Meansdependence

Meansdependence is a term used in statistics to describe a form of dependence between two random variables that is reflected in the behavior of the conditional mean. In brief, Y and X exhibit means dependence if the conditional expectation E[Y | X = x] varies with the value of x rather than remaining constant. Such a pattern indicates that the average outcome of Y depends on the level or state of X.

Formally, meansdependence can be described by a non-constant conditional mean function E[Y | X]. If this function is a linear function of X, the relationship is often referred to as linear mean dependence, commonly modeled by a regression equation Y = β0 + β1 X + ε with E[ε | X] = 0. More complex, nonlinear mean structures are captured by nonparametric regression or generalized additive models.

Meansdependence is related but not identical to other forms of association. Correlation measures linear association and may miss nonlinear mean dependence. Likewise, mean dependence does not by itself imply full independence or identical distributions; higher moments such as variance can still vary with X (a phenomenon known as heteroscedasticity) even when the conditional mean is structured.

Applications include regression diagnostics, econometrics, and quality control, where understanding how the average of a response changes with a predictor informs model specification and inference. The term is not universally standardized; some literature uses mean dependence or dependence in the mean to describe the same concept. See also conditional expectation, regression, and heteroscedasticity.