Review of the literature

The agricultural surveys are very specialized with respect to the other surveys. The agricultural surveys are conducted in order to gather information on the crop area, crop yield, livestock, and other agricultural resources. Apart from the difficulties typical of business data, such as the quantitative nature of many variables and their high concentration, agricultural surveys are indeed characterized by some additional peculiarities. In the case of auxiliary information there are two specific issues that need to be discuses.

First, the definition of the statistical units is not unique. The list of possible statistical units is quite large, and its choice depends not only on the phenomenon for which we are interested in collecting the data, but also on the availability of a frame of units. Second, a rich set of auxiliary variables, other than dimensional variables, is available: consider, for example, the information provided by airplane or satellite remote sensing.

Concerning the first issue, agricultural surveys can be conducted using list frame or spatial reference frame. A list frame is generally based on the agricultural census, farm register or administrative data. A spatial reference frame is defined by a cartographic representation of the territory and by a rule that defines how it is divided into units. According to the available frame we can have different statistical units.

Agricultural holdings are the statistical units of a list frame. Surveys based on agricultural holdings are generally cheaper since it is possible to collect an important amount of information in a single interview. However, they presume that the list is recent and had a good quality, condition that it is not always satisfied.

Points are an example of statistical units of a spatial reference frame. Surveys based on point are often called a point frame survey. Points are in principle dimensionless, but they may be defined as having a certain size for coherence with the observation rules or the location accuracy that can be achieved. Segments are a second typology of statistical units of a spatial reference frame. The choice of the segment size depends on the landscape. Also, segments can be delimited by physical elements.

Two main differences among the statistical units need to be underlined. Information on the positioning (i.e. geo-referencing) of agricultural holdings it is not always available, where instead it is always obtainable for points and segments. Geo-reference is seen as important source for data to be complemented with spatial agricultural information like satellite images, land cover maps or other geo-referenced information layers. As usual in business surveys, the population of agricultural holdings is markedly asymmetric. Usually, asymmetry is positive, as small family-owned holdings coexist with large industrial companies.

With regard to the second issue, the rich set of auxiliary variables in agricultural surveys is mainly available from remote sensing data. Remote sensing can significantly contribute to provide a timely and accurate picture of the agricultural sector, as it is very suitable for gathering information over large areas with high revisit frequency. Indeed, a large range of satellite sensors provides us regularly with data covering a wide spectral range. For deriving the sought information, a large number of spectral analysis tools have been developed. For review of remote sensing applications devoted to the agricultural sector see Section 2 and the references cited therein.

A commonly used auxiliary variable for crop area estimates is the Land Use/Cover (LULC) data. LULC refers to data that is a result of raw satellite data classification into categories based on the return value of the satellite image. LULC data are most commonly in a raster or grid data structure, with each cell having a value that corresponds to a certain classification. LULC have been widely applied to estimate crop area. Hung and Fuller (1987) combine data collected by satellite with data collected with area survey to estimate crop areas. Basic survey regression estimation is compare with two methods of transforming the satellite information prior to regression estimation. González and Cuevas (1993) used thematic map to estimate crop areas. The estimates have been made using regression methods. Pradhan (2001) presents an approach to develop a Geographic Information System (GIS) for crop area estimation to support a crop forecasting systems at regional level. The overall system combines spatial reference frame sampling and remote sensing.

Remote sensing data provide also information on different factors that influence the crop yield. The most popular indicator for studying vegetation health and crop production is the NDVI, which is a normalized arithmetic combination of vegetation reflectance in the red and near infrared. Studies have shown that NDVI values are significantly correlated with crop yields (compare with Section 2). Doraiswamy et al (2005) evaluate the quality of the MODIS 250 m resolution data for retrieval of crop biophysical parameters that could be integrated in crop yield simulation models. For a comprehensive review of different ways to use remote sensing for agricultural statistics see also Gallego (2004) and some references cited in Section 2.

The availability of remote sensing data does not eliminate the need for ground data, since satellite data do not always have the accuracy required. However, this information can be used as auxiliary data to improve the precision of the direct estimates. In this framework, the calibration estimator can improve the efficiency of crop areas and yield estimates for a large geographical area when classified satellite images and NDVI can be used as auxiliary information respectively.

Section 4.2 is devoted to outline the calibration approach. Some possible extensions are presented in Section 4.3. In Section 4.4 we review the issue of model calibration. When complex auxiliary information is available, calibration methods assume special features that are presented in Section 4.5. Section 4.6 describes calibration approach for non-response adjustment. Finally, Section 4.7 contains some remarks about computational issues.

