1. Example For simplicity, assume Z i |F i are independent. Let the relative frame size of the incomplete frame as well as the expected cost vary. Relative AMSE for Decreasing q 2 Relative AMSE for Increasing Second Moment in the Excluded Frame Elements Sampling Perennial Streams An Application in Model Based Optimal Design William Coar and F. Jay Breidt This research is funded by U.S.EPA – Science To Achieve Results (STAR) Program Cooperative Agreement # CR - 829095 Abstract Previous sample designs for perennial streams rely on the River Reach File 3 (RF3) Classification in the National Hydrography Dataset (NHD) to establish a sampling frame of perennial reaches. Recent studies have provided evidence of misclassification resulting in an incomplete frame, biased results and high costs. This application in optimal design investigates the use of auxiliary data to assist in predicting the current status of a stream segment as either perennial or non- perennial. The entire NHD database could then be used as a complete frame. Based on this information, optimal designs for fixed cost are derived for the complete and incomplete frame. Anticipated mean square error under a superpopulation model is then used to compare estimators from the complete and incomplete frame. Future Work Additional numerical investigation of AMSE Investigate optimal design for regression estimator Investigate asymptotic properties of the AMSE ratio under different formulations (fixed frame bias, small frame bias, negligible frame bias). Continue work on estimating p(x i ) Relative Mean Square Error The AMSE of the biased estimator relative to the unbiased estimator can be used to investigate gains or losses with particular model and cost structures. Under equal expected costs for both designs, where Model Define: Assume reach(i) is truly perennial with probability,. For Y i >0, Z i >0 with probability, for known auxiliary. Suppose the total of some attribute about perennial streams is If are a realization from a super- population model, then an anticipated mean squared error of an estimator is defined as Optimal Design for Horvitz- Thompson Estimator The cost of sampling element i is c i. First order inclusion probabilities that minimize AMSE for a fixed expected cost C are said to be optimal. Under the complete frame (entire NHD) Under the incomplete frame (RF3 Classified Perennial) where Results Elimination of the frame bias Complete frame assures positive first order inclusion probabilities for every perennial stream yielding an unbiased estimator. Optimal design gives higher inclusion probabilities to truly perennial streams than to non-perennial. Cost Optimal design sample low cost elements with higher probability. Variability Optimal design samples elements with large second moment with higher probability. Efficiency Relative efficiency of the biased scheme will vary with cost structures, relative frame sizes, and moment structure. DISCLAIMER The work reported here was developed under the STAR Research Assistance Agreement CR-829095 awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This poster has not been formally reviewed by EPA. The views expressed here are solely those of the authors and STARMAP, the Program they represent. EPA does not endorse any products or commercial services mentioned in this poster.