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# Science without sense…double nonsense

### Píldoras sobre medicina basada en pruebas

Archive for the Statistics Category

## Worshipped, but misunderstood

Statistics wears most of us who call ourselves “clinicians” out. The knowledge on the subject acquired during our formative years has long lived in the foggy world of oblivion. We vaguely remember terms such as probability distribution, hypothesis contrast, analysis of variance, regression … It is for this reason that we are always a bit apprehensive when we come to the methods section of scientific articles, in which all these techniques are detailed that, although they are known to us, we do not know with enough depth to correctly interpret their results.

Fortunately, Providence has given us a lifebelt: our beloved and worshipped p. Who has not felt lost with a cumbersome description of mathematical methods to finally breathe a sigh of relieve when finding the value of p? Especially if the p is small and has many zeros.

The problem with p is that, although it is unanimously worshipped, it is also mostly misunderstood. Its value is, very often, misinterpreted. And this is so because many of us harbor misconceptions about what the p-value really means.

Let’s try to clarify it.

Whenever we want to know something about a variable, the effect of an exposure, the comparison of two treatments, etc., we will face the ubiquity of random: it is everywhere and we can never get rid of it, although we can try to limit it and, of course, try to measure its effect.

Let’s give an example to understand it better. Suppose we are doing a clinical trial to compare the effect of two diets, A and B, on weight gain in two groups of participants. Simplifying, the trial will have one of three outcomes: those of diet A gain more weight, those of diet B gain more weight, both groups gain equal weight (there could even be a fourth: both groups lose weight). In any case, we will always obtain a different result, just by chance (even if the two diets are the same).

Imagine that those in diet A put on 2 kg and those in diet B, 3 kg. Is it more fattening the effect of diet B or is the difference due to chance (chosen samples, biological variability, inaccuracy of measurements, etc.)? This is where our hypothesis contrast comes in.

When we are going to do the test, we start from the hypothesis of equality, of no difference in effect (the two diets induce the same increment of weight). This is what we call the null hypothesis (H0) that, I repeat it to keep it clear, we assume that it is the real one. If the variable we are measuring follows a known probability distribution (normal, chi-square, Student’s t, etc.), we can calculate the probability of presenting each of the values of the distribution. In other words, we can calculate the probability of obtaining a result as different from equality as we have obtained, always under the assumption of H0.

That is the p-value: the probability that the difference in the result observed is due to chance. By agreement, if that probability is less than 5% (0.05) it will seem unlikely that the difference is due to chance and we will reject H0, the equality hypothesis, accepting the alternative hypothesis (Ha) that, in this example, will say that one diet better than the other. On the other hand, if the probability is greater than 5%, we will not feel confident enough to affirm that the difference is not due to chance, so we DO NOT reject H0 and we keep with the hypothesis of equal effects: the two diets are similar.

Keep in mind that we always move in the realm of probability. If p is less than 0.05 (statistically significant), we will reject H0, but always with a probability of committing a type 1 error: take for granted an effect that, in reality, does not exist (a false positive). On the other hand, if p is greater than 0.05, we keep with H0 and we say that there is no difference in effect, but always with a probability of committing a type 2 error: not detecting an effect that actually exists (false negative).

We can see, therefore, that the value of p is somewhat simple from the conceptual point of view. However, there are a number of common errors about what p-value represents or does not represent. Let’s try to clarify them.

It is false that a p-value less than 0.05 means that the null hypothesis is false and a p-value greater than 0.05 that the null hypothesis is true. As we have already mentioned, the approach is always probabilistic. The p <0.05 only means that, by agreement, it is unlikely that H0 is true, so we reject it, although always with a small probability of being wrong. On the other hand, if p> 0.05, it is also not guaranteed that H0 is true, since there may be a real effect that the study does not have sufficient power to detect.

At this point we must emphasize one fact: the null hypothesis is only falsifiable. This means that we can only reject it (with which we keep with Ha, with a probability of error), but we can never affirm that it is true. If p> 0.05 we cannot reject it, so we will remain in the initial assumption of equality of effect, which we cannot demonstrate in a positive way.

It is false that p-value is related to the reliability of the study. We can think that the conclusions of the study will be more reliable the lower the value of p, but it is not true either. Actually, the p-value is the probability of obtaining a similar value by chance if we repeat the experiment in the same conditions and it not only depends on whether the effect we want to demonstrate exists or not. There are other factors that can influence the magnitude of the p-value: the sample size, the effect size, the variance of the measured variable, the probability distribution used, etc.

It is false that p-value indicates the relevance of the result. As we have already repeated several times, p-value is only the probability that the difference observed is due to chance. A statistically significant difference does not necessarily have to be clinically relevant. Clinical relevance is established by the researcher and it is possible to find results with a very small p that are not relevant from the clinical point of view and vice versa, insignificant values that are clinically relevant.

It is false that p-value represents the probability that the null hypothesis is true. This belief is why, sometimes, we look for the exact value of p and do not settle for knowing only if it is greater or less than 0.05. The fault of this error of concept is a misinterpretation of conditional probability. We are interested in knowing what is the probability that H0 is true once we have obtained some results with our test. Mathematically expressed, we want to know P (H0 | results). However, the value of p gives us the probability of obtaining our results under the assumption that the null hypothesis is true, that is, P (results | H0).

Therefore, if we interpret that the probability that H0 is true in view of our results (P (H0 | results)) is equal to the value of p (P (results | H0)) we will be falling into an inverse fallacy or transposition of conditionals fallacy.

In fact, the probability that H0 is true does not depend only on the results of the study, but is also influenced by the previous probability that was estimated before the study, which is a measure of the subjective belief that reflects its plausibility, generally based on previous studies and knowledge. Let’s think we want to contrast an effect that we believe is very unlikely to be true. We will value with caution a p-value <0.05, even being significant. On the contrary, if we are convinced that the effect exists, will be settle for with little demands of p-value.

In summary, to calculate the probability that the effect is real we must calibrate the p-value with the value of the baseline probability of H0, which will be assigned by the researcher or by previously available data. There are mathematical methods to calculate this probability based on its baseline probability and the p-value, but the simplest way is to use a graphical tool, the Held’s nomogram, which you can see in the figure.

To use the Held’s nomogram we just have to draw a line from the previous H0 probability that we consider to the p-value and extend it to see what posterior probability value we reach. As an example, we have represented a study with a p-value = 0.03 in which we believe that the probability of H0 is 20% (we believe there is 80% that the effect is real). If we extend the line it will tell us that the minimum probability of H0 is 6%: there is a 94% probability that the effect is real. On the other hand, think of another study with the same p-value but in which we think that the probability of the effect is lower, for example, of 20% (the probability of H0 is 80%). For the same value of p, the minimum posterior probability of H0 is 50%, then there is 50% that the effect is real. As we can see, the posterior probability changes according to the previous probability.

And here we will end for today. We have seen how p-value only gives us an idea of the role that chance may have had in our results and that, in addition, may depend on other factors, perhaps the most important the sample size. The conclusion is that, in many cases, the p-value is a parameter that allows to assess in a very limited way the relevance of the results of a study. To do it better, it is preferable to resort to the use of confidence intervals, which will allow us to assess clinical relevance and statistical significance. But that is another story…

## The cheaters detector

When we think about inventions and inventors, the name of Thomas Alva Edison, known among his friends as the Wizard of Menlo Park, comes to most of us. This gentleman created more than a thousand inventions, some of which can be said to have changed the world. Among them we can name the incandescent bulb, the phonograph, the kinetoscope, the polygraph, the quadruplex telegraph, etc., etc., etc. But perhaps its great merit is not to have invented all these things, but to apply methods of chain production and teamwork to the research process, favoring the dissemination of their inventions and the creation of the first industrial research laboratory.

But in spite of all his genius and excellence, Edison failed to go on to invent something that would have been as useful as the light bulb: a cheaters detector. The explanation for this pitfall is twofold: he lived between the nineteenth and twentieth centuries and did not read articles about medicine. If he had lived in our time and had to read medical literature, I have no doubt that the Wizard of Menlo Park would have realized the usefulness of this invention and would have pull his socks up.

And it is not that I am especially negative today, the problem is that, as Altman said more than 15 years ago, the material sent to medical journals is defective from the methodological point of view in a very high percentage of cases. It’s sad, but the most appropriate place to store many of the published studies is the rubbish can.

In most cases the cause is probably the ignorance of those who write. “We are clinicians”, we say, so we leave aside the methodological aspects, of which we have a knowledge, in general, quite deficient. To fix it, journal editors send our studies to other colleagues, who are more or less like us. “We are clinicians”, they say, so all our mistakes go unnoticed to them.

