Null hypotheses equivalence

Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a significance level, a, that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a. Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10. 

Under the null hypothesis, it is assumed that the sample came from a population whose mean [L is equivalent to some base or reference designated by lo. This can take one of three forms ... 

The next update of the null hypothesis would incorporate a zero-order description of bonding, in terms of a prior prejudice of standard chemical groups. The MaxEnt map then will tell us about the subtle differences induced in formally equivalent chemical bonds by conjugation, stacking, and other intra- and intermolecular interactions. To achieve this degree of accuracy, the refinement of structural parameters... 

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. 

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... 

Another way of thinking about thep-value is as a measure of how consistent the difference is with equal treatments (or equivalently with the null hypothesis). A low p-value says that the difference is not consistent with equal treatments, a high value says that the difference is consistent with equal treatments. The conventional cut-off between low and high is 0.05. 

In this case, the alternative hypothesis states that the two treatments are clinically equivalent the null hypothesis is saying that the two treatments are not equivalent. Note that the alternative encapsulates the objective we are trying to disprove the null in order to establish equivalence. 

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. ... 

With the p-value methodology we are rejecting the null hypothesis Hg in favour of the alternative hypothesis Hj, providing the two (one-sided) p-values are < 2.5 per cent. We have then established equivalence and we can talk in terms of the treatments being significantly equivalent. The terminology sounds almost contradictory, but is a correct statement. If either of the two p-values is above 2.5 per cent then the treatments are not significantly equivalent. 

In general many of the standard parametric tests have non-parametric equivalents. The Mann-Whitney test corresponds to the parametric unpaired t-test. This test is based on rank sums. The combined data are ranked, usually low to high. If the null hypothesis, that the two samples come from identical populations, is true the sum of the ranks assigned to the observations from the two... 

Having calculated the level of significance can be obtained from appropriate tables. The Wilcoxon signed rank test is the non-parametric equivalent of the paired t-test. The Kruskal-Wallis test is another rank sums test that is used to test the null hypothesis that k independent samples come from identical populations against the alternative that the means of the populations are unequal. It provides a non-parametric alternative to the one-way analysis of variance. 

In the case of equivalence, noninferiority, and bioequivalence trials, the null hypotheses established are different from the null hypothesis established in superiority trials. In addition, the null hypothesis in each case is unique, and hence they all differ from each other. However, they share a basic similarity. The null hypothesis for each of these designs states, in effect, that the test drug and the comparator drug do not have similar efficacy. As in all hypothesis testing, the statistical methodologies used look for compelling evidence to reject the respective null hypothesis in each case. 

It is noteworthy that, while the null hypothesis in each of the trial designs discussed (superiority, equivalence, noninferiority, and equivalence) is different, the hypothesis testing approach in each case is fundamentally similar to that in every other case. In each instance it is hoped that the null hypothesis will be rejected in favor of the research hypothesis. 

Concluding equivalence... based on observing a non-significant test result of the null hypothesis that there is no difference between the investigational product and the active comparator is inappropriate (p. 16). 

In this case, the research hypothesis states that the two drugs are equivalent, and the null hypothesis states that they are not equivalent. The locations of the lower and the upper limit of the 95% Cl determine whether or not the null hypothesis is rejected. If both the lower limit and the upper limit lie within the equivalence margin, we reject the null hypothesis and the new drug and the reference drug are declared to be equivalent. If either the lower limit or the upper limit lies outside the equivalence margin or if both limits lie outside, we fail to reject the null hypotheses, and the drugs are not declared to be equivalent. 

First we need a null hypothesis. This will state that the products of the two machines are equivalent and if we went on collecting dodgy canisters for long enough the proportions of returns from each machine would simply be proportional to the numbers they produced. More formally it would be among a large population of returned canisters 61 per cent would be from the Allegro machine and 39 per cent from the Andante . 

Null hypothesis The average effect of the investigational drug on SBP is equivalent to the average effect of the placebo on SBP. 

Following the logic of our memory lip, you will see that the alternate hypothesis in this case, just like in the case of a superiority trial, expresses what we are hoping to find, while the null hypothesis states what we are hoping not to find. The actual natures of the null and alternate hypotheses in an equivalence trial are... 

The probability of committing a type I error is the probability of rejecting the null hypothesis when it is true (for example, claiming that the new treatment is superior to placebo when they are equivalent in terms of the outcome). The probability of committing a type I error is called a, which is sometimes referred to as the size of the test. The probability of committing a type II error is the probability of failing to reject the null hypothesis when it is false. This probability is also called beta (P). The quantity (1 - P) is referred to as the power of the statistical test. It is the probability of rejecting the null hypothesis (in favor of the alternate) when the alternate is true. As stated earlier it is desirable to have low error probabilities associated with a test. As we would like a and p to be as low as possible the quanti-... 

In this instance, independence means that the probability of the response is no more or less likely for one group versus the other. In his original paper, Fisher stated the null hypothesis slightly differently (although equivalent mathematically). The null hypothesis, after Fisher, can be stated in this form The population odds ratio of response to nonresponse for one group versus the other is equal to one. 

Given that the research questions in these trials are different from those used in superiority trials, the formats of the null and alternate hypotheses are also different. The research question associated with an equivalence trial is Does the test drug demonstrate equivalent efficacy compared with the comparator drug The null hypothesis,... 

In either case, if the null hypothesis H0 is rejected the generic should be considered equivalent or better than the innovator. 

In 1 of the 5 reference-controlled trials [253] formal statistical equivalence tests were applied. In this trial Hypericum extract LI 160 (300 mg t.i.d.) was compared with imipramine (50 mg t.i.d.) in patients suffering from a severe depressive episode. The null-hypothesis of non-... 

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