
Stimulus-response model

The stimulus?response model describes a statistical unit as making a quantitative response to a quantitative stimulation administered by the researcher....
.

One of the most basic models is the simple linear regression

Linear regression

In statistics, linear regression is used for two things;Linear regression is a form of regression analysis in which the relationship between one or more independent variables and another variable, called the dependent variable, is modeled by a least squares function, called linear regression equation....
model which assumes a relationship between two random variables Y and X. For instance, one may want to linearly explain child mortality in a given country by its GDP. This is a statistical model because the relationship need not to be perfect and the model includes a disturbance term which accounts for other effects on child mortality other than GDP.

As a second example, Bayes theorem in its raw form may be intractable, but assuming a general model H allows it to become which may be easier. Models can also be compared using measures such as Bayes factor

Bayes factor

In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing....
s or mean square error.

Statistical model

See also