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Generalizing to a Population: CONFIDENCE LIMITS continued

P VALUES AND STATISTICAL SIGNIFICANCE
The traditional approach to reporting a result requires you lot to say whether it is statistically pregnant. You are supposed to do information technology past generating a p value from a test statistic. Y'all then betoken a significant result with "p<0.05". So let's detect out what this p is, what's special well-nigh 0.05, and when to use p. I'll as well deal with the related topics of one-tailed vs ii-tailed tests, and hypothesis testing.

What is a P Value?
It's hard, this one. P is curt for probability: the probability of getting something more extreme than your issue, when at that place is no effect in the population. Bizarre! And what'due south this got to do with statistical significance? Let's run into.

I've already defined statistical significance in terms of conviction intervals. The other approach to statistical significance--the one that involves p values--is a flake convoluted. First you assume there is no effect in the population. Then yous see if the value you go for the result in your sample is the sort of value you would await for no effect in the population. If the value you get is unlikely for no upshot, y'all conclude there is an effect, and you say the result is "statistically significant".

Let's take an example. Yous are interested in the correlation betwixt 2 things, say top and weight, and you lot take a sample of 20 subjects. OK, assume there is no correlation in the population. At present, what are some unlikely values for a correlation with a sample of 20? It depends on what we hateful by "unlikely". Permit's make it mean "extreme values, 5% of the fourth dimension". In that case, with twenty subjects, all correlations more positive than 0.44 or more negative than -0.44 volition occur merely 5% of the fourth dimension. What did you get in your sample? 0.25? OK, that'due south not an unlikely value, then the effect is not statistically significant. Or if you got -0.63, the result would exist statistically pregnant. Like shooting fish in a barrel!

Merely wait a minute. What almost the p value? Yeah, umm, well... The problem is that stats programs don't give you the threshold values, ±0.44 in our example. That's the mode it used to be done before computers. You lot looked up a tabular array of threshold values for correlations or for some other statistic to encounter whether your value was more or less than the threshold value, for your sample size. Stats programs could exercise it that way, just they don't. You lot desire the correlation corresponding to a probability of 5%, just the stats program gives yous the probability corresponding to your observed correlation--in other words, the probability of something more farthermost than your correlation, either positive or negative. That's the p value. A fleck of thought will satisfy yous that if the p value is less than 0.05 (5%), your correlation must exist greater than the threshold value, and then the result is statistically significant. For an observed correlation of 0.25 with 20 subjects, a stats parcel would return a p value of 0.30. The correlation is therefore not statistically meaning.

Phew! Hither's our instance summarized in a diagram:

The curve shows the probability of getting a detail value of the correlation in a sample of twenty, when the correlation in the population is zero. For a particular observed value, say 0.25 as shown, the p value is the probability of getting annihilation more positive than 0.25 and anything more negative than -0.25. That probability is the sum of the shaded areas under the probability curve. It'due south virtually thirty% of the area, or a p value of 0.3. (The total expanse under a probability curve is ane, which means absolute certainty, because you have to get a value of some kind.)

Results falling in that shaded area are non really unlikely, are they? No, nosotros need a smaller surface area before we get excited near the event. Unremarkably it'south an area of 5%, or a p value of 0.05. In the example, that would happen for correlations greater than 0.44 or less than -0.44. And so an observed correlation of 0.44 (or -0.44) would accept a p value of 0.05. Bigger correlations would have fifty-fifty smaller p values and would be statistically pregnant.

Test Statistics
The stats programme works out the p value either directly for the statistic y'all're interested in (e.g. a correlation), or for a test statistic that has a 1:1 human relationship with the effect statistic. A test statistic is but another kind of upshot statistic, 1 that is easier for statisticians and computers to handle. Common test statistics are t, F, and chi-squared. You lot don't always need to know how these statistics are divers, or what their values are. All you need is the p value, or amend nevertheless, the conviction limits or interval for your result statistic.

P Values and Conviction Intervals
Southpeaking of confidence intervals, allow'south bring them back into the picture. It's possible to testify that the two definitions of statistical significance are compatible

--that getting a p value of less than 0.05 is the aforementioned equally having a 95% confidence interval that doesn't overlap zero. I won't endeavour to explain information technology, other than to say that you lot have to slide the conviction interval sideways to prove information technology. But make certain you are happy with this figure, which shows some examples of the relationship between p values and 95% confidence intervals for observed correlations in our case of a sample of 20 subjects.