Consider a probability sample s selected from a finite population using a probability sampling p(.). The first and second order inclusion probabilities, and respectively, are assumed to be strictly positive. Let be the study variable. Suppose we are interested in estimating the population total . An HT estimator of t_y is where is the sampling weight for unit k. The HT estimator is guaranteed to be unbiased regardless of the sampling design p(.). Its variance under p(.) is given as .

Now let us assume that J auxiliary variables are available. Let , k=1,...,N be a J-dimensional vector of auxiliary variables associated with unit k. The totals of the J auxiliary variables are known.

The link between the variables of interest and the auxiliary information is very important for a successful use of the auxiliary information. In agricultural surveys, there are differences among the statistical units regarding the use of the auxiliary variables available.

When the statistical units are the agricultural holdings, the use of the auxiliary information depends on availability of the positioning of agricultural holdings. If the agricultural holdings are geo-referenced the vector of auxiliary information for crop area estimates related to the farm k is given by with containing the number of pixels classified in crop type j according to the satellite data in the farm k. When the statistical units are points, the vector of auxiliary information for crop area estimates related to the point k is given by k=1,...,N, where is an indicator variable with value if the point k is classified in crop type j and otherwise.

The location accuracy between the ground survey and satellite images and the difficulties in improving this accuracy through geometrical correction have been considered one of the main problems in relating remote sensing satellite data to crop areas or yields, mainly in point frame sample surveys where the sampled point represents a very small portion of the territory.

When the statistical units are regular or irregular polygons, similarly to agricultural holdings, the vector of auxiliary information for crop area related to the point k is given by k=1,...,N, with containing the number of pixels classified in crop type j according to the satellite data in the point k.

Ideally, we would like that , but often this is not true. Roughly speaking, the methodology proposed by Deville and Särndal (1992) finds weights by means of a distance measure and a system of calibration equations. The procedures can be summarized as follows:

1. 
   Compute the initial design weight , directly obtained from the sampling design.
   
2. 
   Compute the quantities to correct as little as possible the initial weights for consistency with the auxiliary variables.
   
3. 
   Compute the final weight as .
   

Formally, the class of calibration estimators, calibrated to , is the class of estimators of the form:

1. 
   Define a distance function .
   
2. 
   Given a sample s and the function F(.) chosen at the preceding step, solve with respect to λ the calibration equations (4.7) where the quantity on the right-hand side is known.
   
3. 
   Compute the calibration estimator of t_y, that is,
   

This estimator will give closer estimated of t_y as the relationship between x and y gets stronger. Examples of distances function G are presented in Deville and Särndal (1992):

1. 
   Chi-squared distance:
   
2. 
   Logarithm distance:
   
3. 
   Hellinger distance:
   
4. 
   Minimum entropy distance:
   
5. 
   Modified chi-squared distance:
   
6. 
   Truncated (L,U) logarithm distance or Logit:
   
7. 
   Truncated (L,U) chi-square distance:
   

where q_k is a tuning parameter that can be manipulated to achieve the optimal minimum, L and U are two constants such that L<1<U and A=(U-L)/((1-L)(U-1)). The choice of distance function depends on the statistician and the problem.

It is possible to show that most of the traditional estimators are a special case of the calibration estimator.

For example, the GREG estimator is a special case of the calibration estimator when the chosen distance function is the Chi-square distance.

Consider the chi-square distance function, , then leads to the calibration weight where the vector of Lagrange multipliers is determined from the (4.7) as and where assuming that the inverse exists and that is the Horvitz-Thompson estimator for x. The resulting calibration estimator is:
(4.9)
where . Written in this form, we see that the calibration estimator is the same as the GREG estimator.

If we take x_k=x_k and consider the Chi-square distance function, with q_k=1/x_k, then , . From (4.9) the calibration estimator is the ratio estimator.

Deville et al (1993), Zhang (2000), and Breidt and Opsomer (2008) explain the post-stratified estimator and the raking as a special case of calibration estimation, when the available information consists of known cell counts or known marginal counts in a contingency table, respectively. For simplicity let consider a two-way consistency table with R tows and C columns, and thus RxC=J cells. The cell (r,c), r=1,...,R; c=1,...,C contains N_rc elements. Then . In the case of complete post stratification the vector of auxiliary information is composed of J elements indicating the cell to which the unit k belongs, i.e. if the unit k belongs to the cell j and otherwise.

