The analysis of Zeeman effect data in radio astronomy is discussed; in particular, previous techniques are extended to include the case of low signal-to-noise ratios. Three statistical techniques for estimating the line-of-sight magnetic field are considered: maximum likelihood, least-squares, and Wiener filters. For high signal-to-noise ratios, all three estimators are essentially unbiased. It is concluded that, in the poor to moderate signal-to-noise ratio regime, all three estimators are biased; the maximum likelihood technique yields results that are, in general, substantially less biased than least-squares and Wiener filters. However, it is possible to debias the least-squares results and obtain estimates that are as good as maximum likelihood under a restricted set of conditions. 20 refs.