Patent Document:

let k denote the number of survey questions . since internet can essentially reach almost everyone without major cost , the survey sample could be very large . let n denote the survey sample ( which is generally in the magnitude of hundreds of thousands or millions ). we call a person who receive the survey as a tester . instead of sending all survey questions to each tester , only a subset of survey questions are randomly sent to the tester . for example , if there are a total of 20 questions and each tester only receives 2 questions , there are )( 2 20 = 190 possible ways of selecting 2 questions . if a million people are surveyed , approximately each pair of questions can be surveyed from 1000000 / 190 = 5263 testers , which is still a very large sample . let m denote the number of partial survey question and we call the survey method as partial survey with m questions ( psm ). the purpose of the survey . if the purpose of the survey is only for the mean response , then m = 1 can meet the need . we use “ mean response ” as a general term for the parameter of first moment . for a continuous variable , it is the mean value ; for categorical variable , it is the proportions . if the purpose of the survey is for the mean response and the linear regression between survey questions , m = 2 is the minimal . the targeted survey sample . the smaller the survey sample , the less likelihood a small m can achieve the necessary number of survey responders for each question . the steps are psm method can be outline as follows ( see fig1 ): 1 . for each variable , the mean can be estimated just based on the non - missing response , denoted by { circumflex over ( μ )} y . 2 . the pairwise covariance can be constructed for each pair only using the subsamples that are surveyed for this pair of questions . let □ denote the variance - covariance matrix of the response variable y . the variance - covariance matrix can be estimated by pairwise covariance of non - missing values for each pair , say , { circumflex over ( σ )}. for each pair of questions , the probability of one tester receiving the pair is is the number of possible ways of selecting m questions from k questions . assume the non - response rate p m is the same for all testers , regardless of the questions they received and it can be estimated by the proportion of non - responders . then , the expected number of responders for each pair is 3 . let say one intends to regress y k on y a , where y a is a subset of questions not including y k . let σ a denote the variance - covariance matrix for variables y a and { circumflex over ( σ )} a is the estimator for σ a . since the estimated variance - covariance matrix { circumflex over ( σ )} a may not be positive definite , a small sample modification can be applied to ensure the coefficients can be estimated without modifying the large sample proprieties . let λ min be the minimum eigenvalue of { circumflex over ( σ )} a . a modified estimator for { circumflex over ( σ )} a is where i k is the identify matrix with k dimension . note the choice of small sample modification factor ( n e − 1 − λ min ) can be changed to balance the bias and variance of the estimation for β a . the smaller the modification factor , the less bias for the estimator but larger variance . 4 . the regression coefficient β a can be constructed as where { circumflex over ( σ )} ak is the estimated covariance between y k and y a . the intercept is estimated as { circumflex over ( β )} 0 ={ circumflex over ( μ )} k − y ′ a { circumflex over ( β )} a ( 4 ) generally , the mean response and the relationship between these survey questions through second moments of statistics are sufficient to meet the objectives of the survey . therefore , we will focus on the method of partial survey with 2 questions ( ps2 ) in the simulation . surveys with m ≧ 2 questions allow estimation of higher order of moments , which for example , can be used to estimate the coefficients for polynomial regressions . a drawback for ps with m & gt ; 2 questions is that ( 1 ) the possible combination of m variables is which is large when m is large , and ( 2 ) the proportion of non - response rate increases . if one is especially interested in the relationship among a few key questions , one possible way to do a partial survey where testers may receive survey questions with different number of questions , and the probabilities to distribute various combinations of questions may be different , depending on the importance of variables . when the probabilities