Klecka's tau (''τ'') is a statistic which is used to test whether a given
classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood.
Classification is the grouping of related facts into classes.
It may also refer to:
Business, organizat ...
analysis improves one's classification to groups over a random allocation to the various groups under consideration.
[Klecka, WR (1980) Discriminant analysis. Sage Publications, Beverly Hills] 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
Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features ...
. 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
:
where ''n''
corr is the number of cases correctly classified, ''n''
i is the number of cases in the ''i''
th group, ''N'' is the total number of cases, ''T'' is the number of groups and ''p''
i is the probability of a case being allocated to that group by chance (''p''
i = 1 / ''T'' ).
Uses
In addition to its use in discriminant analysis
[Murphy AMC (2002) The calcaneus: sex assessment of prehistoric New Zealand Polynesian skeletal remains. Forensic Sci Int 129(3) 205–208][Murphy AMC (1986) Determination of sex by discriminant function analysis of New Zealand Polynesian pectoral girdles: forensic science applications. J Anat 149, 249-268][Taylor JV, Dibennardo R (1984) Discriminant function analysis of the central portion of the innominate. Am J Phys Anthropol 64 (3) 315–320][Stromberg MR (1986) Systematics and conservation of the swift fox, ''Vulpes velox'', in North America. Biolog Conservation 35(2) 97–110] it has been used in
multiple regression
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one o ...
analysis,
[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] probit regression
In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from ''probability'' + ''unit''. The purpose of the model is to est ...
,
[Khemani RS, Shapiro DM (1993) An empirical analysis of Canadian merger policy. J Indust Econ 41 (2) 161-177 ] logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear function (calculus), linear combination of one or more independent var ...
[Dattaloa P (1995) A comparison of discriminant analysis and logistic regression. J Social Service Res 19 (3-4): 121-144][Biggerstaff MA (1992) Evaluating the oral examination in Virginia's licensing of clinical social workers. Res Social Work Practice 2(2) 184-197 ] and
image analysis
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as sophi ...
.
[Jiang S, Liu D (2011) On chance-adjusted measures for accuracy assessment in remote sensing image classification. ASPRS 2011 Annual Conference Milwaukee, Wisconsin]
References
{{Statistics, analysis, state=collapsed
Statistical classification