• Survey Research
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• v.7 no.2
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• pp.53-69
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• 2006

Weights can be made and imposed in both sample design stage and analysis stage in a sample survey. While in design stage weights are related with sample data acquisition quantities such as sample selection probability and response rate, in analysis stage weights are connected with external quantities, for instance population quantities and some auxiliary information. The final weight is the product of all weights in both stage. In the present paper, we focus on the weight in analysis stage and investigate the effect of such weights imposed on the weighted mean when estimating the population mean. We consider a finite population with a pair of fixed survey value and weight in each unit, and suppose equal selection probability designs. Under the condition we derive the formulas of the bias as well as mean square error of the weighted mean and show that the weighted mean is biased and the direction and amount of the bias can be explained by the correlation between survey variate and weight: if the correlation coefficient is positive, then the weighted mein over-estimates the population mean, on the other hand, if negative, then under-estimates. Also the magnitude of bias is getting larger when the correlation coefficient is getting greater. In addition to theoretical derivation about the weighted mean, we conduct a simulation study to show quantities of the bias and mean square errors numerically. In the simulation, nine weights having correlation coefficient with survey variate from -0.2 to 0.6 are generated and four sample sizes from 100 to 400 are considered and then biases and mean square errors are calculated in each case. As a result, in the case or 400 sample size and 0.55 correlation coefficient, the amount or squared bias of the weighted mean occupies up to 82% among mean square error, which says the weighted mean might be biased very seriously in some cases.