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14-September-2008 18:02:38 - Statistical significance In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. A statistically significant difference simply means there is statistical evidence that there is a difference; it does not mean the difference is necessarily large, important, or significant in the common meaning of the word. The significance level of a test is a traditional frequentist statistical hypothesis testing concept. In simple cases, it is defined as the probability of making a decision to reject the null hypothesis when the null hypothesis is actually true a decision known as a Type I error, or false positive determination. The decision is often made using the p-value: if the p-value is less than the significance level, then the null hypothesis is rejected. The smaller the p-value, the more significant the result is said to be. In more complicated, but practically important cases, the significance level of a test is a probability such that the probablility of making a decision to reject the null hypothesis when the null hypothesis is actually true is no more than the stated probability. This allows for those applications where the probability of deciding to reject may be much smaller than the significance level for some sets of assumptions encompassed within the null hypothesis. Contents 1 Use in practice 2 Pitfalls 3 Signal-noise ratio conceptualisation of significance 4 See also 5 References 6 External links Use in practice The significance level is usually represented by the Greek symbol, α alpha. Popular levels of significance are 5%, 1% and 0.1%. If a test of significance gives a p-value lower than the α-level, the null hypothesis is rejected. Such results are informally referred to as 'statistically significant'. For example, if someone argues that there's only one chance in a thousand this could have happened by coincidence, a 0.1% level of statistical significance is being implied. The lower the significance level, the stronger the evidence. In some situations it is convenient to express the statistical significance as 1 - α. In general, when interpreting a stated significance, one must be careful to note what, precisely, is being tested statistically. Different α-levels have different advantages and disadvantages. Smaller α-levels give greater confidence in the determination of significance, but run greater risks of failing to reject a false null hypothesis a Type II error, or false negative determination, and so have less statistical power. The selection of an α-level inevitably involves a compromise between significance and power, and consequently between the Type I error and the Type II error. In some fields, for example nuclear and particle physics, it is common to express statistical significance in units of σ sigma, the standard deviation of a Gaussian distribution. A statistical significance of nσ can be converted into a value of α via use of the error function: \alpha = 1 - \operatornameerfn/\sqrt2 The use of σ is motivated by the ubiquitous emergence of the Gaussian distribution in measurement uncertainties. For example, if a theory predicts a parameter to have a value of, say, 100, and one measures the parameter to be 109 ± 3, then one might report the measurement as a 3σ deviation from the theoretical prediction. In terms of α, this statement is equivalent to saying that assuming the theory is true, the likelihood of obtaining the experimental result by coincidence is 0.27% since 1 - erf3/√2 = 0.0027. Fixed significance levels such as those mentioned above may be regarded as useful in exploratory data analyses. However, modern statistical advice is that, where the outcome of a test is essentially the final outcome of an experiment or other study, the p-value should be quoted explicitly. And, importantly, it should be quoted whether or not the p-value is judged to be significant. This is to allow maximum information to be transferred from a summary of the study into meta-analyses. Pitfalls A common misconception is that a statistically significant result is always of practical significance, or demonstrates a large effect in the population. Unfortunately, this problem is commonly encountered in scientific writing. Given a sufficiently large sample, extremely small and non-notable differences can be found to be statistically significant, and statistical significance says nothing about the practical significance of a difference. One of the more common problems in significance testing is the tendency for multiple comparisons to yield spurious significant differences even where the null hypothesis is true. For instance, in a study of twenty comparisons, using an α-level of 5%, one comparison will likely yield a significant result despite the null hypothesis being true. In these cases p-values are adjusted in order to control either the false discovery rate or the familywise error rate. An additional problem is that frequentist analyses of p-values are considered by some to overstate statistical significance.12 See Bayes factor for details. Yet another common pitfall often happens when a researcher writes the ambiguous statement we found no statistically significant difference, which is then misquoted by others as they found that there was no difference. Actually, statistics cannot be used to prove that there is exactly zero difference between two populations. Failing to find evidence that there is a difference does not constitute evidence that there is no difference. This principle is sometimes described by the maxim Absence of evidence is not evidence of absence. According to J. Scott Armstrong, attempts to educate researchers on how to avoid pitfalls of using statistical significance