The relationship between p values and conviction intervals also provides us with a more sensible way to think nigh what the "p" in "p value" stands for. I've already said that it'southward the probability of a more extreme (positive or negative) result than what you lot observed, when the population value is null. Just hey, what does that really mean? I get lost every time I endeavour to wrap my brain effectually it. Here'southward something much better: if your observed effect is positive, then half of the p value is the probability that the true effect is negative. For instance, you observed a correlation of 0.25, and the p value was 0.30. OK, the chance that the true value of the correlation is negative (less than zero) is 0.15 or 15%; or you can say that the odds of a negative correlation are 0.xv:0.85, or about 1 to 6 (1 to 0.85/0.15). Maybe it's amend to it turn around and talk virtually a probability of 0.85 (= ane - p/2), or odds of six to 1, that the truthful effect is positive. Here's another instance: you lot observed an increase in performance of two.6%, and the p value was 0.04, so the probability that operation really did increase is 0.98, or 49 to 1. Check your understanding by working out how to interpret a p value of exactly 1.

Then, if y'all desire to include p values in your side by side newspaper, hither is a new fashion to describe them in the Methods section: "Each p value represents twice the probability that the truthful value of the result has any value with sign reverse to that of the observed value." I wonder if reviewers will accept information technology. In plain linguistic communication, if you observe a positive effect, i - p/2 is the probability that the true effect is positive. But even with this estimation, p values are non a corking mode to generalize an effect from a sample to a population, because what matters is clinical significance, not statistical significance.

Clinical vs Statistical Significance
As nosotros've just seen, the p value gives you a manner to talk about the probability that the issue has any positive (or negative) value. To recap, if y'all observe a positive effect, and it's statistically meaning, then the true value of the effect is likely to be positive. But if you're going to all the trouble of using probabilities to describe magnitudes of effects, information technology'due south ameliorate to talk near the probability that the effect is essentially positive (or negative). Why? Because we want to know the probability that the true value is big enough to count for something in the world. In other words, nosotros want to know the probability of clinical or applied significance. To work out that probability, you will have to think about and accept into account the smallest clinically important positive and negative values of the event; that is, the smallest values that matter to your subjects. (For more on that topic, come across the page nigh a scale of magnitudes.) Then it's a relatively simple matter to calculate the probability that the truthful value of the effect is greater than the positive value, and the probability that the true value is less than the negative value.

I take now included the calculations in the spreadsheet for confidence limits and likelihoods. I've chosen the smallest clinically of import value a "threshold value for chances [of a clinically important issue]". You take to choose a threshold value on the footing of experience or understanding. You besides have to include the observed value of the statistic and the p value provided past your stats program. For changes or differences betwixt means you also accept to provide the number of degrees of freedom for the effect, merely the exact value isn't crucial. The spreadsheet then gives yous the chances (expressed equally probabilities and odds) that the true value is clinically positive (greater than the smallest positive clinically of import value), clinically negative (less than the negative of the smallest important value), and clinically trivial (between the positive and negative smallest important values). The spreadsheet likewise works out confidence limits, as explained in the adjacent section below.

Apply the spreadsheet to play around with some p values, observed values of a statistic, and smallest clinically of import values to come across what the chances are like. I've got an instance there showing that a p value of 0.20 can give chances of lxxx%, 15% and 5% for clinically positive, trivial, and negative values. Wow! It's clear from data like these that editors who stick to a policy of "publishable if and only if p<0.05" are preventing clinically useful findings from seeing the light of day.

I have written two short articles on this topic at the Sportscience site. The commencement article introduces the topic, pretty much as above. The second article summarizes a Powerpoint slide show I have been using for a seminar with the title Statistical vs Clinical or Practical Significance, in which I explicate hypothesis testing, P values, statistical significance, confidence limits, probabilities of clinical significance, a qualitative scale for interpreting clinical probabilities, and some examples of how to use the probabilities in practise. Download the presentation (91 KB) by (right-)clicking on this link. View it equally a full slide show then you run across each slide build.

Confidence Limits from a P Value
Stats programs often don't give you confidence limits, but they always give you the p value. So hither'due south a clever way to derive the conviction limits from the p value. It works for differences between means in descriptive or experimental studies, and for whatsoever normally distributed statistic from a sample. Best of all, information technology's on a spreadsheet! I explain how it works in the adjacent paragraph, but it's a flake tricky and y'all don't accept to understand it to use the spreadsheet. Link back to the previous folio to download the spreadsheet.