Then is the vector of known population cell counts. Regardless of the F function, from the calibration equations (4.7) where and s_rc denote the sample in the cell (r,c). The resulting calibration estimator is that is the calibration estimator is the same as the post-stratified estimator. The post-stratified estimator is the calibration estimator when the statistical units are points, and the vector of auxiliary information is given by crop type.

When the marginal cell count N_r. and N_.c , r=1,...,R; c=1,...,C are known, but the cell count N_rc are not, we denote the procedure to estimate as the cell count raking ratio procedure. Deville et al (1993) obtained the raking ratio weights by minimizing the distance function 2, the logarithm distance.

Among all these distance functions, Andersson and Thorburn (2005) consider the issue of the determination of optimal estimator, and they found that is based on the distance closely related to (but not identical to) the one generating the GREG estimator, i.e. the chi-square distance.

A limitation of the calibration estimator with the Chi-square distance function is that the weights can assume negative and/or extremely large values. Deville and Särndal (1992) recognized this issue and showed how to restrict the weights to fall within a certain range. The distance functions 2,3,4 and 5 guarantee positive weight. However, in each of the aforementioned cases the weights can be unacceptably large with respect to the initial weights. They therefore consider the two additional function 6 and 7 that have the attractive property of yielding weights restricted too an interval that a statistician can specify in advance.

It is important to note that depending on the chosen distance function, there may not exist a closed form solution to Equation (4.7). Indeed, when the model for the correcting factors is a linear function of x, it is possible to rewrite the equation (4.7) in the form where T_s is a symmetric positive definite (J x J) matrix. The solution is therefore given by . When the function is non-linear, the solution can be found using iterative techniques, usually based on the Newton–Raphson algorithm.

Deville and Särndal (1992) state that for any function F_k(u) satisfying certain conditions, the calibration estimator is asymptotically equivalent to the regression estimator given in (4.9). Then, the two estimators have the same asymptotic variance (AV), namely as:

where , with B solution of the equation .

The asymptotic variance of can be estimated as:

Estevao and Särndal (2004), for a fixed set of auxiliary variables and a sampling design find an asymptotically optimal z vector:

The idea of model calibration is proposed in Wu and Sitter (2001), Wu (2003), and Montanari and Ranalli (2005). The motivating factor is that, when the auxiliary information x_k is known for all the population units, this should be used in a more effective way than what it is possible in model free calibration, where a known total is sufficient. Wu and Sitter (2001) considered the following non-linear assisting model

Wu and Sitter (2001) show that the estimator, whatever the choice of the model, is nearly design unbiased under minor conditions on the assisting model and on the sampling design. They compare, also the linear calibration estimator with the model calibration estimator . The linear calibration estimator is less efficient than the model calibration, but has a lot of practical advantage over the model calibration estimator . In fact, in this case, the auxiliary information is not required for all the population units, but it is sufficient the population total . The same weights can be applied to all the y-variables because they don’t depend on y; the estimator is identical to the linear GREG estimator. Moreover, in an empirical study, Wu and Sitter (2001) compare with the non-linear GREG for the same non-linear assisting model. The study shows that the non-linear GREG is in general less efficient than the model calibration estimator. Demnati and Rao (2010) analyze the estimator of the total variance of non-linear population parameters when model calibration is used for estimation.

Montanari and Ranalli (2005) provide further evidence. In an empirical study they compare with the non-linear where the assisting model is fitted via non-parametric regression (local polynomial smoothing). The model calibration estimator achieves only marginal improvement over the non-linear GREG. In model calibration the auxiliary information is required for all the population units. When such information is not available Wu and Luan (2003) propose a two-phase sample, where a large first phase sample measure over the auxiliary variables.

In agricultural surveys, where complete auxiliary information is from satellite data (then available for the population), the non-linear assisting model may give a considerably reduced variance. The relationship between x and y can have many forms, producing a great variety of possible assisting models that generate a wide family model calibration estimator of the form (4.10). Cicchitelli and Montanari (2012) deal with the estimation of the mean of a spatial population, using a model assisted approach that considers semi-parametric methods. The idea is to assume a spline regression model that uses the spatial coordinates as auxiliary information. With a simulation study, they show a significant gain in efficiency with respect to the HT estimator, under a spatially stratified design. They also suggest, when available, the use of quantitative and qualitative covariate, other than the spatial coordinates, to increase the precision of the estimator and to capture the spatial dependence of the target variables.

