Protocol Calibration in the National Resources

Inventory

Cindy Yu 

Jason C. Legg 

August 23, 2007 

ABSTRACT 

The National Resources Inventory (NRI) is a large-scale longitudinal survey conducted by the National Resource Conservation Service in cooperation 

with the Center for Survey Statistics and Methodology (CSSM) at Iowa State 

University since 1982. A key NRI estimate is year-to-year change in acres 

of developed land, where developed land includes roads and urban areas. 

Since 2004, a fully digital data collection procedure has been used. The data

from the NRI calibration experiment are used to estimate the relationship 

between data collected under the old and new protocols. A measurement 

error model is postulated for the relationship, where the duplicate measurements are used to estimate the error variances. If any significant discrepancy 

is detected between new and old measures, some parameters that govern 

the algorithm under new protocol can be changed to alter the relationship. 

The data analyses suggest that the relationship is a line with an intercept 

of zero and a slope of one, therefore the parameters in current use are acceptable. The paper also provides a way to model the measurement error 

variances as functions of the proportion of developed land, which is essential 

for estimating the effect of measurement error for the whole NRI data. 

1 Introduction

During a long-term monitoring study, advances in theory and methodology 

for collecting data occur. Changing data collection procedures can reduce

1 
measurement error and other nonsampling errors. For example, the introduction of computer assisted self-administered interviewing (CASI) has been 

shown to increase reporting accuracy for studies involving sensitive subjects 

(Tourangeau and Smith, 1996). A survey begun in the 1980s likely did not 

use CASI, but rather relied on telephone or in-person interviewing. Converting the survey from in-person interviewing to CASI could reduce bias in 

estimators. However, since measuring change is one of the primary objectives 

of longitudinal surveys, the effect of a change in survey mode needs to be 

measured. The data or survey instrument can be calibrated so that comparisons between data collected under current and prior modes are possible. 

The National Resources Inventory (NRI) is a large-scale monitoring program designed to assess status, condition, and trends of soil, water, and 

related resources (Nusser and Goebel, 1997). Much of the NRI data are 

observed via photograph interpretation. Prior to 2000, photograph interpretation was performed on a transparent overlay on an aerial photograph. 

Now, the photographs are digitized and photograph interpretation is conducted on a computer. Along with the change to digital imagery, a new 

protocol has been created for determining area devoted to developed land in 

a land segment. A calibration study was conducted using 2003 data to assess 

the impact of the protocol change and whether adjustments are needed. 

The NRI survey has a stratified two-stage design. For central (excluding 

Texas) and western states, the strata are defined by the Public Land Survey (PLS) System. For states under the PLS, a stratum is defined to be a 

two mile by six mile block, which is one-third of a township. Typically, two 

half-mile by half-mile blocks, called segments, are selected within a stratum. 

Within each selected segment, three points are selected using a restricted 

randomization procedure to ensure geographic spread. Segment level observations are made on the areas devoted to built-up zones, roads, streams, and 

small water bodies. We refer to structures and the maintained area around 

structures as urban land, roads as roads or railroads, and developed land 

as the combination of urban land and roads. Land use, composition, and 

erosion data are observed at the point level. The NRI protocol change is 

related to segment level observations. 

The NRI longitudinal observation scheme was a pure panel from 1982 

through 1997. Observations were made every five years. In 2000, the NRI 

began using a supplemented panel design with yearly data collection. A supplemented panel design has a pure panel component, called the core, and a 

rotating panel component composed of supplemental panels or supplements. 

2 
Segments were selected into the core and supplements using sampling rates 

determined by 1997 segment compositions. Segments containing points associated with wetlands, high erosion, or adjacent to urban areas were more 

likely to be selected into the core than other segments. We only considered 

segments in the core for the calibration study. 