I'll explain with an example. Suppose you lot've done a controlled experiment on the upshot of a drug on time to run 10,000 m. Suppose the overall deviation between the means you're interested in is 46 seconds, with a p value of 0.26. From the definition of the p value (meet summit effigy on this page), we can describe a normal probability distribution centered on a difference of 0 seconds, such that there is an expanse of 0.26/2 = 0.13 to the right of 46 and a like expanse to the left of -46. Or to put information technology another way, the surface area between -46 and 46 is 1-0.26 = 0.74. If we at present shift that distribution until information technology's centered over 46, it represents the probability distribution for the true value. Nosotros know that the chance of the true value being between 0 and 92 is 0.74, so now all nosotros need is the range that will make the run a risk 0.95, and that will be our 95% confidence interval. To piece of work it out, we use the fact that the distribution is normal. That allows us to summate how many standard deviations (also known equally the z score) nosotros have to go on each side of the hateful to enclose 0.74 of the area nether the normal curve. Nosotros get that from tables of the cumulative normal distribution, or the function NORMSINV in an Excel spreadsheet. Answer: 1.thirteen standard deviations. Ah, but we know that 1.96 standard deviations encloses 95% of the area, and because the one.13 standard deviations represents 46 seconds, our confidence interval must exist -46(1.96/1.13) to +46(1.96/1.13), i.due east. -34 to +126.

Fine, except that it's non really a normal distribution. With a finite number of subjects, it's actually a t distribution, so we take to use TINV in Excel. What's more, the 95% confidence limits are actually a titch more than 1.96 standard deviations each side of the hateful. Exactly how much more depends on the number of subjects, or more precisely, the number of degrees of freedom. With your own data, search effectually in the output from the analysis until you find the degrees of freedom for the error term or the residuals. Put it into the spreadsheet, along with the observed value of the effect statistic, and its p value (not the p value for the model or for an effect in the model, unless information technology is the statistic). If you lot can't find the number of degrees of freedom on the output, the spreadsheet tells you how to summate it. And if you don't get it exactly correct, don't worry: the conviction limits inappreciably change for more than than 20 degrees of liberty.

One Tail or 2?
Notice in the first figure on this page that the p value is calculated for both tails of the distribution of the statistic. That follows naturally from the meaning of statistical significance, and information technology's why tests of significance are sometimes called 2 tailed. In principle you could eliminate one tail, double the area of other tail, and then declare statistical significance if the observed value fell within the one-tailed area. The result would be a 1-tailed test. Your Type I error rate would still be 5%, but a smaller effect would turn out to be statistically meaning. In other words, you would accept more power to detect the event.

And so how come nosotros don't exercise all tests as one-tailed tests? Hmm... The people who support the thought of such tests--and they are a vanishing brood--argue that you can use it to exam for, say, a positive result only if you lot have a good reason for believing beforehand that the event will be positive. I hope I am characterizing their position correctly, because I don't empathize it. What is a "proficient reason"? It seems to me that you would have to be admittedly sure that the outcome would be positive, but in that case running the examination for statistical significance is pointless! I therefore don't purchase into one-tailed tests. If you have any doubts, revert to the conviction-interval view of significance: one-sided confidence intervals just don't brand sense, but confidence limits as placed on each side of the observed value is unquestionably a correct view.

Except that... there is a justification for 1-tailed tests afterwards all. Yous just interpret the p value differently. P values for ane-tailed tests are half those for two-tailed tests. It follows that the p value from a one-tailed test is the exact probability that the true value of the consequence has reverse sign to what you take observed, and one - p is the probability that the true value of the issue has the same sign, as I explained above. Hey, we don't have to muck around with p/2. So here's what you could write in the Methods section of your paper: "All tests of significance are 1-tailed in the direction of the observed effect. The resulting p values stand for the probability that the truthful value of the outcome is of sign opposite to the observed value." Give it a go and run across what happens. Such a argument would be anathema to reviewers or statisticians who affirm that an observed positive issue is not a justification for doing a 1-tailed test for a positive effect. They would contend that you are downgrading the criterion for deciding what is "statistically significant", because you are effectively performing tests with a Type I error of ten%. Fair enough, so don't mention statistical significance at all. Just show 95% confidence limits, and simply say in the Methods: "Our p values, derived from one-tailed tests, stand for the probability that the true value of the effect is of sign opposite to the observed value."

But equally I discussed above, the probability that an outcome has a essentially positive (or negative) value is more useful than the probability that the effect has any positive (or negative) value. Confidence limits are better than one-tailed p values from that point of view, which is why you should always include confidence limits.