Although this is, in itself, serious, it can be remedied by studying. But it is an even more serious fact that, sometimes, these errors can be intentional with the aim of inducing the reader to reach a certain conclusion after reading the article. The remedy for this problem is to make a critical appraisal of the study, paying attention to its internal validity. In this sense, perhaps the most difficult aspect to assess for the clinician without methodological training is that related to the statistics used to analyze the results of the study. It is in this, undoubtedly, that most can be taken advantage of our ignorance using methods that provide more striking results, instead of the right methods.

As I know that you are not going to be willing to do a master’s degree in biostatistics, waiting for someone to invent the cheaters detector, we are going to give a series of clues so that non-expert readers can suspect the existence of these cheats.

The first may seem obvious, but it is not: has a statistical method been used? Although it is exceptionally rare, there may be authors who do not consider using any. I remember a medical congress that I could attend in which the values of a variable were exposed throughout the study that, first, went up and then went down, which allowed the speaker to conclude that the result was not “on the blink”. As it is logical and evident, any comparison must be made with the proper hypotheses contrast and the level of significance and the statistical test used have to be specified. Otherwise, the conclusions will lack any validity.

A key aspect of any study, especially those with an intervention, is the previous calculation of the necessary sample size. The investigator must define the clinically relevant effect that he wants to be able to detect with his study and then calculate what sample size will provide the study with enough power to prove it. The sample of a study is not large or small, but sufficient or insufficient. If the sample is not sufficient, an existing effect may not be detected due to lack of power (type 2 error). On the other hand, a larger sample than necessary may show an effect that is not relevant from the clinical point of view as statistically significant. Here are two very common cheats. First, the study that does not reach significance and its authors say it is due to lack of power (insufficient sample size), but do not make any effort to calculate the power, which can always be done a posteriori. In that case, we can calculate it using statistical programs or any of the calculators available on the internet, such as GRANMO. Second, the sample size is increased until the difference observed is significant, finding the desired p <0.05. This case is simpler: we only have to assess whether the effect found is relevant from the clinical point of view. I advise you to practice and compare the necessary sample sizes of the studies with those defined by the authors. Maybe you’ll have some surprise.

Once the participants have been selected, a fundamental aspect is that of the homogeneity of the basal groups. This is especially important in the case of clinical trials: if we want to be sure that the observed difference in effect between the two groups is due to the intervention, the two groups should be the same in everything, except in the intervention.

For this we will look at the classic table I of the trial publication. Here we have to say that, if we have distributed the participants at random between the two groups, any difference between them will be due, one way or another, to random. Do not be fooled by the p, remember that the sample size is calculated for the clinically relevant magnitude of the main variable, not for the baseline characteristics of the two groups. If you see any difference and it seems clinically relevant, it will be necessary to verify that the authors have taken into account their influence on the results of the study and have made the appropriate adjustment during the analysis phase.

The next point is that of randomization. This is a fundamental part of any clinical trial, so it must be clearly defined how it was done. Here I have to tell you that chance is capricious and has many vices, but rarely produces groups of equal size. Think for a moment if you flip a coin 100 times. Although the probability of getting heads in each throw is 50%, it will be very rare that by throwing 100 times you will get exactly 50 heads. The greater the number of participants, the more suspicious it should seem to us that the two groups are equal. But beware, this only applies to simple randomization. There are methods of randomization in which groups can be more balanced.

Another hot spot is the misuse that can sometimes be made with qualitative variables. Although qualitative variables can be coded with numbers, be very careful with doing arithmetic operations with them. Probably it will not make any sense. Another cheat that we can find has to do with the fact of categorizing a continuous variable. Passing a continuous variable to a qualitative one usually leads to loss of information, so it must have a clear clinical meaning. Otherwise, we can suspect that the reason is the search for a p value less than 0.05, always easier to achieve with the qualitative variable.

Going into the analysis of the data, we must check that the authors have followed the a priori designed protocol of the study. Always be wary of post hoc studies that were not planned from the beginning. If we look for enough, we will always find a group that behaves as we want. As it is said, if you torture the data long enough, it will confess to anything.

Another unacceptable behavior is to finish the study ahead of time for good results. Once again, if the duration of the follow-up has been established during the design phase as the best time to detect the effect, this must be respected. Any violation of the protocol must be more than justified. Logically, it is ethical to finish the study ahead of time due to security reasons, but it will be necessary to take into account how this fact affects the evaluation of the results.

Before performing the analysis of the results, the authors of any study have to debug their data, reviewing the quality and integrity of the values collected. In this sense, one of the aspects to pay attention to is the management of outliers. These are the values that are far from the central values of the distribution. In many occasions they can be due to errors in the calculation, measurement or transcription of the value of the variable, but they can also be real values that are due to the special idiosyncrasy of the variable. The problem is that there is a tendency to eliminate them from the analysis even when there is no certainty that they are due to an error. The correct thing to do is to take them into account when doing the analysis and use, if necessary, robust statistical methods that allow these deviations to be adjusted.

Finally, the aspect that can be more strenuous to those not very expert in statistics is knowing if the correct statistical method has been used. A frequent error is the use of parametric tests without previously checking if the necessary requirements are met. This can be done by ignorance or to obtain statistical significance, since parametric tests are less demanding in this regard. To understand each other, the p-value will be smaller than if we use the equivalent non-parametric test.

Also, with certain frequency, other requirements needed to be able to apply a certain contrast test are ignored. As an example, in order to perform a Student’s t test or an ANOVA, homoscedasticity (a very ugly word that means that the variances are equal) must be checked, and that check is overlooked in many studies. The same happens with regression models that, frequently, are not accompanied by the mandatory diagnosis of the model that allows and justify its use.

Another issue in which there may be cheating is that of multiple comparisons. For example, when the ANOVA reaches significant, the meaning is that there are at least two means that are different, but we do not know which, so we start comparing them two by two. The problem is that when we make repeated comparisons the probability of type I error increases, that is, the probability of finding significant differences only by chance. This may allow finding, if only by chance, a p <0.05, what improves the appearance of the study (especially if you spent a lot of time and / or money doing it). In these cases, the authors must use some of the available corrections (such as Bonferroni’s, one of the simplest) so that the global alpha remains below 0.05. The price to pay is simple: the p-value has to be much smaller to be significant. When we see multiple comparisons without a correction, it will only have two explanations: the ignorance of the one who made the analysis or the attempt to find a statistical significance that, probably, would not support the decrease in p-value that the correction would entail.

Another frequent victim of misuse of statistics is the Pearson’s correlation coefficient, which is used for almost everything. The correlation, as such, tells us if two variables are related, but does not tell us anything about the causality of one variable for the production of the other. Another misuse is to use the correlation coefficient to compare the results obtained by two observers, when probably what should be used in this case is the intraclass correlation coefficient (for continuous variables) or the kappa index (for dichotomous qualitative variables). Finally, it is also incorrect to compare two measurement methods (for example, capillary and venous glycaemia) by correlation or linear regression. For these cases the correct thing would be to use the Passing-Bablok’s regression.

Another situation in which a paranoid mind like mine would suspect is one in which the statistical method employed is not known by the smartest people in the place. Whenever there is a better known (and often simpler) way to do the analysis, we must ask ourselves why they have used such a weird method. In these cases, we will require the authors to justify their choice and provide a reference where we can review the method. In statistics, you have to try to choose the right technique for each occasion and not the one that gives us the most appealing result.

In any of the previous contrast tests, the authors usually use a level of significance for p <0.05, as usual, but the contrast can be done with one or two tails. When we do a trial to try a new drug, what we expect is that it works better than the placebo or the drug with which we are comparing it. However, two other situations can occur that we cannot disdain: that it works the same or, even, that it works worse. A bilateral contrast (with two tails) does not assume the direction of the effect, since it calculates the probability of obtaining a difference equal to or greater than that observed, in both directions. If the researcher is very sure of the direction of the effect, he can make a unilateral contrast (with one tail), measuring the probability of the result in the direction considered. The problem is when he does it for another reason: the p-value of a bilateral contrast is twice as large as that of the unilateral contrast, so it will be easier to achieve statistical significance with the unilateral contrast. The wrong thing is to do the unilateral contrast for that reason. The correct thing, unless there are well-justified reasons, is to make a bilateral contrast.

To go finishing this tricky post, we will say a few words about the use of appropriate measures to present the results. There are many ways to make up the truth without getting to lie and, although basically all say the same, the appearance can be very different depending on how we say it. The most typical example is to use relative risk measures instead of absolute and impact measures. Whenever we see a clinical trial, we must demand that authors provide the absolute risk reduction and the number needed to treat (NNT). The relative risk reduction gives a greater number than the absolute, so it will seem that the impact is greater. Given that the absolute measures are easier to calculate and are obtained from the same data as the relative ones, we should be suspicious if the authors do not offer them to us: perhaps the effect is not as important as they are trying to make us see.

