Nuisance Parameters

Nuisance parameters are elements in statistical models that are not immediately of interest to a researcher but that are important to specify in order to define the primary effect of interest. For example, if we consider the variance, σ2, and the mean, μ, and the mean is our primary interest, then variance might be the nuisance parameter. Nuisance parameters are often variances; however, this is not a strict rule. In general terms, any parameter that is used on the analysis of another can be considered a nuisance parameter.

Thus, the nuisance parameter is a parameter of secondary interest, which is used to account for the estimated value of a parameter of primary interest. A simple example to illustrate the issue is the following: Assume that we want to determine if there are gender differences in learning foreign language skills. Defining a target group, such as university students, and obtaining a sample size to provide enough evidence, we try to measure the means for both male and female students. If we calculate the variance before determining the means, then the variance is a nuisance parameter.

Theoretically speaking, the treatment of nuisance parameters can be quite similar to Bayesian approaches. Despite model uncertainty, when the focus is on parameter estimation, Bayesian analysis allows the model indicator to be treated as a nuisance parameter. It attempts to derive the partition of the likelihood function, which is divided into components providing information about both the parameters of interest and the nuisance parameters (included in the model). Based on frequentist theory, once these partitions are achieved, then it is easy to develop a general estimation approach by using the nuisance parameters.

Generally, the variance is included as continuous data. Three common cases of other parameters of interest are treatment effect, power, and sample size. The reason that a sample size is selected is to provide evidence that a desired effect of interest can be detected for some fixed level of power and sample size. The power analysis is conducted to determine the power of a study with a fixed sample size and detectable effect of interest. In case of a small effect of interest, then a large sample size will be required to have a high probability of detecting the effect at a specified level of power. The preferred level of power is 80% to 90% for planning purposes.

As an example, consider the 2014 study by Yang Ning and colleagues, which involved reducing sensitivity to nuisance parameters in pseudo-likelihood functions. They considered the following model:

$$Y=Xy+\epsilon ,\epsilon \sim N0,\varnothing ,$$

where:

• Y is an n × 1 vector of response variables,
• X is an n × q design matrix,
• ∅ is an n × n matrix indexed by an unknown parameter of interest ∅ with dimension p, and
• ɣ is the nuisance parameter.

In general, as a standard procedure, ɣ is estimated by the conventional least square estimator ɣ = XTX – 1XTY, and inference about ∅ is based on L∅, ɣ, where L∅, ɣ is the log likelihood based on data Y = y. The pseudo-likelihood approach is used mostly in the field of genetic epidemiology.

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