Spatially balanced two-stage equal probability sampling
4.2 Comparative study of spatially balanced sampling methods under the spatial super-population model
4.2.2 Simulation study for comparison of spatially balanced sam- pling methods
4.2. COMPARATIVE STUDY OF SPATIALLY BALANCED SAMPLING METHODS
There were also cases where measure of spatial balance did not lead to the most efficient spatially balanced design. In order to investigate these results with respect to another aspect of sampling design called anticipated mean squared error (AMSE), a simulation study is conducted in the following section.
4.2.2 Simulation study for comparison of spatially balanced sam-
4.2. COMPARATIVE STUDY OF SPATIALLY BALANCED SAMPLING METHODS
of scale parameter σ = 2 is fixed for all scenarios. Three variables with no spatial trend, yij =ϵij,j = 1,2,3, and three variables with a linear spatial trendyij =β(cxi+cyi) +ϵij, j = 4,5,6, are obtained, where cx and cy denotes spatial x- and y-coordinates respec- tively, andβ = 6.5 is chosen which explain 75% to 85% variation in the response variables.
All the of variables are standardized with mean µ= 5 and standard deviation σ = 1, i.e.
zij = 5 + (yij −y¯j)/sj, j = 1, ...,6, where ¯yj and sj are mean and standard deviation of variables yj respectively. Three binary variables y(0.1), y(0.3) and y(0.5) are computed by assign value 1 to 10%, 30% and 50% largest values of responses variables z4, z6 and z7 respectively. For all the spatial variables, values of Moran’s index averaged of 5000 realisations are reported in Table 4.1.
Table 4.1: Values of Moran’s index I for nine populations and three spatial frames aver- aged over 5000 realizations of populations.
No spatial trend Linear spatial trend Binary responses
z1 z2 z3 z4 z5 z6 y(0.1) y(0.3) y(0.5)
Highly clustered 0.006 0.079 0.304 0.401 0.420 0.474 0.176 0.225 0.237 Clustered -0.001 0.037 0.210 0.397 0.406 0.445 0.112 0.152 0.161 Sparse -0.002 0.016 0.141 0.299 0.304 0.333 0.073 0.097 0.102 Equal probability samples of different sizes n = (4,8,20,32,44) are selected which cor- respond to sampling fractions f = (0.01,0.03,0.05,0.08,0.11). Random samples are se- lected using SRS as benchmark method and eight spatially balanced sampling methods:
GRTS, BSS, LPM1, LPM2, SCPS, DBSS and PWD. R-packages spsurvey (Dumelle et al., 2021), BalancedSampling (Grafstr¨om and Lisic, 2019), and Spbsampling (Pan- talone et al.,2019) are used for the selection of random samples.
For each of the six scenario of spatial super-population model (3 spatial autocorrela- tions × 2 spatial trends), nrp = 5000 finite spatial populations were generated. From each realization of the finite spatial population, nrs = 5000 samples are selected under equal probability; mean squared error (MSE) of HT-estimator for population total was calculated using 5000 samples, as given in Eq. (4.1). In order to obtain AMSE of the HT-estimator, MSEs are averaged over 5000 realizations of the spatial populations, as given in Eq. (4.2).
MSE( ˆYHT) = 1 nrs
∑nrs s=1
(YˆHT(s)−Y )2
(4.1)
4.2. COMPARATIVE STUDY OF SPATIALLY BALANCED SAMPLING METHODS
AMSE( ˆYHT) = 1 nrp
∑nrp p=1
MSE( ˆYHT)(p) (4.2)
where MSE( ˆYHT)(p) is MSE of HT-estimator for pth spatial population out of 5000. Let AMSEsrsand AMSEspdenote AMSE’s of HT-estimator under SRS and spatially balanced sampling designs respectively. Relative values of AMSE’s, given by
AM SEsp AM SEsrs
are reported in Tables 4.2, 4.3 and 4.4 for highly clustered, clustered and spars spatial frames respectively. Measure of spatial balance for spatially balanced sampling methods are not reported here, they are expected be same as in simulation study of Benedetti et al. (2017b), because it only depends on the spatial configuration of population units i.e. spatial frame.
