A Search for OH Megamasers at z > 0.1. III. The Complete Survey

We present the final results from the Arecibo Observatory OH megamaser survey. We discuss in detail the properties of the remaining 18 OH megamasers detected in the survey, including 3 redetections. We place upper limits on the OH emission from 85 nondetections and examine the properties of 25 ambiguous cases for which the presence or absence of OH emission could not be determined. The complete survey has discovered 50 new OH megamasers (OHMs) in (ultra)luminous infrared galaxies ([U]LIRGs) which doubles the sample of known OHMs and increases the sample at z>0.1 sevenfold. The Arecibo OH megamaser survey indicates that the OHM fraction in LIRGs is an increasing function of the far-IR luminosity (L_{FIR}) and far-IR color, reaching a fraction of roughly one third in the warmest ULIRGs. Significant relationships between OHMs and their hosts are few, primarily due to a mismatch in size scales of measured properties and an intrinsic scatter in OHM properties roughly equal to the span of the dataset. We investigate relationships between OHMs and their hosts with a variety of statistical tools including survival analysis, partial correlation coefficients, and a principal component analysis. There is no apparent OH megamaser ``fundamental plane.'' We compile data on all previously known OHMs and evaluate the possible mechanisms and relationships responsible for OHM production in merging systems. The OH-FIR relationship is reexamined using the doubled OHM sample and found to be significantly flatter than previously thought: L_{OH} ~ L_{FIR}^{1.2 +/- 0.1}. This near-linear dependence suggests a mixture of saturated and unsaturated masers, either within individual galaxies or across the sample.