Working–Hotelling procedure

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. As such, when only a small subset of the possible values of the independent variable is considered, it is more conservative and yields wider intervals than competitors like the Bonferroni correction at the same level of confidence. It outperforms the Bonferroni correction as more values are considered.

Consider a simple linear regression model ${\displaystyle Y=\beta _{0}+\beta _{1}X+\varepsilon }$, where ${\displaystyle Y}$ is the response variable and ${\displaystyle X}$ the explanatory variable, and let ${\displaystyle b_{0}}$ and ${\displaystyle b_{1}}$ be the least-squares estimates of ${\displaystyle \beta _{0}}$ and ${\displaystyle \beta _{1}}$ respectively. Then the least-squares estimate of the mean response ${\displaystyle E(Y_{i})}$ at the level ${\displaystyle X=x_{i}}$ is ${\displaystyle {\hat {Y_{i}}}=b_{0}+b_{1}x_{i}}$. It can then be shown, assuming that the errors independently and identically follow the normal distribution, that an ${\displaystyle 1-\alpha }$ confidence interval of the mean response at a certain level of ${\displaystyle X}$ is as follows:

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