Augmented Half Normal Effects Plots in the Presence of a Few Error Degrees of Freedom

Fractional factorial 2‐level experiments are often conducted without any error degrees of freedom. In such cases, a half‐normal effects plot–also called Daniel plot according to its inventor Cuthbert Daniel–can be used for assessing effect significance. Half‐normal effects plots are often accompanied by a numeric method for assessing effect significance, most prominently Lenth's method. There are, however, also situations for which a few error degrees of freedom are available, for example, from a replicated center point run. For such cases, besides the obvious possibilities of either ignoring the few replicates (i.e., using half‐normal effects plot and Lenth's method, as if they were not there) or using analysis of variance with the replicates for estimating the error variance, several further proposals for assessing effect significance exist. This paper compares the published methods for significance testing, proposes an additional one (that might be very close to what JMP does) and advocates the use of an augmented half‐normal effects plot that shows error points and a null reference line along with the effects. It is argued that such a plot can be useful in a fully replicated experiment for assessing whether the replication process was of adequate quality. The method is available in the author's R package DoE.base.