Special article
Importance of statistical power and hypothesis in P value
Importancia de la potencia y la hipótesis en el valor p
a Departamento de Estadística e Investigación Operativa, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain
b Institut Universitari d’Investigació en Atenció Primària Jordi Gol (IDIAP Jordi Gol), Barcelona, Spain
c Servei d’Epidemiologia, Agència de Salut Pública de Barcelona, Barcelona, Spain
d Centro de Investigación Biomédica en Red de Epidemiología y Salud Pública (CIBERESP), Barcelona, Spain
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See text for explanations. <span class="elsevierStyleItalic">Source</span>: compiled by the authors from program R.</p>" ] ] ] "textoCompleto" => "<span class="elsevierStyleSections"><p id="par0005" class="elsevierStylePara elsevierViewall">Readers of Clinical Medicine know the importance of defining the denominator of a ratio to estimate a probability: “There is not the same probability in a Catholic being Pope than a Pope being Catholic”.<a class="elsevierStyleCrossRef" href="#bib0080"><span class="elsevierStyleSup">1</span></a> Similarly, for a diagnosis, there is not the same probability that a sick-person tests positive (<span class="elsevierStyleItalic">sensitivity</span>) than a case that has tested positive is sick (<span class="elsevierStyleItalic">positive predictive value</span>)<span class="elsevierStyleItalic">.</span></p><p id="par0010" class="elsevierStylePara elsevierViewall">The <span class="elsevierStyleItalic">p</span> value (or value of <span class="elsevierStyleItalic">p</span>, or <span class="elsevierStyleItalic">p</span>-value, or simply <span class="elsevierStyleItalic">p</span>) bears some analogy with the diagnostic probabilities, as it is defined as the probability of obtaining an equally or more significant result than the observed – testing positive in the diagnostic test–assuming a certain hypothesis H: the patient is healthy. However, a researcher or a clinician may find it more interesting to know the positive value of a test: how likely a hypothesis H is–that the patient is sick – having seen extreme results.</p><p id="par0015" class="elsevierStylePara elsevierViewall">In research, replicability and transparency of an experiment is crucial because it brings us closer to scientific rigour: any independent investigator should be able to replicate our results.<a class="elsevierStyleCrossRefs" href="#bib0085"><span class="elsevierStyleSup">2–4</span></a> We will see that this replicability is closely related to the prior probability that the alternative hypothesis is true, with potency and multiplicity.</p><p id="par0020" class="elsevierStylePara elsevierViewall">A low replicability has important implications. For example, if previous research in animals had not been done with the utmost rigour, subsequent studies with human volunteers will be of no value.<a class="elsevierStyleCrossRef" href="#bib0100"><span class="elsevierStyleSup">5</span></a> The same applies to material investment: <span class="elsevierStyleItalic">The Lancet</span> recently devoted a series of 5 articles and 2 editorials denouncing the waste of resources in research.<a class="elsevierStyleCrossRef" href="#bib0105"><span class="elsevierStyleSup">6</span></a> The indiscriminate use of the <span class="elsevierStyleItalic">p</span> value contributes to all this: Motulsky<a class="elsevierStyleCrossRef" href="#bib0110"><span class="elsevierStyleSup">7</span></a> lists some of the abuses: (1) reanalysis of the data in different ways or increasing the sample size to obtain a significant result; (2) over-interpreting the <span class="elsevierStyleItalic">p</span> value regardless of the estimated effect size; (3) conducting multiple tests for the same hypothesis uncorrected for multiplicity, and (4) over-reliance on standard errors that have often been misinterpreted. All this means that the <span class="elsevierStyleItalic">p</span> value is not necessarily an indicator of the replicability of results.<a class="elsevierStyleCrossRef" href="#bib0115"><span class="elsevierStyleSup">8</span></a></p><p id="par0025" class="elsevierStylePara elsevierViewall">So how do we know that a study is replicable? Replicability and consistency will be higher the greater the: (1) theoretical and empirical basis for the hypothesis; (2) potency of the study, and (3) control of multiplicity. The following shows 2 applications of this.</p><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0005">Analogy</span><p id="par0030" class="elsevierStylePara elsevierViewall">We will begin with an analogy of diagnostic tests: What influences the predictive value of a positive result? The reader can find at this website <a id="intr0010" class="elsevierStyleInterRef" href="http://shiny-eio.upc.edu/bne/VPs">http://shiny-eio.upc.edu/bne/VPs</a> an application that provides the predictive values of test results. Suppose an elderly, 78-year-old woman has a suspected pulmonary embolism after abdominal surgery.