Equivalence confidence intervals

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... 

The statistical significance of a computed difference is best quantified in terms of confidence intervals for the means (CLM). If the mean of a profile T falls into the CLM of profile R , both may be regarded as equivalent. For in vivo data, an acceptance limit of 20% seems to be generally accepted for in vitro data, this would be unnecessarily wide and 5% appears more reasonable. 

If this procedure is followed, then a reaction order will be obtained which is not masked by the effects of the error distribution of the dependent variables If the transformation achieves the four qualities (a-d) listed at the first of this section, an unweighted linear least-squares analysis may be used rigorously. The reaction order, a = X + 1, and the transformed forward rate constant, B, possess all of the desirable properties of maximum likelihood estimates. Finally, the equivalent of the likelihood function can be represented b the plot of the transformed sum of squares versus the reaction order. This provides not only a reliable confidence interval on the reaction order, but also the entire sum-of-squares curve as a function of the reaction order. Then, for example, one could readily determine whether any previously postulated reaction order can be reconciled with the available data. 

III the equivalence approach, which typically compares a statistical parameters confidence interval versus pre-defined acceptance limits (Schuirmann, 1987 Hartmann et al., 1995 Kringle et al., 2001 Hartmann et al., 1994). This approach assesses whether the true value of the parameter(s) are included in their respective acceptance limits, at each concentration level of the validation standards. The 90% 2-sided Cl of the relative bias is determined at each concentration level and compared to the 2% acceptance limits. For precision measurements, if the upper limit of the 95% Cl of the RSDn> is <3% then the method is acceptable (Bouabidi et al., 2010) or,... 

Fig. 6.1 Relationship between significance tests and confidence intervals for the comparison between a new treatment and control. The treatment differences A and B are in favour of the new treatment but superiority is shown only in A. in B, the outcome may meet criteria for equivalence or non-inferiority as defined in the protocol.

A test of the null h)rpothesis that the rates of infection are equal - Hq x jii/hnj = 1 gives a p-value of 0.894 using a chi-squared test. There is therefore no statistical evidence of a difference between the treatments and one is unable to reject the null hypothesis. However, the contrary statement is not true that therefore the treatments are the same. As Altman and Bland succinctly put it, absence of evidence is not evidence of absence. The individual estimated infection rates are jTi = 0.250 and = 0.231 that gives an estimated RR of 0.250/0.231 = 1.083 with an associated 95% confidence interval of 0.332-3.532. In other words, inoculation can potentially reduce the infection by a factor of three, or increase it by a factor of three with the implication that we are not justified in claiming that the treatments are equivalent. 

What was missing in the previous section was a definition of what is meant by equivalence. Since it is imlikely that two treatments wiU have exactly the same effect we will need to consider how big a difference between the treatments would force us to choose one in preference to the other. In the t)q)hoid example there was a difference in rates of 1.9% and we may well believe that such a small difference would justify us in claiming that the treatment effects were the same. But had the difference been 5% would we still have thought them to be the same Or 10 There will be a difference, say S %, for which we are no longer prepared to accept the equivalence of the treatments. This is the so-called equivalence boimdary. If we want then to have a high degree of confidence that two treatments are equivalent it is logical to require that an appropriately chosen confidence interval (say 95%) for the treatment differences should have its extremes within the boundaries of equivalence. 

In Figure 8.9, we illustrate various cases that can arise from studies intended to show equivalence and the relationship between significance in the traditional sense and clinical significance as determined by the confidence interval and the boundaries of equivalence. In case (A), the 95% confidence interval includes both the null hypothesis of no difference and is within the boundaries of equivalence and from both a statistical and clinical perspective there is no evidence of a difference between the treatments. In case (B), in contrast, the confidence interval is still within the boundaries, but does include the null hypothesis, so from a statistical perspective there is a difference between the treatments but it is not clinically relevant. Case (C) shows both statistical and clinical significance, as the confidence interval lies outside the equivalence boundaries and therefore cannot include the null hypothesis. In the final case, (D), the confidence interval includes... 

If in Figure 8.9 a positive difference between treatments were indicative of a benefit for the test treatment then case (C) would indicate significant superiority of the new treatment. In such circumstances, we would not wish to conclude that only the treatments were not equivalent. In such circumstances, we can use a single boundary and such studies are called non-inferiority studies in which the objective is to show that the new treatment is no more than a small amount worse than the standard. The conduct of the inference remains similar if the confidence interval is to the right of the non-inferiority boundary, we can conclude that the new treatment is non-inferior to the standard. 

