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Table 1 Sample size and significance cut-points for (GLL-RUL) differences for a one-sample test, power (80%, 90%, 95%) and 1-tailed 5% significance level based on normal approximation (with continuity correction)

From: Determining sample size for progression criteria for pragmatic pilot RCTs: the hypothesis test strikes back!

RUL

GLL

α (0.05), β (0.2)

α (0.05), β (0.1)

α (0.05), β (0.05)

%

%

n

Ac (%)

n

Ac (%)

n

Ac (%)

10

20

79

12.3 (15.6)

111

16.3 (14.7)

143

20.2 (14.1)

15

25

101

21.1 (20.8)

140

28.0 (20.0)

179

34.7 (19.4)

15

30

49

11.5 (23.4)

68

15.0 (22.1)

87

18.5 (21.3)

20

30

119

31.0 (26.0)

165

41.5 (25.1)

209

51.3 (24.6)

20

35

57

16.4 (28.7)

78

21.4 (27.5)

99

26.3 (26.6)

20

40

34

10.6 (31.3)

46

13.7 (29.7)

59

16.9 (28.6)

25

35

134

41.7 (31.2)

185

55.9 (30.2)

234

69.4 (29.7)

25

40

63

21.4 (34.0)

86

28.1 (32.7)

109

34.7 (31.8)

25

45

37

13.6 (36.7)

51

17.8 (35.0)

64

21.7 (33.9)

25

50

25

9.8 (39.2)

33

12.3 (37.4)

42

15.1 (36.0)

30

40

146

52.9 (36.2)

201

71.0 (35.3)

253

87.9 (34.7)

30

45

68

26.6 (39.1)

93

35.2 (37.8)

117

43.3 (37.0)

30

50

39

16.4 (42.1)

54

21.7 (40.3)

67

26.3 (39.2)

30

55

26

11.6 (44.8)

35

15.0 (42.7)

44

18.2 (41.4)

30

60

18

8.6 (47.8)

25

11.3 (45.1)

30

13.1 (43.8)

35

45

155

64.0 (41.3)

212

85.6 (40.4)

267

106.3 (39.8)

35

50

71

31.5 (44.3)

97

41.7 (43.0)

121

51.0 (42.1)

35

55

41

19.4 (47.3)

56

25.5 (45.5)

69

30.7 (44.4)

35

60

27

13.5 (50.1)

36

17.3 (48.1)

44

20.6 (46.8)

35

65

19

10.1 (53.0)

25

12.7 (50.7)

31

15.2 (49.1)

40

50

160

74.2 (46.4)

219

99.5 (45.4)

275

123.4 (44.9)

40

55

73

36.1 (49.4)

99

47.6 (48.1)

124

58.6 (47.2)

40

60

42

22.0 (52.4)

56

28.4 (50.8)

70

34.7 (49.6)

40

65

27

15.0 (55.5)

36

19.2 (53.4)

44

22.9 (52.1)

40

70

19

11.1 (58.5)

25

14.0 (56.1)

30

16.4 (54.7)

45

55

163

83.8 (51.4)

222

112.1 (50.5)

278

138.7 (49.9)

45

60

74

40.3 (54.5)

100

53.2 (53.2)

124

64.9 (52.3)

45

65

42

24.2 (57.6)

56

31.3 (55.9)

69

37.8 (54.9)

45

70

27

16.4 (60.7)

36

21.1 (58.6)

44

25.2 (57.3)

45

75

19

12.1 (63.8)

24

14.8 (61.7)

29

17.5 (60.2)

50

60

162

91.5 (56.5)

220

122.2 (55.5)

275

151.5 (55.0)

50

65

73

43.5 (59.6)

98

57.1 (58.3)

121

69.5 (57.5)

50

70

41

25.8 (62.8)

55

33.6 (61.1)

67

40.2 (60.0)

50

75

27

17.8 (65.8)

34

21.8 (64.1)

42

26.3 (62.7)

55

65

159

97.8 (61.5)

214

129.7 (60.6)

267

160.2 (60.0)

55

70

71

45.9 (64.7)

94

59.6 (63.4)

117

73.2 (62.6)

55

75

40

27.2 (67.9)

52

34.5 (66.3)

64

41.7 (65.2)

60

70

152

101.1 (66.5)

204

133.9 (65.6)

253

164.6 (65.1)

60

75

68

47.4 (69.8)

89

61.0 (68.5)

109

73.8 (67.7)

60

80

38

27.8 (73.1)

48

34.4 (71.6)

59

41.6 (70.5)

65

75

142

101.6 (71.6)

189

133.6 (70.7)

234

164.1 (70.1)

65

80

63

47.2 (74.9)

81

59.7 (73.7)

99

72.2 (72.9)

65

85

34

26.7 (78.5)

44

33.8 (76.8)

52

39.5 (75.9)

70

80

129

98.9 (76.6)

170

128.8 (75.8)

209

157.2 (75.2)

70

85

56

44.8 (80.1)

72

56.8 (78.9)

87

67.9 (78.1)

75

85

113

92.3 (81.7)

147

118.9 (80.9)

179

143.8 (80.3)

75

90

48

40.9 (85.3)

60

50.5 (84.2)

71

59.3 (83.5)

80

90

93

80.7 (86.8)

119

102.4 (86.0)

143

122.3 (85.5)

  1. RUL upper limit of RED zone (expressed as percentage of total sample), GLL lower limit of GREEN zone (expressed as percentage of total sample), AC AMBER-statistical significance threshold (within the AMBER zone) where an observed estimate below the cut-point will result in a non-significant result (p ≥ 0.05) and figures at or above the cut-point will be significant (p < 0.05) (%, as a percentage of n)
  2. Sample sizes were derived using the normal approximation to the binomial distribution (with continuity correction) formula given in the Appendix, which by convention is stable for np > 5 and n(1 − p) > 5.
  3. For this approach, AC% is calculated from the 1-sided upper 95% confidence limit for the null proportion: 100% × (RUL + z1−α√((RUL(1 − RUL))/n)) [e.g. for RUL = 20% v GLL = 35%, n = 78, power 90%: AC% = 100% × (0.2 + 1.645√((0.2(1 − 0.2))/78)) = 27.5%. In the example this is expressed as a proportion (0.275)]
  4. The AC values do not account for the continuity correction (− 0.5 deduction) which would need to be applied to the observed count from a study prior to cross-checking against the AC cut-offs provided here