Parameter Invariance

According to Webster's New Universal Unabridged Dictionary, invariant simply means “constant,” and, in mathematics more specifically, “a quantity or expression that is constant throughout a certain range of conditions” (p. 1003). Furthermore, a parameter in a statistical model is an explicit component of a mathematical model that has a value in one or multiple populations of interest; its value is estimated with a certain estimation routine based on sample data to calibrate the model. Therefore, parameter invariance simply means that a certain parameter in a certain statistical model is constant across different measurement conditions such as examinee subgroups, time points, or contexts.

From a decision-making viewpoint, parameter invariance is a desirable characteristic of a statistical model, because it implies that identical statistical decisions can be made across different measurement conditions. In this sense, parameter invariance is one of the preconditions for the generalizability of inferences.

Four aspects of parameter invariance warrant a separate highlighting because they are easily overlooked by practitioners. First, parameter invariance is an abstract ideal state, because parameters are either invariant or not. If a parameter varies across measurement conditions, then a lack of invariance is present, which is a continuum of differing values. However, it is always open to debate and secondary assessment whether the difference between parameter values across conditions affects practical decision making. Second, parameter invariance is always tied to a specific statistical model that contains parameters as elements. Therefore, the issue of whether parameters are invariant cannot be answered abstractly without reference to a certain statistical model. Third, parameter invariance implies a comparison of parameter values across measurement conditions, making discussions about invariance without a reference frame for what constitutes different conditions similarly irrelevant.