From Tables 4.2, 4.3 and 4.4, results show that PWD design is the most efficient design in terms of AMSE of HT-estimator under all the scenarios of equal probability sampling considered in this simulation study. For comparison of other sampling methods, let us consider first the populations with linear spatial trend. GRTS is always least efficient among GRTS, LPMs, SCPS and DBSS. It gains efficiency with sample size and spatial correlation and it is more efficient than BSS for some cases with large values of sample size and spatial correlation in this study. For the highly clustered spatial frame with populations having medium and high spatial autocorrelation, DBSS is the second most efficient design (after PWD) followed by SCPS design, while BSS is the second most effi- cient for low spatial autocorrelation followed by DBSS. LPMs tend to be less efficient than SCPS, they gain efficiency relative to SCPS as sample size increases. Moving from highly clustered to clustered spatial frame, BSS takes over the place of DBSS as second most efficient design for populations with medium (in addition to low) spatial autocorrelation for smaller sample sizes, and for all the sample sizes when moving further from clustered to sparse spatial frame. Similarly, when spatial clustering of population units decreases, LPMs gain efficiency against SCPS more quickly; efficiency of LPM2 against LM1 in- creases, even LPM2 is more efficient than LPM1 for smaller sample sizes (n = 4,12,20) under sparse spatial frame. Whereas LPM1 and LPM2 tend to be equally efficient for larger sample size.
Now, let us consider populations with no spatial trend. For highly clustered spatial frame, LPM1 tend to be the second most efficient design (after PWD) for smaller sample sizes
4.2. COMPARATIVE STUDY OF SPATIALLY BALANCED SAMPLING METHODS
(n= 4,12), whereas DBSS takes over this place as sample size and spatial autocorrelation become larger. LPM1 tends to be better than SCPS for medium values of spatial auto- correlation and less efficient for high values. The three designs, LPMs, SCPS and DBSS tend to be equally efficient for low values of spatial autocorrelation, regardless of samples size. When moving from highly clustered to clustered and sparse spatial frames, GRTS, LPMs, SCPS and DBSS are approximately equally efficient for populations with low and medium spatial autocorrelation, whereas for populations with high spatial autocorrela- tion SCPS design tend to be second most efficient design (after PWD) and it becomes approximately equally efficient to DBSS as the sample size increases.
For binary responses, the three methods LPM1, SCPS and DBSS are more efficient than GRTS and BSS, these three methods are approximately equally efficient. However as sample size gets larger, SCPS and LPM1 tend to loose their efficiencies as compared to DBSS for highly clustered and clustered spatial frames and SCPS looses its efficiency more quickly as compared to LPM1. For sparse spatial frame, only LPM1 tends to loose its efficiency as compared to both SCPS and DBSS as sample size increases.
To summarise, spatially balanced sampling is better choice against SRS when there is knowledges about presence of positive spatial autocorrelation in study population. Fur- ther information about spatial structure of population units and study variable can be helpful in choosing an appropriate spatially balanced design for the given study popula- tion. In case of equal probability sampling, PWD is the most efficient design which is not applicable whenπps sampling is required. For very large populations, this design may lead to computational disadvantage over other designs, seeBenedetti and Piersimoni(2017) for details. Apart from PWD, other spatially balanced sampling methods considered here are able to select samples with unequal fixed inclusion probabilities. While comparing these methods under equal probability sampling, DBSS outperforms others when there exists spatial trend with strong spatial correlation in the population, whereas BSS is better than DBSS when spatial correlation is weak. For populations with no spatial trend, when there exist strong spatial correlation and sample size is large (e.g. such that f ≥ 0.05) DBSS is better than others; when sample size is not large LPM1 and SCPS are more efficient than DBSS for clustered and sparse popuations respectively.
4.2. COMPARATIVE STUDY OF SPATIALLY BALANCED SAMPLING METHODS