The change in the developed land observation protocol deals with assigning area to residences. The protocol for residential areas from 1982 through 

2003 involved the data gatherers delineating the area around residences considered as urban. The delineation process involves tracing the boundary of 

a polygon using a hand planimeter on a transparent overlay on top of an 

aerial photograph. Under the new protocol, data gatherers place a cross using a mouse click on the roof of all of the residences on a digital photograph 

displayed on a computer monitor. A computer program places a hexagon centered on each cross on the digitized photograph. Two hexagons are linked if 

the distance between their boundaries is below a specified threshold. If four 

or more polygons are linked, the area of the polygons are considered developed land. An area entirely closed in by linked hexagons or other delineated 

built-up areas is considered built-up if the enclosed area is below another 

specified threshold. Roads are delineated by choosing a line thickness and 

tracing the road or by delineating the area around the road boundary. Nonresidential urban areas are delineated using the vertex method. small water 

bodies are delineated like non-residential urban areas and small streams are 

delineated like roads. The protocols for collecting road, non-residential urban area, small water body, and small stream data are the same for previous 

and current data collection except that delineation is done on a computer 

rather than on a transparent overlay. In 2003, data were collected using both 

the new and old protocols, but only the new protocol has been used in data 

collection beginning in 2004. 

The intent of the protocol change is to reduce the measurement error in 

urban area determinations. Marking residences is a more repeatable process 

than delineation, because the boundary of a delineated area is subject to the 

data gatherer’s discretion. Roads and non-residential urban areas involve a 

decision on what portion of the land is maintained. Therefore, roads and 

non-residential determinations remain at the discretion of the data gatherer. 

Any change in the measurement error distribution for delineations of roads

and non-residential determinations is due to changes in the quality of data 

collection materials. 

The data from the NRI calibration experiment are used to estimate the 

3 
relationship between data collected under the old and new protocols. If the 

relationship is not a line with an intercept of zero and a slope of one, parameters in the program that translates crosses into areas will be modified. 

The size of the hexagons and linking rules can be changed to alter the relationship between observations made under the different protocols. A second 

objective of the experiment is to provide an estimate of the relative contribution of the measurement error variance to the total variance of the estimator. 

The calibration study involves repeated observations, which allows estimates 

of measurement error variances for developed land. We will model the measurement error variances as a function of developed land. 

2 Experiment Design

NRI observations are taken with measurement error. Therefore, the regression of measurements under the new protocol on measurements under the 

prior protocol is biased (Fuller 1987). In order to correct the bias, estimators 

of the measurement error variances under the two protocols are necessary. 

The calibration experiment was designed with replicates for measurement

error variance estimation. Individual segments were selected based on geography and 2003 measurements under the previous protocol. 

The NRI data gatherers have access to previously collected data. Therefore, the measurement error is assumed to be correlated over time. The data 

collection procedure was designed to reduce the correlation between two observations made on the same segment in 2003. Four people are involved in 

data collection under the new protocol. The first two people make 2001 observations using the available 1997 materials. The third and fourth person 

make 2003 observations, where the third person uses materials from the first 

2001 data gatherer and the fourth person uses materials from the second 2001 

data gatherer. A fifth person has made a determination for 2003 under the 

old protocol previously. Eight data collectors are grouped together. For each 

eight segments, four data collectors are randomly selected to work on the first 

four segments and the complement set of data collectors are assigned to work 

on the second set of four segments. A Latin square design assigns the four 

segments to the four data collectors such that each data collector performs 

each of the four observation types once. Some control is made across groups 

of eight segments to ensure mixing of data collectors into the groups of four. 

A working assumption under this design is that the two observations under 

4 
the new protocol made in 2003 are independent and are also independent of 

the observation made under the old protocol. The independence assumption 

is justified by the inclusion of the intermediate data collector between the 

2003 data collector and the original 1997 data collector. 

Photograph interpretation occurs at three Remote Sensing Laboratories 

(RSL). The RSLs are termed West, Central, and East. Each RSL collects 

data on states in the region of the RSL. Data gatherers at each RSL receive 

support first from leaders within the RSL. Training of data gatherers also 

occurs at each RSL. Differences between data collection techniques can arise 

due to differences in leadership at the RSLs. Therefore, the segment selection 

occurred within RSL regions. 

