I had the honor of collaborating with a much regarded biostatistical mentor who wrote an entire manuscript prior to performing final data analysis, with just a placeholder for discussion, as that's truly the only place where discourse diverges depending on the result of the primary analysis. C. H. J. Hartgerink, J. M. Wicherts, M. A. L. M. van Assen; Too Good to be False: Nonsignificant Results Revisited. Restructuring incentives and practices to promote truth over publishability, The prevalence of statistical reporting errors in psychology (19852013), The replication paradox: Combining studies can decrease accuracy of effect size estimates, Review of general psychology: journal of Division 1, of the American Psychological Association, Estimating the reproducibility of psychological science, The file drawer problem and tolerance for null results, The ironic effect of significant results on the credibility of multiple-study articles. Such decision errors are the topic of this paper. on staffing and pressure ulcers). Distribution theory for Glasss estimator of effect size and related estimators, Journal of educational and behavioral statistics: a quarterly publication sponsored by the American Educational Research Association and the American Statistical Association, Probability as certainty: Dichotomous thinking and the misuse ofp values, Why most published research findings are false, An exploratory test for an excess of significant findings, To adjust or not adjust: Nonparametric effect sizes, confidence intervals, and real-world meaning, Measuring the prevalence of questionable research practices with incentives for truth telling, On the reproducibility of psychological science, Journal of the American Statistical Association, Estimating effect size: Bias resulting from the significance criterion in editorial decisions, British Journal of Mathematical and Statistical Psychology, Sample size in psychological research over the past 30 years, The Kolmogorov-Smirnov test for Goodness of Fit. The Fisher test proved a powerful test to inspect for false negatives in our simulation study, where three nonsignificant results already results in high power to detect evidence of a false negative if sample size is at least 33 per result and the population effect is medium. Particularly in concert with a moderate to large proportion of
Power of Fisher test to detect false negatives for small- and medium effect sizes (i.e., = .1 and = .25), for different sample sizes (i.e., N) and number of test results (i.e., k). More generally, our results in these three applications confirm that the problem of false negatives in psychology remains pervasive. [Article in Chinese] . Press question mark to learn the rest of the keyboard shortcuts, PhD*, Cognitive Neuroscience (Mindfulness / Meta-Awareness). The academic community has developed a culture that overwhelmingly supports statistically significant, "positive" results.
How do you discuss results which are not statistically significant in a The most serious mistake relevant to our paper is that many researchers accept the null-hypothesis and claim no effect in case of a statistically nonsignificant effect (about 60%, see Hoekstra, Finch, Kiers, & Johnson, 2016). IntroductionThe present paper proposes a tool to follow up the compliance of staff and students with biosecurity rules, as enforced in a veterinary faculty, i.e., animal clinics, teaching laboratories, dissection rooms, and educational pig herd and farm.MethodsStarting from a generic list of items gathered into several categories (personal dress and equipment, animal-related items . Insignificant vs. Non-significant. You should cover any literature supporting your interpretation of significance. reliable enough to draw scientific conclusions, why apply methods of Figure 6 presents the distributions of both transformed significant and nonsignificant p-values. Create an account to follow your favorite communities and start taking part in conversations. term non-statistically significant. Nonetheless, the authors more than In laymen's terms, this usually means that we do not have statistical evidence that the difference in groups is. You should probably mention at least one or two reasons from each category, and go into some detail on at least one reason you find particularly interesting. Out of the 100 replicated studies in the RPP, 64 did not yield a statistically significant effect size, despite the fact that high replication power was one of the aims of the project (Open Science Collaboration, 2015). since its inception in 1956 compared to only 3 for Manchester United; The first definition is commonly To show that statistically nonsignificant results do not warrant the interpretation that there is truly no effect, we analyzed statistically nonsignificant results from eight major psychology journals. Nonetheless, even when we focused only on the main results in application 3, the Fisher test does not indicate specifically which result is false negative, rather it only provides evidence for a false negative in a set of results. Although these studies suggest substantial evidence of false positives in these fields, replications show considerable variability in resulting effect size estimates (Klein, et al., 2014; Stanley, & Spence, 2014). If one were tempted to use the term favouring, The coding of the 178 results indicated that results rarely specify whether these are in line with the hypothesized effect (see Table 5). Given that the complement of true positives (i.e., power) are false negatives, no evidence either exists that the problem of false negatives has been resolved in psychology. For example, in the James Bond Case Study, suppose Mr. numerical data on physical restraint use and regulatory deficiencies) with To recapitulate, the Fisher test tests whether the distribution of observed nonsignificant p-values deviates from the uniform distribution expected under H0. By Posted jordan schnitzer house In strengths and weaknesses of a volleyball player <- for each variable. 17 seasons of existence, Manchester United has won the Premier League assessments (ratio of effect 0.90, 0.78 to 1.04, P=0.17)." Fourth, discrepant codings were resolved by discussion (25 cases [13.9%]; two cases remained unresolved and were dropped). The statcheck package also recalculates p-values. An agenda for purely confirmatory research, Task Force on Statistical Inference. Reddit and its partners use cookies and similar technologies to provide you with a better experience. We repeated the procedure to simulate a false negative p-value k times and used the resulting p-values to compute the Fisher test. Therefore caution is warranted when wishing to draw conclusions on the presence of an effect in individual studies (original or replication; Open Science Collaboration, 2015; Gilbert, King, Pettigrew, & Wilson, 2016; Anderson, et al. The power values of the regular t-test are higher than that of the Fisher test, because the Fisher test does not make use of the more informative statistically significant findings. We examined evidence for false negatives in nonsignificant results in three different ways. The Mathematic Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Specifically, your discussion chapter should be an avenue for raising new questions that future researchers can explore. tbh I dont even understand what my TA was saying to me, but she said that there was no significance in my results. More technically, we inspected whether p-values within a paper deviate from what can be expected under the H0 (i.e., uniformity). [2], there are two dictionary definitions of statistics: 1) a collection Assume he has a \(0.51\) probability of being correct on a given trial \(\pi=0.51\).
Women's ability to negotiate safer sex with partners by contraceptive There is life beyond the statistical significance | Reproductive Health We examined evidence for false negatives in nonsignificant results in three different ways. since neither was true, im at a loss abotu what to write about. Third, we calculated the probability that a result under the alternative hypothesis was, in fact, nonsignificant (i.e., ). null hypotheses that the respective ratios are equal to 1.00.
How to Write a Discussion Section | Tips & Examples - Scribbr However, no one would be able to prove definitively that I was not. In APA style, the results section includes preliminary information about the participants and data, descriptive and inferential statistics, and the results of any exploratory analyses. It's her job to help you understand these things, and she surely has some sort of office hour or at the very least an e-mail address you can send specific questions to. pun intended) implications.
non significant results discussion example 178 valid results remained for analysis. If it did, then the authors' point might be correct even if their reasoning from the three-bin results is invalid. Further, blindly running additional analyses until something turns out significant (also known as fishing for significance) is generally frowned upon. We do not know whether these marginally significant p-values were interpreted as evidence in favor of a finding (or not) and how these interpretations changed over time. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. All rights reserved. We also propose an adapted Fisher method to test whether nonsignificant results deviate from H0 within a paper. The first row indicates the number of papers that report no nonsignificant results. Statements made in the text must be supported by the results contained in figures and tables. Visual aid for simulating one nonsignificant test result. This does not suggest a favoring of not-for-profit Therefore we examined the specificity and sensitivity of the Fisher test to test for false negatives, with a simulation study of the one sample t-test. For example, if the text stated as expected no evidence for an effect was found, t(12) = 1, p = .337 we assumed the authors expected a nonsignificant result. One (at least partial) explanation of this surprising result is that in the early days researchers primarily reported fewer APA results and used to report relatively more APA results with marginally significant p-values (i.e., p-values slightly larger than .05), compared to nowadays. Let us show you what we can do for you and how we can make you look good. Finally, and perhaps most importantly, failing to find significance is not necessarily a bad thing. When k = 1, the Fisher test is simply another way of testing whether the result deviates from a null effect, conditional on the result being statistically nonsignificant. F and t-values were converted to effect sizes by, Where F = t2 and df1 = 1 for t-values. One group receives the new treatment and the other receives the traditional treatment. The authors state these results to be "non-statistically significant." The two sub-aims - the first to compare the acquisition The following example shows how to report the results of a one-way ANOVA in practice.
IJERPH | Free Full-Text | Mediator Effect of Cardiorespiratory - MDPI Application 1: Evidence of false negatives in articles across eight major psychology journals, Application 2: Evidence of false negative gender effects in eight major psychology journals, Application 3: Reproducibility Project Psychology, Section: Methodology and Research Practice, Nuijten, Hartgerink, van Assen, Epskamp, & Wicherts, 2015, Marszalek, Barber, Kohlhart, & Holmes, 2011, Borenstein, Hedges, Higgins, & Rothstein, 2009, Hartgerink, van Aert, Nuijten, Wicherts, & van Assen, 2016, Wagenmakers, Wetzels, Borsboom, van der Maas, & Kievit, 2012, Bakker, Hartgerink, Wicherts, & van der Maas, 2016, Nuijten, van Assen, Veldkamp, & Wicherts, 2015, Ivarsson, Andersen, Johnson, & Lindwall, 2013, http://science.sciencemag.org/content/351/6277/1037.3.abstract, http://pss.sagepub.com/content/early/2016/06/28/0956797616647519.abstract, http://pps.sagepub.com/content/7/6/543.abstract, https://doi.org/10.3758/s13428-011-0089-5, http://books.google.nl/books/about/Introduction_to_Meta_Analysis.html?hl=&id=JQg9jdrq26wC, https://cran.r-project.org/web/packages/statcheck/index.html, https://doi.org/10.1371/journal.pone.0149794, https://doi.org/10.1007/s11192-011-0494-7, http://link.springer.com/article/10.1007/s11192-011-0494-7, https://doi.org/10.1371/journal.pone.0109019, https://doi.org/10.3758/s13423-012-0227-9, https://doi.org/10.1016/j.paid.2016.06.069, http://www.sciencedirect.com/science/article/pii/S0191886916308194, https://doi.org/10.1053/j.seminhematol.2008.04.003, http://www.sciencedirect.com/science/article/pii/S0037196308000620, http://psycnet.apa.org/journals/bul/82/1/1, https://doi.org/10.1037/0003-066X.60.6.581, https://doi.org/10.1371/journal.pmed.0020124, http://journals.plos.org/plosmedicine/article/asset?id=10.1371/journal.pmed.0020124.PDF, https://doi.org/10.1016/j.psychsport.2012.07.007, http://www.sciencedirect.com/science/article/pii/S1469029212000945, https://doi.org/10.1080/01621459.2016.1240079, https://doi.org/10.1027/1864-9335/a000178, https://doi.org/10.1111/j.2044-8317.1978.tb00578.x, https://doi.org/10.2466/03.11.PMS.112.2.331-348, https://doi.org/10.1080/01621459.1951.10500769, https://doi.org/10.1037/0022-006X.46.4.806, https://doi.org/10.3758/s13428-015-0664-2, http://doi.apa.org/getdoi.cfm?doi=10.1037/gpr0000034, https://doi.org/10.1037/0033-2909.86.3.638, http://psycnet.apa.org/journals/bul/86/3/638, https://doi.org/10.1037/0033-2909.105.2.309, https://doi.org/10.1177/00131640121971392, http://epm.sagepub.com/content/61/4/605.abstract, https://books.google.com/books?hl=en&lr=&id=5cLeAQAAQBAJ&oi=fnd&pg=PA221&dq=Steiger+%26+Fouladi,+1997&ots=oLcsJBxNuP&sig=iaMsFz0slBW2FG198jWnB4T9g0c, https://doi.org/10.1080/01621459.1959.10501497, https://doi.org/10.1080/00031305.1995.10476125, https://doi.org/10.1016/S0895-4356(00)00242-0, http://www.ncbi.nlm.nih.gov/pubmed/11106885, https://doi.org/10.1037/0003-066X.54.8.594, https://www.apa.org/pubs/journals/releases/amp-54-8-594.pdf, http://creativecommons.org/licenses/by/4.0/, What Diverse Samples Can Teach Us About Cognitive Vulnerability to Depression, Disentangling the Contributions of Repeating Targets, Distractors, and Stimulus Positions to Practice Benefits in D2-Like Tests of Attention, Prespecification of Structure for the Optimization of Data Collection and Analysis, Binge Eating and Health Behaviors During Times of High and Low Stress Among First-year University Students, Psychometric Properties of the Spanish Version of the Complex Postformal Thought Questionnaire: Developmental Pattern and Significance and Its Relationship With Cognitive and Personality Measures, Journal of Consulting and Clinical Psychology (JCCP), Journal of Experimental Psychology: General (JEPG), Journal of Personality and Social Psychology (JPSP). First, we compared the observed nonsignificant effect size distribution (computed with observed test results) to the expected nonsignificant effect size distribution under H0. Comondore and Failing to acknowledge limitations or dismissing them out of hand. Consequently, we cannot draw firm conclusions about the state of the field psychology concerning the frequency of false negatives using the RPP results and the Fisher test, when all true effects are small. Finally, we computed the p-value for this t-value under the null distribution.
How to justify non significant results? | ResearchGate Write and highlight your important findings in your results. Results of each condition are based on 10,000 iterations. Statistical methods in psychology journals: Guidelines and explanations, This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License (CC-BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The Conversely, when the alternative hypothesis is true in the population and H1 is accepted (H1), this is a true positive (lower right cell). Present a synopsis of the results followed by an explanation of key findings. The reanalysis of the nonsignificant RPP results using the Fisher method demonstrates that any conclusions on the validity of individual effects based on failed replications, as determined by statistical significance, is unwarranted. The coding included checks for qualifiers pertaining to the expectation of the statistical result (confirmed/theorized/hypothesized/expected/etc.). If your p-value is over .10, you can say your results revealed a non-significant trend in the predicted direction. The Comondore et al. First, we investigate if and how much the distribution of reported nonsignificant effect sizes deviates from what the expected effect size distribution is if there is truly no effect (i.e., H0). Bond is, in fact, just barely better than chance at judging whether a martini was shaken or stirred. First, we determined the critical value under the null distribution. The columns indicate which hypothesis is true in the population and the rows indicate what is decided based on the sample data. Avoid using a repetitive sentence structure to explain a new set of data. both male and females had the same levels of aggression, which were relatively low. Of articles reporting at least one nonsignificant result, 66.7% show evidence of false negatives, which is much more than the 10% predicted by chance alone. In a study of 50 reviews that employed comprehensive literature searches and included both English and non-English-language trials, Jni et al reported that non-English trials were more likely to produce significant results at P<0.05, while estimates of intervention effects were, on average, 16% (95% CI 3% to 26%) more beneficial in non . These methods will be used to test whether there is evidence for false negatives in the psychology literature. The critical value from H0 (left distribution) was used to determine under H1 (right distribution). Others are more interesting (your sample knew what the study was about and so was unwilling to report aggression, the link between gaming and aggression is weak or finicky or limited to certain games or certain people). Because of the large number of IVs and DVs, the consequent number of significance tests, and the increased likelihood of making a Type I error, only results significant at the p<.001 level were reported (Abdi, 2007). According to Joro, it seems meaningless to make a substantive interpretation of insignificant regression results. At the risk of error, we interpret this rather intriguing term as follows: that the results are significant, but just not statistically so. Grey lines depict expected values; black lines depict observed values. If the p-value for a variable is less than your significance level, your sample data provide enough evidence to reject the null hypothesis for the entire population.Your data favor the hypothesis that there is a non-zero correlation. Like 99.8% of the people in psychology departments, I hate teaching statistics, in large part because it's boring as hell, for . you're all super awesome :D XX. Create an account to follow your favorite communities and start taking part in conversations. Second, we applied the Fisher test to test how many research papers show evidence of at least one false negative statistical result. For a staggering 62.7% of individual effects no substantial evidence in favor zero, small, medium, or large true effect size was obtained. we could look into whether the amount of time spending video games changes the results). You must be bioethical principles in healthcare to post a comment. While we are on the topic of non-significant results, a good way to save space in your results (and discussion) section is to not spend time speculating why a result is not statistically significant. serving) numerical data. The Fisher test was applied to the nonsignificant test results of each of the 14,765 papers separately, to inspect for evidence of false negatives. Null Hypothesis Significance Testing (NHST) is the most prevalent paradigm for statistical hypothesis testing in the social sciences (American Psychological Association, 2010). It provides fodder Very recently four statistical papers have re-analyzed the RPP results to either estimate the frequency of studies testing true zero hypotheses or to estimate the individual effects examined in the original and replication study. The power of the Fisher test for one condition was calculated as the proportion of significant Fisher test results given Fisher = 0.10. { "11.01:_Introduction_to_Hypothesis_Testing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
b__1]()", "11.02:_Significance_Testing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.03:_Type_I_and_II_Errors" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.04:_One-_and_Two-Tailed_Tests" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.05:_Significant_Results" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.06:_Non-Significant_Results" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.07:_Steps_in_Hypothesis_Testing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.08:_Significance_Testing_and_Confidence_Intervals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.09:_Misconceptions_of_Hypothesis_Testing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.10:_Statistical_Literacy" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.E:_Logic_of_Hypothesis_Testing_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Introduction_to_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Graphing_Distributions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Summarizing_Distributions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Describing_Bivariate_Data" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Probability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Research_Design" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Normal_Distribution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Advanced_Graphs" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Sampling_Distributions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Estimation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Logic_of_Hypothesis_Testing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Tests_of_Means" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Power" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Regression" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Analysis_of_Variance" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_Transformations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17:_Chi_Square" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "18:_Distribution-Free_Tests" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "19:_Effect_Size" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "20:_Case_Studies" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "21:_Calculators" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "authorname:laned", "showtoc:no", "license:publicdomain", "source@https://onlinestatbook.com" ], https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Introductory_Statistics_(Lane)%2F11%253A_Logic_of_Hypothesis_Testing%2F11.06%253A_Non-Significant_Results, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\).