# Klecka's tau

Klecka's tau (τ) is a statistic which is used to test whether a given classification analysis improves one's classification to groups over a random allocation to the various groups under consideration.[1] The maximum value of τ is 1.0 indicating no errors in the prediction. A value of zero indicates no improvement over a random assignment.

The distribution of τ is not presently known and it is used as a descriptive rather than as an analytic statistic.

## Rationale for use

Klecka's τ was developed for use with discriminant analysis. The raw accuracy of discriminant analysis the sum of correct predictions divided by the total number of cases. Klecka noted that although the percentage of cases predicted accurately is the most intuitive measure of discrimination, the magnitude of this percentage should be judged in relation to the expected percentage of correct classifications made by random assignment. A proportional reduction in error statistic (τ) can be calculated giving a standard measure of improvement over a random assignment regardless of the number of groups.

## Mathematical formulation

τ is defined as[citation needed]

${\displaystyle \tau ={\frac {n_{corr}-\sum _{i=1}^{T}p_{i}n_{i}}{N-\sum _{i=1}^{T}p_{i}n_{i}}}}$

where ncorr is the number of cases correctly classified, ni is the number of cases in the ith group, N is the total number of cases, T is the number of groups and pi is the probability of a case being allocated to that group by chance (pi = 1 / T ).

## Uses

In addition to its use in discriminant analysis[2][3][4][5] it has been used in multiple regression analysis,[6] probit regression,[7] logistic regression[8][9] and image analysis.[10]

## References

1. ^ Klecka, WR (1980) Discriminant analysis. Sage Publications, Beverly Hills
2. ^ Murphy AMC (2002) The calcaneus: sex assessment of prehistoric New Zealand Polynesian skeletal remains. Forensic Sci Int 129(3) 205–208
3. ^ Murphy AMC (1986) Determination of sex by discriminant function analysis of New Zealand Polynesian pectoral girdles: forensic science applications. J Anat 149, 249-268
4. ^ Taylor JV, Dibennardo R (1984) Discriminant function analysis of the central portion of the innominate. Am J Phys Anthropol 64 (3) 315–320
5. ^ Stromberg MR (1986) Systematics and conservation of the swift fox, Vulpes velox, in North America. Biolog Conservation 35(2) 97–110
6. ^ Closea ME & Davies‐Colley RJ (1990) Baseflow water chemistry in New Zealand rivers 2. Influence of environmental factors. NZ J Marine Freshwater Res 24(3) 343-356
7. ^ Khemani RS, Shapiro DM (1993) An empirical analysis of Canadian merger policy. J Indust Econ 41 (2) 161-177
8. ^ Dattaloa P (1995) A comparison of discriminant analysis and logistic regression. J Social Service Res 19 (3-4): 121-144
9. ^ Biggerstaff MA (1992) Evaluating the oral examination in Virginia's licensing of clinical social workers. Res Social Work Practice 2(2) 184-197 doi: 10.1177/104973159200200205
10. ^ Jiang S, Liu D (2011) On chance-adjusted measures for accuracy assessment in remote sensing image classification. ASPRS 2011 Annual Conference Milwaukee, Wisconsin