Segments were divided based on the area of developed land and area of 

small water and small streams. Segments completely covered with water, 

federal land, or developed land are not highly interesting for the experiment because the residence protocol will not need to be applied. Therefore, 

segments classified as 100% urban, 100% federal, or 100% water were not 

included in the study. Alaska and Hawaii were not included in the selection. 

Segments were selected from the remaining pool to have a spread of land 

features. Segments with a change in urban, water, or road determinations

from 2001 to 2003 under the old protocol were selected with certainty for the 

Central and West RSLs. A subset of segments in these categories were selected with certainty for the East RSL. The remaining segments were divided 

into strata defined by presence of developed land with no change, presence 

of water with no change and no developed land, and no water or developed 

land in segment with no change from 2003. Within each category, segments 

were sorted by a geographic code and a systematic sample was selected. A 

total of 2699 segments were selected into the study. The West RSL had 608 

segments, the Central RSL had 1055 segments, and the East RSL had 1036 

segments. 

The sample selection is biased, since segments were selected on the basis 

of 2003 observations. Segments without developed land in 2003 under the 

old protocol were not selected at a high rate and those selected contained 

water features. Therefore, the occurrence of a segment without developed 

land under the old protocol and developed land under the new protocol is 

much less in our sample than the occurrence of a segment with developed 

land under the old protocol and no developed land under the new protocol. 

This sampling bias affects the estimated relationship between old and new 

protocol observations near a true developed land value of zero acres. 

5 
The data used in this paper are observations made using 2003 photography from the West RSL. Developed land areas were converted into proportions by dividing built-up determinations by digitized segment size. Segments containing federal land were removed from the analysis dataset, because the boundary of federal land within a segment cannot be determined. 

Twenty-seven calibration segments contain some federal land. An additional 

seventy-seven segments where all three 2003 observations have no developed 

land and one segment with 100% developed land were removed from the 

analysis dataset. Including segments with no or all developed land would 

increase the evidence that observations under the new and old protocol estimate the same quantity, possibly masking some match departures away from 

the extremes. Some data were modified after initial data collection through 

a review process. The review process ensures that the new protocol is calibrated to a properly applied old protocol. The dataset used for this analysis 

contains 503 segments. 

3 Estimation of the mean function

We define developed land to be the sum of segment areas reported for large 

urban, small urban, and public roads. The variable of interest is the proportion of developed land in a segment in 2003. Let Bj,i,03 be the developed 

land area observation for segment i made by observer j using 2003 materials, 

where j can be 1 or 2 for observations made under the new protocol or o for 

the observation made under the old protocol. Let Si be the digitized size of 

segment i. The three estimated proportions of developed land in segment i 

are 

Y1i = B1,i,03/Si

, (1) 

Y2i = B2,i,03/Si

, (2) 

and 

Xi = Bo,i,03/Si

. (3) 

The mean of Xiis 0.231 (0.011) and the mean of 2−1(Y1i + Y2i) is 0.232 

(0.011). The correlation between X1i and 2−1

(Y1i + Y2i) is 0.954. 

A proposed model for the 2003 data is 

Xi = xi + ui

, (4) 

Yji = β0 + β1xi + eji, (5) 

6 
xi  ̆ (μx, σx

2

), (6) 

and 

2 3 2 2 33 ui σ

2 0 0 ui

4 e1i

5|xi  ̆ 40,4 0 σ

2 0 55, (7) ei

e2i 0 0 σ

2

ei

for all i and j = 1, 2, where xiis the true proportion for segment i, and 

ui and eji are measurement errors on segment i under the old and the new 

protocols, respectively. The errors are assumed independent across different 

segments. By the experiment design, we assume e1i

, e2i and ui are conditionally independent from each other for each segment i. 

