Many recent analyses have indicated that large scale Wilkinson Microwave Anisotropy Probe (WMAP) data display anomalies that appear inconsistent with the standard cosmological paradigm. However, the effects of foreground contamination, which require elimination of some fraction of the data, have not been fully investigated due to the complexity in the analysis. Here we develop a general formalism of how to incorporate these effects into any analysis of this type. Our approach is to compute the full multidimensional probability distribution function of all possible sky realizations that are consistent with the data and with the allowed level of contamination. Any statistic can be integrated over this probability distribution to assess its significance. As an example we apply this method to compute the joint probability distribution function for the possible realizations of quadrupole and octopole using the WMAP data. This 12 dimensional distribution function is explored using the Markov Chain Monte Carlo technique. The resulting chains are used to assess the statistical significance of the low quadrupole using frequentist methods, which we find to be $3-4\%$. Octopole is normal and the probability of it being anomalously low or as low as WMAP reported value is very small. We address the quadrupole-octopole alignment using several methods that have been recently used to argue for anomalies, such as angular momentum dispersion, multipole vectors and a new method based on feature matching. While we confirm that the full-sky map Internal Linear Combination Map (ILC) suggest an alignment, we find that removing the most contaminated part of the data also removes any evidence of alignment: the probability distributions strongly disfavor the alignment. This suggests that most of the evidence for it comes from non-Gaussian features in the part of the data most contaminated by the foregrounds. We also present an example, that of octopole alignment with the ecliptic, where the statistical significance can be enhanced by removing the contamination.

• Received 1 May 2004

DOI:https://doi.org/10.1103/PhysRevD.70.083002