A method to correct for frame membership error in dual frame estimators
Section 6. Example: Angler survey

We illustrate the use of the proposed bias correction procedure with its application to a dual frame mail survey of anglers in North Carolina (NC) in 2009. It was a pilot survey testing several changes to an ongoing program collecting recreational marine angler effort by NOAA, where effort is defined as the number of fishing trips during a specified time period. The two frames were a NC address frame and a license frame, which included the names and addresses of anglers who had any of several types of licenses. The target population of the pilot survey was recreational anglers who fished in NC saltwater, regardless of where they lived. The target time period of fishing was Wave 6 of 2009 (November - December). These two frames together had some undercoverage because unlicensed anglers whose home address was outside NC were not included in the union of the two frames. 

6.1  Sample design

The address frame was obtained from the US Postal Service and covered all households in NC. The license frame included all persons listed on the NC database of licensed anglers as of the date of the license pull, which was several days before the mailing of the surveys. Independent samples of addresses were drawn from the two frames. Estimates were made of the fishing effort in NC during Wave 6, 2009 using the Hartley estimator with θ = 1 / 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0ZaaSGbaeaacaaIXaaabaGaaGOmaaaacaGGUaaaaa@3AF2@ The sample from the address frame was a complex sample, and itself involved two phases. The sample from the license frame required only one phase. In this application, units were at risk of misclassification only if they were chosen from the larger of the two frames, the address frame, since it was not known whether the persons in those households owned fishing licenses. The analysts did know that all persons selected from the license frame had a valid license during the wave and also knew whether or not they had an address in North Carolina, since the their address was available from the frame.

The sample design for the address frame was conducted in two phases. A random sample of 1,800 addresses was selected first, stratified by geography. The strata were defined as addresses in coastal and non-coastal counties of NC, with samples of 900 each. A screening questionnaire asked whether any household member fished in saltwater in the last 12 months. The second phase sample consisted of one randomly chosen angler from every household that reported fishing by any household member in the first phase. One additional angler was selected from households reporting more than one active adult angler. The reason for this two-phase construction was to avoid sending a lenthy questionnaire to non-angling households, in order to decrease cost and increase response rate.

The license frame was obtained from the NC license database. All individuals who were listed on the database on the day the frame was pulled and were licensed to fish during the target period (Wave 6, 2009) were included. The license frame was preprocessed to make it suitable for sampling. Multiple records with the same core data (name, date of birth and address) were deleted, as were anglers identified as being under 18. The license frame was divided into three strata: coastal, non-coastal, and out-of-state. The file was sorted by address, and a systematic sample of 450 anglers was selected from each stratum. Sampling in the license frame was conducted in a single phase, and used a questionnaire identical to the second phase questionnaire for the address frame sample. As in the address frame, a supplemental sample was selected from addresses with more than one licensed angler present on the frame.

The common questionnaire used for both frames included an item that asked whether the respondent had a NC marine recreational fishing license. This question was included to determine domain membership for those chosen from the address frame. However, analysts observed that some respondents from the license frame reported they did not have a license, which alerted them to the possible presence of domain misclassification error. As a result, an operation was undertaken to determine true domain membership for respondents from the address frame. We attempted to match 100% of the sampled addresses to the license frame. The last part of this process involved a human matcher trying to identify if a particular angler within a matched address appeared to be the licensed angler, based on available data from the license frame and survey responses. This was a time-consuming operation, which motivated the search for alternatives. The goal was to develop methods for the operational survey that allowed for determinination of true domain status for only a subset of the sample. However, since we did have access to the true domain status for the entire sample, we were able to examine misclassification probabilities and subdomain means, as well as to compare Y ^ ^ BC s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaajy aajaWaaSbaaSqaaiaabkeacaqGdbaabeaaieaakiaa=LbicaqGZbaa aa@3A6E@ results with an estimate made from “true” data.

