In the theory of

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, a nuisance variable is a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

that is fundamental to the

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

, but that is of no particular interest in itself or is no longer of any interest: one such usage arises for the Chapman–Kolmogorov equation. For example, a model for a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

may be defined conceptually using intermediate variables that are not observed in practice. If the problem is to derive the theoretical properties, such as the mean, variance and covariances of quantities that would be observed, then the intermediate variables are nuisance variables. The related term nuisance factor has been used in the context of block experiments, where the terms in the model representing block-means, often called "factors", are of no interest. Many approaches to the analysis of such experiments, particularly where the

experimental design The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...

is subject to randomization, treat these factors as random variables. More recently, "nuisance variable" has been used in the same context. "Nuisance variable" has been used in the context of statistical surveys to refer information that is not of direct interest but which needs to be taken into account in an analysis.{{Cite journal , last1 = Sanderman , first1 = R. , last2 = Coyne , first2 = J. C. , last3 = Ranchor , first3 = A. V. , doi = 10.1016/j.pec.2006.04.002 , title = Age: Nuisance variable to be eliminated with statistical control or important concern? , journal = Patient Education and Counseling , volume = 61 , issue = 3 , pages = 315–316 , year = 2006 , pmid = 16731313 In the context of stochastic models, the treatment of nuisance variables does not necessarily involve working with the full joint distribution of all the random variables involved, although this is one approach. Instead, an analysis may proceed directly to the quantities of interest. The term nuisance variable is sometimes also used in more general contexts, simply to designate those variables that are marginalized over when finding a

marginal distribution In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...

References

Stochastic processes Latent variable models