of each possible combination of m questions to be surveyed are not equal , the n e for each pair can be calculated by the number of responders for the pair that are used to estimate σ a , and the small modification factor in equation ( 2 ) can be adapted using the minimum of the n e &# 39 ; s or the average of the n e &# 39 ; s . the above estimators for mean response and regression coefficients should perform excellent when the non - response rate is low for partial survey . however , it is possible that even with the fewest number of questions ( e . g ., ps2 ), the non - response rate is still high . in this case , we propose a new estimation method called partial survey extrapolation ( pse ) estimation to reduce the bias in the estimation for mean response and coefficients . let 1 ≦ m 1 & lt ; m 2 & lt ; . . . & lt ; m d ≦ k be d ≧ 3 integers between 1 and k . the targeted testers can be divided into d groups randomly with each group receiving partial survey with m d questions [ ps ( m d )]. then , the mean response can be estimated for each group of testers . let { circumflex over ( μ )} d denote the estimator for group d , and r d be the proportion of missing survey responses for group d , d = 1 , 2 , . . . , d . the mean response estimator be can constructed by extrapolating then { circumflex over ( μ )} d to the ideal case of no missing survey response . the extrapolation idea , combined with simulation , is called simulation extrapolation , has been used for estimation of parameters in measurement error models simulation extrapolation ( cook and stefanski , 1994 ) and in data with missing observations ( hsu , 2013 ). here , we only need extrapolation without simulation . typically , a quadratic extrapolation function can be used to achieve good results . for example , if using a quadratic extrapolation function ƒ ( t )= α 0 + α 1 t + α 2 t 2 , the parameters ( α 0 , α 1 , α 2 ) can be estimated through a linear regression of { circumflex over ( μ )} d on ( 1 , r d , r d 2 ). the extrapolation estimator { circumflex over ( μ )}* is the estimator for ƒ ( t ) when t = 0 ( i . e ., when the proportion of missing is equal of 0 ): in this section , we conduct monte carlo simulations to compare the performance of 4 survey methods : full survey with no missing response ( fsnm ), full survey ( fs ), ps2 and pse . fsnm is an ideal but unrealistic case which is used to benchmark the performance of other methods . for fs and ps2 , the probability of non - response depends on an unobserved latent variable modelled as where a and b are constants to control the rate of missing survey responses . the larger the number of survey questions ( m ) is , the higher probability of non - response . therefore , the number of missing responses for ps2 is much compared to fs . this makes sense as the non - response rate increases as the survey becomes lengthier . the response variables y and the latent variable z are generated as the following : 1 . generate k + 1 variables from multivariate normal distribution with correlation r = 0 . 5 2 . transform the data by the cdf of standard normal distribution to uniform distribution 3 . categorize each variable into a ordinal variable of 5 scales ( 1 to 5 ) with equal probability to simulate the case that the survey questions are often ordinal variables 4 . the first k ordinal variables are yk and the ( k + 1 ) th variable is z we study 4 scenarios with various a , b , k and n ( table 0 ). for each scenario , 10 , 000 simulations are performed . we only present the simulation results for the mean response for y 1 , y 2 and y 3 , and the regression coefficients of y 3 on y 1 and y 2 ( say β 0 , β 1 and β 2 ) as results for other mean responses or regression coefficients should be similar . notation : a and b are used to control the survey nonresponse rate in equation ( 5 ), k is the number of full survey questions , n is the number of testers are surveyed , and ρ m , is the survey in the first two scenarios , we assume a =− 3 , b = 1 . the non - response rate is approximately 50 % for fs , and 92 % ( k = 10 ) to 94 % ( k = 20 ) for ps2 . although one could argue the response rate for ps2 should not depend on k , this difference in the response rate between k = 10 and k = 20 is small and this should not impact the validity of the simulation results . for the first 2 scenarios , the non - response rate is low for ps2 , so no pse estimator is constructed . in scenario 1 , we choose k = 10 and n = 2 , 000 ; and in scenario 2 , we choose k = 20 and n = 10 , 000 . the results