Another example is the use of odds ratio versus risk ratio (when both can be calculated). The odds ratio tends to magnify the association between the variables, so its unjustified use can also make us to be suspicious. If you can, calculate the risk ratio and compare the two measures.

Likewise, we will suspect of studies of diagnostic tests that do not provide us with the likelihood ratios and are limited to sensitivity, specificity and predictive values. Predictive values can be high if the prevalence of the disease in the study population is high, but it would not be applicable to populations with a lower proportion of patients. This is avoided with the use of likelihood ratios. We should always ask ourselves the reason that the authors may have had to obviate the most valid parameter to calibrate the power of a diagnostic test.

And finally, be very careful with the graphics representations of results: here the possibilities of making up the truth are only limited by our imagination. You have to look at the units used and try to extract the information from the graph beyond what it might seem to represent at first glance.

And here we leave the topic for today. We have not spoken in detail about another of the most misunderstood and manipulated entities, which is none other than our p. Many meanings are attributed to p, usually erroneously, as the probability that the null hypothesis is true, probability that has its specific method to make an estimate. But that is another story…

## Pairing

You will all know the case of someone who, after carrying out a study and collecting several million variables, addressed the statistician of his workplace and, demonstrating in a reliable way his clarity of ideas regarding his work, he said: please (You have to be educated), crosscheck everything with everything, to see what comes out.

At this point, several things can happen to you. If the statistician is an unscrupulous soulmate, he will give you a half smile and tell you to come back after a few days. Then, you will be provided with several hundred sheets with graphics, tables and numbers with which you will not know what to do. Another thing that can happen to you is to send to hell, tired as she will be to have similar requests made.

But you can be lucky and find a competent and patient statistician who, in a self-sacrificing way, will explain to you that the thing should not work like that. The logical thing is that you, before collecting any data, have prepared a report of the project in which it is planned, among other things, what is to be analyzed and what variables must be crossed between them. She can even suggest you that, if the analysis is not very complicated, you can try to do it yourself.

The latter may seem like the delirium of a mind disturbed by mathematics but, if you think about it for a moment, it is not such a bad idea. If we do the analysis, at least the preliminary, of our results, it can help us to better understand the study. Also, who can know what we want better than ourselves?

With the current statistical packages, the simplest bivariate statistics can be within our reach. We only have to be careful in choosing the right hypothesis test, for which we must take into account three aspects: the type of variables that we want to compare, if the data are paired or independent and if we have to use parametric or non-parametric tests. Let’s see these three aspects.

Regarding the type of variables, there are multiple denominations according to the classification or the statistical package that we use but, simplifying, we will say that there are three types of variables. First, there are the continuous variables. As the name suggests, they collect the value of a continuous variable such as weight, height, blood glucose concentration, etc. Second, there are the nominal variables, which consist of two or more categories that are mutually excluding. For example, the variable “hair color” can have the categories “brown”, “blonde” and “red hair”. When these variables have two categories, we call them dichotomous (yes / no, alive / dead, etc.). Finally, when the categories are ordered by rank, we speak of ordinal variables: ” do not smoke “, ” smoke little “, ” smoke moderately “, ” smoke a lot “. Although they can sometimes use numbers, they indicate the position of the categories within the series, without implying, for example, that the distance from category 1 to 2 is the same as that from 2 to 3. For example, we can classify vesicoureteral reflux in grades I, II, III and IV (having a degree IV is more than a II, but it does not mean that you have twice as much reflux).

Knowing what kind of variable we are dealing with is simple. If we doubt, we can follow the following reasoning based on the answer to two questions:

1. Does the variable have infinite theoretical values? Here we have to do a bit of abstraction and think about what “theoretical values” really means. For example, if we measure the weight of the subjects of the study, theoretical values will be infinite although, in practice, this will be limited by the precision of our scale. If the answer to this first question is “yes” we will be before a continuous variable. If it is not, we move on to the next question.
2. Are the values sorted in some kind of rank? If the answer is “yes”, we will be dealing with an ordinal variable. If the answer is “no”, we will have a nominal variable.

The second aspect is that of paired or independent measures. Two measures are paired when a variable is measured twice after having applied some change, usually in the same subject. For example: blood pressure before and after a stress test, weight before and after a nutritional intervention, etc. On the other hand, independent measures are those that are not related to each other (they are different variables): weight, height, gender, age, etc.

Finally, we mentioned the possibility of using parametric or non-parametric tests. We are not going to go into detail now, but in order to use a parametric test the variable must fulfill a series of characteristics, such as following a normal distribution, having a certain sample size, etc. In addition, there are techniques that are more robust than others when it comes to having to meet these conditions. When in doubt, it is preferable to use non-parametric techniques unnecessarily (the only problem is that it is more difficult to achieve statistical significance, but the contrast is just as valid) than using a parametric test when the necessary requirements are not met.

Once we have already answered these three aspects, we can only make the pairs of variables that we are going to compare and choose the appropriate statistical test. You can see it summarized in the attached table.The type of independent variable is represented in the rows, which is the one whose value does not depend on another variable (it is usually on the x axis of the graphic representations) and which is usually the one that we modified in the study to see the effect on another variable (the dependent). In the columns, on the other hand, we have the dependent variable, which is the one whose value is modified with the changes of the independent variable. Anyway, do get muddled: the statistical software will make the hypothesis contrast without taking into account which is the dependent and which the independent, only taking into account the types of variables.

The table is self-explanatory, so we will not give it much time. For example, if we have measured blood pressure (contiuous variable) and we want to know if there are differences between men and women (gender, nominal dichotomous variable), the appropriate test will be Student’s t test for independent samples. If we wanted to see if there is a difference in pressure before and after a treatment, we would use the same Student’s t test but for paired samples.

Another example: if we want to know if there are significant differences in the color of hair (nominal, polytomous: “blond”, “brown” and “redhead) and if the participant is from the north or south of Europe (nominal, dichotomous), we could use a Chi-square’s test.

And here we will end for today. We have not talked about the peculiarities of each test that we have to take into account, but we have only mentioned the test itself. For example, the chi-square’s has to meet minimums in each box of the contingency table, in the case of Student’s t we must consider whether the variances are equal (homoscedasticity) or not, etc. But that is another story…

## Achilles and Effects Forest

Achilles. What a man! Definitely, one of the main characters among those who were in that mess that was ensued in Troy because of Helena, a.k.a. the beauty. You know his story. In order to make him invulnerable his mother, who was none other than Tetis, the nymph, bathed him in ambrosia and submerged him in the River Stix. But she made a mistake that should not have been allowed to any nymph: she took him by his right heel, which did not get wet with the river’s water. And so, his heel became his only vulnerable part. Hector didn’t realize it in time but Paris, totally on the ball, put an arrow in Achilles’ heel and sent him back to the Stix, but not into the water, but rather to the other side. And without Charon the Ferryman.

This story is the origin of the expression “Achilles’ heel”, which usually refers to the weakest or most vulnerable point of someone or something that, otherwise, is usually known for its strength.

For example, something as robust and formidable as meta-analysis has its Achilles heel: the publication bias. And that’s because in the world of science there is no social justice.

All scientific works should have the same opportunities to be published and achieve fame, but the reality is not at all like that and they can be discriminated against for four reasons: statistical significance, popularity of the topic they are dealing with, having someone to sponsor them and the language in which they are written.

These are the main factors that can contribute to publication bias. First, studies with more significant results are more likely to be published and, within these, they are more likely to be published when the effect is greater. This means that studies with negative results or effects of small magnitude may not be published, which will draw a biased conclusion from the analysis only of large studies with positive results. In the same way, papers on topics of public interest are more likely to be published regardless of the importance of their results. In addition, the sponsor also influences: a company that finances a study with a product of theirs that has gone wrong, probably is not going to publish it so that we all know that their product is not useful.

Secondly, as is logical, published studies are more likely to reach our hands than those that are not published in scientific journals. This is the case of doctoral theses, communications to congresses, reports from government agencies or, even, pending studies to be published by researchers of the subject that we are dealing with. For this reason it is so important to do a search that includes this type of work, which is included within the grey literature term.

Finally, a series of biases can be listed that influence the likelihood that a work will be published or retrieved by the researcher performing the systematic review such as language bias (the search is limited by language), availability bias ( include only those studies that are easy for the researcher to recover), the cost bias (studies that are free or cheap), the familiarity bias (only those from the researcher’s discipline are included), the duplication bias (those that have significant results are more likely to be published more than once) and citation bias (studies with significant results are more likely to be cited by other authors).

