The Combination of Estimates from Different Experiments

When we are trying to make the best estimate of some quantity A that is available from the research conducted to date, the problem of combining results from different experiments is encountered. The problem is often troublesome, particularly if the individual estimates were made by different workers using different procedures. This paper discusses one of the simpler aspects of the problem, in which there is sufficient uniformity of experimental methods so that the ith experiment provides an estimate xi of u, and an estimate si of the standard error of xi . The experiments may be, for example, determinations of a physical or astronomical constant by different scientists, or bioassays carried out in different laboratories, or agricultural field experiments laid out in different parts of a region. The quantity xi may be a simple mean of the observations, as in a physical determination, or the difference between the means of two treatments, as in a comparative experiment, or a median lethal dose, or a regression coefficient. The problem of making a combined estimate has been discussed previously by Cochran (1937) and Yates and Cochran (1938) for agricultural experiments, and by Bliss (1952) for bioassays in different laboratories. The last two papers give recommendations for the practical worker. My purposes in treating the subject again are to discuss it in more general terms, to take account of some recent theoretical research, and, I hope, to bring the practical recommendations to the attention of some biologists who are not acquainted with the previous papers. The basic issue with which this paper deals is as follows. The simplest method of combining estimates made in a number of different experiments is to take the arithmetic mean of the estimates. If, however, the experiments vary in size, or appear to be of different precision, the investigator may wonder whether some kind of weighted meani would be more precise. This paper gives recommendations about the kinds of weighted mean that are appropriate, the situations in which they