The Bayesian equivalent to the frequentist 90% confidence interval is delineated by the 5th and 95th percentiles of the posterior distribntion. Bayesian confidence intervals for SSD (Figures 5.4 to 5.5), 5th percentile, i.e., HC5 and fraction affected (Figures 5.4 to 5.6) were calculated from the posterior distribution. Thns, the nncer-tainties of both HC and FA are established in 1 consistent mathematical framework FA estimates at the logio HC lead to the intended protection percentage, i.e., M °(logio HCf) = p where p is a protection level. Further full distribution of HC and FA uncertainty can be very easily extracted from posterior distribntion for any level of protection and visualized (Figures 5.5 to 5.7). 

It is all too common to see a conclusion that treatments are the same (or similar) simply on the back of a large p-value this is not necessarily the correct conclusion. Presentation of the 95 per cent confidence interval will provide a statement about the possible magnitude of the treatment difference. This can be inspected and only then can a conclusion of similarity be made if this interval is seen to exclude clinically important differences. We will return to a more formal approach to this in Chapter 12 where we will discuss equivalence and non-inferiority. 

As seen in Figure 12.1, this confidence interval is completely contained between the equivalence margins —151/min to 151/min and all of the values for the treatment difference supported by the confidence interval are compatible with the definition of clinical equivalence we have established equivalence as defined. 

In contrast, suppose that the 95 per cent confidence interval had turned out to be (—171/min, 121/min). This interval is not entirely within the equivalence margins and the data are supporting potential treatment differences below the lower equivalence margin. In this case we have not established equivalence. 

Note that there are no conventional p-values here. Such p-values have no role in the evaluation of equivalence establishing equivalence is based entirely on the use of confidence intervals. 

Statistical analysis is generally based on the use of confidence intervals. For equivalence trials, two-sided confidence intervals should be used. Equivalence is inferred when the entire confidence interval falb within the equivalence margins. ... 

Although conventional p-values have no role to play in equivalence or noninferiority trials there is a p-value counterpart to the confidence intervals approach. The confidence interval methodology was developed by Westlake (1981) in the context of bioequivalence and Schuirmann (1987) developed a p-value approach that was mathematically connected to these confidence intervals, although much more difficult to understand It nonetheless provides a useful way of thinking, particularly when we come later to consider type I and type II errors in this context and also the sample size calculation. We will start by looking at equivalence and use A to denote the equivalence margins. 

In practice I would always recommend using confidence intervals for evaluating equivalence and non-inferiority rather than these associated p-values. This is because the associated p-values tend to get mixed up with conventional p-values for detecting differences. The two are not the same and are looking at quite different things. The confidence interval approach avoids this confusion and provides a technique that is easy to present and interpret. 

Analysis of variance appropriate for a crossover design on the pharmacokinetic parameters using the general linear models procedures of SAS or an equivalent program should be performed, with examination of period, sequence and treatment effects. The 90% confidence intervals for the estimates of the difference between the test and reference least squares means for the pharmacokinetic parameters (AUCo-t, AUCo-inf, Cmax should be calculated, using the two one-sided t-test procedure). 

The confidence interval that is equivalent to the two-sided test is obtained from the critical regions ... 

Using log-transformed data, bioequivalence is established by showing that the 90% confidence interval of the ratio of geometric mean responses (usually AUC and Cmax) of the two formulations is contained within the limits of 0.8 to 1.25 [22]. Equivalently, it could be said that bioequivalence is established if the hypothesis that the ratio of geometric means is less than or equal to 0.8 is rejected with... 

An equivalence approach has been and continues to be recommended for BE comparisons. The recommended approach relies on (1) a criterion to allow the comparison, (2) a confidence interval (Cl) for the criterion, and (3) a BE limit. Log-transformation of exposure measures before statistical analysis is recommended. BE studies are performed as single-dose, crossover studies. To compare measures in these studies, data have been analyzed using an average BE criterion. This guidance recommends continued use of an average BE criterion to compare BA measures for replicate and nonreplicate BE studies of both immediate- and modihed-release products. 

This chapter introduces basic concepts in statistical analysis that are of relevance to describing and analyzing the data that are collected in clinical trials, the hallmark of new drug development. (Statistical analysis in nonclinical studies was addressed earlier in Chapter 4.) This chapter therefore sets the scene for more detailed discussion of the determination of statistical significance via the process of hypothesis testing in Chapter 7, evaluation of clinical significance via the calculation of confidence intervals in Chapter 8, and discussions of adaptive designs and of noninferiority/equivalence trials in Chapter 11. 

To demonstrate practical superiority, equivalence or non-inferiority, we must inspect the 95 per cent confidence interval for the difference between mean values. The P value is completely irrelevant to all these questions and is potentially misleading. 

Inns et al. (1990) had also determined the LD50 of i.v. lewisite administration at 1.8 mg/kg (1.6-2.1 mg/kg 95% confidence interval). Thus, it can be concluded that by exposing 2 cm of rabbit skin to a dose of 5.3 mg/kg for 6 h, a dose producing the equivalent effect of 1.8 mg/kg is absorbed. No further calculations that might exaggerate the reliabihty of available data shall be conducted here. 

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