One may think that this loss of studies during the review cannot be so serious, since it could be argued, for example, that studies not published in peer-reviewed journals are usually of poorer quality, so they do not deserve to be included in the meta-analysis However, it is not clear either that scientific journals ensure the methodological quality of the study or that this is the only method to do so. There are researchers, like those of government agencies, who are not interested in publishing in scientific journals, but in preparing reports for those who commission them. In addition, peer review is not a guarantee of quality since, too often, neither the researcher who carries out the study nor those in charge of reviewing it have a training in methodology that ensures the quality of the final product.

All this can be worsened by the fact that these same factors can influence the inclusion and exclusion criteria of the meta-analysis primary studies, in such a way that we obtain a sample of articles that may not be representative of the global knowledge on the subject of the systematic review and meta-analysis.

If we have a publication bias, the applicability of the results will be seriously compromised. That is why we say that the publication bias is the true Achilles’ heel of meta-analysis.

If we correctly delimit the inclusion and exclusion criteria of the studies and do a global and unrestricted search of the literature we will have done everything possible to minimize the risk of bias, but we can never be sure of having avoided it. That is why techniques and tools have been devised for its detection.

The most used has the sympathetic name of funnel plot. It shows the magnitude of the measured effect (X axis) versus a precision measurement (Y axis), which is usually the sample size, but which can also be the inverse of the variance or the standard error. We represent each primary study with a point and observe the point cloud.

In the most usual way, with the size of the sample on the Y axis, the precision of the results will be higher in the larger sample studies, so that the points will be closer together in the upper part of the axis and will be dispersed when approaching the origin of the axis Y. In this way, we observe a cloud of points in the form of a funnel, with the wide part down. This graphic should be symmetrical and, if that is not the case, we should always suspect a publication bias. In the second example attached you can see how there are “missing” studies on the side of lack of effect: this may mean that only studies with positive results are published.

This method is very simple to use but, sometimes, we can have doubts about the asymmetry of our funnel, especially if the number of studies is small. In addition, the funnel can be asymmetrical due to quality defects in the studies or because we are dealing with interventions whose effect varies according to the sample size of each study. For these cases, other more objective methods have been devised, such as the Begg’s rank correlation test and the Egger’s linear regression test.

The Begg’s test studies the presence of association between the estimates of the effects and their variances. If there is a correlation between them, bad going. The problem with this test is that it has little statistical power, so it is not reliable when the number of primary studies is small.

Egger’s test, more specific than Begg’s, consists of plotting the regression line between the precision of the studies (independent variable) and the standardized effect (dependent variable). This regression must be weighted by the inverse of the variance, so I do not recommend that you do it on your own, unless you are consummate statisticians. When there is no publication bias, the regression line originates at the zero of the Y axis. The further away from zero, the more evidence of publication bias.

As always, there are computer programs that do these tests quickly without having to burn your brain with the calculations.

What if after doing the work we see that there is publication bias? Can we do something to adjust it? As always, we can.

The simplest way is to use a graphic method called trim and fill. It consists of the following: a) we draw the funnel plot; b) we remove the small studies so that the funnel is symmetrical; c) the new center of the graph is determined; d) we recover the previously removed studies and we add their reflection to the other side of the center line; e) we estimate again the effect.Another very conservative attitude that we can adopt is to assume that there is a publication bias and to ask how much it affects our results, assuming that we have left studies not included in the analysis.

The only way to know if the publication bias affects our estimates would be to compare the effect in the retrieved and unrecovered studies but, of course, then we would not have to worry about the publication bias.

To know if the observed result is robust or, on the contrary, it is susceptible to be biased by a publication bias, two methods of the fail-safe N have been devised.

The first is the Rosenthal’s fail-safe N method. Suppose we have a meta-analysis with an effect that is statistically significant, for example, a risk ratio greater than one with a p <0.05 (or a 95% confidence interval that does not include the null value, one). Then we ask ourselves a question: how many studies with RR = 1 (null value) will we have to include until p is not significant? If we need few studies (less than 10) to make the value of the effect null, we can worry because the effect may in fact be null and our significance is the product of a publication bias. On the contrary, if many studies are needed, the effect is likely to be truly significant. This number of studies is what the letter N of the name of the method means.

The problem with this method is that it focuses on the statistical significance and not on the relevance of the results. The correct thing would be to look for how many studies are needed so that the result loses clinical relevance, not statistical significance. In addition, it assumes that the effects of the missing studies is null (one in case of risk ratios and odds ratios, zero in cases of differences in means), when the effect of the missing studies can go in the opposite direction than the effect that we detect or in the same sense but of smaller magnitude.

To avoid these disadvantages there is a variation of the previous formula that assesses the statistical significance and clinical relevance. With this method, which is called the Orwin’s fail-safe N, it is calculated how many studies are needed to bring the value of the effect to a specific value, which will generally be the least effect that is clinically relevant. This method also allows to specify the average effect of the missing studies.

To end the meta-analysis explanation, let’s see what is the right way to express the results of data analysis. To do it well, we can follow the recommendations of the statement, which devotes seven of its 27 items to give us advice on how to present the results of a meta-analysis.

First, we must inform about the selection process of studies: how many we have found and evaluated, how many we have selected and how many rejected, explaining in addition the reasons for doing so. For this, the flowchart that should include the systematic review from which the meta-analysis proceeds if it complies with the PRISMA statement is very useful.

Secondly, the characteristics of the primary studies must be specified, detailing what data we get from each one of them and their corresponding bibliographic citations to facilitate that any reader of the review can verify the data if he does not trust us. In this sense, there is also the third section, which refers to the evaluation of the risk of study biases and their internal validity.

Fourth, we must present the results of each individual study with a summary data of each intervention group analyzed together with the calculated estimators and their confidence intervals. These data will help us to compile the information that PRISMA asks us in its fifth point referring to the presentation of results and it is none other than the synthesis of all the meta-analysis studies, their confidence intervals, homogeneity study results, etc.

This is usually done graphically by means of an effects diagram, a graphical tool popularly known as forest plot, where the trees would be the primary studies of the meta-analysis and where all the relevant results of the quantitative synthesis are summarized.

The Cochrane’s Collaboration recommends structuring the forest plot in five well differentiated columns. Column 1 lists the primary studies or the groups or subgroups of patients included in the meta-analysis. They are usually represented by an identifier composed of the name of the first author and the date of publication.Column 2 shows the results of the measures of effect of each study as reported by their respective authors.

Column 3 is the actual forest plot, the graphic part of the subject. It shows the measures of effect of each study on both sides of the zero effect line, which we already know is zero for mean differences and one for odds ratios, risk ratios, hazard ratios, etc. Each study is represented by a square whose area is usually proportional to the contribution of each one to the overall result. In addition, the square is within a segment that represents the extremes of its confidence interval.

These confidence intervals inform us about the accuracy of the studies and tell us which are statistically significant: those whose interval does not cross the zero effect line. Anyway, do not forget that, although crossing the line of no effect and being not statistically significant, the interval boundaries can give us much information about the clinical significance of the results of each study. Finally, at the bottom of the chart we will find a diamond that represents the global result of the meta-analysis. Its position with respect to the null effect line will inform us about the statistical significance of the overall result, while its width will give us an idea of its accuracy (its confidence interval). Furthermore, on top of this column will find the type of effect measurement, the analysis model data is used (fixed or random) and the significance value of the confidence intervals (typically 95%).

This chart is usually completed by a fourth column with the estimated weight of each study in per cent format and a fifth column with the estimates of the weighted effect of each. And in some corner of this forest will be the measure of heterogeneity that has been used, along with its statistical significance in cases where relevant.

To conclude the presentation of the results, PRISMA recommends a sixth section with the evaluation that has been made of the risks of bias in the study and a seventh with all the additional analyzes that have been necessary: stratification, sensitivity analysis, metaregression, etc.

As you can see, nothing is easy about meta-analysis. Therefore, the Cochrane’s recommends following a series of steps to correctly interpret the results. Namely:

1. Verify which variable is compared and how. It is usually seen at the top of the forest plot.
2. Locate the measure of effect used. This is logical and necessary to know how to interpret the results. A hazard ratio is not the same as a difference in means or whatever it was used.
3. Locate the diamond, its position and its amplitude. It is also convenient to look at the numerical value of the global estimator and its confidence interval.
4. Check that heterogeneity has been studied. This can be seen by looking at whether the segments that represent the primary studies are or are not very dispersed and whether they overlap or not. In any case, there will always be a statistic that assesses the degree of heterogeneity. If we see that there is heterogeneity, the next thing will be to find out what explanation the authors give about its existence.
5. Draw our conclusions. We will look at which side of the null effect line are the overall effect and its confidence interval. You already know that, although it is significant, the lower limit of the interval should be as far as possible from the line, because of the clinical relevance, which does not always coincide with statistical significance. Finally, look again at the study of homogeneity. If there is a lot of heterogeneity, the results will not be as reliable.

