Duncan's New Multiple Range Test
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A Duncan's New Multiple Range Test is a post-hoc multiple comparison procedure that compares multiple pairs of means based on the Student-Newman-Keuls method.
- AKA: MRT.
- Context:
- It was developed by David B. Duncan in 1955.
- See: Statistical Test, Multiple Comparisons Problem, Student-Newman-Keuls Method.
References
2016
- (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Duncan's_new_multiple_range_test Retrieved 2016-08-28
- 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.
- 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 is commonly used in agronomy and other agricultural research.
- The result of the test is a set of subsets of means, where in each subset means have been found not to be significantly different from one another.
- Definition
- Assumptions:
- 1. A sample of observed means [math]\displaystyle{ m_{1},m_{2},...,m_{n} }[/math] , which have been drawn independently from n normal populations with "true" means, [math]\displaystyle{ \mu_{1},\mu_{2},...,\mu_{n} }[/math] respectively.
- 2. A common standard error [math]\displaystyle{ \sigma }[/math]. This standard error is unknown, but there is available the usual estimate [math]\displaystyle{ s_{m} }[/math] , which is independent of the observed means and is based on a number of degrees of freedom, denoted by [math]\displaystyle{ n_{2} }[/math] . (More precisely, [math]\displaystyle{ S_{m} }[/math], has the property that [math]\displaystyle{ \frac{n_{2}\cdot S_{m}^2}{\sigma^2_{m}} }[/math] is distributed as [math]\displaystyle{ \chi^2 }[/math] with [math]\displaystyle{ n_2 }[/math] degrees of freedom, independently of sample means).
- The exact definition of the test is:
- The difference between any two means in a set of n means is significant provided the range of each and every subset which contains the given means is significant according to an [math]\displaystyle{ \alpha_{p} }[/math] level range test where [math]\displaystyle{ \alpha_p=1-\gamma_p }[/math] , [math]\displaystyle{ \gamma_p =(1-\alpha)^{(p-1)} }[/math] and [math]\displaystyle{ p }[/math] is the number of means in the subset concerned.
- The exact definition of the test is:
- 2. A common standard error [math]\displaystyle{ \sigma }[/math]. This standard error is unknown, but there is available the usual estimate [math]\displaystyle{ s_{m} }[/math] , which is independent of the observed means and is based on a number of degrees of freedom, denoted by [math]\displaystyle{ n_{2} }[/math] . (More precisely, [math]\displaystyle{ S_{m} }[/math], has the property that [math]\displaystyle{ \frac{n_{2}\cdot S_{m}^2}{\sigma^2_{m}} }[/math] is distributed as [math]\displaystyle{ \chi^2 }[/math] with [math]\displaystyle{ n_2 }[/math] degrees of freedom, independently of sample means).
- Exception: The sole exception to this rule is that no difference between two means can be declared significant if the two means concerned are both contained in a subset of the means which has a non-significant range.
1955
- (Duncan, 1955) ⇒ David B. Duncan (1955). “Multiple range and multiple F tests". Biometrics, 11(1), 1-42. doi:10.2307/3001478
- The common practice for testing the homogeneity of a set of treatment means in an analysis of variance is to use an F (or z) test. This procedure has special desirable properties for testing the homogeneity hypothesis that the n population means concerned are equal. An F-test alone, however, generally falls short of satisfying all of the practical requirements involved. When it rejects the homogeneity hypothesis, it gives no decisions as to which of the differences among the treatment means may be considered significant and which may not. To illustrate, Table I shows results of a barley grain yield experiment conducted by E. Shulkcum of this Institute at Accomac, Virginia, in 1951. Seven varieties, A, B, … , G, were replicated six times in a randomised block design. The F-ratio (in section b) for testing the homogeneity of the varietal means is highly significant. This indicates that one or more of the differences among the means are significant but it does not specify which ones.