<a class="elsevierStyleCrossRef" href="#bib0120"><span class="elsevierStyleSup">9</span></a> Doctors agree that the probability of the disorder for her age range is around 70%. This value represents the probability <span class="elsevierStyleItalic">a priori</span>. And they recommend a pulmonary ventilation-perfusion scintigraphy that has a sensitivity of 40% and a specificity of 98%. By using the application with this data (<a class="elsevierStyleCrossRef" href="#fig0005">Fig. 1</a>A) it follows that the probability <span class="elsevierStyleItalic">a posterior</span> of the disease is 97.9%. The test has provided additional information and increased the likelihood of the disorder by 70% prior (<span class="elsevierStyleItalic">a priori</span>) to 97.9%, after the positive result (<span class="elsevierStyleItalic">a posteriori</span>)<span class="elsevierStyleItalic">.</span> In another example, a 28-year-old man with the <span class="elsevierStyleItalic">a priori</span> probability of 20% would have a <span class="elsevierStyleItalic">a posterior</span> probability of 83.3% (<a class="elsevierStyleCrossRef" href="#fig0005">Fig. 1</a>B). In both cases, the test provides considerable evidence. But what would happen if the specificity was 66% instead of 98% (<a class="elsevierStyleCrossRef" href="#fig0005">Fig. 1</a>C and D)? In this case, the <span class="elsevierStyleItalic">a posterior</span> probabilities of the elderly lady and the young man have increased only from 70 to 73.3% and from 20 to 22.7%. That is, the test would have added very little information.</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0010">What influences replicability?</span><p id="par0035" class="elsevierStylePara elsevierViewall">What happens to the credibility that deserves a scientific result? The application is now at <a id="intr0015" class="elsevierStyleInterRef" href="http://shiny-eio.upc.edu/bne/efectos">http://shiny-eio.upc.edu/bne/efectos</a>. Let's suppose that in a given intervention, the pilot studies – Phase II, proof of concept or feasibility, according to terminology – have been allowed to progress from the R&D stage. Imagine that these previous studies have allowed the selection of interventions so that 3 out of 4 are really effective (<span class="elsevierStyleItalic">a priori</span> expectation of real effect: 75%). Within this scenario, the sample size necessary to guarantee, for example, a potency of 80% to detect such an effect by limiting the <span class="elsevierStyleItalic">α</span> risk is calculated (concluding that there is actually an effect when in fact there is not) to 5% unilateral (1<span class="elsevierStyleHsp" style=""></span>−<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">α</span><span class="elsevierStyleHsp" style=""></span>=<span class="elsevierStyleHsp" style=""></span>0.95). If the experiment reached statistical significance, the application generates that confidence in a positive outcome (ratio of actual effects among the significant results) is 98% (<a class="elsevierStyleCrossRef" href="#fig0010">Fig. 2</a>A), a figure that anticipates future replication.</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia><p id="par0040" class="elsevierStylePara elsevierViewall">Now let's see 3 ways in which we can compromise this good result. First, imagine that the research team has not pre-developed their hypothesis well and aims to directly conduct the confirmatory study on which they will base subsequent decisions. Let's assume that these false short-cuts lower expectation of the effect by 10%. In this case, confidence in a positive result decreases to 64% (<a class="elsevierStyleCrossRef" href="#fig0010">Fig. 2</a>B). Secondly, if these authors do not have many cases, the study will have little potency. Again, the application shows that for a 30% potency, the confidence in a positive result decreases to 40% (<a class="elsevierStyleCrossRef" href="#fig0010">Fig. 2</a>C). And now it is more likely that this significant result actually comes from an intervention without effect. Finally, let's verify the consequences of violating the third parameter <span class="elsevierStyleItalic">α</span>. Imagine a researcher decides to do many hypothesis tests (what is known as a “fishing expedition”) on different variables, or on the same variable repeated on different occasions, either over time or in different measurement conditions. This multiplicity will result in the loss of control of the <span class="elsevierStyleItalic">α</span> risk and would distort the interpretation of the results. If, for example, the researcher calculates 14 <span class="elsevierStyleItalic">p</span> values in separate independent tests, the probability that simply by chance at least one is significant, is slightly greater than 50%. In this situation, confidence in a significant result decreases to 6.2% (<a class="elsevierStyleCrossRef" href="#fig0010">Fig. 2</a>D). Thus, (1) without a progressive advancement of R&D (expectation of the effect of 10%); (2) without an appropriate sample design and size (30% potency), and (3) without control of the <span class="elsevierStyleItalic">α</span> risk (50%), only 6 out of 100 significant interventions have a real effect–that is, they can be replicated in the future.