Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations
If exact p-value is reported, then the relationship between confidence intervals and hypothesis testing is very close. However, the objective of the two methods is . In statistics, a confidence interval or compatibility interval (CI) is a type of interval estimate, Philosophical issues; Relationship with other statistical topics . Confidence intervals are closely related to statistical significance testing. of the notions of confidence intervals and of statistical hypothesis testing are distinct . How Hypothesis Tests Work: Confidence Intervals and Confidence Levels . The relationship between the confidence level and the significance level for a.
One should always use two-sidedPvalues. Two-sided P values are designed to test hypotheses that the targeted effect measure equals a specific value e. When, however, the test hypothesis of scientific or practical interest is a one-sided dividing hypothesis, a one-sided P value is appropriate. For example, consider the practical question of whether a new drug is at least as good as the standard drug for increasing survival time.
- Which one?
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This question is one-sided, so testing this hypothesis calls for a one-sided P value. Nonetheless, because two-sided P values are the usual default, it will be important to note when and why a one-sided P value is being used instead.
The disputed claims deserve recognition if one wishes to avoid such controversy. For example, it has been argued that P values overstate evidence against test hypotheses, based on directly comparing P values against certain quantities likelihood ratios and Bayes factors that play a central role as evidence measures in Bayesian analysis [ 377277 — 83 ].
Nonetheless, many other statisticians do not accept these quantities as gold standards, and instead point out that P values summarize crucial evidence needed to gauge the error rates of decisions based on statistical tests even though they are far from sufficient for making those decisions.
Briefly, explain the relationship between confidence interval and hypothesis testing?
See also Murtaugh [ 88 ] and its accompanying discussion. Common misinterpretations of P value comparisons and predictions Some of the most severe distortions of the scientific literature produced by statistical testing involve erroneous comparison and synthesis of results from different studies or study subgroups.
Among the worst are: This belief is often used to claim that a literature supports no effect when the opposite is case. In reality, every study could fail to reach statistical significance and yet when combined show a statistically significant association and persuasive evidence of an effect.
Thus, lack of statistical significance of individual studies should not be taken as implying that the totality of evidence supports no effect. When the same hypothesis is tested in two different populations and the resultingPvalues are on opposite sides of 0. Statistical tests are sensitive to many differences between study populations that are irrelevant to whether their results are in agreement, such as the sizes of compared groups in each population. As a consequence, two studies may provide very different P values for the same test hypothesis and yet be in perfect agreement e.
For example, suppose we had two randomized trials A and B of a treatment, identical except that trial A had a known standard error of 2 for the mean difference between treatment groups whereas trial B had a known standard error of 1 for the difference. Differences between results must be evaluated by directly, for example by estimating and testing those differences to produce a confidence interval and a P value comparing the results often called analysis of heterogeneity, interaction, or modification.
When the same hypothesis is tested in two different populations and the samePvalues are obtained, the results are in agreement. Again, tests are sensitive to many differences between populations that are irrelevant to whether their results are in agreement. Two different studies may even exhibit identical P values for testing the same hypothesis yet also exhibit clearly different observed associations.
For example, suppose randomized experiment A observed a mean difference between treatment groups of 3. If one observes a smallPvalue, there is a good chance that the next study will produce aPvalue at least as small for the same hypothesis. This is false even under the ideal condition that both studies are independent and all assumptions including the test hypothesis are correct in both studies.
In general, the size of the new P value will be extremely sensitive to the study size and the extent to which the test hypothesis or other assumptions are violated in the new study [ 86 ]; in particular, P may be very small or very large depending on whether the study and the violations are large or small. Finally, although it is we hope obviously wrong to do so, one sometimes sees the null hypothesis compared with another alternative hypothesis using a two-sided P value for the null and a one-sided P value for the alternative.
This comparison is biased in favor of the null in that the two-sided test will falsely reject the null only half as often as the one-sided test will falsely reject the alternative again, under all the assumptions used for testing.
Common misinterpretations of confidence intervals Most of the above misinterpretations translate into an analogous misinterpretation for confidence intervals. A reported confidence interval is a range between two numbers.
The frequency with which an observed interval e. These further assumptions are summarized in what is called a prior distribution, and the resulting intervals are usually called Bayesian posterior or credible intervals to distinguish them from confidence intervals [ 18 ]. Symmetrically, the misinterpretation of a small P value as disproving the test hypothesis could be translated into: As with the P value, the confidence interval is computed from many assumptions, the violation of which may have led to the results.
Even then, judgements as extreme as saying the effect size has been refuted or excluded will require even stronger conditions. If two confidence intervals overlap, the difference between two estimates or studies is not significant.
As with P values, comparison between groups requires statistics that directly test and estimate the differences across groups. Finally, as with P values, the replication properties of confidence intervals are usually misunderstood: This statement is wrong in several ways. When the model is correct, precision of statistical estimation is measured directly by confidence interval width measured on the appropriate scale. It is not a matter of inclusion or exclusion of the null or any other value.
The first interval excludes the null value of 0, but is 30 units wide.
The second includes the null value, but is half as wide and therefore much more precise. Nonetheless, many authors agree that confidence intervals are superior to tests and P values because they allow one to shift focus away from the null hypothesis, toward the full range of effect sizes compatible with the data—a shift recommended by many authors and a growing number of journals.
Another way to bring attention to non-null hypotheses is to present their P values; for example, one could provide or demand P values for those effect sizes that are recognized as scientifically reasonable alternatives to the null. As with P values, further cautions are needed to avoid misinterpreting confidence intervals as providing sharp answers when none are warranted.