To estimate the parameters, we define the observation vector 

p

Z = (Z1, Z2, Z3) = (Xi

, 2

−1

[Y1i + Y2i], 2 

−1

[Y1i − Y2i]). (8) 

Let the sample covariance matrix of Z be 

X m = (n − 1)−1

(Zi − Z)

0

(Zi − Z). (9) 

i2A

Under the model, the sample covariance matrix has expectation 

2 3 σ

2 + σ2 β1σ2

x a,u x

0 

E(m) = 4 β1σ

2 β1

2σ

2 + 0.5σ2 0 5, (10) x x a,e

0 0 σ

2

a,e

where σ

2 and σ2 denote the averages of σ2 and σ2

, respectively, and σ

2

a,u a,e ui ei x

is the variance of the targets of the old procedure for the calibration sample 

and does not have an interpretation for the whole NRI. Method of moments 

estimators are 

βb0 = Z2 − βb1Z1, (11) 

b −1 β1 = m12 (m22 − 0.5m33), (12) 

σ

2 2 bx = (m22 − 0.5m33)−1m12, (13) 

σ

2 b = (14) a,e m33, 

σb

2 = σ2

, (15) a,u m11 − bx

and 

σb

2

b a,u θ = . (16) 

σb

2

a,e

7 
Table 1: Parameter Estimates from the Original Model for West RSL Data 

σb

2 σb2 σb2 βb0 βb1 θ x a,e a,u

b

0.0011 0.998 0.0556 0.00086 0.00498 5.8 

(0.0032) (0.015) (0.0044) (0.00022) (0.00058) (2.0) 

The function θb estimates the ratio of the average error variance under 

the old and new protocols. Parameter estimates using estimators (11)-(16) 

are in Table 1 for data from the West RSL. Standard errors are estimated 

using a delete-1 jackknife, where the jackknife weights are those of simple 

random sampling. From the global parameter estimates, the intercept is not 

statistically significant from 0 and the slope is not statistically significantly 

different from 1. The error variance under the old protocol is estimated to 

be 5.8 times the error variance under the new protocol. 

Nonlinearity of the fixed relationship between Y and x needs to be investigated in case the actual relationship is not a straight line. Therefore, we 

propose a second model that allows for a shift in the slope. Our new model 

specifies a slope change at x = 0.5. 1

Because xiis not directly observed, we construct an estimator for xi

using an estimated generalized least squares (EGLS) procedure. That is, 

(Z1i

, Z2i − βb0) is regressed on (1, βb1)

0 using relative weights equal to the 

inverses of the estimated average variances, where (βb0, βb1) are the estimated 

coefficients in Table 1. That is 

xbi = w1Z1i + w2(Z2i − βb0)/βb1 (17) 

where w1 = σb

−2/(σb−2 + 2βb2σb−2

), and w2 = 2βb2σb

−2/(σb−2 + 2βb2σb−2

). a,u a,u 1 a,e 1 a,e a,u 1 a,e

To ease the computation of parameter estimates, we write the inverted 

model as a regression of Xi on yi

, the true measurement on segment i under 

the new protocol. The model with a slope change at x = 0.5 is 

Yji = yi + eji, (18) 

Xi = δ0 + δ1yivi + 0.5δ1(1 − vi) + δ2(yi − 0.5)(1 − vi) + ui

, (19) 

1We could choose other points to split the line. However, plots of the average of new 

measurements versus the old suggest x = 0.5 gives a good chance to detect a trajectory 

difference. 

8 
ˆ 1 xi < 0.5 

vi = (20) 0 xi  0.5, 

and the moments of the errors, eji and ui

, are defined as before. The indicator 

variable, vi

, is estimated by vbi by replacing xi with xbiin (20). To estimate δ0, 

δ1 and δ2, we regress Z1i (i.e. Xi) on Gi = (1, Z2ivbi+0.5(1−vbi),(Z2i−0.5)(1− 

vbi)). Since are taken with measurement error, the above regression is Z2i

biased (Fuller 1987). We adjust the regression estimators to account for the 

effect of measurement error. By writing the model as regression of Xi on 

yi

, the complexity in computing the correction matrices is reduced. Let A1

denote the part of the sample where vbi = 1 and A2 denote the part of the 

sample where vbi = 0. The bias corrected regression estimator is 

2 3 δb0

6 7 4 δb1 5 = (G

0G − C)−1

(G

0Z1), (21) 

