If there are a large number of ties, then the denominator in the definition of Kendall’s tau (see Kendall’s Tau Basic Concepts) should be replaced by

where *n _{x}* is the number of pairs with a tie in variable

*x*and

*n*

_{y}is the number of pairs with a tie in variable y. This version of Kendall’s Tau is often called

**tau-b**(instead of the previous version which is called

**tau-a**).

The calculation of *n*_{y} is similar to that of *D* described in Kendall’s Tau Hypothesis Testing, namely for each *i*, count the number of *j > i* for which *x _{i} = x_{j}*. This sum is

*n*

_{y}. Calculating

*n*is similar, although potentially easier since the

_{x}*x*are in ascending order.

_{i}Since in general *C*(*m*, 2) = 1 + 2 +⋯+ (*m*–1), it follows that

where* t _{i}* = the number of elements in the

*i*th group of ties among the

*x*values and

*u*= the number of elements in the

_{j}*j*th group of ties among the y values.

**Example 1**: Repeat the analysis for Example 1 of Kendall’s Tau Normal Approximation using Kendall’s tau for the data in range A3:B18 of Figure 1.

**Figure 1 – Hypothesis testing for Kendall’s tau (with ties)**

As we did in Example 1 of Kendall’s Tau Hypothesis Testing, we first sort the data, placing the results in range D3:E18. This time we can see that there are ties.

The calculation is similar to that used for Example 1 of Kendall’s Tau Normal Approximation, except that we need to account for the ties. In particular, the formula for inversions (*D*) needs to be modified. E.g. cell F4 in Figure 1 contains the formula =COUNTIFS(E5:E$19,”<“&E4, D5:D$19,”>” & D4).

In order to calculate the modified denominator for tau we need to calculate *n _{x}* and

*n*

_{y}. E.g. the calculation for

*n*is carried out by putting the formula =COUNTIF(D5:D$19,”=”&D4) in cell H4. We next highlight the range H4:H18 and press

_{x}**Ctrl-D**to copy this formula into all the relevant cells in column H. Placing the formula = SUM(H4:H18) in cell H19 yields the value for

*n*. The value

_{x}*n*

_{y}(cell I19) can be calculated in a similar way.

We can calculate the value of *C* as the sum of the concordance elements in a fashion similar to that used to calculate *D*.

E.g. cell G4 contains the formula =COUNTIFS(E5:E$19,”>”&E4, D5:D$19,”>” & D4). Alternatively we note that *C* = *C*(*n*, 2) – *D* – *T.* Now *C*(*n*, 2) = *C*(15, 2) = 105 (cell M5), *D* = 72 (cell F19) and *T* = *n _{x}* +

*n*

_{y}–

*n*

*= 7 + 4 – 1 = 10. The number of ties is equal to the number of ties in*

_{x&y}*x*plus the number of ties y minus the number of ties for both

*x*and y,

*n*

*. We calculate*

_{x&y}*n*

*as the sum of the cells in column J where for example cell J4 contains the formula =COUNTIFS(D5:D$19,”=”&D4,E5:E$19,”=”&E4).*

_{x&y}Kendall’s tau (cell M8) is calculated by the formula =(M7-M6)/SQRT((M5-H19)*(M5–I19)).

**Observation**: If there are a lot of ties we also need to modify the calculation of the standard error as follows:

**Example 2**: Repeat the analysis for Example 1 using the improved version of the standard error and *z* described above.

We show the analysis in Figure 2.

**Figure 2 – Hypothesis testing for Kendall’s tau: improved version**

*C* and *D* are calculated as before, but this time we handle the ties using the formulas

Column H contains a non-zero value only for those values in column D (the *x* values) which are the first one of a group of ties. This value is one less than the number of ties in that group. Similarly column I handles the ties from column E (the y values). E.g. the value 78 occurs 4 times in column D, the first of these in cell D12, and so cell H12 contains the value 4 – 1 = 3. This is done using the formula

=IF(COUNTIF(D$3:D11,D12)=0,COUNTIF(D13:D$19,D12),0)

Thus there are *C*(4, 2) = 6 pairs with value 78.

Since for any *m*, *C*(*m*, 2) = *m*(*m–*1)/2, we can calculate the number of ties for *x*, *n _{x}* =

*C*(

*t*, 2) by the formula =SUMPRODUCT(H4:H18,H4:H18+1)/2, and similarly for

_{i}*n*

_{y}. In a similar fashion we can calculate the values of all the formulas in the previous observation, as shown in Figure 3.

**Figure 3 – Calculation of standard error**

From *m* (cell Q9) we calculate the standard error (cell L9) and the z-score (cell L12) as shown in Figure 2.

**Observation**: The KCORREL function with *ties* = TRUE (default) can be employed to carry out all these calculations automatically (see above). Alternatively, the **Kendall’s Tau** option of the **Correlation** data analysis tool can also be employed to carry out all these calculations (see Correlation Data Analysis Tool).

For example, the key results for Example 2 can be calculated using the formula =KCORREL(A4:A18,B4:B18,TRUE), as shown in Figure 4.