Why 0.05?
Due westhat's so special about a p value of 0.05, or a confidence interval of 95%? Nothing really. Someone decided that it was reasonable, so we're now stuck with it. P < 0.01 has also go a bit of a tradition for declaring significance. Both are hangovers from the days earlier computers, when it was difficult to summate exact p values for the value of a examination statistic. Instead, people used tables of values for the exam statistic respective to a few arbitrarily called p values, namely 0.05, 0.01, and sometimes 0.001. These values have now become enshrined as the threshold values for declaring statistical significance. Journals ordinarily want you to state which 1 you're using. For instance, if you land that your level of significance is 5% (also called an alpha level), then you're allowed to call any effect with a p value of less than 0.05 significant. In many journals results in figures are marked with ane asterisk (*) if p<0.05 and two (**) if p<0.01.

Some researchers and statisticians claim that a decision has to exist made almost whether a result is statistically significant. According to this logic, if p is less than 0.05 you have a publishable result, and if p is greater than 0.05, you don't. Hither's a diagram showing the folly of this view of the earth. One of these results is statistically significant (p<0.05), and the other isn't (p>0.05). Which is publishable? Answer: both are, although you'd have to say in both cases that more subjects should have been tested to narrow downwards the likely range of values for the correlation. And in case y'all missed the bespeak, the exact p values are 0.049 and 0.051. Don't ask me which is which!

Some journals persist with the former-fashioned exercise of allowing authors to evidence statistically significant results with p<0.05 or p<0.01, and not-pregnant results with p>0.05. Verbal p values convey more information, but confidence intervals give a much better idea of what could be going on in the population. And with confidence intervals you lot don't get hung upward on p values of 0.06.

Hypothesis Testing
The philosophy of making a decision about statistical significance also spawned the practice of hypothesis testing, which has grown to the extent that some departments make their enquiry students list the hypotheses to exist tested in their projects. The idea is that you state a null hypothesis (i.e. that at that place is no effect), then see if the information you become allow you to pass up it. Which ways there is no effect until proved otherwise--like being innocent until proved guilty. This philosophy comes through clearly in such statements every bit "let'south meet if at that place is an outcome".

What's wrong hither? Well, people may be truly innocent, but in nature effects are seldom truly zero. Y'all probably wouldn't investigate something if you actually believed there was nothing going on. So what really matters is estimating the magnitude of effects, not testing whether they are zippo. But that's simply a philosophical outcome. At that place are more important practical issues. Getting students to test hypotheses diverts their attention from the magnitude of the effect to the magnitude of the p value. Read that previous judgement once again, delight, it's that important. And then when a student researcher gets p>0.05 and therefore "accepts the null hypothesis", s/he commonly concludes erroneously that there is no effect. And if s/he gets p<0.05 and therefore "rejects the zip hypothesis", s/he still has little thought of how big or how small the effect could be in the population. In fact, virtually research students don't even know they are supposed to exist making inferences about population values of a statistic, even afterwards they have washed statistics courses. That's how hopelessly confusing hypothesis testing and p values are.

"Permit's see if there is an consequence" isn't also bad, if what you mean is "let's come across if there is a not-trivial consequence". That's what people really intend. But a test for statistical significance does non address the question of whether the upshot is non-trivial; instead, it's a test of whether the upshot is greater than zip (for an observed positive effect). And information technology's easy to get a statistically significant effect that could exist trivial, and so hypothesis testing doesn't exercise a proper job. With conviction limits you tin can see immediately whether the effect could exist footling

Enquiry questions are more important than research hypotheses. The correct question is "how big is the effect?" And I don't only mean the upshot you discover in your sample. I mean the outcome in the population, so you will accept to bear witness confidence limits to delimit the population effect.

Using P Values
Westwardhen I first published this book, I was prepared to concede that p values have a apply when you study lots of effects. For example, with 20 correlations in a table, the ones marked with asterisks stand out from the rest. Now I'm non so sure near the utility of those asterisks. The not-pregnant results might be but as interesting. For example, if the sample size is large plenty, a non-meaning result means the outcome tin only exist petty, which is just as of import as the upshot existence substantial. And if the sample size isn't large enough, a non-significant upshot with the lower conviction limit in the niggling region (due east.g. r = 0.34, 95%CL = -0.03 to 0.62) is arguably only a tad less interesting than a statistically significant result with the lower confidence limit still in the fiddling region (east.g. r = 0.38, 95%CL = 0.02 to 0.65). So I think I'll harden my attitude. No more than p values.

By the way, if you lot do study p values with your outcome statistics, there is no point in reporting the value of the examination statistic also. It'south superfluous information, and few people know how to interpret the magnitude of the test statistic anyway. But you lot must brand certain yous requite confidence limits or verbal p values, and describe the statistical modeling procedure in the Methods department.