Even though we observed that some on the license frame made errors concerning their license status, this did not cause a domain misclassification error for the license frame because the true license status was known. For the license frame, domain misclassfication could occur only if the in or out-of-state status of the household could not be determined accurately. It is possible such errors could occur. For example, if a household with an out-of-state address on the license frame were sampled, but it had a second in-state address that appeared on the address frame, then the domain assignment would be incorrect. However, the incidence of such cases was believed to be small enough that it could be ignored, so we treated the misclassification probabilities as if they were known to be 0 for the units on the license frame in our analysis. 

6.2  Sample analysis

The domain misclassification rates for the sample from the address frame are shown separately by stratum in Table 6.1. In this case domain a b * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gadaahaaWcbeqaaiaacQcaaaaaaa@389F@ contains those respondents from the address frame who report that they are licensed, while domain a * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaiOkaaaaaaa@37B8@ contains those reporting they are not licensed. Anglers who reported that they are unlicensed have about a 5% error rate in both strata. Those who reported they are licensed have extremely high error rates, with those in non-coastal counties more likely to be wrong than right! We point out that the address frame respondents from which these estimates were reported are those who were in the second phase of the address frame sample. This means that they had screened in because their household had at least one person who had fished in the last 12 months. As a result, a very high fraction of these respondents were anglers compared to the general population.


Table 6.1
Misclassification rates calculated from full sample (Address frame, Wave 6, 2009)
Table summary
This table displays the results of Misclassification rates calculated from full sample (Address frame. The information is grouped by (appearing as row headers), Proportion of those who report not being licensed who are ( p ^ ab| a * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaace WGWbGbaKaadaWgaaWcbaWaaqGaaeaacaWGHbGaamOyaiaaykW7aiaa wIa7aiaaykW7caWGHbWaaWbaaWqabeaacaGGQaaaaaWcbeaaaOGaay jkaiaawMcaaaaa@432E@ and Proportion of those who report being licensed who are not ( p ^ a|a b * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaace WGWbGbaKaadaWgaaWcbaWaaqGaaeaacaWGHbGaaGPaVdGaayjcSdGa aGPaVlaadggacaWGIbWaaWbaaWqabeaacaGGQaaaaaWcbeaaaOGaay jkaiaawMcaaaaa@432E@ (appearing as column headers).
Proportion of those who report not being
licensed who are  ( p ^ ab| a * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaace WGWbGbaKaadaWgaaWcbaWaaqGaaeaacaWGHbGaamOyaiaaykW7aiaa wIa7aiaaykW7caWGHbWaaWbaaWqabeaacaGGQaaaaaWcbeaaaOGaay jkaiaawMcaaaaa@432E@
Proportion of those who report being licensed
who are not  ( p ^ a|a b * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaace WGWbGbaKaadaWgaaWcbaWaaqGaaeaacaWGHbGaaGPaVdGaayjcSdGa aGPaVlaadggacaWGIbWaaWbaaWqabeaacaGGQaaaaaWcbeaaaOGaay jkaiaawMcaaaaa@432E@
Coastal Stratum 0.04 0.46
Non-coastal Stratum 0.06 0.63

We also examined the equal means assumptions using data from the address frame sample. The estimated mean effort in each of the four categories of domain and perceived domain membership are shown in Table 6.2. The columns classify respondents into perceived domains, while the rows classify according to their true domain. The table shows that respondents’ fishing behavior is consistent with what they report their license status to be rather than what their true status is. Thus, we believe that the equal means assumption of our proposed method is more reasonable for the angler survey data than Lohr’s equal mean assumption.


Table 6.2
Estimated mean #of fishing trips (SE) by subdomain for Wave 6 2009 NC Address frame
Table summary
This table displays the results of Estimated mean #of fishing trips (SE) by subdomain for Wave 6 2009 NC Address frame. The information is grouped by y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaGabmyEayaara aaaa@393A@ for subdomains (appearing as row headers), reported no license ( a * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaGGQaaaaaGccaGLOaGaayzkaaaaaa@3B78@ and reported license ( a b * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaaca WGHbGaamOyamaaCaaaleqabaGaaiOkaaaaaOGaayjkaiaawMcaaaaa @3C5F@ (appearing as column headers).
y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaGabmyEayaara aaaa@393A@ for subdomains reported no license ( a * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaGGQaaaaaGccaGLOaGaayzkaaaaaa@3B78@ reported license ( a b * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeWaaeaaca WGHbGaamOyamaaCaaaleqabaGaaiOkaaaaaOGaayjkaiaawMcaaaaa @3C5F@
true no license ( a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbaacaGLOaGaayzkaaaaaa@3A89@ 0.34 (0.14) 0.88 (0.41)
true license ( ab ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaamOyaaGaayjkaiaawMcaaaaa@3B70@ 0.35 (0.46) 0.98 (0.24)

