Introduction
An Analysis of Variance (ANOVA) indicates whether
there is at least one significant difference between group means, but it does
not specify which groups differ from each other. Therefore, ANOVA is considered
a global or omnibus test. To identify the specific
differences, it is necessary to perform comparisons between the group means
after the global test.
The methods used to compare means after an ANOVA
are called a posteriori tests or post
hoc comparisons. Among the most well-known tests—and widely
available in statistical packages—are the Tukey test (covered
previously) and the Duncan test, which will be detailed here.
The Duncan test, also known as Duncan’s
Multiple Range Test (MRT or DMRT), is applied after a significant
ANOVA to identify which pairs of means differ statistically in experiments with
three or more groups. Unlike the Tukey test, which performs independent
pairwise comparisons, the Duncan test is a sequential procedure based
on the calculation of minimum significant ranges.
Although more laborious to perform manually—as it
requires calculating different critical ranges—the Duncan test is widely used
in some fields. Its operation will be demonstrated with an example. However, it
should be noted that DMRT is not widely accepted as Tukey or SNK procedures and
is declared for some statisticians to perform poorly.
Example
Consider the fictional data on blood pressure
reduction (in mmHg) presented in Table 1. This data was subjected to a
one-way ANOVA, the results of which are in Table 2. Since the F-value
was significant at the 5% significance level, the null hypothesis is rejected,
concluding that there is at least one difference between the group means. The
sample means for each group are in Table 3.
Table 1 –
Blood pressure reduction (mmHg) by treatment group
Table 3 –
Mean blood pressure reduction (mmHg) by group
Question: Which
means are statistically different?
To answer this question, the Duncan test is
applied. First, it's important to understand its logic.
How the Duncan Test
Works
The Duncan test compares the range (the difference)
between sets of ordered sample means with a calculated minimum
significant range (Rm).
· If the observed difference between the largest and smallest mean in a set is greater than the calculated minimum range (Rm), it is concluded that the corresponding population means are different at the chosen significance level.
·
The test is sequential (stepwise).
It begins by comparing the largest and smallest mean of the entire set (k means).
If the difference is significant, the algorithm proceeds, comparing subsets of
means. Crucially, if a difference at a particular step is not
significant, all comparisons within that specific subset are considered not
significant and are halted.
Steps of the Test
1. Order the means in ascending or descending order.
2. Calculate the minimum
significant range (Rm) for different
values of m, where m is the number of means spanned
in the interval being compared (e.g., when comparing the 1st and 5th mean in a
list of 6, m=5). The formula is:
Where:
· Rm=
critical value of the studentized range for a given m and for the residual
degrees of freedom, found in specific tables (e.g., Harter, 1960);
· MSE = mean square error (residual) from the
ANOVA;
· r = number of replications per group (assuming
groups of equal size).
Application to the Example
·
Number of treatments (means): k = 6
· MSE =
36.00 (Table 2)
· n =
5 replications (Table 1)
· Residual degrees of freedom
= 24
Standard Error Calculation:
Comparisons:
1.
m = 6: Compare the
largest (D, 29.0) and smallest (Control, 2.0) mean.
Comparison:
Difference: 29.0
- 2.0 = 27.0
27.0 >
8.79 → Significant.
2. m = 5: Compare the ranges spanning 5 means.
Comparisons:
Difference: D (29.0) vs. B (9.0): 20.0 > 8.66 → Significant.
Difference: A (17.0) vs. Control (2.0): 15.0 > 8.66 → Significant.
3.
m = 4: Compare the
ranges spanning 4 means.
Comparisons:
Difference: D (29.0) vs. C (13.2): 15.8 > 8.48 → Significant.
Difference: A (17.0) vs. B (9.0): 8.0 < 8.48 → Not
Significant.
*Since A and B do not differ, their subset
(A, C, E, B) is homogeneous. Internal comparisons (e.g., C vs. B, E vs. B) are
not performed for m=4, m=3, m=2.
Difference: E (12.6) vs. Control (2.0): 10.6 > 8.48 → Significant.
4.
m = 3: Compare the remaining
ranges spanning 3 means.
Comparisons:
Difference: D (29.0) vs. E (12.6): 16.4 > 8.23 → Significant.
Difference: A (17.0) vs.C(13.2): 3.8<8.23 → Not Significant. (Consistent with the
previous halt).
5.
m= 2: Compare the remaining
intervals for adjacent pairs or pairs of interest.
Difference: D (29.0) vs. A (17.0): 12.0 > 7.83 → Significant.
*Other comparisons within the homogeneous block (A,
C, E, B) are not necessary.
Presentation of
Results
The standard way to present the results is to
assign the same letter to means that do not differ significantly from each
other.
Table 6 –
Mean comparison by Duncan's test
Alternatively, means can be presented in order with an underscore connecting those that do not differ.
Control B E C A D
2.0
d 9.0 c 12.6 bc
13.2 bc 17.0 b 29.0 a
Final Notes
1.If the groups have unequal sizes (different
number of replications), n in the formula is replaced by the harmonic
mean of the group sizes.
2. DMRT is not widely accepted as Tukey or SNK procedures;
it is declared for some statisticians to perform poorly. We will discuss these
issues in a next post.
1. Montgomery
DC. Design and Analysis of Experiments. 4ed. New York: Wiley; 1997.
2. Zar,
J.H. 4ed.Upper Saddle River, Prentice Hall.
3. University of York. Tables for Duncan's multiple
range tests Critical values https://www.york.ac.uk › maths › tables › duncan.
4. Critical care. National
Library of Medicine. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC420045/
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