And with this we end the topic of meta-analysis. In fact, the forest plot is not exclusive to meta-analyzes and can be used whenever we want to compare studies to elucidate their statistical or clinical significance, or in cases such as equivalence studies, in which the null effect line is joined of the equivalence thresholds. But it still has one more utility. A variant of the forest plot also serves to assess if there is a publication bias in the systematic review, although, as we already know, in these cases we change its name to funnel graph. But that is another story…

## Apples and pears

You all sure know the Chinese tale of the poor solitary rice grain that falls to the ground and nobody can hear it. Of course, if instead of falling a grain it falls a sack full of rice that will be something else. There are many examples of union making strength. A red ant is harmless, unless it bites you in some soft and noble area, which are usually the most sensitive. But what about a marabout of millions of red ants? That is what scares you up, because if they all come together and come for you, you could do little to stop their push. Yes, the union is strength.

And this also happens with statistics. With a relatively small sample of well-chosen voters we can estimate who will win an election in which millions vote. So, what could we not do with a lot of those samples? Surely the estimate would be more reliable and more generalizable.

Well, this is precisely one of the purposes of meta-analysis, which uses various statistical techniques to make a quantitative synthesis of the results of a set of studies that, although try to answer the same question, do not reach exactly to the same result. But beware; we cannot combine studies to draw conclusions about the sum of them without first taking a series of precautions. This would be like mixing apples and pears which, I’m not sure why, should be something terribly dangerous because everyone knows it’s something to avoid.

Think that we have a set of clinical trials on the same topic and we want to do a meta-analysis to obtain a global result. It is more than convenient that there is as little variability as possible among the studies if we want to combine them. Because, ladies and gentlemen, here also rules the saying: alongside but separate.

Before thinking about combining the results of the studies of a systematic review to perform a meta-analysis, we must always make a previous study of the heterogeneity of the primary studies, which is nothing more than the variability that exists among the estimators that have been obtained in each of those studies.

First, we will investigate possible causes of heterogeneity, such as differences in treatments, variability of the populations of the different studies and differences in the designs of the trials. If there is a great deal of heterogeneity from the clinical point of view, perhaps the best thing to do is not to do meta-analysis and limit the analysis to a qualitative synthesis of the results of the review.

Once we come to the conclusion that the studies are similar enough to try to combine them we should try to measure this heterogeneity to have an objective data. For this, several privileged brains have created a series of statistics that contribute to our daily jungle of acronyms and letters.

Until recently, the most famous of those initials was the Cochran’s Q, which has nothing to do either with James Bond or our friend Archie Cochrane. Its calculation takes into account the sum of the deviations between each of the results of primary studies and the global outcome (squared differences to avoid positives cancelling negatives), weighing each study according to their contribution to overall result. It looks awesome but in reality, it is no big deal. Ultimately, it’s no more than an aristocratic relative of ji-square test. Indeed, Q follows a ji-square distribution with k-1 degrees of freedom (being k the number of primary studies). We calculate its value, look at the frequency distribution and estimate the probability that differences are not due to chance, in order to reject our null hypothesis (which assumes that observed differences among studies are due to chance). But, despite the appearances, Q has a number of weaknesses.

First, it’s a very conservative parameter and we must always keep in mind that no statistical significance is not always synonymous of absence of heterogeneity: as a matter of fact, we cannot reject the null hypothesis, so we have to know that when we approved it we are running the risk of committing a type II error and blunder. For this reason, some people propose to use a significance level of p < 0.1 instead of the standard p < 0.5. Another Q’s pitfall is that it doesn’t quantify the degree of heterogeneity and, of course, doesn’t explain the reasons that produce it. And, to top it off, Q loses power when the number of studies is small and doesn’t allow comparisons among different meta-analysis if they have different number of studies.

This is why another statistic has been devised that is much more celebrated today: I2. This parameter provides an estimate of total variation among studies with respect to total variability or, put it another way, the proportion of variability actually due to heterogeneity for actual differences among the estimates compared with variability due to chance. It also looks impressive, but it’s actually an advantageous relative of the intraclass correlation coefficient.

Its value ranges from 0 to 100%, and we usually consider the limits of 25%, 50% and 75% as signs of low, moderate and high heterogeneity, respectively. I2 is not affected either by the effects units of measurement or the number of studies, so it allows comparisons between meta-analysis with different units of effect measurement or different number of studies.

If you read a study that provides Q and you want to calculate I2, or vice versa, you can use the following formula, being k the number of primary studies:

$I^{2}=&space;\frac{Q-k+1}{Q}$

There’s a third parameter that is less known, but not less worthy of mention: H2. It measures the excess of Q value in respect of the value that we would expect to obtain if there were no heterogeneity. Thus, a value of 1 means no heterogeneity and its value increases as heterogeneity among studies does. But its real interest is that it allows calculating I2 confidence intervals.

Other times, the authors perform a hypothesis contrast with a null hypothesis of non-heterogeneity and use a ji-square or some similar statistic. In these cases, what they provide is a value of statistical significance. If the p is <0.05 the null hypothesis can be rejected and say that there is heterogeneity. Otherwise we will say that we cannot reject the null hypothesis of non-heterogeneity.

In summary, whenever we see an indicator of homogeneity that represents a percentage, it will indicate the proportion of variability that is not due to chance. For their part, when they give us a “p” there will be significant heterogeneity when the “p” is less than 0.05.

Do not worry about the calculations of Q, I2 and H2. For that there are specific programs as RevMan or modules within the usual statistical programs that do the same function.

A point of attention: always remember that not being able to demonstrate heterogeneity does not always mean that the studies are homogeneous. The problem is that the null hypothesis assumes that they are homogeneous and the differences are due to chance. If we can reject it we can assure that there is heterogeneity (always with a small degree of uncertainty). But this does not work the other way around: if we cannot reject it, it simply means that we cannot reject that there is no heterogeneity, but there will always be a probability of committing a type II error if we directly assume that the studies are homogeneous.

For this reason, a series of graphical methods have been devised to inspect the studies and verify that there is no data of heterogeneity even if the numerical parameters say otherwise.

The most employed of them is, perhaps, the , with can be used for both meta-analysis from trials or observational studies. This graph represents the accuracy of each study versus the standardize effects. It also shows the adjusted regression line and sets two confidence bands. The position of each study regarding the accuracy axis indicates its weighted contribution to overall results, while its location outside the confidence bands indicates its contribution to heterogeneity.

Galbraith’s graph can also be useful for detecting sources of heterogeneity, since studies can be labeled according to different variables and see how they contribute to the overall heterogeneity.

Another available tool you can use for meta-analysis of clinical trials is L’Abbé’s plot. It represents response rates to treatment versus response rates in control group, plotting the studies to both sides of the diagonal. Above that line are studies with positive treatment outcome, while below are studies with an outcome favorable to control intervention. The studies usually are plotted with an area proportional to its accuracy, and its dispersion indicates heterogeneity. Sometimes, L’Abbé’s graph provides additional information. For example, in the accompanying graph you can see that studies in low-risk areas are located mainly below the diagonal. On the other hand, high-risk studies are mainly located in areas of positive treatment outcome. This distribution, as well as being suggestive of heterogeneity, may suggest that efficacy of treatments depends on the level of risk or, put another way, we have an effect modifying variable in our study. A small drawback of this tool is that it is only applicable to meta-analysis of clinical trials and when the dependent variable is dichotomous.

Well, suppose we study heterogeneity and we decide that we are going to combine the studies to do a meta-analysis. The next step is to analyze the estimators of the effect size of the studies, weighing them according to the contribution that each study will have on the overall result. This is logical; it cannot contribute the same to the final result a trial with few participants and an imprecise result than another with thousands of participants and a more precise result measure.

The most usual way to take these differences into account is to weight the estimate of the size of the effect by the inverse of the variance of the results, subsequently performing the analysis to obtain the average effect. For these there are several possibilities, some of them very complex from the statistical point of view, although the two most commonly used methods are the fixed effect model and the random effects model. Both models differ in their conception of the starting population from which the primary studies of meta-analysis come.

The fixed effect model considers that there is no heterogeneity and that all studies estimate the same effect size of the population (they all measure the same effect, that is why it is called a fixed effect), so it is assumed that the variability observed among the individual studies is due only to the error that occurs when performing the random sampling in each study. This error is quantified by estimating intra-study variance, assuming that the differences in the estimated effect sizes are due only to the use of samples from different subjects.

On the other hand, the random effects model assumes that the effect size varies in each study and follows a normal frequency distribution within the population, so each study estimates a different effect size. Therefore, in addition to the intra-study variance due to the error of random sampling, the model also includes the variability among studies, which would represent the deviation of each study from the mean effect size. These two error terms are independent of each other, both contributing to the variance of the study estimator.

