According to current policy, chemicals are evaluated for possible cancer risk to humans at low dose by testing in bioassays, where high doses of the chemical are given to rodents. Thus, risk is extrapolated from high dose in rodents to low dose in humans. The accuracy of these extrapolations is generally unverifiable, since data on humans are limited. However, it is feasible to examine the accuracy of extrapolations from mice to rats. If mice and rats are similar with respect to carcinogenesis, this provides some evidence in favor of interspecies extrapolations; conversely, if mice and rats are different, this casts doubt on the validity of extrapolations from mice to humans. One measure of interspecies agreement is concordance, the percentage of chemicals that are classified the same way as to carcinogenicity in mice and rats. Observed concordance in National Cancer Institute/National Toxicology Program bioassays is around 75%, which may seem on the low side--because mice and rats are closely related species tested under the same experimental conditions. Theoretically, observed concordance could underestimate true concordance, due to measurement error in the bioassays. Thus, bias in concordance is of policy interest. Expanding on previous work by Piegorsch et al., we show that the bias in observed concordance can be either positive or negative: an observed concordance of 75% can arise if the true concordance is anything between 20 and 100%. In particular, observed concordance can seriously overestimate true concordance. A variety of models more or less fit the data, with quite different implications for bias. Therefore, given our present state of knowledge, it seems unlikely that true concordance can be determined from bioassay data.

The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a summary of recent invariant-measure characterizations. Then a new statistical derivation of the law in the form of a CLT-like theorem for significant digits is presented. If distributions are selected at random (in any "unbiased" way) and random samples are then taken from each of these distributions, the significant digits of the combined sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant-digit phenomenon in many different empirical contexts and helps justify its recent application to computer design, mathematical modelling and detection of fraud in accounting data.

Karl Pearson and G. Udny Yule developed the main interpretations of correlation used by statisticians for the past century or so. They also examined a number of situations in which the correlation inference was unsatisfactory. This paper considers the development of their ideas on both genuine and spurious correlations and makes some reference to related modern work.