### Statistical significance - Wikipedia

However, correlation studies the relationship between one variable and another, Finally, the test of significance may show that the two methods are related, but it is . Bland and Altman plot for data from the table 1, with the. Statistical “significance tests” based on this concept have been a central part of . but also obscures the close relationship between P values and confidence intervals, In fact, any P value less than 1 implies that the test hypothesis is not the Hoenig JM, Heisey DM. The abuse of power: the pervasive fallacy of power. With pearson correlation analysis, I got the significant pcorrelation (r=). Popular Answers (1) For one thing, the effect size that you test comes at the design stage. .. Prairie () tried to fix the problem of low p values from large ecological datasets (especially from field.

### Correlation Coefficients

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.

**Basic Excel Business Analytics #51: Testing Significance of Regression Relationship with 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. 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.

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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. These assumptions have gradually come under scrutiny via sensitivity and bias analysis [ 98 ], but such methods remain far removed from the basic statistical training given to most researchers.

Less often stated is the even more crucial assumption that the analyses themselves were not guided toward finding nonsignificance or significance analysis biasand that the analysis results were not reported based on their nonsignificance or significance reporting bias and publication bias. Selective reporting renders false even the limited ideal meanings of statistical significance, P values, and confidence intervals.

Because author decisions to report and editorial decisions to publish results often depend on whether the P value is above or below 0. Although this selection problem has also been subject to sensitivity analysis, there has been a bias in studies of reporting and publication bias: It is usually assumed that these biases favor significance. Addressing such problems would require far more political will and effort than addressing misinterpretation of statistics, such as enforcing registration of trials, along with open data and analysis code from all completed studies as in the AllTrials initiative, http: In the meantime, readers are advised to consider the entire context in which research reports are produced and appear when interpreting the statistics and conclusions offered by the reports.

Conclusions Upon realizing that statistical tests are usually misinterpreted, one may wonder what if anything these tests do for science. They were originally intended to account for random variability as a source of error, thereby sounding a note of caution against overinterpretation of observed associations as true effects or as stronger evidence against null hypotheses than was warranted. We have no doubt that the founders of modern statistical testing would be horrified by common treatments of their invention.

But it has long been asserted that the harms of statistical testing in more uncontrollable and amorphous research settings such as social-science, health, and medical fields have far outweighed its benefits, leading to calls for banning such tests in research reports—again with one journal banning P values as well as confidence intervals [ 2 ].

## Correlation Coefficients

Given, however, the deep entrenchment of statistical testing, as well as the absence of generally accepted alternative methods, there have been many attempts to salvage P values by detaching them from their use in significance tests. One approach is to focus on P values as continuous measures of compatibility, as described earlier.

Although this approach has its own limitations as described in points 1, 2, 5, 9, 15, 18, 19it avoids comparison of P values with arbitrary cutoffs such as 0. Another approach is to teach and use correct relations of P values to hypothesis probabilities.

For example, under common statistical models, one-sided P values can provide lower bounds on probabilities for hypotheses about effect directions [ 4546, ]. Whether such reinterpretations can eventually replace common misinterpretations to good effect remains to be seen. When two methods are compared, neither provides an unequivocally correct measurement, so it could be interesting trying to assess the degree of agreement. To assess this degree of agreement, the correct statistical approach is not obvious.

Many studies give the product—moment correlation coefficient r between the results of two measurement methods as an indicator of agreement. In Altman and Bland re-proposed an alternative analysis, firstly presented by Eksborg in 1based on the quantification of the agreement between two quantitative measurements by studying the mean difference and constructing limits of agreement 2. Correlation and linear regression Correlation is a statistical technique that can show whether, and how strongly, pairs of variables are related.

There are several different correlation techniques, including the Pearson or product-moment correlation, probably the most common one. It is computed as the ratio of covariance between the variables to the product of their standard deviations. The numerical value of r ranges from This enables us to get an idea of the strength of relationship - or rather the strength of linear relationship between the variables.

Usually, a linear regression study is performed together with correlation measurement.

### What statistical analysis should I use? Statistical analyses using Stata

Actually, linear regression can be calculated only if the correlation exists and correlation coefficient can be interpreted only if the P value is significant. However, P is significant and regression can be calculated for most cases of method comparison. Linear regression finds the best line that predicts one variable from the other one.

Linear regression quantifies goodness of fit with r2, the coefficient of determination. Correlation describes linear relationship between two sets of data but not their agreement 3. Moreover, frequently a null hypothesis is used to verify if the two methods are not linearly related. With even a minimal trend, the probability of null hypothesis is very small and it can be safely, but sometimes erroneously, concluded that the two measurement methods are indeed related.

However, the two methods that are designed to measure the same variable should have good correlation when a set of samples are chosen in such manner that the property to be determined varies considerably.

In the case of method comparison, this means that samples should cover a wide concentration range. A high correlation for any two methods designed to measure the same property could thus, in itself just be a sign that one has chosen a widespread sample.

Correlation quantifies the degree to which two variables are related. But a high correlation does not automatically imply that there is good agreement between the two methods. The correlation coefficient and regression technique are sometimes inadequate and can be misleading when assessing agreement, because they evaluate only the linear association of two sets of observations. The r measures the strength of a relation between two variables, not the agreement between them. Similarly, r2, named the coefficient of determination, only tells us the proportion of variance that the two variables have in common.

Finally, the test of significance may show that the two methods are related, but it is obvious that two methods designed to measure the same variable are related.