In summary, the fixed effect model incorporates only one error term for the variability of each study, while the random effects model adds, in addition, another error term due to the variability among the studies.

You see that I have not written a single formula. We do not actually need to know them and they are quite unfriendly, full of Greek letters that no one understands. But do not worry. As always, statistical programs like RevMan from the Cochrane Collaboration allow you to do the calculations in a simple way, including and removing studies from the analysis and changing the model as you wish.

The type of model to choose has its importance. If in the previous homogeneity analysis we see that the studies are homogeneous we can use the fixed effect model. But if we detect that heterogeneity exists, within the limits that allow us to combine the studies, it will be preferable to use the random effects model.

Another consideration is the applicability or external validity of the results of the meta-analysis. If we have used the fixed effect model, we will be committed to generalize the results out of populations with characteristics similar to those of the included studies. This does not occur with the results obtained using the random effects model, whose external validity is greater because it comes from studies of different populations.

In any case, we will obtain a summary effect measure along with its confidence interval. This confidence interval will be statistically significant when it does not cross the zero effect line, which we already know is zero for mean differences and one for odds ratios and risk ratios. In addition, the amplitude of the interval will inform us about the precision of the estimation of the average effect in the population: how much wider, less precise, and vice versa.

If you think a bit, you will immediately understand why the random effects model is more conservative than the fixed effect model in the sense that the confidence intervals obtained are less precise, since it incorporates more variability in its analysis. In some cases it may happen that the estimator is significant if we use the fixed effect model and it is not significant if we use the random effect model, but this should not condition us when choosing the model to use. We must always rely on the previous measure of heterogeneity, although if we have doubts, we can also use the two models and compare the different results.

Having examined the homogeneity of primary studies we can come to the grim conclusion that heterogeneity dominates the situation. Can we do something to manage it? Sure, we can. We can always not to combine the studies, or combine them despite heterogeneity and obtain a summary result but, in that case, we should also calculate any measure of variability among studies and yet we could not be sure of our results.

Another possibility is to do a stratified analysis according to the variable that causes heterogeneity, provided that we are able to identify it. For this we can do a sensitivity analysis, repeating calculations once removing one by one each of the subgroups and checking how it influences the overall result. The problem is that this approach ignores the final purpose of any meta-analysis, which is none than obtaining an overall value of homogeneous studies.

Finally, the brainiest on these issues can use meta-regression. This technique is similar to multivariate regression models in which the characteristics of the studies are used as explanatory variables, and effect’s variable or some measure of deviation of each study with respect to global result are used as dependent variable. Also, it should be done a weighting according to the contribution of each study to the overall result and try not to score too much coefficients to the regression model if the number of primary studies is not large. I wouldn’t advise you to do a meta-regression at home if it is not accompanied by seniors.

And we only need to check that we have not omitted studies and that we have presented the results correctly. The meta-analysis data are usually represented in a specific graph that is known as forest plot. But that is another story…

## Three feet of a cat

To look for three legs of a cat, or splitting hairs, is a popular Spanish saying. It seems that when one looks for three feet of a cat he tries to demonstrate something impossible, generally with tricks and deceptions. As the English speakers say, if it ain’t broke, don’t fix it. In fact, the initial saying referred to looking for five feet instead of three. This seems more logical, since as cats have four legs, finding three of them is easy, but finding five is impossible, unless we consider the tail of the cat as another foot, which does not make much sense.

But today we will not talk about cats with three, four or five feet. Let’s talk about something a little more ethereal, such as multivariate multiple linear regression models. This is a cat with a lot of feet, but we are going to focus only on three of them that are called collinearity, tolerance and inflation factor (or increase) of the variance. Do not be discouraged, it’s easier than it may seem.

We saw in ahow simple linear regression models related two variables to each other, so that the variations of one of them (the independent variable or predictor) could be used to calculate how the other variable would change (the dependent variable). These models were represented by the equation y = a + bx, where x is the independent variable and y the dependent variable.

However, multiple linear regression adds more independent variables, so that it allows to make predictions of the dependent variable according to the values of the predictor or independent variables. The generic formula would be as follows:

y = a + bx1 + cx2 + dx3 + … + nxn, where n is the number of independent variables.

One of the conditions for the multiple linear regression models to work properly is that the independent variables are actually independent and uncorrelated.

Imagine an absurd example in which we put in the model the weight in kilograms and the weight in pounds. Both variables will vary in the same way. In fact the correlation coefficient, R, will be 1, since practically the two represent the same variable. Such foolish examples are difficult to see in scientific work, but there are others less obvious (including, for example, height and body mass index, which is calculated from weight and height) and others that are not at all evident for the researcher. This is what is called collinearity, which is nothing more than the existence of a linear association between the set of independent variables.

Collinearity is a serious problem for the multivariate model, since the estimates obtained by it are very unstable, as it becomes more difficult to separate the effect of each predictor variable.

Well, to determine if our model suffers from collinearity we can construct a matrix where the coefficients of correlation, R, of some variables with others are shown. In those cases in which we observe high R, we can suspect that there is collinearity. However, if we want to quantify this we will resort to the other two feet of the cat that we mentioned at the beginning: tolerance and inflation factor of variance.

If we square the coefficient R we obtain the coefficient of determination (R2), which represents the percentage of the variation (or variance) of a variable that is explained by the variation in the other variable. Thus, we find the concept of tolerance, which is calculated as the complement of R2 (1-R2) and represents the proportion of the variability of that variable that is not explained by the rest of the independent variables included in the regression model.

In this way, the lower the tolerance, the more likely there is collinearity. Collinearity is generally considered to exist when R2 is greater than 0.9 and therefore the tolerance is below 0.1.

We only have to explain the third foot, which is the inflation factor of the variance. This is calculated as the inverse of the tolerance (1 / T) and represents the proportion of the variability (or variance) of the variable that is explained by the rest of the predictor variables of the model. Of course, the greater the inflation factor of the variance, the greater the likelihood of collinearity. Generally, collinearity is considered to exist when the inflation factor between two variables is greater than 10 or when the mean of all inflation factors of all independent variables is much greater than one.

And here we are going to leave the multivariate models for today. Needless to say, everything we have told is done in practice using computer programs that calculate these parameters in a simple way.

We have seen here some aspects of multiple linear regression, perhaps the most widely used multivariate model. But there are others, such as multivariate analysis of variance (MANOVA), factors analysis, or clusters analysis. But that is another story…

## In search of causality

In medicine we often try to look for cause-effect relationships. If we want to show that the drug X produces an effect, we have only to select two groups of people, one group we give the drug, the other group we do not give it and see if there are differences.

But it is not so simple, because we can never be sure that differences in effect between the two groups actually are due to other factors than the treatment we have used. These factors are the so-called confounding factors, which may be known or unknown and which may bias the results of the comparison.

To resolve this problem a key element of a clinical trial, randomization, was invented. If we divide the participants in the trial between the two branches randomly we will get these confounding variables to be distributed homogeneously between the two arms of the trial, so any difference between the two will have to be due to the intervention. Only in this way can we establish cause-effect relationships between our exposure or treatment and the outcome variable we measure.

The problem of quasi-experimental and observational studies is that they lack randomization. For this reason, we can never be sure that the differences are due to exposure and not to any confounding variable, so we cannot safely establish causal relationships.

This is an annoying inconvenience, since it will often be impossible to carry out randomized trials either for ethical, economic reasons, the nature of the intervention or whatever. That is why some tricks have been invented in order to establish causal relations in the absence of randomization. One of these techniques is the propensity score we saw in an earlier post. Another is the one we are going to develop today, which has the nice name of regression discontinuity.

Regression discontinuity is a quasi-experimental design that allows causal inference in the absence of randomization. It can be applied when the exposure of interest is assigned, at least partially, according to the value of a continuous random variable if this variable falls above or below a certain threshold value.

Consider, for example, a hypocholesterolemic drug that we will use when LDL cholesterol rises above a given value, or an antiretroviral therapy in an AIDS patient that we will indicate when his CD4 count falls below a certain value. There is a discontinuity in the threshold value of the variable that produces a sudden change in the probability of assignment to the intervention group, as I show in the attached figure.

In these cases where the allocation of treatment depends, at least in part, on the value of a continuous variable, the allocation in the vicinity of the threshold is almost as if it were random. Why? Because determinations are subject to random variability by sampling error (in addition to the variability of biological variables themselves), which makes individuals very close to the threshold, above or below, very similar in terms of the variables that may act as confounders (being above or below the threshold may depend on the random variability of the result of the measurement of the variable), similar to what happens in a clinical trial. At the end of the day, we may think that a clinical trial is nothing more than a discontinuity design in which the threshold is a random number.