Two-phase sampling is very frequent in the agricultural surveys. Generally, in the first phase a systematic sample is selected. Each point is then classified, using orthophotos or satellite images, in land use categories. In the second phase a subsample is selected for the ground survey. The auxiliary information has a composite structure that the calibration estimator has to take into account.

A two-phase sampling design, in its simplest form, is as follows. A sample s₁ is selected first, and then a sample s₂ is chosen among those members of the population selected in sample s₁. The design weights are , for the sample s₁ and , for the sample s₂. The basic unbiased estimator is given by with . The auxiliary information may be available for the entire population unit and the units belonging to the first phase sample. That is, two different kinds of auxiliary variables may be available:

• 
  Population level. The variables are known for , thus the total is known.
  
• 
  Sample level. The variables are known only for the units in the sample s₁. The total is estimated by .
  

Alternative formulations of the calibration problem are possible. Estevao and Särndal (2006, 2009) illustrate some possibilities of how to use the composite information: one-step or two-step calibration option. In the single step option we determine the calibration weights w_k that satisfy the following condition, where and . In the two-step option, first we find the weights w_1k such that , then we compute the final calibration weights w_k that satisfy , where . The efficiency of different options depends on the pattern of correlation among y_k, x_1k, and x_2k.

Like all the surveys also agricultural surveys deal with non- response, where the reasons are different according the statistical units. Indeed, if the statistical units are farms the total non-response is generally due to non-contact or refusal from agricultural holdings; If the statistical units are point or areas the total non-response are due to the inability to observe the selected point/area.

Consider a probability sample s selected from a finite population ; the known inclusion probability of the unit k is and the design weight is . If non-response occurs, the response set and the study variable y_k are observed only for . The classical procedures dealing with non-response consist in adjusting the design weight for non-response based on non-response modeling.

If we define the unknown response probability of element k as , the unbiased estimator . Standard statistical techniques such as logistic modelling or response homogeneous groups are often used to estimate response propensities on the basis of auxiliary covariates available both for respondents and non-respondent.

Calibration can also be used to construct adjusted weights for unit non-response (Särndal and Lundström 2005, 2008). The calibration approach for non-response consists of a reweighting scheme, which makes the distinction between two different kinds of auxiliary variables:

• 
  Sample level variables, which aim to remove non-response bias in survey estimates. The variables must be known only for the units in the sample s. Contrary to simple calibration, their control totals are not required. The total is estimate without bias by .
  
• 
  Population level variables, which aim to reduce sampling variance. Like any usual calibration variable, the benchmark totals must be known from other sources. The variables must be known for , thus the total is known.
  

The calibration can be done considering the combined auxiliary vectors and total information:

Here, estimate the inverse response probability .

In agricultural surveys there are many potential auxiliary variables. A decision then has to be made which of these variables should be selected for inclusion in the auxiliary vector to make it as effective as possible, for bias reduction in particular. Särndal and Lundström (2010) develop a bias indicator useful to select auxiliary variables effective to reduce the non-response bias. The main advantage of using calibration to deal with unit non-response is that auxiliary variables no longer need to be available for the population. In addition, as there is no need for explicit response modelling, the calibration approach is simple and flexible.


4.7 Computational issues
A great number of software packages are available for computing calibrated weights, such as the SAS macro CALMAR (Deville et al 1993), the SPSS program G-CALIB, and the functions offered by the R packages Survey. These packages, in different ways, try to resolve computational issues like: exclude negative weights satisfying the given calibration equations, keep the computed weights within desirable bounds, drop some x variables to remove near linear dependencies, down weight outlying values in the auxiliary variables that may be a cause of extreme weights. In particular, calibration in the presence of outliers is discussed in Beaumont and Alavi (2004). They present a practical ways of implementing M-estimators for multipurpose surveys where the weights of influential units are modified and a calibration approach is used to obtain a single set of robust estimation weights.

To conclude, the reader can find excellent reviews about the calibration estimator in Zhang (2000), Estevao and Särndal (2006), Särndal (2007), and Kim and Park (2010). Zhang (2000) presents a synthesis of the relations between post-stratification and calibration. Estevao and Särndal (2006) describe some recent progress, and offer new perspectives in several non-standard set-ups, including estimation for domains in one-phase sampling, and estimation for two-phase sampling. Särndal (2007) reviews the calibration approach, with an emphasis on progress achieved in the past decade. Kim and Park (2010) present a review of the class of calibration estimator considering the functional form of the calibration weight.

The main advantages and drawbacks of the methods described in this topic are summarized in the following Table 4.1.