for estimation of the mean response ( μ 1 , μ 2 , μ 3 ) are presented in table 1 and the results for the estimation of regression coefficients ( β 1 , β 2 , β 3 ) are presented in table 2 . table 1 summarizes the simulation results for the estimation of the mean response for y 1 , y 2 and y 3 based on 10 , 000 simulations . fsnm , as an ideal but unrealistic case , unsurprisingly performs best with essentially no bias and minimum standard deviations . fs is seriously biased , as expected . ps2 shows little bias but had the larger standard deviation than fsnm and fs . ps2 also has much smaller mean squared errors ( mse ) than the fs method . table 2 summarizes the simulation results for the estimation of the regression coefficients based on 10 , 000 simulations . since the true regression coefficients are difficult to calculate analytically , we use the mean of the 10 , 000 simulations based on fsnm method to estimate the true mean . the estimated true coefficients are β 0 = 1 . 13027 , β 1 = 0 . 31185 , β 2 = 0 . 31143 for k = 10 β 0 = 1 . 13115 , β 1 = 0 . 31150 , β 2 = 0 . 31141 for k = 20 ps estimator has smaller bias , but larger standard deviation and mse than the fs method . in order to understand the performance of ps2 and psee when the non - response rate is high , we simulate 2 additional scenarios . in both scenarios , we choose k = 10 and n = 10 , 000 . in scenario 3 , a =− 2 . 5 , b = 2 , which gives non - response rate of 83 % for fs and 23 % for ps2 . in scenario 4 , a =− 2 , b = 2 . 5 , which gives non - response rate of 91 % for fs and 39 % for ps2 . for pse method , 30 % testers were distributed ps2 , 35 % testers were distributed the partial survey with 3 questions ( ps3 ) and 35 % testers were distributed the partial survey with 5 questions ( ps5 ). table 3 provides the simulation results for estimation of mean response for scenarios 3 and 4 . the fs method has the largest bias and smallest standard deviation , and pse method has the smallest bias but largest standard deviation . the bias based on ps2 method is slightly larger than pse but is much smaller than fs , and the standard deviation from ps2 method is slightly larger than fs , but much smaller than pse . as a result , ps2 method has the smallest mse while fs method has the largest mse . table 4 provides the simulation results for estimation of regression coefficients for scenarios 3 and 4 . fs method has the largest bias in both scenarios for all coefficients . the biases for ps2 and psee methods are similar and smaller than fs . however , fs method has the smallest standard deviation and mse . the standard deviation for pse is much larger than ps2 . since the bias does not change , but the standard deviation decreases when the total of number testers ( n ) increase . we expect the mse of ps2 will be smaller than fs when n is large enough . for example , the standard deviation for n = 1 , 000 , 000 would be 100 − 1 / 2 = 10 − 1 of the standard deviation for n = 10 , 000 . the mse for ps2 estimator of □ 1 would be approximately 0 . 00104 , which would be smaller than 0 . 00274 , the mse of fs estimator . in summary , based on the simulation results from tables 1 - 4 , it is clear that for mean response and coefficient estimation , ps2 and pse have the smaller bias and larger standard deviation than fs . the mse for the mean response estimation based on ps2 and pse methods is much smaller than fs . for regression coefficients , the mse based on ps2 and pse was larger than fs , based on the simulations . however , we expect the mse for ps2 would be smaller than fs when the survey sample is large enough . archer , t . m . ( 2008 ). response rates to expect from web - based surveys and what to do about it . journal of extension [ online ], 46 ( 3 ) article 3rib3 . available at : http :// www . joe . org / joe / 2008june / rb3 . php cook j . r . and stefanski l . a . ( 1994 ). simulation - extrapolation estimation in parametric measurement error models . journal of the american statistical association 89 : 1314 - 1328 . monroe , m . c . and adams , d . c . ( 2012 ). increasing response rates to web - based surveys . journal of extension [ online ], 46 ( 3 ) article 6tot7 . available at http :// www . joe . org / joe / 2012december / tt7 . php yu - yi hsu ( 2013 ). reducing parameter estimation bias for data with missing values using simulation extrapolation . phd dissertation . http :// lib . dr . iastate . edu / cgi / viewcontent . cgi ? article = 4448 & amp ; context = etd

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