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How to Determine a p-Value When Testing a Null Hypothesis

407) suggested that “an investigator would be misled less frequently and would be more likely to obtain the information he seeks were he to formulate his experimental problems in terms of the estimation of population parameters, with the establishment of confidence intervals about the estimated values, rather than in terms of a null hypothesis against all possible alternatives.”Many other critics have echoed that advice, to which we also subscribe, especially when the outcome measure is well defined.

Dallal, PhDOne common criticism of significance tests is that no null hypothesis is ever true.

There are several approaches that can be used to test hypotheses concerning two independent proportions. Here we present one approach - the chi-square test of independence is an alternative, equivalent, and perhaps more popular approach to the same analysis. Hypothesis testing with the chi-square test is addressed in the third module in this series: BS704_HypothesisTesting-ChiSquare.

One-sided tests have been proposed for such circumstances.

One-tailed tests make it easier to reject the null hypothesis when the alternative is true.

Since the purpose of a questionnaire is to obtain data about individuals, questionnaire design must be guided by established standards for ethical treatment of human subjects. These guidelines apply to acquisition of questionnaire data just as they do for biological samples such as blood and urine, or to genetic testing. In the United States and many other countries, no studies involving humans may be conducted with public funds unless approval of questionnaire language and content is first obtained from an appropriate Institutional Review Board. Such approval is intended to assure that questions are confined to legitimate study purposes, and that they do not violate the rights of study participants to answer questions voluntarily. Participants must be assured that their participation in the study is entirely voluntary, and that refusal to answer questions or even to participate at all will not subject them to any penalties or alter their relationship with their employer or medical practitioner.

While the questionnaire is often the most visible part of an epidemiological study, particularly to the workers or other study participants, it is only a tool and indeed is often called an “instrument” by researchers. depicts in a very general way the stages of survey design from conception through data collection and analysis. The figure shows four levels or tiers of study operation which proceed in parallel throughout the life of the study: sampling, questionnaire, operations, and analysis. The figure demonstrates quite clearly the way in which stages of questionnaire development are related to the overall study plan, proceeding from an initial outline to a first draft of both the questionnaire and its associated codes, followed by pretesting within a selected subpopulation, one or more revisions dictated by pretest experiences, and preparation of the final document for actual data collection in the field. What is most important is the context: each stage of questionnaire development is carried out in conjunction with a corresponding stage of creation and refinement of the overall sampling plan, as well as the operational design for administration of the questionnaire.

Null and Alternative Hypotheses for a Mean

A large sample, two-sided, 0.05 level t test puts aprobability of 0.025 in each tail.

Hypothesis testing applications with a dichotomous outcome variable in a single population are also performed according to the five-step procedure. Similar to tests for means, a key component is setting up the null and research hypotheses. The objective is to compare the proportion of successes in a single population to a known proportion (p0). That known proportion is generally derived from another study or report and is sometimes called a historical control. It is important in setting up the hypotheses in a one sample test that the proportion specified in the null hypothesis is a fair and reasonable comparator.

This example raises an important issue in terms of study design. In this example we assume in the null hypothesis that the mean cholesterol level is 203. This is taken to be the mean cholesterol level in patients without treatment. Is this an appropriate comparator? Alternative and potentially more efficient study designs to evaluate the effect of the new drug could involve two treatment groups, where one group receives the new drug and the other does not, or we could measure each patient's baseline or pre-treatment cholesterol level and then assess changes from baseline to 6 weeks post-treatment. These designs are also discussed here.

The test statistic for examining hypotheses about one population mean:
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A one-sided test puts all of the probability into a single tail.

Would you reject the hypothesis H(0):MU = 72 versus the alternative H(1):MU =/= 72 on the basis of the observations, when testing at level ALPHA = .05?

Note: If you had been testing the two-sided alternative,

Testing for statistical significance is often performed on measures of effect to evaluate the likelihood that the effect observed differs from the null hypothesis (i.e., no effect). While many studies, particularly in other areas of biomedical research, may express significance by p-values, epidemiological studies typically present confidence intervals (CI) (also called confidence limits). A 95% confidence interval, for example, is a range of values for the effect measure that includes the estimated measure obtained from the study data and that which has 95% probability of including the true value. Values outside the interval are deemed to be unlikely to include the true measure of effect. If the CI for a rate ratio includes unity, then there is no statistically significant difference between the groups being compared.

Some analysts have proposed two-sided tests with unequal tail areas.

Once the type of test is determined, the details of the test must be specified. Specifically, the null and alternative hypotheses must be clearly stated. The null hypothesis always reflects the "no change" or "no difference" situation. The alternative or research hypothesis reflects the investigator's belief. The investigator might hypothesize that a parameter (e.g., a mean, proportion, difference in means or proportions) will increase, will decrease or will be different under specific conditions (sometimes the conditions are different experimental conditions and other times the conditions are simply different groups of participants). Once the hypotheses are specified, data are collected and summarized. The appropriate test is then conducted according to the five step approach. If the test leads to rejection of the null hypothesis, an approximate p-value is computed to summarize the significance of the findings. When tests of hypothesis are conducted using statistical computing packages, exact p-values are computed. Because the statistical tables in this textbook are limited, we can only approximate p-values. If the test fails to reject the null hypothesis, then a weaker concluding statement is made for the following reason.

Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72

Here we presented hypothesis testing techniques for means and proportions in one and two sample situations. Tests of hypothesis involve several steps, including specifying the null and alternative or research hypothesis, selecting and computing an appropriate test statistic, setting up a decision rule and drawing a conclusion. There are many details to consider in hypothesis testing. The first is to determine the appropriate test. We discussed Z and t tests here for different applications. The appropriate test depends on the distribution of the outcome variable (continuous or dichotomous), the number of comparison groups (one, two) and whether the comparison groups are independent or dependent. The following table summarizes the different tests of hypothesis discussed here.

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