Two-sided significance levels

The proposed sample size was 600 patients and two interims were planned after 200 and 400 patients (completing 3 months follow-up) using the O Brien and Fleming scheme with adjusted two-sided significance levels of 0.00052, 0.014 and 0.045. A futility rule was also introduced, based on conditional power (under the current trend) being below 30 per cent for the trial to be stopped. 

It is desired to run a placebo-controlled parallel group trial in asthma. The target variable is forced expiratory volume in one second (FEVi). The clinically relevant difference is presumed to be 200 ml and the standard deviation 450 ml. A two-sided significance level of 0.05 (or 5%) is to be used and the power should be 0.8 (or 80%). What should the sample size be. ... 

Figure 14.2 Probability of at least one effect reversal as a function of the number of centres given 80% overall power for a two-sided significance level of 5%.

The limit for statistically significant changes then is fc- /2 SD-r, where k depends on the desired probability level. Considering a two-sided 5% level, k is 1.96. The corresponding one-sided factor is 1.65. If a higher probability level is desired, k should be increased. 

In addition. Figure 9 shows the goodness of fit of each model, separately. Also, we plot two-sided confidence levels at a significance level of 95% for our case study. The width of the confidence interval... 

Three days after the beginning of high level Rn/Rn-d exposure TMRP-values of the irradiated cells decreased significantly (level of significance oc (two-sided) 0.05) below the control value as an example series 2 is shown in Figure 3. This is followed by a further decrease to a minimum on the 5th-6th day of exposure. After re-incubating the cells in a normal atmosphere for about two days, TMRP-values reached again control levels. 

TABLE II Statistical Comparison of the TMRP-Mean-Values of Control and Irradiated Cells <(two sided) = Level of Significance... 

If a random variable X is defined over a continuous domain Q in 91, the unknown mean p of a sample lies in a known two-sided confidence interval o = [x , x6] at 100(1 — a) percent, or, equivalently, is known at the a significance level, if... 

Fig. 1.3. Signal/noise ratios [(treated mean-control mean)]/SD for eight haematological traits in the CD-1 outbred stock. The dotted line represents the signal/noise ratio that should be detectable in a future experiment with a group size of 20 animals and a 90% power using a two-sided f-test and a 5% significance level.

The tabulated value of t (from a t-table for a double-sided or two-tailed test which effectively treats the observed difference between x and m of 0.1 as 0.1) for a 95 percent significance level and four degrees of freedom is 2.78. Since the calculated value of / is less than 2.78, the hypothesis that m = 92.5 is not rejected. 

The tabulated t-value ( double-sided test) for a 95 percent significance level and six degrees of freedom is 2.45. Thus, there is greater than a 95 percent probability that the two analyses are not the same, i.e., the hypothesis may be rejected with a likelihood of less than 5 percent error. Note that this test has not indicated whether one analysis was better than the other, but rather whether or not there was any significant difference between the two. 

The r-statistic is (207.6 - 199.6)/3.78 = 2.12. Entering Table 5 with 60 degrees of freedom, we obtain a critical t-value for a one-sided test at the 0.025 level of significance equal to 2.00. As the alternative hypothesis is that the two QC specimens differ, without regard to which has the higher and which has the lower value, the test is a two-sided test. The critical table value at 0.025, one-sided, is the critical value to use for a two-sided test at 0.05. The calculated statistic exceeds this critical value, and we reject the hypothesis that the two sets of QC samples were prepared identically. If we have the stock solutions used to prepare the two QC specimens, we would probably analyze them to see whether they have identical concentrations. 

In figure 9.2, the mean square error of two estimates of the treatment effect for females is given as a frinction of the trial demographic fraction (females as a proportion of all patients in the trial). It is assumed that it had been suspected that the treatment effect might be less for women than for men and that, to err on the side of caution, the trial has been designed with total numbers equal to the numbers which would be required to achieve 80% power for a 5% significance level (two-sided) for a trial... 

Now, whereas (1 - (3)/a is a monotonically increasing function of the sample size n, Li(P)/Lq(P) is not. It may increase at first, but eventually it will decline. The situation is illustrated in Figure 13.2, which takes the particular example of the trial is asthma considered in section 13.1 above and shows the likelihood for the null hypothesis (scaled to equal 1 in the case where the observed treatment difference is zero) for all possible observed treatment differences and also for the alternative hypothesis where the true treatment effect is equal to the clinically relevant difference. The situations for trials with 10 and 200 patients per group are illustrated. The critical value of the observed treatment difference for a two-sided test at the 5% level is marked in each case. For the smaller trial, a larger difference is required for significance. In the larger trial, a smaller difference is adequate. 

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