have had little success. In the papers Significance Tests Harm Progress in Forecasting,3 and Statistical Significance Tests are Unnecessary Even When Properly Done,4 Armstrong makes the case that even when done properly, statistical significance tests are of no value. A number of attempts failed to find empirical evidence supporting the use of significance tests. Tests of statistical significance are harmful to the development of scientific knowledge because they distract researchers from the use of proper methods. Armstrong suggests authors should avoid tests of statistical significance; instead, they should report on effect sizes, confidence intervals, replications/extensions, and meta-analyses. Signal-noise ratio conceptualisation of significance Statistical significance can be considered to be the confidence one has in a given result. In a comparison study, it is dependent on the relative difference between the groups compared, the amount of measurement and the noise associated with the measurement. In other words, the confidence one has in a given result being non-random i.e. it is not a consequence of chance depends on the signal-to-noise ratio SNR and the sample size. Expressed mathematically, the confidence that a result is not by random chance is given by the following formula by Sackett:5 \mathrmconfidence = \frac\mathrmsignal\mathrmnoise \times \sqrt\mathrmsample\ size. For clarity, the above formula is presented in tabular form below. Dependence of confidence with noise, signal and sample size tabular form Parameter Parameter increases Parameter decreases Noise Confidence decreases Confidence increases Signal Confidence increases Confidence decreases Sample size Confidence increases Confidence decreases In words, the dependence of confidence is high if the noise is low and/or the sample size is large and/or the effect size signal is large. The confidence of a result and its associated confidence interval is not dependent on effect size alone. If the sample size is large and the noise is low a small effect size can be measured with great confidence. Whether a small effect size is considered important is dependent on the context of the events compared. In medicine, small effect sizes reflected by small increases of risk are often considered clinically relevant and are frequently used to guide treatment decisions if there is great confidence in them. Whether a given treatment is considered a worthy endeavour is dependent on the risks, benefits and costs. See also Statistics portal A/B testing ABX test Fisher's method for combining independent tests of significance Reasonable doubt References ^ Goodman S 1999. Toward evidence-based medical statistics. 1: The P value fallacy.. Ann Intern Med 130 12: 995-1004. PMID 10383371. ^ Goodman S 1999. Toward evidence-based medical statistics. 2: The Bayes factor.. Ann Intern Med 130 12: 1005-13. PMID 10383350. ^ Armstrong, J. Scott 2007. Significance tests harm progress in forecasting. International Journal of Forecasting 23: 321-327. doi:10.1016/j.ijforecast.2007.03.004. Full Text ^ Armstrong, J. Scott 2007. Statistical Significance Tests are Unnecessary Even When Properly Done. International Journal of Forecasting 23: 335-336. doi:10.1016/j.ijforecast.2007.01.010. Full Text ^ Sackett DL. Why randomized controlled trials fail but needn't: 2. Failure to employ physiological statistics, or the only formula a clinician-trialist is ever likely to need or understand!. CMAJ. 2001 Oct 30;1659:1226-37. PMID 11706914. Free Full Text. External links Raymond Hubbard, M.J. Bayarri, P Values are not Error Probabilities. A working paper that explains the difference between Fisher's evidential p-value and the Neyman-Pearson Type I error rate α. The Concept of Statistical Significance Testing - Article by Bruce Thompon of the ERIC Clearinghouse on Assessment and Evaluation, Washington, D.C. v d e Statistics Experimental design Design of experiments - Sampling - Stratified sampling - Replication - Blocking - Statistical power - Sample size Descriptive statistics Continuous data Mean Arithmetic, Geometric, Harmonic - Median - Mode - Range - Variance - Standard deviation - Skewness - Kurtosis - Effect size Categorical data Frequency - Contingency table Inferential statistics Bayesian inference - Hypothesis testing - Significance - Power - Null hypothesis/Alternative hypothesis/Error - Z-test - Student's t-test - Chi-square test - F-test - P-value - Interval estimation - Maximum likelihood - Analysis of variance - Meta-analysis - Confidence interval Survival analysis Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models Correlation Confounding variable - Pearson product-moment correlation coefficient - Rank correlation Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient Regression analysis Linear regression - Nonlinear regression - Logistic regression - Maximum Likelihood Estimation Statistical graphics Bar chart - Biplot - Box plot - Control chart - Histogram - Q-Q plot - Run chart - Scatterplot - Stemplot Category · Project · Portal · List of topics Retrieved from http://en..org/wiki/Statistical_significance Categories: Hypothesis testing Views Article Discussion this page History Personal tools Log in / create account Navigation Main page Contents Featured content Current events Random article Search Go Search Interaction Community portal Recent changes Contact Donate to Help Toolbox What links here Related changes Upload file Special pages Printable version Permanent link Cite this page Languages Deutsch Español Français í•œêµì–´ Italiano עברית Lietuvių Nederlands 日本語 Polski Português РуÑ?Ñ?кий Suomi Svenska ä¸æ–‡ This page was last modified on 20 August 2008, at 14:00
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