The P values will vary greatly, however, among hypotheses inside the interval, as well as among hypotheses on the outside. Also, two hypotheses may have nearly equal P values even though one of the hypotheses is inside the interval and the other is outside.
Thus, if we use P values to measure compatibility of hypotheses with data and wish to compare hypotheses with this measure, we need to examine their P values directly, not simply ask whether the hypotheses are inside or outside the interval.
This need is particularly acute when as usual one of the hypotheses under scrutiny is a null hypothesis. Common misinterpretations of power The power of a test to detect a correct alternative hypothesis is the pre-study probability that the test will reject the test hypothesis e.
The corresponding pre-study probability of failing to reject the test hypothesis when the alternative is correct is one minus the power, also known as the Type-II or beta error rate [ 84 ] As with P values and confidence intervals, this probability is defined over repetitions of the same study design and so is a frequency probability.
One source of reasonable alternative hypotheses are the effect sizes that were used to compute power in the study proposal. Pre-study power calculations do not, however, measure the compatibility of these alternatives with the data actually observed, while power calculated from the observed data is a direct if obscure transformation of the null P value and so provides no test of the alternatives.
Thus, presentation of power does not obviate the need to provide interval estimates and direct tests of the alternatives. For these reasons, many authors have condemned use of power to interpret estimates and statistical tests [ 4292 — 97 ], arguing that in contrast to confidence intervals it distracts attention from direct comparisons of hypotheses and introduces new misinterpretations, such as: If you accept the null hypothesis because the nullPvalue exceeds 0.
It does not refer to your single use of the test or your error rate under any alternative effect size other than the one used to compute power.
It can be especially misleading to compare results for two hypotheses by presenting a test or P value for one and power for the other.
Thus, claims about relative support or evidence need to be based on direct and comparable measures of support or evidence for both hypotheses, otherwise mistakes like the following will occur: If the nullPvalue exceeds 0. This claim seems intuitive to many, but counterexamples are easy to construct in which the null P value is between 0.
We will however now turn to direct discussion of an issue that has been receiving more attention of late, yet is still widely overlooked or interpreted too narrowly in statistical teaching and presentations: That the statistical model used to obtain the results is correct.
Too often, the full statistical model is treated as a simple regression or structural equation in which effects are represented by parameters denoted by Greek letters. Yet these tests of fit themselves make further assumptions that should be seen as part of the full model. For example, all common tests and confidence intervals depend on assumptions of random selection for observation or treatment and random loss or missingness within levels of controlled covariates. For example, a confidence interval can be used to describe how reliable survey results are.
A major factor determining the length of a confidence interval is the size of the sample used in the estimation procedure, for example, the number of people taking part in a survey. Meaning and interpretation[ edit ] See also: The confidence interval can be expressed in terms of samples or repeated samples: This considers the probability associated with a confidence interval from a pre-experiment point of view, in the same context in which arguments for the random allocation of treatments to study items are made.
Here the experimenter sets out the way in which they intend to calculate a confidence interval and to know, before they do the actual experiment, that the interval they will end up calculating has a particular chance of covering the true but unknown value.
The explanation of a confidence interval can amount to something like: In each of the above, the following applies: Consider now the case when a sample is already drawn, and the calculations have given [particular limits]. The answer is obviously in the negative.
The parameter is an unknown constant, and no probability statement concerning its value may be made Seidenfeld's remark seems rooted in a not uncommon desire for Neyman-Pearson confidence intervals to provide something which they cannot legitimately provide; namely, a measure of the degree of probability, belief, or support that an unknown parameter value lies in a specific interval. Following Savagethe probability that a parameter lies in a specific interval may be referred to as a measure of final precision.
While a measure of final precision may seem desirable, and while confidence levels are often wrongly interpreted as providing such a measure, no such interpretation is warranted. Admittedly, such a misinterpretation is encouraged by the word 'confidence'.
This will be discussed in the examples that follow. In one sample tests for a continuous outcome, we set up our hypotheses against an appropriate comparator. We select a sample and compute descriptive statistics on the sample data - including the sample size nthe sample mean and the sample standard deviation s. We then determine the appropriate test statistic Step 2 for the hypothesis test.
Briefly, explain the relationship between confidence interval and hypothesis testing?
The formulas for test statistics depend on the sample size and are given below. Test Statistics for Testing H0: Data are provided for the US population as a whole and for specific ages, sexes and races. An investigator hypothesizes that in expenditures have decreased primarily due to the availability of generic drugs.
To test the hypothesis, a sample of Americans are selected and their expenditures on health care and prescription drugs in are measured. The sample data are summarized as follows: Is there statistical evidence of a reduction in expenditures on health care and prescription drugs in ? We will run the test using the five-step approach.
Set up hypotheses and determine level of significance H0: Select the appropriate test statistic. Set up decision rule. Compute the test statistic. We now substitute the sample data into the formula for the test statistic identified in Step 2.
We do not reject H0 because In summarizing this test, we conclude that we do not have sufficient evidence to reject H0. We do not conclude that H0 is true, because there may be a moderate to high probability that we committed a Type II error.
It is possible that the sample size is not large enough to detect a difference in mean expenditures. The NCHS reported that the mean total cholesterol level in for all adults was Total cholesterol levels in participants who attended the seventh examination of the Offspring in the Framingham Heart Study are summarized as follows: Is there statistical evidence of a difference in mean cholesterol levels in the Framingham Offspring?
Here we want to assess whether the sample mean of We reject H0 because Because we reject H0, we also approximate a p-value. Statistical Significance versus Clinical Practical Significance This example raises an important concept of statistical versus clinical or practical significance.