The Selection Process Between the Sampling Probabilities Plans

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POSDEM

Example for experts 1: different sample sizes and variances

Example for experts 2: graphic of a superpopulation model

Example for experts 3: superpopulation model: variance of the estimator

Example for experts 1: different sample sizes and variances

For balanced sampling, we want to calculate the variance of the estimator for different sample sizes, and their comparison with stratified sampling and random sampling without replacement.

Solution

In the menu option "Calculations", select "Correlogram" and then type the parameters: initial sample size, final sample size and the interval between them. For example: 2, 200 and 4.

Results

Sample size  V(x)_eql  VC_eq%  Var_st  CV_st%  Var_wr  VC_wr%  Corr. intra 
2  275.95  18.26%  454.76  23.44%  748.06  30.06%  -0.6320 
4  158.33  13.83%  171.26  14.38%  372.15  21.20%  -0.1926 
8  56.87  8.29%  71.52  9.30%  184.20  14.92%  -0.0995 
10  48.03  7.62%  55.45  8.18%  146.61  13.31%  -0.0755 
16  26.67  5.68%  33.76  6.39%  90.22  10.44%  -0.0477 
20  30.42  6.06%  26.75  5.68%  71.42  9.29%  -0.0313 
25  16.92  5.12%  21.08  5.05%  56.39  8.25%  -0.0299 
40  8.08  3.12%  12.91  3.95%  33.83  6.39%  -0.0201 
50  7.75  3.06%  9.86  3.45%  26.31  5.64%  -0.0151 
80  2.32  1.67%  5.71  2.63%  15.04  4.26%  -0.0111 
100  5.78  2.64%  4.35  2.29%  11.28  3.69%  -0.0062 
200  0.54  0.81%  1.50  1.35%  3.76  2.13%  -0.0047 

Remarks

The balanced sampling method presents more precision than other methods for little sample sizes. We can see too a characteristic of systematic sampling: due to the internal structure of the population, it is possible that the variance of the estimator increases with a higher sample size.

Example for experts 2: graphic of a superpopulation model

Find, for the natural population of the previous example a mathematical model, that represents the population structure.

Solution

Once selected a data base, select the option "Generate population structure". In this window, type a degree of 5 for the polinom and then press enter to draw the model. Try some values for the error term until achieve an structure similar to the original.

Xi=1.8335E(+01)+1.0158E(+00)i+-7.1831E(-03)*i2+3.0846E(-05)i3+- +-7.0717E(-08)i4+7.0852E(-11)i5+-e

where e is the error term that represents the dispersion due to randomness. E(e) = 0 and Var(e) = 5*(1+i*0.1)

Superpopulation

Superpopulation model

Example for experts 3: superpopulation model and variance of the estimator

Generate the structure of the population a high number of times and, for each of them, calculate the variance of the estimator regarding to the model.

Solution

In the menu "Calculations", select "Number of populations" and fix a number higher than 40. Then select "Superpopulation" and "Variance of the estimator".

Esperances and variances regarding the model of the variance of the estimator

Sampling methods

E*Vp V*Vp CVS. E*+(2*D*)
Systematic with constant interval (SIC) 17.54 33.46 0.33 29.11
Systematic centered(CLS) 7.17 100.9 1.4 27.26
Systematic with corrections of Yates (CY) 12.97 20.56 0.35 22.04
Systematic balanced (BSS) 12.25 19.11 0.36 20.99
Systematic modified (MSS) 11.68 17.67 0.36 20.09
Unequal probabilities with replacement and pps (RPPS) 12.84 13.86 0.29 20.28
Unequal probabilities without replacement and pps (WRPPS) 12.43 14.01 0.3 19.92

Remarks

The main result that we can see in the table is the low value of the centered methods and, by contrast, their high variability. This fact means that, althought the expected error is low, it is very variable between different trials of population. So that, this method presents little precision, in comparison with the other methods.