The math of regression discontinuity is not for beginners so I do not intend to explain it here (I would have to understand it first), so we will settle for knowing some terms that will help us to understand the works that use this methodology.

Regression discontinuity may be sharp or fuzzy. In the sharp one, the probability of assignment changes from zero to one at the threshold (the allocation of treatment follows a deterministic rule). For example, treatment is initiated when the threshold is crossed, regardless of other factors. On the other hand, in the fuzzy regression there are other factors at stake that make the probability of allocation change in the threshold, but not from zero to one, but may depend on those other factors added.

Thus, the result of the regression model varies somewhat depending on whether it is a sharp or fuzzy regression discontinuity. In the case of sharp regression, the so-called average causal effect is calculated, according to which participants are assigned to the intervention with certainty if they cross the threshold. In the case of fuzzy regression, the allocation is no longer performed according to a deterministic model, but according to a probabilistic one (according to the threshold value and other factors that the researcher may consider important). In these cases, an intention-to-treat analysis should be done according to the difference in the probability of allocation near the cut-off point (some may not exceed the threshold but be assigned to the intervention because the investigator considers the other factors).

Thus, the probabilistic model will have to measure the effect on the compliers (those assigned to the intervention), so the regression model will give us the complier average causal effect, which is the typical measure of fuzzy regression discontinuity.

And I think we’re going to leave it for today. We have not said anything about the regression equation, but suffice it to say that it takes into account the slopes of the probability function of allocation before and after the threshold and an interaction variable for the possibility that the effects of the treatment are heterogeneous on both sides of the threshold. As you see, everything is quite complicated, but for that are the statistical packages like R or Stata that implement these models with little effort.

Finally, to say only that it is usual to see models that use linear regression for quantitative outcome variables, but there are extensions of the model that use dichotomous variables and logistic regression techniques, and even models with survival studies and time-to-event variables. But that is another story…

## Censorship

In the best-known sense, censorship is the action of examining a work intended for the public, suppressing or modifying the part that does not fit certain political, moral or religious aspect, to determine whether or not it can be published or exhibited. So what do we mean in statistics when we talk about censored data? Nothing to do with politics, morality or religion. In order to explain what a censored data is, we must first discuss the time-to-event variables and survival analyzes.

In general, we can say that there are three types of variables: quantitative, qualitative and time-to-event. The first two are fairly well understood in general, but the time-to-event are a little more complicated to understand.-

Imagine that we want to study the mortality of that terrible disease that fildulastrosis is. We could count the number of deaths at the end of the study period and divide them by the total population at the beginning. For example, if at the beginning there are 50 patients and four die during follow-up, we could calculate the mortality as 4/50 = 0.08, or 8%. Thus, if we have followed the population for five years, we can say that the survival of the disease at five years is 92% (100-8 = 92).

Simple, isn’t it? The problem is that this is only valid when all subjects have the same follow-up period and no losses or dropouts occur throughout the study, a situation that is often far from the reality in most cases.

In these cases, the correct thing to do is to measure not only if death occurs (which would be a dichotomous variable), but also when it occurs, also taking into account the different follow-up period and the losses. Thus, we would use a time-to-event variable, which is composed of a dichotomous variable (the event being measured) and a continuous variable (the follow-up time when it occurs).

Following the example above, participants in the study could be classified into three types: those who die during follow-up, those who remain alive at the end of the study, and those who are lost during follow-up.

Of those who die we can calculate their survival but, what is the survival of those who are alive at the end of the study? And what is the survival of those who are lost during follow-up? It is clear that some of the lost may have died at the end of the study without us detecting it, so our measure of mortality will not be accurate.

And this is where we find the censored data. All those who do not present the event during the survival study are called censored (losses and those who finish the study without presenting the event). The importance of these censored data is that they must be taken into account when doing the survival study, as we will see below.

The methodology to be followed is to create a survival table that takes into account the events (in this case the deaths) and the censored data, as we can see in the attached table.

The columns of the table represent the following: x, the year number of the follow-up; Nx, the number of participants alive at the beginning of that year; Cx, the number of losses of that year (censored); Mx, the number of deaths during that period; PD, probability of dying in that period; PPS, the probability of surviving in that period (the probability of not presenting the event); And PGS, the global probability of survival up to that point.As we see, the first year we started with 50 participants, one of whom died. The probability of dying in that period is 1/50 = 0.02, so the probability of survival in the period (which is equal to the global since it is the first period) is 1-0.02 = 0, 98.

In the second period we start with 49 and no one dies or is lost. The PD in the period is zero and survival one. Thus, the overall probability will be 1×0.98 = 0.98.

In the third period we continue with 49. Two are lost and one dies. The PD is 1/49 = 0.0204 and the PPS is 1-0.0204 = 0.9796. If we multiply the PSP by the global of the previous period, we obtain the overall survival of this period: 0.9796×0.98 = 0.96.

In the fourth period we started with 46 participants, resulting in five losses and two deaths. The PD will be 2/46 = 0.0434, the PPS of 1-0.0434 = 0.9566 and the PGS of 0.9566×0.96 = 0.9183.

And last, in the fifth period we started with 39 participants. We have two censored and no event (death). PD is zero, PPS is equal to one (no one dies in this period) and PGS 1×0.9183 = 0.9183.

Finally, taking into account the censored data, we can say that the overall survival at five years of fildulastrosis is 91.83%.

And with this we are going to leave it for today. We have seen how a survival table with censored data is constructed to take into account unequal follow-up of participants and losses during follow-up.

Only two thoughts before finishing. First, even if we talk about survival analysis, the event does not have to be the death of the participants. It can be any event that occurs throughout the study follow-up.

Second, the time-to-event and censored data are the basis for performing other statistical techniques that estimate the probability of occurrence of the event under study at a given time, such as the Cox regression models. But that is another story…

## A case of misleading probability

Today we are going to see another of those examples where intuition about the value of certain probabilities plays tricks on us. And, for that, we will use nothing less than Bayes’ theorem, playing a little with conditioned probabilities. Let’s see step by step how it works.

What is the probability of two events occurring? The probability of an event A occurring is P(A) and that of B, P(B). Well, the probability of the two occurring is P(A∩B), which, if the two events are independent, is equal to P(A) x P(B).

Imagine that we have a die with six faces. If we throw it once, the probability of taking out, for example, a five is 1/6 (one result among the six possible). The probability to draw a four is also 1/6. What will be the probability of getting a four, once in the first roll we get a five? Since the two runs are independent, the probability of the combination five followed by four will be 1/6 x 1/6 = 1/36.

Now let’s think of another example. Suppose that in a group of 10 people there are four doctors, two of whom are surgeons. If we take one at random, the probability of being a doctor is 4/10 = 0.4 and that of a surgeon is 2/10 = 0.2. But if we get one and know that he is a doctor, the probability that he is a surgeon will no longer be 0.2, because the two events, being a doctor and a surgeon, are not independent. If you are a doctor, the probability that you are a surgeon will be 0.5 (half the doctors in our group are surgeons).

When two events are dependent, the probability of occurrence of the two will be the probability of occurrence of the first, once the second occurs, by the probability of occurrence of the second. So the P(surgeon) = P(surgeon|doctor) x P(doctor). We can generalize the expression as follows:

P(A∩B) = P(A|B) x P(B), and changing the order of the components of the expression, we obtain the so-called Bayes rule, as follows:

P(A|B) = P(A∩B) / P(B).

The P(A∩B) will be the probability of B, once A is produced, by the probability of A = P(B|A) x P(A). On the other hand, the probability of B will be equal to the sum of the probability of occurrence B once A is produced plus the probability of occurring B without occurring A, which put in mathematical form is of the following form:

P(B|A) x P(A) + P(B|Ac) x P(Ac), being P(Ac) the probability of not occurring A.

If we substitute the initial rule for its developed values, we obtain the best known expression of the Bayes theorem:

Let’s see how the Bayes theorem is applied with a practical example. Consider the case of acute fildulastrosis, a serious disease whose prevalence in the population is, fortunately, quite low, one per 1000 inhabitants. Then, the P(F) = 0.001.

Luckily we have a good diagnostic test, with a sensitivity of 98% and a specificity of 95%. Suppose now that I take the test and it gives me a positive result. Do I have to scare myself a lot? What is the probability that I actually have the disease? Do you think it will be high or low? Let’s see.

A sensitivity of 98% means that the probability of giving positive when having the disease is 0.98. Mathematically, P(POS|F) = 0,98. On the other hand, a specificity of 95% means that the probability of a negative result being healthy is 0.95. That is, P(NEG|Fc) = 0.95. But what we want to know is neither of these two things, but we really look for the probability of being sick once we test positive, that is, P (F|POS).

