Proportions hypothesis testing

Ford 1, Norrie J and Ahmedi S (1995) Model inconsistency, illustrated by the Cox Proportional Flazards model Statistics in Medicine, 14, 735-746 Gardner MJ and Altman DG (1989) Estimation rather than hypothesis testing confidence intervals rather than p-values In Statistics with Confidence (eds MJ Gardner and DG Altman), Fondon British Medical Journal, 6-19... 

We have already met P values in the context of significance testing and it is singularly unfortunate that the same letter should be introduced for a second important function, but then that is statistics for you. (To achieve some clarity. I will use lower case p for proportion and upper case P in hypothesis testing.)... 

Hypothesis tests and interval estimates of proportions are frequently presented in clinical study reports, especially in earlier studies of development when late phase studies are being planned. Accordingly, discussion now turns to analysis methods that can be used to account for sampling variation and, therefore, determine if the results observed are likely due to chance alone. 

In the case of a hypothesis test for two proportions the null and alternate statistical hypotheses can be stated as follows ... 

As an Illustration of this hypothesis test, consider the following hypothetical data from a confirmatory study of a new antihypertensive. In a randomized, double-blind, 12-week study, the test treatment was compared with placebo. The primary endpoint of the study was the proportion of participants who attained an SBP goal <140 mmHg. Of 146 participants assigned to placebo, 34 attained an SBP < 140 mmHg at week 12. Of 1S4 assigned to test treatment, 82 attained the goal. Let us look at how these results can help us to make a decision based on the... 

Hypothesis test for two (or more) proportions y test of homogeneity... 

Smith et al. (2000) argue that it is insufficient to simply hypothesis test for dose proportionality using regression methods because an imprecise study may lead to large confidence intervals around the model parameters that indicate dose proportionality but are in fact meaningless. If Y(h) and Y(l) denote the value of the dependent variable, like Cmax, at the highest (h) and lowest (1) dose tested, respectively, and the drug is dose proportional then... 

We want to decide if the proportion of markings in the SSp with the same cardinality of rrii is greater than the estimated proportion 0. For that we conduct the following hypothesis test for proportions ... 

Chemistry consisted of trying to alter the proportion of these elements in what wasn t wanted to make what was wanted. But by the l600s the chemists had tried about all the combinations they could, and they found that they still did not have much control over the end product. They were starting to see that the Aristotelean hypothesis would fail. But before it was thrown out completely, it underwent one last resurgence in the theory of phlogiston. This time however hypothesis testing and rejection took about a hundred years instead of 2000. 

An important example of a hypothesis testing of the last-mentioned kind is the analysis of the influence of varying the values of individual input variables (e.g., catalyst components and their proportions) on the performance of a catalytic material by means of an approach called analysis of variance. The approach assumes that each output variable... 

One-sided hypothesis test. To test Ho 0 < 0o Hi 0 > 0q v/e calculate the proportion of the random sample from the posterior that satisfies the null hypothesis. If this proportion is less than a, we reject the null hypothesis at that level. 

Ohm s law the statement that the current moving through a circuit is proportional to the applied potential and inversely proportional to the circuit s resistance (E = iR). (p. 463) on-column injection the direct injection of thermally unstable samples onto a capillary column, (p. 568) one-taUed significance test significance test in which the null hypothesis is rejected for values at only one end of the normal distribution, (p. 84)... 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

The assumption in step 1 would first he tested hy obtaining a random sample. Under the assumption that p <. 02, the distrihiition for a sample proportion would he defined hy the z distrihiition. This distrihiition would define an upper hound corresponding to the upper critical value for the sample proportion. It would he unlikely that the sample proportion would rise above that value if, in fact, p <. 02. If the observed sample proportion exceeds that limit, corresponding to what would he a very unlikely chance outcome, this would lead one to question the assumption that p <. 02. That is, one would conclude that the null hypothesis is false. To test, set... 

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