OPTIMAL PERIODICITY OF A SURVEY: EXTENSIONS OF PROBABLE-ERROR MODELS

This paper extends prior work on the problem of ch~x~sing optimal periodicity (and associated sample sizes) for repeated surveys of public and private sch(×~ls with joint consideration of data deterioration (resulting from unobserved year-to-year changes in the underlying process variables), sampling error, and cost. The family of "probable-error models" that was first described in Ghosh et al. (1994) has been extended and empirical results obtained for state-level as well as national-level estimates using data from three rounds of the Schools and Staffing Survey (SASS). As noted in the 1994 paper, the models provide "a direct approximate method for characterizing the decision problem of making a joint choice of inter-survey intervals and sample sizes under a fixed cost constraint. " The extensions reviewed in the present paper assume, for the most part, that conventional direct estimation methods will be used by the data user. In the case of a proposed alternative sampling design suggested by the modeling results, the data user may wish to consider the use of an indirect estimation (time series modeling) approach along the lines discussed in Smith et al. (1995). SASS was conducted at three-year intervals for school years 1987-88, 1990-91, and 1993-94. Future rounds may be conducted at intersurvey intervals of 4, 5, or 6 years. The modeling extensions are illustrated here in a review of two of several new models that were formulated as modifications of the earlier models. The two models provide alternative formulations to account for the approximate average errors incurred by a data user within successive 12month periods following a SASS data collection and up to the time of the next data collection. Projected absolute errors have been estimated for future national-level and typical state-level data collections for selected policy variables and a range of fixed-tovariable cost ratios for each possible periodicity. The two illustrative models, denoted as Model 3A and Model 4M, are modifications of Model 3 and Model 4 of the 1994 paper. They combine a sampling absolute error (s.a.e.) and a process shift D over time in different ways to obtain, for different periodicities, estimates of the year-by-year projected absolute errors that would be incurred by a data user as well as average pr~jected absolute errors for each multi-year periodicity. Annual dollar resources tor SASS are assumed to be fixed. For each of several scenarios this assumption constrains the total annualized cost to a fixed amount and hence determines the sample size for each combination of a periodicity (4, 5, or 6 years) and a fixed-to-variable cost ratio. We assume that data users will keep on using the data obtained from the most recent past survey until a new survey is undertaken and the newly collected data are processed and released to data users. Thus, if the inter-survey period is long, "deterioration" of the data could affect the quality of decisions made by users. On the other hand, if the survey is undertaken very frequently, the costs of conducting the survey and analyzing the data and the indirect costs of the response burden may be judged to have costs that exceed the benefits achieved in using ti'esh data. In the context of repeated surveys, it is useful to distinguish both opportunities and problems presented by different designs. Typical analyses of cost-benefit tradeoffs tend to focus oil the best use of a fixed resource amount over a time period that would include two or more survey data collections. The present budgetary restrictions for the 1990s are such that the "fixed" resource amount may be arbitrarily depressed and may overconstrain any realistic formulation of the optimization problem. In fact, the "truly optimal" formulation may be precluded by external constraints. The usual cost model for a smnple survey assumes a start-up cost Co and a per unit (ultimate sample unit) cost C1. Thus, the total cost is represented as C = Co + nC1. However, the start-up cost may depend on the periodicity. If so we represent the startu-up cost as C0,k (where k is the periodicity), which may be regarded as increasing with increasing periodicity; that is, the start-up cost may be more if the periodicity is five years compared to the start-up cost for a periodicity of four years and so on. On the other hand, the start-up cost may be considered to be constant; that is, it may not depend on the periodicity of the survey. Further details are given in Ghosh et al. (1994). We assume that the true value of a variable remains constant for a year after the survey date. This is an appropriate assumption for the SASS survey system since nearly all of the observed variables under the various SASS questionnaires have an annual