The sample data contained weights provided by the survey designers that accounted for the complex design and nonresponse adjustment. Because the domain misclassification probabilities differed by stratum, we adjusted the weights as described in Section 5 separately by stratum, using individual estimates of misclassification for each address frame domain. We assumed no domain misclassification for the license frame. Six estimates of effort were computed and are shown in Table 6.3:

  1. Uncorrected Hartley estimator (labeled Unadj. in table): The perceived domain membership was used to estimate the total, using the Hartley estimator as in (3.3);
  2. 20%, 40%, 100%-subsampled estimator: Units from each stratum of the phase 1 address frame sample were subsampled, and their true domains were used to estimate the misclassification probabilities. The weight adjustments were calculated based on the estimated misclassification probabilities;
  3. Corrected Hartley estimator (True): The true domain membership ascertained from the matching operation was used to estimate the total number using the Hartley estimator with the original weights, as in (3.1). This is considered the best available estimate since it requires no assumptions for unbiasedness.

The first row contains the five estimates, the second row contains an estimate of bias for each, and the third row shows the square root of the sum of estimated variance and squared bias. The bias displayed in row 2 is the difference between each estimate and the corrected Hartley estimate (True column). We acknowledge that the address matching algorithm is undoubtedly not perfect, which means that the “True” estimator may still contain bias in addition to its sampling variability. Still, taking this as our best assessment of bias, we see that after applying the bias correction method, the estimated bias is reduced by using the bias-corrected estimator from 211 K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaig dacaaIXaGaam4saaaa@38F9@ to 40 K 80 K . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaaic dacaWGlbGaaGjbVlabgkHiTiaaysW7caaI4aGaaGimaiaadUeacaGG Uaaaaa@3F44@ The difference between Y ^ H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadIeaaeqaaaaa@37DE@ and Y ^ ^ BC MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaajy aajaWaaSbaaSqaaiaabkeacaqGdbaabeaaaaa@38AB@ with 100% subsampling may reflect failure of the required equal mean assumptions. The estimated RMSE is reduced by using a bias-adjusted method instead of the unadjusted Hartley estimator by about 70 K . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaic dacaWGlbGaaiOlaaaa@38F4@


Table 6.3
Estimated total fishing trips (Address frame, Wave 6, 2009)
Table summary
This table displays the results of Estimated total fishing trips (Address frame. The information is grouped by (appearing as row headers), Unadj., 20% sub. m A =36 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamyBamaaBa aaleaacaWGbbaabeaakiaaysW7caaI9aGaaGjbVlaaiodacaaI2aaa aa@3F70@ , 40% sub. m A =71 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamyBamaaBa aaleaacaWGbbaabeaakiaaysW7caaI9aGaaGjbVlaaiEdacaaIXaaa aa@3F6F@ , 100% sub. and True (appearing as column headers).
Unadj. 20% sub. m A =36 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamyBamaaBa aaleaacaWGbbaabeaakiaaysW7caaI9aGaaGjbVlaaiodacaaI2aaa aa@3F70@ 40% sub. m A =71 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamyBamaaBa aaleaacaWGbbaabeaakiaaysW7caaI9aGaaGjbVlaaiEdacaaIXaaa aa@3F6F@ 100% sub. True
Estimate 731,430 889,860 863,488 905,947 942,360
Bias 210,930 52,500 78,872 36,413 0
RMSE 244,531 181,809 180,311 176,954 213,966

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