Fixed-Sample-Size Analysis of Sequential Observations

Abstract

The methods most commonly employed for the statistical analysis of observations are based on the assumption that the number of observations was decided on in advance. The number of observations is indeed chosen in advance in many types of experiment or observational inquiry. In an agricultural field experiment, the number of plots and their treatments must be completely specified long before any observations can be taken; and (apart from possible failures which will be recorded as missing observations) the number of observations eventually obtained is precisely the number chosen at the outset. Many chemical determinations are made in triplicate or quadruplicate or some other fixed number of times, according to a definite rule; and so the number of readings obtained in each determination is fixed in advance. A sample survey of households in a town may be based on inquiries at every hundredth house in a list; here again the number of observations is fixed in advance, provided we include the instances of non-response. There are other sorts of inquiry where the number of observations is not fixed in advance, and where the experimenter, if asked just before he began, would not be able to say how many observations he would take. He may, for example, be following some recognized type of sequential sampling rule, such as olne of A. Wald's sequential tests, or the inverse sampling of J. B. S. Haldane and M. C. K. Tweedie. In that case, when reporting his observations he will presumably state what the sampling rule was, and he will use a method of statistical analysis specially designed for that sampling rule. Most commonly, however, when the number of observations is not fixed in advance, this is because at the outset the experimenter has not fully made up his mind as to his requirements or resources or the nature of the material being studied; and so he does not decide to make a certain fixed number of observations nior to follow a definite sequential sampling rule, but proposes simply to take observations until such time as it shall seem appropriate to stop. S;ometimes indeed