Null hypothesis definition

Significance tests, however, also are subject to type 2 errors in which the null hypothesis is falsely retained. Consider, for example, the situation shown in Figure 4.12b, where S is exactly equal to (Sa)dl. In this case the probability of a type 2 error is 50% since half of the signals arising from the sample s population fall below the detection limit. Thus, there is only a 50 50 probability that an analyte at the lUPAC detection limit will be detected. As defined, the lUPAC definition for the detection limit only indicates the smallest signal for which we can say, at a significance level of a, that an analyte is present in the sample. Failing to detect the analyte, however, does not imply that it is not present. 

Our data were 15 heads and 5 tails, so how do we calculate the p-value Well, remember the earlier definition and translate that into the current setting the probability of getting the observed data or more extreme data in either direction with a fair coin. To get the p-value we add up the probabilities (calculated when the null hypothesis is true - coin fair) associated with our data (15 heads, 5 tails) and more extreme data (a bigger dilference between the number of heads and the number of tails) in either direction ... 

Usually the type I error is fixed at 0.05 (5 per cent). This is because we use 5 per cent as the significance level the cut-off between significance (p < 0.05) and non-significance (p > 0.05). The null distribution tells us precisely what will happen when the null hypothesis is true we veill get extreme values in the tails of that distribution, even when pj = p2. However, when we do see a value in the extreme outer 5 per cent, we declare significant differences and by definition this will occur 5 per cent of the time when Hg is true. 

Too often a p-value less than 0.05 is seen as definitive proof that the treatments are different while a p-value above 0.05 is seen as no proof at all. The p-value is a measure of the compatibility of the data with equal treatments, the smaller the p-value the stronger the evidence against the null hypothesis. The p-value is a measure of evidence in relation to the null hypothesis treating p < 0.05, p > 0.05 in a binary way as proof/no proof is a gross over-simplification and we must never lose sight of that. 

Let / denote predicted by the model to be inactive and I denote observed to be inactive in the assay by exceeding the decision threshold , with analogous definitions for A, predicted to be a hit , and A, observed to be a hit . With the null hypothesis that a compound is inactive, we have ... 

Regulatory agencies have traditionally accepted only two-sided hypotheses because, theoretically, one could not rule out harm (as opposed to simply no effect) associated with the test treatment. If the value of a test statistic (for example, the Z-tesl statistic) is in the critical region at the extreme left or extreme right of the distribution (that is, < -1.96 or > 1.96), the probability of such an outcome by chance alone under the null hypothesis of no difference is 0.05. However, the probability of such an outcome in the direction indicative of a treatment benefit is half of 0.05, that is, 0.025. This led to a common statistical definition of "firm" or "substantial" evidence as the effect was unlikely to have occurred by chance alone, and it could therefore be attributed to the test treatment. Assuming that two studies of the test treatment had two-sided p values < 0.05 with the direction of the treatment effect in favor of a benefit, the probability of the two results occurring by chance alone would be 0.025 X 0.025, that is, 0.000625 (which can also be expressed as 1/1600). 

There are a number of values of the treatment effect (delta or A) that could lead to rejection of the null hypothesis of no difference between the two means. For purposes of estimating a sample size the power of the study (that is, the probability that the null hypothesis of no difference is rejected given that the alternate hypothesis is true) is calculated for a specific value of A. in the case of a superiority trial, this specific value represents the minimally clinically relevant difference between groups that, if found to be plausible on the basis of the sample data through construction of a confidence interval, would be viewed as evidence of a definitive and clinically important treatment effect. 

Alternative hypothesis Usually comes about from the logic of statistical testing. One example of an alternative hypothesis (refer to the definition of a null hypothesis) is to state that the precision of population A is not... 

We now calculate the mean and standard deviation for these differences using Equation [8.2], to give d] 2 = 0.126, SEj = 0.00894. This gives an experimental tj-value for the five individual paired differences of dj2/Sd= 14.1. Now we test the null hypothesis [H, d] 2 = zero] with do/ = (b — 1) = 4 for p = 0.05, for which t = 2.776 (Table 8.1). Clearly tj > /, and the two analytical methods are definitely not statistically indistinguishable at the 95 % confidence level, unlike the conclusion drawn from the (Bl) evaluation of the same data. This is an excellent example of the need to fit the specifics of a t-test to include all of the available information about the raw experimental data. 

The Minitab macro tintegraimac is used to evaluate the denominator numerically. We want to do an inference such as finding a credible interval for the parameter, or calculating the posterior probability of a one-sided null hypothesis about the parameter. The cumulative distribution function (CDF) of the posterior is the definite integral of the numerical posterior density. It is given by... 

Response Surface Model A dose-response surface is an extension of dose-response lines (isobols) to three dimensions. In this representation there can be a dose-response surface representing additivity and surfaces above and below suggesting deviation from additivity. Tam et al. [90] studied the combined pharmacodynamic interactions of two antimicrobial agents, meropenem and tobramycin. Total bacterial density data, expressed as CFU (colony forming units), were modeled using a three-dimensional surface. Effect summation was used as the definition of additivity (null interaction hypothesis) and the pharmacodynamic model was assumedi to take the functional form... 

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