# Confidence interval and hypothesis testing relationship quotes

All of these things "happened" due to hypothesis tests. used interval estimation with no recognition of any of the relationships to these things. There is an extremely close relationship between confidence intervals and hypothesis testing. When a 95% confidence interval is constructed, all values in the. Many published statistical analyses quote P-values as ≥ (not Statistical significance of the 95% confidence interval when P difference between sample means clinically significant.

In this article, we consider this further and discuss how to interpret the results. More on choosing an appropriate statistical test Deciding which statistical test to use to analyse a set of data depends on the type of data interval or categorical, paired vs unpaired being analysed and whether or not the data are normally distributed. Interpretation of the results of statistical analysis relies on an appreciation and consideration of the null hypothesis, P-values, the concept of statistical vs clinical significance, study power, types I and II statistical errors, the pitfalls of multiple comparisons, and one vs two-tailed tests before conducting the study.

Assessing whether a data set follows a normal distribution It may be apparent from constructing a histogram or frequency curve that the data follow a normal distribution. The data may be subjected to formal statistical analysis for evidence of normality using one or more specific tests usually included in computer software packages, such as the Shapiro—Wilkes test.

However, the choice between parametric and non-parametric statistical analysis is less important with samples of this size as both analyses are almost equally powerful and give similar results. When in doubt as to the type of distribution that the sample data follow, particularly when the sample size is small, non-parametric analysis should be undertaken, accepting that the analysis may lack power. The best solution to avoiding mistakes in choosing the appropriate statistical test for analysis of data is to design a study with sufficiently large numbers of subjects in each group.

Unpaired vs paired data When comparing the effects of an intervention on sample groups in a clinical study, it is essential that the groups are as similar as possible, differing only in respect of the intervention of interest. One common method of achieving this is to recruit subjects into study groups by random allocation. All subjects recruited should have an equal chance of being allocated into any of the study groups. Provided the sample sizes are large enough, the randomization process should ensure that group differences in variables that may influence outcome of the intervention of interest e.

These variables may themselves be subjected to statistical analysis and the null hypothesis that there is no difference between the study groups tested.

Such a study contains independent groups and unpaired statistical tests are appropriate. An example would be a comparison of the efficacy of two different drugs for the treatment of hypertension.

The data obtained in this study would be paired and subject to paired statistical analysis. The effectiveness of the pairing may be determined by calculating the correlation coefficient and the corresponding P-value of the relationship between data pairs. A third method involves defining all those characteristics that the researcher believes may influence the effect of the intervention of interest and matching the subjects recruited for those characteristics.

This method is potentially unreliable, depending as it does on ensuring that key characteristics are not inadvertently overlooked and therefore not controlled. The main advantage of the paired over the unpaired study design is that paired statistical tests are more powerful and fewer subjects need to be recruited in order to prove a given difference between the study groups. Against this are pragmatic difficulties and additional time needed for crossover studies, and the danger that, despite a washout period, there may still be an influence of the first treatment on the second.

The pitfalls of matching patients for all important characteristics also have to be considered. The null hypothesis and P-values Before undertaking statistical analysis of data, a null hypothesis is proposed, that is, there is no difference between the study groups with respect to the variable s of interest i.

Once the null hypothesis has been defined, statistical methods are used to calculate the probability of observing the data obtained or data more extreme from the prediction of the null hypothesis if the null hypothesis is true.

For example, we may obtain two sample data sets which appear to be from different populations when we examine the data. Hence, the specified confidence level is called the coverage probability. As Neyman stressed repeatedly, this coverage probability is a property of a long sequence of confidence intervals computed from valid models, rather than a property of any single confidence interval.

Many journals now require confidence intervals, but most textbooks and studies discuss P values only for the null hypothesis of no effect. This exclusive focus on null hypotheses in testing not only contributes to misunderstanding of tests and underappreciation of estimation, but also obscures the close relationship between P values and confidence intervals, as well as the weaknesses they share. Therefore, based on the articles in our reference list, we review prevalent P value misinterpretations as a way of moving toward defensible interpretations and presentations.

