page_title: Working–Hotelling procedure
section_title: Graphical representation
hierarchical_section_title: Working–Hotelling procedure / Graphical representation
context_page_description: In statistics, particularly regression analysis, the Working–Hotelling procedure, named after Holbrook Working and Harold Hotelling, is a method of simultaneous estimation in linear regression models. One of the first developments in simultaneous inference, it was devised by Working and Hotelling for the simple linear regression model in 1929. It provides a confidence region for multiple mean responses, that is, it gives the upper and lower bounds of more than one value of a dependent variable at several levels of the independent variables at a certain confidence level. The resulting confidence bands are known as the Working–Hotelling–Scheffé confidence bands.
  Like the closely related Scheffé's method in the analysis of variance, which considers all possible contrasts, the Working–Hotelling procedure considers all possible values of the independent variables; that is, in a particular regression model, the probability that all the Working–Hotelling confidence intervals cover the true value of the mean response is the confidence coefficient.
context_section_description: In the simple linear regression case, Working–Hotelling–Scheffé confidence bands, drawn by connecting the upper and lower limits of the mean response at every level, take the shape of hyperbolas. In drawing, they are sometimes approximated by the Graybill–Bowden confidence bands, which are linear and hence easier to graph:
  where denotes the upper  percentile of the Studentized maximum modulus distribution with two means and  degrees of freedom.
caption: The simple linear regression model with a Working–Hotelling confidence band.