Phi coefficient


Phi coefficient


In statistics, the phi coefficient (also referred to as the "mean square contingency coefficient" and denoted by φ or rφ) is a measure of association for two binary variables introduced by Karl Pearson[1]. This measure is similar to the Pearson correlation coefficient in its interpretation. In fact, a Pearson correlation coefficient estimated for two binary variables will return the phi coefficient.[2] The square of the Phi coefficient is related to the chi-squared statistic for a 2×2 contingency table (see Pearson's chi-squared test)[3]

\phi^2 = \frac{\chi^2}{n}

where n is the total number of observations. Two binary variables are considered positively associated if most of the data falls along the diagonal cells. In contrast, two binary variables are considered negatively associated if most of the data falls off the diagonal. If we have a 2×2 table for two random variables x and y

y = 1 y = 0 total
x = 1 n11 n10 n_{1\bullet}
x = 0 n01 n00 n_{0\bullet}
total n_{\bullet1} n_{\bullet0} n

where n11, n10, n01, n00, are non-negative "cell cell counts" that sum to n, the total number of observations. The phi coefficient that describes the association of x and y is

\phi = \frac{n_{11}n_{00}-n_{10}n_{01}}{\sqrt{n_{1\bullet}n_{0\bullet}n_{\bullet0}n_{\bullet1}}}

Phi is related to the point-biserial correlation coefficient and Cohen's d and estimates the extent of the relationship between two variables (2 x 2).[4]

Maximum values

Although computationally the Pearson correlation coefficient reduces to the phi coefficient in the 2×2 case, the interpretation of a Pearson correlation coefficient and phi coefficient must be taken cautiously. The Pearson correlation coefficient ranges from −1 to +1, where ±1 indicates perfect agreement or disagreement, and 0 indicates no relationship. The phi coefficient has a maximum value that is determined by the distribution of the two variables. If both have a 50/50 split, values of phi will range from −1 to +1. See Davenport El-Sanhury (1991) [5] for a thorough discussion.

See also

References

  1. ^ Cramer, H. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, p282 (second paragraph). ISBN 0691080046
  2. ^ Guilford, J. (1936). Psychometric Methods. New York: McGraw–Hill Book Company, Inc.
  3. ^ Everitt B.S. (2002) The Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-x
  4. ^ Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353)
  5. ^ Davenport, E., & El-Sanhury, N. (1991). Phi/Phimax: Review and Synthesis. Educational and Psychological Measurement, 51, 821–828.



Wikimedia Foundation. 2010.

Look at other dictionaries:

  • phi coefficient — noun an index of the relation between any two sets of scores that can both be represented on ordered binary dimensions (e.g., male female) • Syn: ↑phi correlation, ↑fourfold point correlation • Topics: ↑statistics • Hypernyms: ↑ …   Useful english dictionary

  • phi correlation — noun an index of the relation between any two sets of scores that can both be represented on ordered binary dimensions (e.g., male female) • Syn: ↑phi coefficient, ↑fourfold point correlation • Topics: ↑statistics • Hypernyms: ↑ …   Useful english dictionary

  • Phi (letter) — Phi (uppercase Φ, lowercase φ or Unicode|ϕ), pronounced [IPA|fī] in modern Greek and as [IPA|faɪ] in English, is the 21st letter of the Greek alphabet. In modern Greek, it represents [IPA|f] , a voiceless labiodental fricative. In Ancient Greek… …   Wikipedia

  • Coefficient de marée — Calcul de marée Le calcul de marée est la méthode utilisée en navigation maritime pour estimer la hauteur d eau, dans un lieu et à un instant donné, en prenant en compte l influence de la marée. Les marées sont le résultat de l attraction de la… …   Wikipédia en Français

  • Matthews correlation coefficient — The Matthews correlation coefficient is used in machine learning as a measure of the quality of binary (two class) classifications. It takes into account true and false positives and negatives and is generally regarded as a balanced measure which …   Wikipedia

  • Activity coefficient — An activity coefficient [ [http://www.iupac.org/goldbook/A00116.pdf Gold Book definition] ] is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. In an ideal mixture the… …   Wikipedia

  • Effect size — In statistics, an effect size is a measure of the strength of the relationship between two variables in a statistical population, or a sample based estimate of that quantity. An effect size calculated from data is a descriptive statistic that… …   Wikipedia

  • Contingency table — In statistics, a contingency table (also referred to as cross tabulation or cross tab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. It is often used to record and analyze the… …   Wikipedia

  • Cramér's V —     Cramér s V (φc) In statistics, Cramér s V (sometimes referred to as Cramér s phi and denoted as φc) is a popular[citation needed] measure of association between two nominal variables, giving a value between 0 and +1 (inclusive). It… …   Wikipedia

  • distribution free statistic — noun a statistic computed without knowledge of the form or the parameters of the distribution from which observations are drawn • Syn: ↑nonparametric statistic • Topics: ↑statistics • Hypernyms: ↑statistic • Hyponyms: ↑ …   Useful english dictionary


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.