We adopt the format of Goodman [ 40 ] in providing a list of misinterpretations that can be used to critically evaluate conclusions offered by research reports and reviews. The P value assumes the test hypothesis is true—it is not a hypothesis probability and may be far from any reasonable probability for the test hypothesis. The P value simply indicates the degree to which the data conform to the pattern predicted by the test hypothesis and all the other assumptions used in the test the underlying statistical model.

ThePvalue for the null hypothesis is the probability that chance alone produced the observed association; for example, if thePvalue for the null hypothesis is 0. This is a common variation of the first fallacy and it is just as false. To say that chance alone produced the observed association is logically equivalent to asserting that every assumption used to compute the P value is correct, including the null hypothesis. Thus to claim that the null P value is the probability that chance alone produced the observed association is completely backwards: The P value is a probability computed assuming chance was operating alone.

The absurdity of the common backwards interpretation might be appreciated by pondering how the P value, which is a probability deduced from a set of assumptions the statistical modelcan possibly refer to the probability of those assumptions. A small P value simply flags the data as being unusual if all the assumptions used to compute it including the test hypothesis were correct; it may be small because there was a large random error or because some assumption other than the test hypothesis was violated for example, the assumption that this P value was not selected for presentation because it was below 0.

A large P value only suggests that the data are not unusual if all the assumptions used to compute the P value including the test hypothesis were correct. The same data would also not be unusual under many other hypotheses. Furthermore, even if the test hypothesis is wrong, the P value may be large because it was inflated by a large random error or because of some other erroneous assumption for example, the assumption that this P value was not selected for presentation because it was above 0.

A largePvalue is evidence in favor of the test hypothesis. In fact, any P value less than 1 implies that the test hypothesis is not the hypothesis most compatible with the data, because any other hypothesis with a larger P value would be even more compatible with the data.

A P value cannot be said to favor the test hypothesis except in relation to those hypotheses with smaller P values. Furthermore, a large P value often indicates only that the data are incapable of discriminating among many competing hypotheses as would be seen immediately by examining the range of the confidence interval.

A null-hypothesisPvalue greater than 0. If the null P value is less than 1 some association must be present in the data, and one must look at the point estimate to determine the effect size most compatible with the data under the assumed model. Statistical significance indicates a scientifically or substantively important relation has been detected. Especially when a study is large, very minor effects or small assumption violations can lead to statistically significant tests of the null hypothesis.

Again, a small null P value simply flags the data as being unusual if all the assumptions used to compute it including the null hypothesis were correct; but the way the data are unusual might be of no clinical interest.

One must look at the confidence interval to determine which effect sizes of scientific or other substantive e. Lack of statistical significance indicates that the effect size is small. A large null P value simply flags the data as not being unusual if all the assumptions used to compute it including the test hypothesis were correct; but the same data will also not be unusual under many other models and hypotheses besides the null.

Again, one must look at the confidence interval to determine whether it includes effect sizes of importance. And again, the P value refers to a data frequency when all the assumptions used to compute it are correct.

In addition to the test hypothesis, these assumptions include randomness in sampling, treatment assignment, loss, and missingness, as well as an assumption that the P value was not selected for presentation based on its size or some other aspect of the results. To see why this description is false, suppose the test hypothesis is in fact true. It does not refer to your single use of the test, which may have been thrown off by assumption violations as well as random errors.

This is yet another version of misinterpretation 1. Pvalues are properly reported as inequalities e. This is bad practice because it makes it difficult or impossible for the reader to accurately interpret the statistical result.

Only when the P value is very small e. There is little practical difference among very small P values when the assumptions used to compute P values are not known with enough certainty to justify such precision, and most methods for computing P values are not numerically accurate below a certain point. Statistical significance is a property of the phenomenon being studied, and thus statistical tests detect significance.

## Confidence Intervals & Hypothesis Testing (1 of 5)

The effect being tested either exists or does not exist. 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. 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.

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.

### Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations

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.

### Statistics IV: Interpreting the results of statistical tests | BJA Education | Oxford Academic

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.

**Hypothesis Test and Confidence Interval - Statistics Tutorial #15 - MarinStatsLectures**

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: