That might not seem entirely negligible. At this point, you might be puzzled. After all, we did not put any trend into this series!
It is simply a random realization of a red noise process. The problem is that our residuals are not uncorrelated. They are red noise. In fact, the residuals looks a lot like the original series itself:. So How do we determine if this autocorrelation coefficient is statistically significant?
Well, we can treat it like it were a correlation coefficient. The only catch is that we have to use N-1 in place of N , because there are only N-1 values in the series when we offset it by one time step to form the lagged series required to estimate a lag-one autocorrelation.
So, clearly, we cannot ignore it. We have to take it into account. At this point, you might be getting a bit exasperated. When, if ever, can we conclude there is a trend? Well, why don't we now consider the case where we know we added a real trend in with the noise, i. If we apply our linear regression machinery to this example, we do detect a notable trend:.
Now, that's a trend - your eye isn't fooling you. Finally, let's look at what happens when the same trend 0. What result does the regression analysis give now? We still recover a similar trend, although it's a bit too large. We know that the true trend is 0. So, are we done? Not quite. So, we will have to use the reduced degrees of freedom N'. So, the trend is still found to be statistically significant, but the significance is no longer at the astronomical level it was when the residuals were uncorrelated white noise.
The effect of the "redness" of the noise has been to make the trend less statistically significant because it is much easier for red noise to have produced a spurious apparent trend from random chance alone. Often, residuals have so much additional structure — what is sometimes referred to as heteroscedasticity how's that for a mouthful? In this case, the basic assumptions of linear regression are called into question and any results regarding trend estimates, statistical significance, etc.
In this case, more sophisticated methods that are beyond the scope of this course are required. Now, let us look at some real temperature data! We will use our very own custom online Linear Regression Tool written for this course.
What is the trend in statistics? How do you explain a trend? How do you find the trend in statistics? What is a trend in time series analysis? What is a positive trend? What makes trends more popular than fad? Why do trends emerge? What Characterises a mega trend? Why is time frame important in a trend? What is the difference between a trend and an issue? Previous Article What are the different types of trends?
Next Article What does blood sacrifice mean? Back To Top. JAMA ; We would like to thank Doleman et al 1 for their interest in our editorial 2 , but disagree with many of their comments. We do not agree that the misuse of 'trend' to describe almost significant differences is a semantic issue. We feel that it is an error. Also, we cannot see how drawing attention to this relatively rare error clouds the 'greater issue in the reporting of statistics in the scientific literature'.
Doleman et al suggest that P values are outdated and should be avoided. However, their arguments are based on the misuse and misinterpretation of P values, not on their correct use and interpretation. Moreover, many of their arguments apply equally to the use of confidence intervals CI. For example, both CI width and P value are heavily dependent on sample size 3. More importantly, both approaches require reference to a pre- determined clinically significant effect size for a correct interpretation, not only CIs 3,4,5,6.
Another fundamental issue, which Doleman et al do not mention, is the requirement for adequate a priori power whether P values or CIs are used 3,4,5,6. In the examples they provide, the correct interpretation of the P values would be the same as the correct interpretation of the CIs.
In other words, it is not the use of P values that result in 'widely different and erroneous conclusions'; it is the misuse. Nevertheless, we are all too aware of the widespread misinterpretation of P values in the anaesthesia literature, as has been reported by Gibbs and Weightman recently 6,7 , and previously by many other authors 8.
Where we differ from Doleman et al is that we would encourage the correct use of P values, rather than their abandonment. The correct use would include avoiding the term 'trend' to describe 'almost significant' differences. We agree that CIs provide valuable information on the range of likely true effect sizes and the precision of estimates in the population being studied.
However, we would caution that to be used and interpreted correctly, CIs are as dependent as P values on reference to the pre- specified minimum clinically important effect size and adequate a priori power 3,4,5,6. We suspect that there are many examples of CI use in the anaesthesia literature where these conditions are not met.
Re: Misuse of 'trend' to describe 'almost significant' differences in anaesthesia research. Br J Anaesth eLetter. Daly LE. Confidence intervals and sample sizes. Statistics with confidence. Bristol: BMJ Books, The use of predicted confidence intervals when planning experiments and the misuse of power when interpreting results. Ann Intern Med ; Katz MH. Study design and statistical analysis. A practical guide for clinicians.
Cambridge: Cambridge University Press, Beyond effect size: consideration of the minimum effect size of interest in anesthesia trials. Anesth Analg ; An audit of the statistical validity of conclusions of clinical superiority in anaesthesia journals.
Anaesth Intensive Care ; Could do better: statistics in anaesthesia research. We would like to thank Dr Smith 1 for his interest and comments on our editorial 2.
We agree with his remarks about medical conservatism and appreciate the desire of many authors to highlight results that are almost, but not quite, statistically significant. However, this can be achieved without using the potentially misleading term 'trend'.
Authors can legitimately describe such findings as 'encouraging', 'of interest', or 'worthy of further investigation', so long as the lack of statistical significance is clear, and the same interpretation is given to all P values in their study that fall within this 'almost statistically significant' range, and not only those that support their test hypothesis.
Alternatively, authors are at liberty to accept a larger alpha error from the outset e. We agree that P values present a continuum and that actual P values contain valuable information, whether or not they are 'significant'.
For this reason, we would encourage reporting of actual P values under all circumstances i. We consider that when interpreted correctly, P values already 'speak for themselves', although the effect size, pre-specified minimum clinically important effect size, and beta error must also be considered 3.
We feel that it is not necessary 'to drop the concept of significance'. It is necessary only to use it correctly and to know its limitations.
Smith I. I thank Drs Gibbs and Gibbs for highlighting the linguistic and statistical inaccuracies in the use of "trends" in anaesthesia research literature BJA ; 3 : As a teacher and reviewer, I have argued against this practice, while as an author I have almost certainly fallen into the same trap myself!
While I cannot disagree with a word of the editorial, I can perhaps offer a little more insight into why this trend might be developing.
The authors correctly describe how a predefined p value is used to determine whether the null hypothesis is rejected or accepted and hence whether results are statistically significant or not.
Because of medical conservatism, we are reluctant to replace a tried and tested therapy with something relatively new, and so the threshold for rejecting the null hypothesis is set high deliberately, typically at a level where the observed result would only have occurred by chance less that five times in a hundred.
Conversely this means that, in the area in which "trends" tend to be mentioned, that the observed result is likely to represent a genuine difference between treatments approximately ninety to just under ninety five percent of the time. In other circumstances, these would still be very good odds indeed! It is not surprising that authors want to highlight these results, even if trend is not the correct word to use. Statistical tests provide a precise estimate of the likelihood that the results are due to random chance, or to a genuine effect, along a continuum extending from near certainty to almost infinite improbability.
Perhaps the time has come to drop the concept of significance and simply let the p-values talk for themselves. We read with interest the article by Gibbs and Gibbs on the misuse of the word trend to describe 'almost' significant p values [1]. However, we believe this discussion of semantics is clouding a somewhat greater issue in the reporting of statistics in the scientific literature.
As Gibbs and Gibbs noted, p values are inferential statistics that give the probability of obtaining a value the same or greater than that found if the null hypothesis was true. However, we argue that p values are outdated and for clinical studies in particular, their use should be avoided.
This is not a new concept, however, we hope this letter will serve to remind the readership of the flaws in null hypothesis statistical testing. We will first highlight the problems related with the [mis]use of p values, and then discuss the advantages of estimation-based methods, illustrating this with a theoretical example. Firstly, the use of p values can often detract from a more important issue when conducting a clinical study, the assessment of clinical significance.
The misinterpretation of p values means that readers may mistake 'statistical significance' as 'clinical significance'. As the calculation of p values is heavily dependent on sample size, large studies may demonstrate very small p values that are not clinically significant.
In fact, such small p values may in fact be evidence against the use of a particular treatment, as it can make us more confident that a treatment will not have a clinically significant effect see later example.
Such a level may have originated from statistician Ronald Fisher who suggested this as a sensible cut-off, although he never advocated this as an absolute rule [2]. It is absurd to suggest a study that reports a p value of 0. Such 'negative' p values promote fear in researchers and students while also rendering a study less likely to be published [4]. Ever since the end of the 's, the use of confidence intervals CI was proposed as an alternative to p values [5].
Confidence intervals present a range of values with which the population mean is likely to lie. Such an approach is advocated by the International Committee of Medical Journal Editors who state 'When possible, quantify findings and present them with appropriate indicators of measurement error or uncertainty such as confidence intervals. Avoid relying solely on statistical hypothesis testing, such as P values, which fail to convey important information about effect size and precision of estimates'.
Confidence intervals contain a wealth of information as well as including all the information of a traditional p value. Moreover, confidence intervals can be better used to assess the likelihood an intervention has a clinically significant effect.
Consider an example, we wish to know the efficacy of two different analgesic agents x and y for treating postoperative pain. We undertake two randomised controlled trials RCT with both agents and pre-determine a clinically significant reduction in pain as 15mm on a mm VAS [8]. Although the p value of the first study is very low, the results indicate that we can be confident this agent does not produce a clinically significant effect and should therefore not be used. The second study, although not statistically significant, does not exclude a clinically significant effect and requires more studies to be conducted in order to increase power and narrow the confidence interval.
If we had relied solely on statistical significance, widely different and erroneous conclusions would be made. We accept that this argument is not new and we applaud the British Journal of Anaesthesia for promoting confidence intervals in their instructions for authors page.
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