Testing one-sided

Essential Statistics for the Pharmaceutical Sciences Philip Rowe 

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Interpretation If the alternate hypothesis had been stated as //i Xmean is different from /r, a two-sided test is applied with 2.5% probability being provided for each possibility Xmean smaller than p" resp. Xmean larger than /i . Because 1.92 is smaller than 2.45, the test criterion is not exceeded, so Hi is rejected. On the other hand, if it was known beforehand that Xmean can only be smaller than p, the one-sided test is conducted under the alternate hypothesis H Xn,ean smaller than p in this case the result is elose, with 1.92 almost exceeding 1.94. 

This amounts to stating the analytical results obtained from HPLC-purity determinations on one batch are not expected to exceed the individual limit AIL more than once in 20 batches. Since a one-sided test is carried out here, the t(a = 0.1,/) for the two-sided case corresponds to the /(a/2 = 0.05,/) value needed. The target level TL is related to the AIL as is the lower end... 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

ANOVA) if the standard deviations are indistinguishable, an ANOVA test can be carried out (simple ANOVA, one parameter additivity model) to detect the presence of significant differences in data set means. The interpretation of the F-test is given (the critical F-value for p = 0.05, one-sided test, is calculated using the algorithm from Section 5.1.3). 

Schuirmann, D. J. A Comparison of the Two One-Sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability. J. Pharmacol. Biopharm., 15, 1987, 657-680. 

In part this is because of the way we set up the hypotheses in our earlier discussion we asked is the coin fair or is the coin not fair We could have asked a different set of questions is the coin fair or are heads more likely than tails in which case we could have been justified in calculating the p-value only in the tail corresponding to heads more likely than tails . This would have given us a one-tailed or a one-sided test. Under these circumstances, had we seen 17 tails and 3 heads then this would not have led to a significant p-value, we would have discounted that outcome as a chance finding, it is not in the direction that we are looking for. 

It is important to clarify whether one- or two-sided tests of statistical significance will be used, and in particular to justify prospectively the use of one-sided tests... The approach of setting type I errors for one-sided tests at half the conventional type I error used in two-sided tests is preferable in regulatory settings. ... 

Operationally, this is equivalent to the method of using two simultaneous one-sided tests to test the (composite) null hypothesis that the treatment difference is outside the equivalence margins versus the (composite) alternative hypothesis that the treatment difference is within the margins. ... 

Using a 2.5 per cent significance level for non-inferiority in this way may initially appear to be out of line with the conventional 5 per cent significance level for superiority. A moments thought should suffice however, to realise that in a test for superiority we would never make a claim if our treatment was significantly worse than placebo, we would only ever make a claim if we were significantly better than placebo, so effectively we are conducting a one-sided test at the 2.5 per cent level to enable a positive conclusion of superiority for the active treatment. 

Schuirmann DJ (1987) A comparison of two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability Journal of Pharmacokinetics and... 

Figure 21.5 The one-tailed or one-sided test, with a normal distribution curve for comparison.

The actual hypothesis to be tested must be carefully chosen. The basic issue is to ensure a safe working environment and we interpret this in the sense that safety must be statistically demonstrated or the environment is considered unsafe. Thus a hypothesis is constructed that the 95th percentile of the distribution of air concentrations is equal to the 95th percentile of the limiting distribution BQ (null hypothesis) and an environment is not considered safe until statistical evidence demonstrates that < BQ(alternate hypothesis) in a lower one-sided test. (Note BQ= ygagd-645) where yg and ag are parameters of the limiting distribution.)... 

Figure 5. Distributions, based upon various sample sizes, of the estimated 95th percentile (xgsgL645) of a limiting distribution (vg° = 2.0 fig° = 3.2 BQ = 10.0) shaded areas represent critical regions (a =. 05) for a lower one-sided test of the hypothesis that the sample values were derived from the limiting distribution.

This example illustrates a one-sided test, that is the critical region is on one side of the probability distribution because of the way the hypotheses are stated. 

Because of the relationship k( - a) < k( 1 - a/2) one-sided tests are sharper , i.e. they will indicate significance earlier. Therefore a null hypothesis is rejected earlier than it would be by use of a two-sided test. 

Let us now compare our experimental result found by AAS with a given critical level for the maximum Zn content in sewage water, e.g. g0 = 200 gg L 1. Because this is a one-sided constraint we have to use a one-sided test and keep the critical error level of a = 0.05 (q = 0.95). We then have to formulate the statistical hypothesis according to ... 

Introduce the use of one-sided tests where an experiment was designed to find out whether an end-point changes in a specified direction and describe the special form of null-hypothesis used in one-sided tests... 

Review the age-old trick of switching from a two-sided to a one-sided test thereby converting disappointing non-significance to the much coveted significance... 

It often perplexes people that they cannot simply consider how an experiment was performed in order to tell whether a one or a two-sided test should be used. The fact is that the decision depends upon what question the experiment was designed to answer. If the purpose was to look for any old change - use a two-sided test. If your only interest lay in checking for a change in a particular, specified direction, a one-sided test should be used. 

If we are going to test a one-sided question, we need to modify our null and alternative hypotheses. For two-sided testing, the null hypothesis would be that there is no difference in clearance and the alternative would be that there is. For a one-sided test (looking for a greater clearance) we want our alternative hypothesis to be there is an increase in clearance . The null hypothesis then has to cover all other possibilities - clearance is either unchanged or reduced . 

Hypotheses for a one-sided test looking for an increase in clearance... 

In Chapter 5, we saw that we can generate a one-sided 95 per cent confidence interval by calculating a 90 per cent confidence interval and then using just one limit and ignoring the other. We can then say that there is only a 5 per cent chance that the true mean value lies beyond the one limit that is being quoted. We do essentially the same to perform a one-sided test for increased clearance. The steps are ... 

Figure 10.1 shows how we would interpret the various possible outcomes of a one-sided test for an increase. 

Figure 10.1 Interpretation of the confidence interval for the difference between mean clearances, when performing a one-sided test for evidence of an increase...

One-sided test for an increase in the measured value- quote the lower confidence limit and declare the result significant if it is above zero (a reduced/unchanged value has been successfully excluded). 

With all statistical tests, one aim is to ensure that, where there is no real effect, we will make false positive claims on only 5 per cent of occasions. Consider what would happen if there was actually no effect on clearance and we carried out 20 trials, each analysed by a one-sided test (testing for an increase). Bearing in mind that the actual procedure is to calculate a 90 per cent Cl, but then convert it to what is effectively a 95 per cent Cl by ignoring one of its limits, we can predict the likely outcomes as in Figure 10.2 ... 

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