Duncan's new multiple range test

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.

This test is often followed by the Compact Letter Display (CLD) methodology that renders the output of such test much more accessible to non-statistician audiences.

Definition

Assumptions:
1.A sample of observed means ${\displaystyle m_1,m_2,...,m_n}$ , which have been drawn independently from n normal populations with "true" means, ${\displaystyle {\mu }_1,{\mu }_2,...,{\mu }_n}$ respectively.
2.A common standard error ${\displaystyle \sigma }$. This standard error is unknown, but there is available the usual estimate ${\displaystyle s_m}$ , which is independent of the observed means and is based on a number of degrees of freedom, denoted by ${\displaystyle n_2}$ . (More precisely, ${\displaystyle S_m}$, has the property that ${\displaystyle \frac{n_2\cdot S_m^2}{{\sigma }_m^2}}$ is distributed as ${\displaystyle {\chi }^2}$ with ${\displaystyle n_2}$ 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 ${\displaystyle {\alpha }_p}$ level range test where ${\displaystyle {\alpha }_p=1-{\gamma }_p}$ , ${\displaystyle {\gamma }_p=(1-\alpha {)}^{(p-1)}}$ and ${\displaystyle p}$ is the number of means in the subset concerned.

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.

Procedure

The procedure consists of a series of pairwise comparisons between means. Each comparison is performed at a significance level ${\displaystyle {\alpha }_p}$ , defined by the number of means separating the two means compared (${\displaystyle {\alpha }_p}$ for ${\displaystyle p-2}$ separating means). The test are performed sequentially, where the result of a test determines which test is performed next.

The tests are performed in the following order: the largest minus the smallest, the largest minus the second smallest, up to the largest minus the second largest; then the second largest minus the smallest, the second largest minus the second smallest, and so on, finishing with the second smallest minus the smallest.

With only one exception, given below, each difference is significant if it exceeds the corresponding shortest significant range; otherwise it is not significant. Where the shortest significant range is the significant studentized range, multiplied by the standard error. The shortest significant range will be designated as ${\displaystyle R_{(p,\alpha )}}$, where ${\displaystyle p}$ is the number means in the subset. 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.

An algorithm for performing the test is as follows:

1.Rank the sample means, largest to smallest. 2. For each <span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt20" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"m_{i}"}}' id="mwOg"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}"/></span> sample mean, largest to smallest, do the following: 2.1 for each sample mean, (denoted <span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt21" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"m_{j}"}}' id="mwOw"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle m_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f2b12b1676ba2f42ad8d2c9ee6aa46e7667b73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.95ex; height:2.343ex;" alt="{\displaystyle m_{j}}"/></span>), for smallest up to <span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt22" data-mw='{"name":"math","attrs":{},"body":{"extsrc":" m_{(i-1)} "}}' id="mwPA"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle m_{(i-1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{(i-1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f513b01d3f524a943b3058307dbcc46baca4b73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.22ex; height:2.509ex;" alt="{\displaystyle m_{(i-1)}}"/></span>. 2.1.1 compare <span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt23" data-mw='{"name":"math","attrs":{},"body":{"extsrc":" m_i -m_j "}}' id="mwPQ"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle m_{i}-m_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}-m_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34159f8f73a80f5624b3ff514216a5d44209a9be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.631ex; height:2.676ex;" alt="{\displaystyle m_{i}-m_{j}}"/></span> to critical value <span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt24" data-mw='{"name":"math","attrs":{},"body":{"extsrc":" \\sigma_m \\cdot R_{(p,\\alpha)} "}}' id="mwPg"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle \sigma _{m}\cdot R_{(p,\alpha )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{m}\cdot R_{(p,\alpha )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff75aacfd3a1ce30db425a0c9c65d86f6d197fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.293ex; height:3.009ex;" alt="{\displaystyle \sigma _{m}\cdot R_{(p,\alpha )}}"/></span>,<span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt25" data-mw='{"name":"math","attrs":{},"body":{"extsrc":" P=i-j, \\alpha=\\alpha_p "}}' id="mwPw"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle P=i-j,\alpha =\alpha _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>i</mi> <mo>−<!-- − --></mo> <mi>j</mi> <mo>,</mo> <mi>α<!-- α --></mi> <mo>=</mo> <msub> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=i-j,\alpha =\alpha _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cfb0397632b1e0b86e39b7a139d0ee00aec186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.612ex; height:2.843ex;" alt="{\displaystyle P=i-j,\alpha =\alpha _{p}}"/></span> 2.1.2 if <span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt26" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"m_i-m_j"}}' id="mwQA"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle m_{i}-m_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}-m_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34159f8f73a80f5624b3ff514216a5d44209a9be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.631ex; height:2.676ex;" alt="{\displaystyle m_{i}-m_{j}}"/></span> does not exceed the critical value, <b id="mwQQ">the subset <span class="mwe-math-element mwe-math-element-inline" typeof="mw:Extension/math" about="#mwt27" data-mw='{"name":"math","attrs":{},"body":{"extsrc":" (m_j , m_{j+1},...,m_{I}) "}}' id="mwQg"><span class="mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="mathjax_ignore" alttext="{\displaystyle (m_{j},m_{j+1},...,m_{I})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m_{j},m_{j+1},...,m_{I})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce50846c268016b37602d3c58454aad5dcdb2cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.115ex; height:3.009ex;" alt="{\displaystyle (m_{j},m_{j+1},...,m_{I})}"/></span> is declared not significantly different</b>: 2.1.2.1 Go to next iteration of loop 2. 2.1.3 Otherwise, keep going with loop 2.1

Critical values

Duncan's multiple range test makes use of the studentized range distribution in order to determine critical values for comparisons between means. Note that different comparisons between means may differ by their significance levels- since the significance level is subject to the size of the subset of means in question.

Let us denote ${\displaystyle Q_{(p,\nu ,{\gamma }_{(p,\alpha )})}}$ as the ${\displaystyle {\gamma }_{\alpha }}$ quantile of the studentized range distribution, with p observations, and ${\displaystyle \nu }$ degrees of freedom for the second sample (see studentized range for more information). Let us denote ${\displaystyle r_{(p,\nu ,\alpha )}}$ as the standardized critical value, given by the rule:

If p=2
${\displaystyle r_{(p,\nu ,\alpha )}=Q_{(p,\nu ,{\gamma }_{(p,\alpha )})}}$
Else
${\displaystyle r_{(p,\nu ,\alpha )}=max(Q_{(p,\nu ,{\gamma }_{(p,\alpha )})},r_{(p-1,\nu ,\alpha )})}$

The shortest critical range, (the actual critical value of the test) is computed as : ${\displaystyle R_{(p,\nu ,\alpha )}={\sigma }_m\cdot r_{(p,\nu ,\alpha )}}$. For ${\displaystyle \nu }$->∞, a tabulation exists for an exact value of Q (see link). A word of caution is needed here: notations for Q and R are not the same throughout literature, where Q is sometimes denoted as the shortest significant interval, and R as the significant quantile for studentized range distribution (Duncan's 1955 paper uses both notations in different parts).