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Table 2 Optimal sampling strategies (n s , n d ) and the resulting mean exposure variance s 2 μ (cf. equation (3)) at different combinations of variance components (s 2 BS ,s 2 BD ,s 2 μWD ; sections a-c), unit costs (π s , π d , c q ), and exponents α and β describing the shape of the relationship between costs and number of measurements (cf. equation (4))

From: Optimizing cost-efficiency in mean exposure assessment - cost functions reconsidered

a. (s 2 BS , s 2 BD , s 2 μWD ) = (2, 10, 10)

 

α:

0.50

0.75

1.00

1.25

1.50

  

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

(π s , π d , c q )

β

          

(2, 10, 10)

0.50

(14, 2)

0.86

(14, 2)

0.86

(10, 3)

0.87

(13, 2)

0.92

(9, 3)

0.96

 

0.75

(24, 1)

0.92

(13, 2)

0.92

(9, 3)

0.96

(12, 2)

1.00

(8, 3)

1.08

 

1.00

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(11, 2)

1.09

(10, 2)

1.20

 

1.25

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

1.50

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

(11, 5.5, 5.5)

0.50

(17, 3)

0.51

(15, 3)

0.58

(11, 4)

0.64

(7, 7)

0.69

(5, 10)

0.80

 

0.75

(22, 2)

0.55

(14, 3)

0.62

(10, 4)

0.70

(6, 7)

0.81

(6, 6)

0.89

 

1.00

(39, 1)

0.56

(18, 2)

0.67

(9, 4)

0.78

(8, 4)

0.88

(6, 5)

1.00

 

1.25

(39, 1)

0.56

(32, 1)

0.69

(14, 2)

0.86

(7, 4)

1.00

(6, 4)

1.17

 

1.50

(39, 1)

0.56

(32, 1)

0.69

(13, 2)

0.92

(6, 4)

1.17

(5, 4)

1.40

(20, 1, 1)

0.50

(49, 5)

0.12

(27, 7)

0.18

(14, 12)

0.26

(8, 23)

0.36

(6, 28)

0.45

 

0.75

(52, 4)

0.13

(21, 9)

0.20

(13, 12)

0.28

(8, 19)

0.38

(6, 23)

0.48

 

1.00

(80, 2)

0.15

(26, 5)

0.23

(13, 9)

0.32

(8, 14)

0.43

(6, 17)

0.53

 

1.25

(74, 2)

0.16

(27, 4)

0.26

(13, 7)

0.37

(8, 10)

0.50

(6, 12)

0.61

 

1.50

(134, 1)

0.16

(29, 3)

0.30

(12, 6)

0.44

(9, 6)

0.59

(5, 12)

0.73

b. ( s 2 BS , s 2 BD , s 2 μWD ) = (11, 5.5, 5.5)

 

α:

0.50

0.75

1.00

1.25

1.50

  

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

(2, 10, 10)

0.50

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

0.75

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

1.00

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

1.25

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

1.50

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

(11, 5.5, 5.5)

0.50

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(12, 2)

1.38

(9, 2)

1.83

 

0.75

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(12, 2)

1.38

(9, 2)

1.83

 

1.00

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(9, 2)

1.83

 

1.25

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(8, 2)

2.06

 

1.50

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(8, 2)

2.06

(20, 1, 1)

0.50

(134, 1)

0.16

(44, 2)

0.38

(19, 4)

0.72

(11, 6)

1.17

(7, 14)

1.68

 

0.75

(134, 1)

0.16

(43, 2)

0.38

(18, 4)

0.76

(11, 5)

1.20

(7, 12)

1.70

 

1.00

(134, 1)

0.16

(42, 2)

0.39

(19, 3)

0.77

(11, 4)

1.25

(7, 9)

1.75

 

1.25

(134, 1)

0.16

(40, 2)

0.41

(18, 3)

0.81

(10, 5)

1.32

(7, 7)

1.80

 

1.50

(134, 1)

0.16

(53, 1)

0.42

(20, 2)

0.83

(11, 3)

1.33

(7, 5)

1.89

c. ( s 2 BS , s 2 BD , s 2 μWD ) = (20, 1, 1)

 

α:

0.50

0.75

1.00

1.25

1.50

  

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

( n s , n d )

s 2 μ

(2, 10, 10)

0.50

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

0.75

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

1.00

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

1.25

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

 

1.50

(24, 1)

0.92

(23, 1)

0.96

(22, 1)

1.00

(20, 1)

1.10

(17, 1)

1.29

(11, 5.5, 5.5)

0.50

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(10, 1)

2.20

 

0.75

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(10, 1)

2.20

 

1.00

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(10, 1)

2.20

 

1.25

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(10, 1)

2.20

 

1.50

(39, 1)

0.56

(32, 1)

0.69

(22, 1)

1.00

(15, 1)

1.47

(10, 1)

2.20

(20, 1, 1)

0.50

(134, 1)

0.16

(53, 1)

0.42

(22, 1)

1.00

(12, 2)

1.75

(8, 3)

2.58

 

0.75

(134, 1)

0.16

(53, 1)

0.42

(22, 1)

1.00

(12, 2)

1.75

(8, 3)

2.58

 

1.00

(134, 1)

0.16

(53, 1)

0.42

(22, 1)

1.00

(12, 2)

1.75

(8, 2)

2.63

 

1.25

(134, 1)

0.16

(53, 1)

0.42

(22, 1)

1.00

(12, 2)

1.75

(8, 2)

2.63

 

1.50

(134, 1)

0.16

(53, 1)

0.42

(22, 1)

1.00

(12, 1)

1.83

(8, 2)

2.63

  1. In all cases, the available budget R was set at 500. The determination of the boldface strategy in section a was illustrated in figure 4.