Tukey Testing Procedure With Family Significance Level R

Statistical test for multiple comparisons

Tukey's range test, also known as Tukey's test, Tukey method, Tukey's honest significance exam, or Tukey's HSD (honestly meaning divergence) test,^[1] is a unmarried-step multiple comparison procedure and statistical test. It can be used to discover means that are significantly different from each other.

Named subsequently John Tukey,^[2] information technology compares all possible pairs of means, and is based on a studentized range distribution (q) (this distribution is similar to the distribution of t from the t-exam. Come across below).^[3]

Tukey's test compares the means of every treatment to the means of every other treatment; that is, it applies simultaneously to the prepare of all pairwise comparisons

${\displaystyle \mu _{i}-\mu _{j}\,}$

and identifies whatsoever difference between two means that is greater than the expected standard fault. The confidence coefficient for the set, when all sample sizes are equal, is exactly ${\displaystyle ane-\alpha }$ for whatsoever ${\displaystyle 0\leq \blastoff \leq 1}$ . For diff sample sizes, the confidence coefficient is greater than 1 − α. In other words, the Tukey method is conservative when at that place are unequal sample sizes.

A common mistaken belief is that the Tukey hsd should but be used following a significant ANOVA. The ANOVA is not necessary considering the Tukey test controls the Type I error rate on its ain.

Assumptions [edit]

1. The observations beingness tested are independent within and among the groups.^{[ citation needed ]}
2. The groups associated with each mean in the examination are normally distributed.^{[ citation needed ]}
3. At that place is equal within-group variance beyond the groups associated with each hateful in the exam (homogeneity of variance).^{[ citation needed ]}

The test statistic [edit]

Tukey's test is based on a formula very similar to that of the t-test. In fact, Tukey's test is substantially a t-test, except that information technology corrects for family unit-wise fault rate.

The formula for Tukey's test is

${\displaystyle q_{s}={\frac {Y_{A}-Y_{B}}{SE}},}$

where Y _A is the larger of the two means being compared, Y _B is the smaller of the ii means beingness compared, and SE is the standard mistake of the sum of the means.

This q_s value can then be compared to a q value from the studentized range distribution. If the q_s value is larger than the critical value q_α obtained from the distribution, the two ways are said to exist significantly different at level ${\displaystyle \alpha :0\leq \blastoff \leq 1}$ .^[3]

Since the null hypothesis for Tukey'due south test states that all means beingness compared are from the same population (i.e. μ _i = μ ₂ = μ ₃ = ... = μ_thou ), the means should be normally distributed (according to the cardinal limit theorem). This gives rise to the normality assumption of Tukey'south exam.

The studentized range (q) distribution [edit]

The Tukey method uses the studentized range distribution. Suppose that nosotros take a sample of size north from each of chiliad populations with the same normal distribution Due north(μ, σ ^two) and suppose that ${\displaystyle {\bar {y}}}$ _min is the smallest of these sample ways and ${\displaystyle {\bar {y}}}$ _max is the largest of these sample means, and suppose S ² is the pooled sample variance from these samples. Then the following random variable has a Studentized range distribution.

${\displaystyle q={\frac {{\overline {y}}_{\max }-{\overline {y}}_{\min }}{S{\sqrt {2/n}}}}}$

This value of q is the basis of the critical value of q, based on three factors:

1. α (the Blazon I error rate, or the probability of rejecting a truthful zippo hypothesis)
2. k (the number of populations)
3. df (the number of degrees of freedom (N –yard) where N is the full number of observations)

The distribution of q has been tabulated and appears in many textbooks on statistics. In some tables the distribution of q has been tabulated without the ${\displaystyle {\sqrt {2}}}$ factor. To understand which table information technology is, nosotros can compute the effect for chiliad = 2 and compare it to the consequence of the Student's t-distribution with the aforementioned degrees of freedom and the aforementionedα. In addition, R offers a cumulative distribution function (ptukey) and a quantile function (qtukey) forq.

Confidence limits [edit]

The Tukey confidence limits for all pairwise comparisons with confidence coefficient of at least one − α are

${\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\pm {\frac {q_{\alpha ;k;Northward-g}}{\sqrt {2}}}{\widehat {\sigma }}_{\varepsilon }{\sqrt {\frac {2}{n}}}\qquad i,j=1,\ldots ,k\quad i\neq j.}$

Detect that the point estimator and the estimated variance are the same as those for a unmarried pairwise comparing. The simply difference between the conviction limits for simultaneous comparisons and those for a single comparison is the multiple of the estimated standard deviation.

Also annotation that the sample sizes must be equal when using the studentized range approach. ${\displaystyle {\widehat {\sigma }}_{\varepsilon }}$ is the standard deviation of the entire blueprint, non but that of the two groups existence compared. It is possible to work with diff sample sizes. In this case, ane has to calculate the estimated standard deviation for each pairwise comparing as formalized by Clyde Kramer in 1956, so the procedure for unequal sample sizes is sometimes referred to as the Tukey–Kramer method which is as follows:

${\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\pm {\frac {q_{\alpha ;thou;North-k}}{\sqrt {2}}}{\widehat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{n}}_{i}+{\frac {1}{n}}_{j}}}\qquad }$

where n _i and due north _j are the sizes of groups i and j respectively. The degrees of freedom for the whole design is likewise applied.

Comparing ANOVA and Tukey-Kramer tests [edit]

Both ANOVA and Tukey-Kramer tests are based on the aforementioned assumptions. However, these two tests for grand groups (i.e. μ ₁ = μ ₂ = ... = μ_k ) may upshot in logical contradictions when one thousand> two, even if the assumptions hold. It is possible to generate a set of pseudorandom samples of strictly positive measure such that hypothesis μ ₁ = μ ₂ is rejected at significance level ${\displaystyle one-\alpha >0.95}$ while μ ₁ = μ _ii = μ _three is not rejected fifty-fifty at ${\displaystyle 1-\alpha =0.975}$ .^[iv]

Encounter too [edit]

• Familywise error rate
• Newman–Keuls method

Notes [edit]

1. ^ Lowry, Richard. "One Mode ANOVA – Contained Samples". Vassar.edu. Archived from the original on October 17, 2008. Retrieved December four, 2008. Also occasionally as "honestly," see e.thou. Morrison, South.; Sosnoff, J. J.; Heffernan, Grand. Due south.; Jae, S. Y.; Fernhall, B. (2013). "Aging, hypertension and physiological tremor: The contribution of the cardioballistic impulse to tremorgenesis in older adults". Periodical of the Neurological Sciences. 326 (1–2): 68–74. doi:ten.1016/j.jns.2013.01.016.
2. ^ Tukey, John (1949). "Comparing Individual Means in the Analysis of Variance". Biometrics. 5 (two): 99–114. JSTOR 3001913.
3. ^ ^{a b} Linton, L.R., Harder, L.D. (2007) Biological science 315 – Quantitative Biological science Lecture Notes. University of Calgary, Calgary, AB
4. ^ Gurvich, 5.; Naumova, Yard. (2021). "Logical Contradictions in the One-Way ANOVA and Tukey–Kramer Multiple Comparisons Tests with More Than Ii Groups of Observations". Symmetry. 13 (8:1387). doi:ten.3390/sym13081387.

Further reading [edit]

• Montgomery, Douglas C. (2013). Design and Analysis of Experiments (8th ed.). Wiley. Section 3.5.seven.

External links [edit]

• NIST/SEMATECH eastward-Handbook of Statistical Methods: Tukey'south method

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Source: https://en.wikipedia.org/wiki/Tukey%27s_range_test

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