# Confidence intervals and value relationship in numbers

### Confidence interval - Wikipedia

In statistics, a confidence interval or compatibility interval (CI) is a type of interval estimate, Confidence intervals consist of a range of potential values of the unknown Philosophical issues; Relationship with other statistical topics. Another way of thinking about a confidence interval is that it is the range of likely values of the parameter (defined as the point estimate + margin of error) with a. How to interpret odds ratios, confidence intervals and p values with a stepwise Contents: Introduction. Odds ratio. Confidence interval. P value .. which indicates no relationship between decreased rates of breast cancer.

Since the interval contains zero no differencewe do not have sufficient evidence to conclude that there is a difference. Confidence Intervals for Matched Samples, Continuous Outcome The previous section dealt with confidence intervals for the difference in means between two independent groups. There is an alternative study design in which two comparison groups are dependent, matched or paired.

Consider the following scenarios: A single sample of participants and each participant is measured twice, once before and then after an intervention.

## The Relationship Between Confidence Intervals and p-values

A single sample of participants and each participant is measured twice under two different experimental conditions e. A goal of these studies might be to compare the mean scores measured before and after the intervention, or to compare the mean scores obtained with the two conditions in a crossover study.

Yet another scenario is one in which matched samples are used. For example, we might be interested in the difference in an outcome between twins or between siblings. Once again we have two samples, and the goal is to compare the two means. However, the samples are related or dependent. In the first scenario, before and after measurements are taken in the same individual. In the last scenario, measures are taken in pairs of individuals from the same family.

**The Normal Distribution and the 68-95-99.7 Rule**

When the samples are dependent, we cannot use the techniques in the previous section to compare means. Because the samples are dependent, statistical techniques that account for the dependency must be used. These techniques focus on difference scores i. The Unit of Analysis This distinction between independent and dependent samples emphasizes the importance of appropriately identifying the unit of analysis, i. In the one sample and two independent samples applications participants are the units of analysis.

Nonetheless, it is also possible to test other effect sizes. We may also test hypotheses that the effect does or does not fall within a specific range; for example, we may test the hypothesis that the effect is no greater than a particular amount, in which case the hypothesis is said to be a one-sided or dividing hypothesis [ 78 ].

Much statistical teaching and practice has developed a strong and unhealthy focus on the idea that the main aim of a study should be to test null hypotheses. This exclusive focus on null hypotheses contributes to misunderstanding of tests. Adding to the misunderstanding is that many authors including R.

The focus of traditional definitions of P values and statistical significance has been on null hypotheses, treating all other assumptions used to compute the P value as if they were known to be correct. Recognizing that these other assumptions are often questionable if not unwarranted, we will adopt a more general view of the P value as a statistical summary of the compatibility between the observed data and what we would predict or expect to see if we knew the entire statistical model all the assumptions used to compute the P value were correct.

Specifically, the distance between the data and the model prediction is measured using a test statistic such as a t-statistic or a Chi squared statistic. The P value is then the probability that the chosen test statistic would have been at least as large as its observed value if every model assumption were correct, including the test hypothesis. This definition embodies a crucial point lost in traditional definitions: In logical terms, the P value tests all the assumptions about how the data were generated the entire modelnot just the targeted hypothesis it is supposed to test such as a null hypothesis.

Furthermore, these assumptions include far more than what are traditionally presented as modeling or probability assumptions—they include assumptions about the conduct of the analysis, for example that intermediate analysis results were not used to determine which analyses would be presented. It is true that the smaller the P value, the more unusual the data would be if every single assumption were correct; but a very small P value does not tell us which assumption is incorrect.

For example, the P value may be very small because the targeted hypothesis is false; but it may instead or in addition be very small because the study protocols were violated, or because it was selected for presentation based on its small size.

Conversely, a large P value indicates only that the data are not unusual under the model, but does not imply that the model or any aspect of it such as the targeted hypothesis is correct; it may instead or in addition be large because again the study protocols were violated, or because it was selected for presentation based on its large size.

## Confidence interval

The general definition of a P value may help one to understand why statistical tests tell us much less than what many think they do: Not only does a P value not tell us whether the hypothesis targeted for testing is true or not; it says nothing specifically related to that hypothesis unless we can be completely assured that every other assumption used for its computation is correct—an assurance that is lacking in far too many studies.

Nonetheless, the P value can be viewed as a continuous measure of the compatibility between the data and the entire model used to compute it, ranging from 0 for complete incompatibility to 1 for perfect compatibility, and in this sense may be viewed as measuring the fit of the model to the data. Their difference is profound: In contrast, the P value is a number computed from the data and thus an analysis result, unknown until it is computed.

Moving from tests to estimates We can vary the test hypothesis while leaving other assumptions unchanged, to see how the P value differs across competing test hypotheses. Confidence intervals are examples of interval estimates. Neyman [ 76 ] proposed the construction of confidence intervals in this way because they have the following property: 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. 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. 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.

- Confidence Intervals and p-Values
- A beginner’s guide to interpreting odds ratios, confidence intervals and p-values
- Confidence Intervals

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.

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

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.