δb2

where 

2 3 0 0 0 

C = 4 5 0 C1 0 , (22) 

0 0 C2

X C1 = 0.25(Y1i − Y2i)

2

, (23) 

i2A1

and 

X C2 = 0.25(Y1i − Y2i)

2

. (24) 

i2A2

Under the model described in (18) and (19), the sample covariance of Z

can be used to estimate the average error variances once the regression coefficients are obtained. The estimator for σ

2

is m33 from (14). An estimator a,e

for σ

2

is obtained by combining estimators from the two parts of the data a,u

set. Let mv be the sample covariance matrix of (Z1, Z2) for data with vbi = 1 

and m1−v be the covariance sample covariance matrix of (Z1, Z2) for data 

with vbi = 0. The expectation of the mean squares is 

2 P δ

2σ2 + n −1 σ2

δ1σ

2

1 vy 1 ui vy

E{m 4 5 v} = 

i2A1 P (25) 

σ

2

δ1σvy

2

vy + (2n1)

−1σ2

ei

i2A1

9 

3 
Table 2: Parameter Estimates from the Split Line Model for West RSL Data 

σb

2 σb2

δb0 δb1 δb2 θ a,e a,u

b

0.0012 0.985 1.045 0.00086 0.00454 5.3 

(0.0033) (0.027) (0.064) (0.00022) (0.00077) (1.7) 

and 

2 −1 P 3 δ

2σ2 + n σ2

δ2σ

2

2 1−v,y 2 ui 1−v,y

E{m 4 P 5 1−v} = 

i2A2

, (26) δ2σ

2 σ2 + (2n2)−1σ2

1−v,y 1−v,y ei

i2A2

where n1 is the size of A1, n2 is the size of A2, σ

2

is the variance of yiin A1, vy

and σ

2

is the variance of yiin A2. Method-of-moments estimators for σ

2

1−v,y vy

and σ

2 are 1−v,y

σ

2

δb−1 b = (27) vy 1 mv,12

and 

σ

2

δb−1 b1−v,y = 2 m1−v,12. (28) 

An estimator for σ

2

is a,u

σ

2

δ

2σ2

δ

2σ2 b = (n1 + n2)−1

(n1{mv,11 − b b } + n2{m1−v,11 − b b }). (29) a,u 1 vy 2 1−v,y

Estimators for the parameters of model (18)-(19) are in Table 2. Standard 

errors were computed using a delete-1 jackknife. 

The intercept is not statistically significantly different from zero and both 

the slopes before x = 0.5 and after x = 0.5 are not statistically significantly 

different from 1 (Figure 1). The values of estimated σb

2 and θb are smaller a,u

than the corresponding estimates from model (4)-(5), but the difference in 

estimates is not large. 

We compute an approximate F-test of 

Ho : (δ0, δ1, δ2) = (0, 1, 1) (30) 

versus 

Ha : (δ0, δ1, δ2) =6 (0, 1, 1). (31) 

The F statistic is 0.52, which when compared to F distribution with 3 and 

497 degrees of freedom results in a p-value of 0.67. Therefore, we accept the 

reduced model of 

Yji = yi + eji, (32) 

10 
Figure 1: Fitted split line model with binned Z1 and Z2 means 

11 
Table 3: Parameter Estimates from the Reduced Model for West RSL Data 

σb

2 σb2 σb2

θb

x a,e a,u

0.0555 0.00086 0.00498 5.8 

(0.0042) (0.00022) (0.00058) (2.0) 

Xi = yi

, (33) 

where the moment and independence assumptions of eji and ui are the same 

as (7). 

Figure 1 contains both the fitted split line (dashed) and the (0, 0) to (1, 1) 

reference line (solid). In order to asses the model fit, we divide the data set 

into 10 bins with equal number of observations up to rounding from data 

sorted by xbi values. Figure 1 shows the mean of Z2 versus Z1 in each bin. 

The binned means lie closely around the lines, indicating the reduced model 

((δ0, δ1, δ2) = (0, 1, 1)) suffices for describing the data. 

Collectively, the result of the F-test and the evidence in Figure 1 suggest 

that the relationship between data collected under the old and new protocols 

is a line with an intercept of zero and a slope of one. Therefore, no additional 

modifications to the program need to be made for the West. Under the 

reduced model, the estimated average error variances can be obtained using 

Equations (13) to (15), where the coefficients β0 and β1 in (10) are replaced 

by 0 and 1, respectively (Table 3). 

In addition, we test whether the mean of Z1 is statistically significantly 

different from the mean of Z2 within each bin using approximate t-test (Table 

4). The t-statistics are constructed as bias adjusted Beale ratios to account 

for skewness (Tin 1965). The t-tests provide evidence that the observations 

under the new and old protocol do not correspond well for segments with 

little developed land. However, misfitting the actual trend near xi = 0 will 

not result in large changes in total estimates. Part of the difference between 

new and old protocols is attributable to the bias in the calibration sample 

selection. Further, several of the differences in the small bins are due to 

differences in delineation of small road pieces. The new protocol is the same

as the old protocol for road measurements. Changing the protocol will not 

affect the differences due to roads. Therefore, parameters in the program 

that translates crosses into areas under new protocol are accepted for the 

West RSL. 

12 
Table 4: Approximate t-test for the Differences Between Z1 and Z2 Over Ten Bins 

Bins 1 2 3 4 5 

mean of Z1 0.008 0.027 0.045 0.083 0.109 

mean of Z2 0.004 0.020 0.037 0.076 0.127 

t-value 4.31 2.70 1.82 1.17 -2.21 

Bins 6 7 8 9 10 

mean of Z1 0.172 0.249 0.350 0.487 0.775 

mean of Z2 0.187 0.255 0.355 0.495 0.756 

t-value -1.49 -0.32 -0.25 -0.55 1.45 

4 Estimation of the variance function

The calibration experiment provides the opportunity to estimate the effect 

of measurement error on NRI estimators. In order to extend the variance results to a larger set of data than the calibration data set, we need a functional 

form for the measurement error variance. The reason for this requirement 

is that the calibration experiment is a biased sample of the NRI segments. 

If the measurement error variance is a function of xi

, then the estimates of 

the average variances depends on the set of xi chosen for the calibration experiment. Modeling the variance functions is difficult due to a few extreme

differences between observations made on the same segment. Model assumptions presented below are made to construct estimators of the measurement 

error variance functions. However, the assumptions are not believed to be 

true for all of the data, nor would many standard diagnostic procedures be 

possible to check the validity of assumptions. 

The expectation of the squared deviations Z3

2

i

and (Z1i − Z2i)

2 are estimated as a function of the true proportion of developed land. As with 

the estimation of the mean function before, the estimated true proportion 

is used as a proxy for the truth. Two constraints are put on the functional 

form of variances. One constraint is that the functions be symmetric around 

0.5. The underlying assumption is that delineations of developed lands when 

the true proportion are 40% and 60% are associated with the same level of 

difficulty. In other words, the delineation of an area in a particular segment

has same effort as the delineation of the complement of the area. The second constraint is that the variance of Z3i

is proportional to the variance of 

(Z1i − Z2i). A plot of Z3

−

i

2

(Z1i − Z2i)

2 versus xi

is flat except near zero and 

13 
one, providing evidence for the second modeling constraint. 

An examination of the data shows that the variance of the squared deviations increases as the squared deviations increase. A working assumption 

for modeling is that the variances of centered Z3

2

i

and (Z1i−Z2i)

2 are proportional to the square of the expectations. This working assumption is that of 

a constant coefficient of variation model, which is commonly used to model 

data with increasing variances. One argument for the constant coefficient of 

variation is that the deviations possess binomial distribution properties as 

they are constructed from proportions. The second moment of a binomial 

distribution is a quadratic function of the true proportion. 

