Assessing the Significance of Apparent Correlations between Radio and Gamma-Ray Blazar Fluxes
Whether or not a correlation exists between the radio and gamma-ray flux densities of blazars is a long-standing question, and one that is difficult to answer confidently because of various observational biases, which may either dilute or apparently enhance any intrinsic correlation between radio and gamma-ray luminosities. We introduce a novel method of data randomization to evaluate quantitatively the effect of these biases and to assess the intrinsic significance of an apparent correlation between radio and gamma-ray flux densities of blazars. The novelty of the method lies in a combination of data randomization in luminosity space (to ensure that the randomized data are intrinsically, and not just apparently, uncorrelated) and significance assessment in flux space (to explicitly avoid Malmquist bias and automatically account for the limited dynamical range in both frequencies). The method is applicable even to small samples that are not selected with strict statistical criteria. For larger samples we describe a variation of the method in which the sample is split in redshift bins, and the randomization is applied in each bin individually; this variation is designed to yield the equivalent to luminosity-function sampling of the underlying population in the limit of very large, statistically complete samples. We show that for a smaller number of redshift bins, the method yields a worse significance, and in this way it is conservative: although it may fail to confirm an existing intrinsic correlation in a small sample that cannot be split into many redshift bins, it will not assign a stronger, artificially enhanced significance. We demonstrate how our test performs as a function of number of sources, strength of correlation, and number of redshift bins used, and we show that while our test is robust against common-distance biases and associated false positives for uncorrelated data, it retains the power of other methods in rejecting the null hypothesis of no correlation for correlated data.
Additional Information© 2012 American Astronomical Society. Received 2011 July 3; accepted 2012 March 27; published 2012 May 17. We thank Andy Strong and the anonymous referees for insightful comments which improved this manuscript. The OVRO 40 M program is supported in part by NASA grants NNX08AW31G and NNG06GG1G, and NSF grant AST-0808050. Support from the Max-Planck Institut für Radioastronomie for upgrading the OVRO 40 M telescope receiver is also acknowledged. We are grateful to Russ Keeney for his tireless efforts in support of observations at the Owens Valley Radio Observatory. V.P. acknowledges support for this work provided by NASA through Einstein Postdoctoral Fellowship grant number PF8-90060 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060, and thanks the Department of Physics at the University of Crete for their hospitality during the completion of part of this work. W.M. acknowledges support from the U.S. Department of State and the Comisiόn Nacional de Investigaciόn Científica y Tecnolόgica (CONICYT) in Chile for a Fulbright-CONICYT scholarship.
Published - Pavlidou2012p18542Astrophys_J.pdf