View source for Normally hyperbolic invariant manifold

The '''Newman–Keuls''' or '''Student–Newman–Keuls (SNK)''' method is a stepwise [[multiple comparisons]] procedure used to identify [[Sample (statistics)|sample]] [[Arithmetic mean|means]] that are [[Statistical significance|significantly]] different from each other.<ref name=Muth>{{cite book |last1 = Muth |first1 = James E. De |title = Basic Statistics and Pharmaceutical Statistical Applications |edition=2nd |publisher = Chapman and Hall/CRC |location = Boca Raton, FL |year = 2006 |isbn = 0-849-33799-2 |pages=229–259}}</ref> It was named after [[William Sealy Gosset|Student]] (1927),<ref name=Student>{{cite journal|author=Student|title=Errors of routine analysis|journal=Biometrika|volume=19|issue=1/2|year=1927|pages=151–164}}</ref> D. Newman,<ref name=Newman>{{cite journal|author=Newman D|title=The distribution of range in samples from a normal population, expressed in terms of an independent estimate of standard deviation|journal=Biometrika|volume=31|issue=1|year=1939|pages=20&ndash;30|doi=10.1093/biomet/31.1-2.20}}</ref> and M. Keuls.<ref name=Keuls>{{cite journal|author=Keuls M|title=The use of the "studentized range" in connection with an analysis of variance|journal=Euphytica|volume=1|year=1952|pages=112&ndash;122}}</ref> This procedure is often used as a [[post-hoc analysis|post-hoc test]] whenever a significant difference between three or more sample means has been revealed by an [[analysis of variance|analysis of variance (ANOVA)]].<ref name=Muth /> The Newman–Keuls method is similar to [[Tukey's range test]] as both procedures use [[Studentized range|Studentized range statistics]].<ref name=Broota>{{cite book |last1 = Broota |first1 = K.D. |title = Experimental Design in Behavioural Research |edition=1st |publisher = New Age International (P) Ltd. |location = New Delhi, India |year = 1989 |isbn = 8-122-40215-1 |pages=81–96}}</ref><ref name=Sheskin>{{cite book |last1 = Sheskin |first1 = David J. |title = Handbook of Parametric and Nonparametric Statistical Procedures |edition=3rd |publisher = CRC Press |location = Boca Raton, FL |year = 1989 |isbn = 1-584-88440-1 |pages=665–756}}</ref> Compared to Tukey's range test, the Newman–Keuls method is more [[Statistical power|powerful]] but less conservative.<ref name="Sheskin"/>

==Procedure==

The Newman–Keuls method employs a stepwise approach when comparing sample means.<ref name=Toothaker>{{cite book |last1 = Toothaker |first1 = Larry E. |title = Multiple Comparison Procedures (Quantitative Applications in the Social Sciences) |edition=2nd |publisher = Chapman and Hall/CRC |location = Newburry Park, CA |year = 1993 |isbn = 0-803-94177-3 |pages=27–45}}</ref> Prior to any mean comparison, all sample means are rank-ordered in ascending or descending order, thereby producing an ordered range of sample means.<ref name=Muth /><ref name=Toothaker /> A comparison is then made between the largest and smallest sample means within the largest range.<ref name=Toothaker /> Assuming that the largest range is four means, a significant difference between the largest and smallest means as revealed by the Newman–Keuls method would result in a rejection of the [[null hypothesis]]. The next largest comparison of two sample means would then be made within a smaller range of three means. Unless there is no significant differences between two sample means within any given range, this stepwise comparison of sample means will continue until a final comparison is made with the smallest range of just two means. If there is no significant difference between the two sample means, then all the null hypotheses within that range would be retained and no further comparisons within smaller ranges are necessary.

{| class="wikitable"
|+ Range of sample means
|-
! 
! <math>\bar{X}_1</math>
! <math>\bar{X}_2</math>
! <math>\bar{X}_3</math>
! <math>\bar{X}_4</math>
|-
| Mean values || 2 || 4 || 6|| 8
|-
| 2
|
| 2
| 4
| 6
|-
| 4
| 
|
| 2
| 4
|-
| 6
|
|
|
| 2
|-
|}
To determine if there is a significant difference between two means with equal sample sizes, the Newman–Keuls method uses a formula that is identical to the [[Tukey's_range_test#The_test_statistic|one]] used in Tukey's range test, which calculates the ''q'' value by taking the difference between two sample means and dividing it by the standard error:

:<math> q = \frac{\bar{X}_A - \bar{X}_B}\sqrt{\frac{MS_E}{n}}, </math>

where <math>q</math> represents the [[Studentized range]] value, <math>\bar{X}_A</math> and <math>\bar{X}_B</math> are the largest and smallest sample means within a range, <math>MS_E</math> is the error variance taken from the ANOVA table, and <math>n</math> is the sample size (number of observations within a sample). The computed ''q'' value is then compared to a ''q'' critical value taken from a ''q'' distribution table. If the computed ''q'' value is equal to or greater than the ''q'' critical value, then the null hypothesis (''H''<sub>0</sub>: ''μ''<sub>A</sub> = ''μ''<sub>B</sub>) can be rejected.<ref name=Zar>{{cite book |last1 = Zar |first1 = Jerrold H. |title = Biostatistical Analysis |edition=4th |publisher = Prentice Hall |location = Newburry Park, CA |year = 1999 |isbn = 0-130-81542-X |pages=208–230}}</ref>

If comparisons are made with means of unequal sample sizes, then the Newman-Keuls formula would be adjusted as follows:

:<math> q = \frac{\bar{X}_A - \bar{X}_B}\sqrt{\frac{MS_E}{2}(\frac{1}{n_A} + \frac{1}{n_B})}, </math>

where <math>n_A</math> and <math>n_B</math> represent the sample sizes of the two sample means.

==See also==
*[[Multiple comparisons]]
*[[Post-hoc analysis]]
*[[Tukey's range test]]

==References==
{{reflist}}

{{DEFAULTSORT:Newman-Keuls method}}
[[Category:Multiple comparisons]]
[[Category:Analysis of variance]]