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Statistical formulae for calculatingsome 95% condence intervals

95% condence interval = effect size 1.96 standard error of the effect size

Single-arm phase II trial

Counting people (single proportion)

Number of responses to treatment =28Number of subjects ( N ) =50Observed proportion ( P ) =28/ 50 =0.56 (or 56%)Standard error of the true proportion (SE) = [P (1 P )]/ N = (0.56 0.44)/ 50 =0.0795%CI

= P

1.96

SE

=0.56

1.96

0.07

=0.42to0.70 ( or 42 to 70%)

For small trials (e.g. N < 30) exact methods provide a more accurate 95%condence interval (Geigy Scientic Tables. Introduction to Statistics, StatisticsTables and Mathematical Formulae , 8th edn. Ciba Geigy, 1982).

Taking measurements on people (single mean value)

Mean value ( x) =34 mm (VAS score)Standard deviation (

s) =18 mmNumber of subjects ( N ) =40

Standard error (SE) =s

n =18/ 40 =2.8 mm95% CI = mean 1.96 SE =34 1.96 2.8 =34 5.5 =28 to 40 mmFor small trials ( N < 30), a different multiplier to 1.96 is used. It comes fromthe t-distribution, and gets larger as the sample size gets smaller

The multiplier of 1.96 is associated with a two-sided condence interval.For a one-sided limit a value of 1.645 could be used, but only the loweror upper limit is needed, depending on whether the proportion or mean

205 A Concise Guide to Clinical Trials Allan Hackshaw 2009 Allan Hackshaw. ISBN: 978-1-405-16774-1

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206 Statistical formulae for 95% CI

associated with the new therapy should be greater or smaller than standardtreatments to indicate improvement.

Randomised phase II or III trial with two groups

Counting people (risk difference or relative risk)Example is serological u (Box 7.1)

P1 =r1/ N 1 =41/ 927 =0.044P2 =r2/ N 2 =80/ 911 =0.088For risk difference

Observed risk difference

= P1

P2

= 0.044(

4.4%)

Standard error (SE) = {[P1 (1 P1)]/ N 1 +[P2 (1 P2)]/ N 2} =0.0115595%CI = difference 1.96 SE = 0.044 1.96 0.01155= 0.066 to 0.021 = 6.6% to 2.1%

For relative risk (RR)

Observed RR = P1 P2 =0.5Take natural logarithm (base e)

= loge (0.5)

= 0.693

Standard error of the log RR (SE) = (1/ r1 +1/ r2 1/ N 1 1/ N 2) =0.18695% CI for the log RR = log RR 1.96 SE

= 0.693 1.96 0.186 = 1.058 to 0.328Transform back (take exponential) =0.35 to 0.72(i.e. e

1.058 to e0.328)(e is the natural number 2.71828)

Converted to a percentage change in risk, 95% CI is 28 to 65% reduction in risk

Taking measurements on people (difference betweentwo mean values)Example is the Atkins diet (Box 7.4)

Change in weight loss at three months

Atkins diet: N 1 =33 Mean 1 = 6.8 kg SD1 =5.0 kgConventional diet: N 2 =30 Mean 2 = 2.7 kg SD2 =3.7 kg

Difference between the two means =Mean 1 Mean 2 = 6.8(2.7)= 4.1 kgStandard error of the mean difference (SE) = (SD 21 / N 1 +SD 22 / N 2)

= (5.02/ 33 +3.72/ 30) =1.195%CI =mean difference 1.96 SE= 4.1 1.96 1.1 = 6.3 to 1.9 kg

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Statistical formulae for 95% CI 207

1.96 is used when each trial group has at least say 30 subjects. For smaller stud-ies, a larger multiplier and the t-distribution are used, and there is a differentformulae depending on whether the standard deviations are similar betweenthe groups.

Time-to-event data (hazard ratio)A statistical package should be used to estimate 95% CIs because the calcula-tion for the standard error is not simple. However, if only the median andnumber of events in each treatment group are available, there is a simplemethod to obtain an approximate estimate of the CI, but only after assumingthat the distribution of the time-to-event measure has an exponential distri- bution (i.e. the event rate is constant over time).Example is early vs late radiotherapy in treating lung cancer (Spiro et al., J Clin Oncol 2006; 24 : 38233830), and the outcome is time to death:Early radiotherapy:Median survival M1 =13.7 months Number of deaths =E1 =135Late radiotherapy:Median survival M2 =15.1 months Number of deaths =E2 =136Hazard ratio (early vs late) HR = M2/ M1 =15.1/ 13.7 =1.10Standard error of the log hazard ratio (SE) = (1/ E1 +1/ E2)

= (1/ 135 +1/ 136) =0.121595% CI for the log HR = loge HR 1.96 SE

= log(1.10) 1.96 0.1215 = 0.143 to 0.333Transform back (take exponential) =0.87 to 1.40(i.e. e

0.143 to e0.333)

These are close, but not identical, to the results calculated using the raw data:HR = 1.16, 95% CI 0.91 to 1.47