C+-probability explained

In statistics, a c⁺-probability is the probability that a contrast variable obtains a positive value.^[1] Using a replication probability, the c⁺-probability is defined as follows: if we get a random draw from each group (or factor level) and calculate the sampled value of the contrast variable based on the random draws, then the c⁺-probability is the chance that the sampled values of the contrast variable are greater than 0 when the random drawing process is repeated infinite times. The c⁺-probability is a probabilistic index accounting for distributions of compared groups (or factor levels).^[2]

The c⁺-probability and SMCV are two characteristics of a contrast variable. There is a link between SMCV and c⁺-probability.^{[1] [2]} The SMCV and c⁺-probability provides a consistent interpretation to the strength of comparisons in contrast analysis.^[2] When only two groups are involved in a comparison, the c⁺-probability becomes d⁺-probability which is the probability that the difference of values from two groups is positive.^[3] To some extent, the d⁺-probability (especially in the independent situations) is equivalent to the well-established probabilistic index P(X > Y). Historically, the index P(X > Y) has been studied and applied in many areas.^{[4] [5] [6] [7] [8]} The c⁺-probability and d⁺-probability have been used for data analysis in high-throughput experiments and biopharmaceutical research.^{[1] [2]}

See also

• Contrast (statistics)
• Effect size
• SSMD
• SMCV
• Contrast variable
• ANOVA

Notes and References

1. Zhang XHD. A method for effectively comparing gene effects in multiple conditions in RNAi and expression-profiling research. Pharmacogenomics . 10 . 3. 345–58. 2009 . 20397965 . 10.2217/14622416.10.3.345 .
2. Book: Zhang XHD. 2011. Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research. Cambridge University Press. 978-0-521-73444-8.
3. Zhang XHD. A new method with flexible and balanced control of false negatives and false positives for hit selection in RNA interference high-throughput screening assays. Journal of Biomolecular Screening . 12 . 5. 645–55. 2007 . 17517904. 10.1177/1087057107300645 . free.
4. Owen DB, Graswell KJ, Hanson DL . Nonparametric upper confidence bounds for Pr(Y < X) and confidence limits for Pr(Y < X) when X and Y are normal . Journal of the American Statistical Association . 59 . 906–24. 1964 . 10.2307/2283110. 2027/mdp.39015094992651. free.
5. Church JD, Harris B . The estimation of reliability from stress-strength relationships. Technometrics . 12 . 49–54. 1970 . 10.1080/00401706.1970.10488633.
6. Downton F. The estimation of Pr(Y < X) in normal case. Technometrics . 15 . 551–8. 1973 . 10.2307/1266860.
7. Reiser B, Guttman I . Statistical inference for of Pr(Y ≤ X) – normal case. Technometrics . 28 . 253–7. 1986 . 10.2307/1269081.
8. Acion L, Peterson JJ, Temple S, Arndt S . Probabilistic index: an intuitive non-parametric approach to measuring the size of treatment effects. Statistics in Medicine . 25 . 4. 591–602. 2006 . 16143965. 10.1002/sim.2256 .