# Duncan's new multiple range test

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In statistics, Duncan's new multiple range test (MRT) is a multiple comparison procedure developed by David B. Duncan in 1955. Duncan's MRT belongs to the general class of multiple comparison procedures that use the studentized range statistic qr to compare sets of means.

Duncan's new multiple range test (MRT) is a variant of the Student–Newman–Keuls method that uses increasing alpha levels to calculate the critical values in each step of the Newman–Keuls procedure. Duncan's MRT attempts to control family wise error rate (FWE) at αew = 1 − (1 − αpc)k−1 when comparing k, where k is the number of groups. This results in higher FWE than unmodified Newman–Keuls procedure which has FWE of αew = 1 − (1 − αpc)k/2.

David B. Duncan developed this test as a modification of the Student–Newman–Keuls method that would have greater power. Duncan's MRT is especially protective against false negative (Type II) error at the expense of having a greater risk of making false positive (Type I) errors.

Duncan's test has been criticised as being too liberal by many statisticians including Henry Scheffé, and John W. Tukey. Duncan argued that a more liberal procedure was appropriate because in real world practice the global null hypothesis H0= "All means are equal" is often false and thus traditional statisticians overprotect a probably false null hypothesis against type I errors. Duncan later developed the Duncan–Waller test which is based on Bayesian principles. It uses the obtained value of F to estimate the prior probability of the null hypothesis being true.

The main criticisms raised against Duncan's procedure are:

• Duncan's MRT does not control family wise error rate at the nominal alpha level, a problem it inherits from Student–Newman–Keuls method.
• The increased power of Duncan's MRT over Newman–Keuls comes from intentionally raising the alpha levels (Type I error rate) in each step of the Newman–Keuls procedure and not from any real improvement on the SNK method.

## ReferencesEdit

• Duncan, D B.; Multiple range and multiple F tests. Biometrics 11:1–42, 1955.