</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0015">Recommendations</span><p id="par0045" class="elsevierStylePara elsevierViewall">This alarming situation of “mistrust” in a significant result was described by John Ioannidis in 2005,<a class="elsevierStyleCrossRef" href="#bib0125"><span class="elsevierStyleSup">10</span></a> where clinical trials and meta-analyses with an <span class="elsevierStyleItalic">a priori</span> evidence of at least 50%, an <span class="elsevierStyleItalic">α</span> of 5% and a potency of 80% involve an <span class="elsevierStyleItalic">a posterior</span> probability of real evidence of 94%. Moreover, exploratory expeditions in search of hypotheses at the beginning of R&D with an <span class="elsevierStyleItalic">a priori</span> evidence that Ioannidis quantifies below 1% and without control of the probability of Type II error (lets suppose <span class="elsevierStyleItalic">β</span><span class="elsevierStyleHsp" style=""></span>=<span class="elsevierStyleHsp" style=""></span>80%) can result in probabilities of 3% of real findings. Along these lines, the journal <span class="elsevierStyleItalic">Basic and Applied Social Psychology</span> has decided to remove the <span class="elsevierStyleItalic">p</span> values from their originals from 2015,<a class="elsevierStyleCrossRef" href="#bib0130"><span class="elsevierStyleSup">11</span></a> causing much controversy in doing so.<a class="elsevierStyleCrossRefs" href="#bib0135"><span class="elsevierStyleSup">12–14</span></a></p><p id="par0050" class="elsevierStylePara elsevierViewall">The methodology of a researcher during their R&D process must be held in absolute transparency, in regard to the hypothesis situation. There is no shame in suggesting, rather than contrasting, a hypothesis. On the contrary, it can be argued that it is more novel to propose a new hypothesis than contrasting one that is already known. After all, although many trips were needed to open a trade route to the Americas, the real news was Christopher Columbus’ first voyage, where a chance finding allowed him to create a truly innovative hypothesis.</p><p id="par0055" class="elsevierStylePara elsevierViewall">In short, should we retire the <span class="elsevierStyleItalic">p</span> value? We believe it is not yet the time to do so as it is still relevant. However, we recommend preparing the ground, and in accordance with transparency,<a class="elsevierStyleCrossRef" href="#bib0150"><span class="elsevierStyleSup">15</span></a> authors should report: (1) whether the hypothesis existed prior to the results; (2) whether they had previous calculated potency, and (3) whether they have respected the <span class="elsevierStyleItalic">α</span> risk – for example, if they have made a single test and have been faithful to the anticipated plan. If either of these points were not met, they should end by saying that their results suggest but do not confirm their hypothesis, exposing what features their studies should include to contrast it, so as to thus be able to progress more quickly down the R&D path.</p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0020">Financing</span><p id="par0060" class="elsevierStylePara elsevierViewall">2014 SGR 464 aid from the <span class="elsevierStyleGrantSponsor" id="gs1">Generalitat de Catalunya</span> (JC and KL).</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025">Conflict of interest</span><p id="par0065" class="elsevierStylePara elsevierViewall">The authors declare no conflict of interest.</p></span></span>" "textoCompletoSecciones" => array:1 [ "secciones" => array:7 [ 0 => array:2 [ "identificador" => "sec0005" "titulo" => "Analogy" ] 1 => array:2 [ "identificador" => "sec0010" "titulo" => "What influences replicability?" ] 2 => array:2 [ "identificador" => "sec0015" "titulo" => "Recommendations" ] 3 => array:2 [ "identificador" => "sec0020" "titulo" => "Financing" ] 4 => array:2 [ "identificador" => "sec0025" "titulo" => "Conflict of interest" ] 5 => array:2 [ "identificador" => "xack227162" "titulo" => "Acknowledgements" ] 6 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2015-07-27" "fechaAceptado" => "2015-10-06" "NotaPie" => array:1 [ 0 => array:2 [ "etiqueta" => "☆" "nota" => "<p class="elsevierStyleNotepara" id="npar0005">Please cite this article as: Cortés J, Casals M, Langohr K, González JA. Importancia de la potencia y la hipótesis en el valor p. Med Clin (Barc). 2016;146:178–181.</p>" ] ] "multimedia" => array:2 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Fig. 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 4350 "Ancho" => 2416 "Tamanyo" => 545151 ] ] "descripcion" => array:1 [ "en" => "<p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">A positive result of a diagnostic test that may increase the probability of being significantly sick (A and B) or irrelevant (C and D). See text for explanations. <span class="elsevierStyleItalic">Source</span>: compiled by the authors from program R.</p>" ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Fig. 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 4333 "Ancho" => 2452 "Tamanyo" => 481626 ] ] "descripcion" => array:1 [ "en" => "<p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">Probability of future replication of a significant <span class="elsevierStyleItalic">p</span> value: in a study based on a solid suitably dimensioned previous research (A); under the same conditions, but without solid previous evidence (B); in addition, without suitable dimensioning of the size to ensure its potency (C), and with multiplicity of analysis (D). 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