Initial models were fit to the squared deviations. However, due to the 

skewness of the data, the fitted functions were poorly estimated. The skewness of the data on the squared scale caused estimators to be determined by 

only a few segments. Transformations of the data were explored to find a 

transformation suitable for using a least squares estimator. The square root 

transformation decreased the effect of skewness in the data enough to make 

the least squares solution reasonable. The working model on the transformed 

scale is 

E|Z3i

| = γ0 + γ1(0.5

2.5 − |xi − 0.5|2.5

) := gi, (34) 

E|Z1i − Z2i

| = κ(γ0 + γ1(0.5

2.5 − |xi − 0.5|2.5

)) = κgi. (35) 

Since the model is not linear in coefficients, the Gauss-Newton algorithm is 

used to obtain the non-linear generalized least squares fit. The estimating 

equations were weighted by an initial estimate of gi and κ. The 2.5 power 

was determined by comparing the fit and the mean squared errors across

several different powers. The distributions of the absolute deviations are well 

approximated by the distributions of a multiples of a χ

2

1

random variables. 

Therefore, we compared the mean squared errors to 2, the variance of a χ

2

1

random variable. The mean squared error from the 2.5 power model fit is 

2.06. Table 5 gives the estimated coefficients of the variance function and 

their delete-1 jackknife standard errors. 

In order to estimate θb, the ratio of the variance of error in the previous 

protocol to the one in the current protocol, the variance functions in (34) 

and (35) need to be converted back to square scale. We ratio adjust the 

squared fitted functions so that the average of the squared fitted functions is 

14 
 

 

Table 5: Parameter Estimates from the Final Model with 2.5 Power for West RSL 

Data 

κb γb b 0 γb1 θ 

3.55 0.00212 0.129 5.8 

(0.31) (0.00030) (0.012) (2.7) 

the same as the average of Z3

2

i

and (Z1i − Z2i)

2

. Let 

n

!−1

X X n

2 Z2 R1 = gbi 3i

(36) 

i=1 i=1

and 

n

!−1

X X n

R2 = κb

2

gbi

2

(Z1i − Z2i)

2

. (37) 

i=1 i=1

Estimators for the mean of the squared deviations are 

Eb(Z3

2

i

) = R1gbi

2

(38) 

and 

b κ

2b2 E(Z1i − Z2i)2 = R2b gi

. (39) 

An estimator of the ratio of measurement error variances is 

b R

−1

θ = 1 R2κb

2 − 0.5. (40) 

Standard errors are computed using a delete-1 jackknife variance estimator 

(Table 5). The estimated θ of 5.8 is near the estimate using the average 

variances when fitting the mean function earlier (Table 3). The estimated 

variance of the ratio of variances is not well estimated in any of our results due 

to the skewness in the distribution, which explains the discrepancy between 

standard errors of θ estimators. In order to see the two fits of (38) and (39) 

on the square scale, we plot the fitted functions of squares and standardize 

them to the same scale (Figure 2). The model fits the data well on the square

scale, indicating that the model furnishes adequate results. 

15 
Figure 2: Fitted Variance Function of 2.5 power with binned Z

2 and (Z1 − Z2)2

3

means 

16 
5 Discussion

The parameters used to translate marked residences into developed area have 

been adjusted during data collection and analysis. The parameters used in 

this article for the West RSL provide encouraging results that estimators 

under the new and old protocol coincide within an acceptable tolerance. 

Some adjustments to the protocol have been attempted for the discrepancy 

between measurements when the proportion of developed land is very low. 

However, the adjustments did not solve the lack of fit problem. Further

analysis related to the effect of segment size and regional differences has 

been considered. Similar procedures are used to examine the relationship 

between observations under the new and old protocols for the Central and 

East RSLs. 

6 References

Fuller, W. A. (1987). Measurement Error Models. New York: John Wiley 

and Sons Inc. 

Nusser, S. M. and J. J. Goebel (1997). The National Resources Inventory: 

a long-term multi-resource monitoring programme. Environmental and

Ecological Statistics 4, 181-204. 

Tin, M. (1965). Comparison of some ratio estimators. Journal of the American Statistical Association 60, 294-307. 

Tourangeau, R. and T. W. Smith (1996). Asking sensitive questions: the 

impact of data collection mode, question format, and question content. 

Public Opinion Quarterly 60, 275-304. 

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