**Figure 4 – Output of KCORREL with ties correction**

In figure 1, your formulas calculating ties-life and ties-cig also count ties-both. In other words you are double-counting ties-both. You can see this because if you add the total Concordances + discordances + ties-life + ties-cig + ties-both you get 107. Cell M5 says you should only have 105 pairs. The 2 extra pairs are the single pair in ties-both being counted twice.

Grant,

I believe that I am using Concordances + discordances + ties-life + ties-cig – ties-both, with the last term subtracted, not added.

Charles

This me again,

The results are correct. I was little confused.

But in Figure 2, the column J is not used. Why we compute Both tied?

Thanks,

Abdelkader,

Column J is used to compute cell L7.

Charles

Hi Dr,

Many thanks for your time.

Cell L7 doen’t has any formula. May be you mean M7, but the formula is “=G19”. Not “J”. No cell is linked to “J”. Pls check & confirm.

Thanks

Are you referring to Figure 1 or Figure 2?

Charles

Figure 1.

Yes, you are correct that column J is never used.

Charles

Hi Dr,

Using Formula as Figure 2 gives different results than your macros in RealStats.xlam. May i download a new one?

Thanks

Hello Dr.,

many thanks for your quick replies.

I used formulas as you indicates.

In H4 the formula is =COUNTIF(D5:D$19,”<=“&D4) . Note "<=" instead of "<". the result is correct. Is it correct??

In I4 i used the formula: =COUNTIF(E5:E$19,”<=“&E4) . I get 14, while you get 0.

Where is the problem?

Thanks

Abdelkader,

The formula in cell H4 is =COUNTIF(D5:D$19,”=“&D4) with an “=” and not “<". I have now corrected this on the referenced webpage. Similarly the formula in cell I4 is =COUNTIF(E5:E$19,”=“&E4). Thanks for identifying this problem. Charles

When I artificially increase the number of ties in my Y variable so that they make up more than %75 of my data it causes an error because the equation for “m” (cell S9) ends up taking the squareroot of a negative number.

Does the calculation not work on data with too many ties, or does the formula need to be changed so that you take the square root of the absolute value instead?

Dugan,

I am not aware of this problem or how to solve it. Can you send me an Excel file with the data you used to create the problem so that i can take a look at it? Please send it to the email address found on Contact Us.

Charles

One thing I’m missing in your pages discussing of Kendall’s tau is how to interpret the importance of the effects. Statistical significance is one thing (i.e. p values), but how high a value of tau would represent substantive, practically significant correspondence?

Specifically, here’s my application: I have two lists of measures taken at different times (around five years apart) for around 20 countries or so. My hypothesis is that the countries have not changed substantively in this five-year period. The exact values of the measures is meaningless in my context (so, differences in means via a t-test is inappropriate); all that matters is that the ordering (ranking) of each country relative to each other has not changed. So I am hypothesizing that the two lists have a Kendall’s tau that is substantively close to +1.0.

I know that this is a case of hypothesis noninferiority/equivalence testing (not standard hypothesis testing), but my question is, what minimal value of tau (e.g. +0.9) would be considered so high that the differences between the two lists would be considered substantively negligeable? How would I even go about finding such a number?

I would appreciate any help or pointers you might give me in answering this question.

This is a subjective judgement. I would say that a correlation coefficient of +.7 or higher would commonly be viewed as high.

When the correlation coefficient r is used as an effect size, a rough estimate of effect size is that r=.5 represents a large effect (explains 25% of the variance), r=.3 represents a medium effect (explains 9% of the variance), and r=.1 represents a small effect (explains 1% of the variance).

Charles

Thanks for your very useful resources. This is just what I was looking for. So, my understanding is that using the adjusted formula will always give the most accurate results, no matter how many ties there are. If there are no ties, then the results are identical to the regular formula. However, if there are ties, then the adjusted formula is more accurate. If there are only a few ties (how few is “few”?), then the differences are negligeable. If there are a large number of ties (how large is “large”?), then the results of the regular formula are unreliable.

So, my question is: what is “a large number of ties”? That is, how large is “large”? 10% (e.g. out of 20 items, 2 are ties)? 5%? In other words, when would you decide you have enough ties to justify using this adjusted formula?

And a related question (which would render the prior question moot): Is there any harm in always using this adjusted formula that accounts for ties instead of the simpler one? If this adjusted formula is always more accurate, then I wouldn’t even bother ever using the simpler one (since Excel is doing the hard work for me).

Yes, I would always use the formula with the ties adjustment.

Charles

Thank you for your help. I think you have provided a very useful resource here

In Figure 2 how are the SUM values in columns H and I, 7 and 4?

Peter,

The “sums” are not simple totals. They are the sums of form f(f+1)/2. In column H the non-zero entries are 1 and 3. The ties contribution is therefore (1*2)/2 + (3*4)/2 = 1 + 6 = 7. This is calculated via the formula =SUMPRODUCT(H4:H18,H4:H18+1)/2.

Charles

Please ignore my previous question I have found my error!

I’m trying to use the formula for calculating Kendall’s tau with ties, but when I enter the modified formula for D in Excel 2011 for Mac I am told there are too many arguments. Do you have a procedure which can be used with this version of Excel? Many thanks.