For each contrast, a t-ratio is computed by dividing the contrast by the PSE. The reference distribution of these t-ratios under the null hypothesis is not computationally tractable, and so it is obtained by simulation. The method, described below, is based on a discussion in Ye and Hamada (2000).

Denote the t-ratio for the ith contrast by ti:

Of primary importance in screening experiments is the individual error rate, namely the probability of declaring a given effect as active when it is not. For the ith effect, this occurs when |ti| is large, falling in the upper tail of it’s reference distribution.

Because the platform constructs a relatively large number of effects, the experimentwise error rate is also of importance. The experimentwise error rate is the probability of declaring any effect as active when no effects are active. An experimentwise error occurs when no effects are active and the maximum of the absolute t-ratios, max|ti|, is large and falls in the upper tail of its reference distribution.

The Fit Two Level Screening platform obtains reference distributions for both types of error rates using Monte Carlo simulation. Consider a set of n - 1 values that is simulated from a normal distribution with mean 0 and standard deviation equal to PSE. This set of values represents potential contrast values for the experiment under the null hypothesis of no active effects. In all, 10,000 sets of contrast values are generated.

Because the contrast distributions are identical, all of the 10,000*(n - 1) values obtained in the simulation are generated from the distribution of values for any specific contrast under the null hypothesis that the contrast is not active.

Consider the ith contrast. Lenth t-ratios are constructed using each simulated value. The reference distribution for the individual error rate is approximated by the absolute values of these t-ratios. The p-value given in the Individual p-Value column of the report is the interpolated fractional position of the observed absolute Lenth t-Ratio among the 10,000*(n - 1) simulated absolute t-ratios arranged in descending order. This approximates the area to the right of the absolute value of the observed absolute Lenth t-Ratio with respect to the reference distribution.

An experimentwise error occurs if any t-ratio leads to rejecting the null hypothesis when all effects are inactive. Equivalently, an experimentwise error occurs if the maximum of the absolute t-ratios, max|ti|, leads to rejecting the null hypothesis.

To obtain a reference distribution in this case, in each of the 10,000 simulations, the maximum of the absolute t-ratios is computed. These 10,000 maximum values form the reference distribution. The p-value given in the Simultaneous p-Value column of the report is the interpolated fractional position of the observed absolute Lenth t-Ratio among the 10,000 simulated maximum absolute t-ratios arranged in descending order. This approximates the area to the right of the absolute value of the absolute Lenth t-Ratio with respect to the reference distribution based on the simulated maximum absolute t-ratios.

To change the number of default sets of simulations from 10,000, you must assign a value to a global JSL variable named LenthSimN. As an example, do the following: