View source for Tie

A group of observations in a sample that have the same value. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928001.png" /> be independent random variables subject to the same absolutely-continuous probability law with probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928002.png" />. Then with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928003.png" />, none of the observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928004.png" /> will be equal, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928005.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928006.png" />, and so every member <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928007.png" /> of the order statistics (cf. [[Order statistic|Order statistic]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928008.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

constructed from the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t0928009.png" /> will be strictly greater than its predecessor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280010.png" />.

However, in practice, because of rounding-off errors in the calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280011.png" />, several groups of observations can arise, in each of which the observations are all equal. Every such group of coincident observations is called a tie. Thus, instead of (*), the experimenter may observe the order statistics

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280012.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280013.png" /></td> </tr></table>

where all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280015.png" />. Thus, when ties occur, that is, when some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280016.png" />, difficulties arise in defining the rank vector, which plays a basic role in the construction of rank statistics (cf. [[Rank statistic|Rank statistic]]). As yet (1992) there are no precise recommendations for defining the ranks of coincident observations. There are two common approaches to the solution of this problem. The first consists of randomization. According to this approach, the ranks of the elements

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280017.png" /></td> </tr></table>

making up the the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280018.png" />-th group are taken to be some permutation of the numbers

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280019.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280020.png" /></td> </tr></table>

each having probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280021.png" />. The merit of this approach consists of its simplicity, but for certain alternatives with respect to the distribution of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280022.png" />, the actual randomization chosen has an effect on the results of the statistical analysis.

In the second approach, all tied observations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280023.png" /></td> </tr></table>

making up the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280024.png" />-th group are assigned the same, so-called midrank

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280025.png" /></td> </tr></table>

equal to the arithmetic mean of the numbers

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280026.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280027.png" /></td> </tr></table>

It is natural that such a procedure also affects the properties of rank statistics, and this must be taken into account in practice. For example, the second approach is recommended in the construction of the statistics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280028.png" /> of the [[Wilcoxon test|Wilcoxon test]] when there are ties. Then the expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280030.png" /> remains the same as in the case when there are no ties, but its variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280031.png" /> decreases to

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and this must be taken into account when normalizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092800/t09280033.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Hájek,   "Theory of rank tests" , Academia  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.N. Bol'shev,   N.V. Smirnov,   "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR></table>