To calculate it, we have only to apply the theorem of Bayes:

Then we replace the symbols with their values and solve the equation:

So we see that, in principle, I do not have to scare a lot when the test gives me a positive result, since the probability of being ill is only 2%. As you see, much lower than intuition would tell us with such a high sensitivity and specificity. Why is this happening? Very simple, because the prevalence of the disease is very low. We are going to repeat the experiment assuming now that the prevalence is 10% (0,1):

As you see, in this case the probability of being ill if I give positive rises to 68%. This probability is known as positive predictive value which, as we can see, can vary greatly depending on the frequency of the effect we are studying.

And here we leave it for today. Before closing, let me warn you not to seek what the fildulastrosis is. I would be very surprised if anyone found it in a medical book. Also, be careful not to confuse P (POS|F) with P (F|POS), since you would make a mistake called reverse fallacy or fallacy of transposition of conditionals, which is a serious error.

We have seen how the calculation of probabilities gets somewhat complicated when the events are not independent. We have also learned how unreliable predictive values are when the prevalence of the disease changes. That is why the likelihood ratios were invented, which do not depend so much on the prevalence of the disease that is diagnosed and allow a better overall assessment of the power of the diagnostic test. But that is another story…

## Do not be misled by the outliers

We saw in a previous post that the extreme values of a distribution, called outliers, can skew statistical estimates we calculate in our sample.

A typical example is the arithmetic mean, which moves in the direction of the extreme values, if any, particularly as more extreme values are. We saw that to avoid this inconvenience, there were a number of relatives of the arithmetic that were considered robust or what is the same, they were less sensitive to the presence of outliers. Of these, the best known is the median, although some more, as the trimmed mean, the winsorized mean, weighted mean, geometric mean, etc.

Well, something like what happens to the mean occurs also with the standard deviation, the statistical of scale or dispersion used more frequently. Standard deviation is biased by the presence of extreme values, obtaining values that are unrepresentative of the actual spread of the distribution.

Consider the example we used when speaking about the robust estimators of the mean. Suppose we measure the levels of serum cholesterol in a group of people and we find the following values (in mg/dl): 166, 143, 154, 168, 435, 159, 185, 155, 167, 152, 152, 168, 177, 171, 183, 426, 163, 170, 152 and 155. As shown, there are two extreme values (426 and 435 mg/dl) that will bias our mean and standard deviation. In our case, we can calculate the standard deviation and see that its value is 83 mg/dl, clearly not adjusted to the dispersion of most of the values with respect to any of the robust centralization measure we can choose.

What we can do in this case? Well, we can use any of the robust estimators of the deviation, there are several. Some of them arise from the robust estimators of the mean. Here are some.

The first, which arises from the median, is the median absolute deviation (MAD). If you remember, the standard deviation is the sum of the differences of each value with mean, squared and divided by the number of elements, n (or n-1 if we want to obtain an unbiased estimator of typical deviation in the population). Well, similarly, we can calculate the median of the absolute deviations of each value with the median of the sample according to the following formula

MAD = Median {|Xi – Me|}, from i=1 to n.

We can calculate it in our example and see that its value is 17.05 mg / dl, rather adjusted than the standard deviation.

The second is calculated from the trimmed mean. This, as the name suggests, is calculated by cutting a certain percentage of the distribution, at its ends (the distribution has to be ordered from smallest to largest). For example, to calculate the 20% trimmed mean in our example, we’d remove 10% per side (two elements per side: 143, 152, 426 and 435) and calculate the arithmetic mean with the other. Well, we can calculate the classical standard deviation using the remaining elements, getting the value of 10.5 mg/dl.

And thirdly, we could follow a similar reasoning used and get the winsorized mean. In this case, instead of eliminating the values, we would replace them with the closest without removing them. Once the distribution is winsorized, we calculate the standard deviation with the new values in the usual way. Its value is 9.3 mg / dl, similar to the above.

What of the three should we use?. Well, we want to use one that behave efficiently when the distribution is normal (in these cases the best is the classical standard deviation) but that is not very sensitive when the distribution is beyond the normal. In this sense, the best is the median absolute deviation, followed by the winsorized standard deviation.

One last advice before finishing. Do not calculate these measures by hand, as it can be very laborious. Statistical programs do the math for us effortlessly.

And here we ended. We have not talked a word about other estimators of the family of the M-estimators, such as the biweighted mean variance or the adjusted mean percentage variance. These averages are much more difficult to understand from the mathematical point of view, although they are very easy to calculate with the appropriate software package. But that is another story…


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# Integration Technique

Greetings,

I know $$\int \frac {1} {{\sqrt{1-x^2}} dx$$ is arcsine(x)

My question is can $$\int \frac {1} {\sqrt{1-x^2}} dx$$
be solved by some technique (parts, substitution...) and the answer be in terms of x and NOT an expression of arcsine?

Meaning, I would like the solution to the integral in terms of x and possibly other functions (ln for example) but not in terms of trig functions.

If so, can you show me?

Thanks
Sparky_

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It can be done by an appropriate trig substitution ie. x=sin(y), but the answer will end up identically. I'm fairly certain that arcsin(x) is the only answer you will ever get though.

Antiderivatives are unique up to a constant. Ie, all antiderivatives of that will be arcsine(x)+C.

Hurkyl
Staff Emeritus
Gold Member
The only two other (sane) ways I can think of to write that antiderivative is as a power series, or in terms of a complex logarithm. But any of those ways will simply be a different way of writing arcsin x.

One could certainly express arcsin in terms of logs using Euler's formula...

Gib Z
Homework Helper
One could certainly express arcsin in terms of logs using Euler's formula...

...or in terms of a complex logarithm

So yes I think the OP is happy,

possibly other functions (ln for example)[\QUOTE]

What I am wanting is an expression for arcsine that uses other functions (hopefully LN and powers of x) but I am hoping to work in the real domain.

Gib Z help me quite a bit with a derivation regarding arcsine.

I was hoping some clever integration could come up with another expression equal to arcsin by integrating the derivative of arcsine.

Any thoughts?

Thanks
Sparky_

HallsofIvy
Homework Helper
If trig substitution, which just about everyone mentioned, is not a "clever integration", I don't know what is!

I agree trig substitution is clever, I was hoping for a solution that doesn't include trig functions.

-Sparky_

Gib Z
Homework Helper
I remember what sparky wanted from his previous thread that i tried to help him in. He had $$\arcsin x = -i \ln ( ix \pm \sqrt{1-x^2})$$ as he should, but could not prove why the expressions were equal.

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Correct Gib_Z and you did help me prove it - you helped me get the true solution and in that I ended up with some other solutions that I assumed involved the other quadrants.

Using that expression (arcsin = -iln(...) has become very involved due to working in the complex domain.

I'm now searching for a similar type of solution in the real domain only. (I suppose this rules out Euler's)

-Sparky_

Gib Z
Homework Helper
Well then the best answer I can give you is the Taylor Series for Arcsin x, which If i remember correctly is this:

$$\arcsin x = \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}$$

Thank you all for your help.

Sparky_

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View other such prime-parametrized groups

Definition

Let $p$ be a (fixed here) prime number. This is a group determined uniquely up to isomorphism based on $p$ and is sometimes denoted $\mathbb{Z}_{(p)}$ (though that notation is also used for other things, and we can best infer meaning from context).

As an inverse limit

The additive group of p-adic integers is a profinite group defined as the inverse limit of the inverse system:

$0 \leftarrow \mathbb{Z}/p\mathbb{Z} \leftarrow \mathbb{Z}/p^2\mathbb{Z} \leftarrow \dots \leftarrow \mathbb{Z}/p^n\mathbb{Z} \leftarrow \dots$

where each of the maps:

$\mathbb{Z}/p^{n-1}\mathbb{Z} \leftarrow \mathbb{Z}/p^n\mathbb{Z}$

reduces an integer mod $p^n$ to its value mod $p^{n-1}$.

Note that this definition also endows the group with a topology as a profinite group. In this topology, two elements are close if they agree mod $p^n$ for large $n$.

As sequences with cumulative information

$(a_0, a_1, a_2, \dots, a_n, \dots)$

where $a_i$ is an integer mod $p^{i+1}$, the addition is coordinate-wise (with each coordinate addition in the integers mod $p^{i+1}$), and for $i < j$, reducing $a_i$ mod $p^{j+1}$ yields $a_j$.

As sequences with carries

The additive group of p-adic integers is the set of formal sums:

$\sum_{i=0}^\infty x_ip^i$

where $x_i \in \{ 0,1,2,\dots,p-1 \}$, and the addition is done with carries, i.e., to add two sequences $\sum_{i=0}^\infty x_ip^i$ and $\sum_{i=0}^\infty y_ip^i$, we add coordinate-wise and if any of the sums is $p$ or more, we take a carry of 1 to the next sum.

As the additive group of the ring of Witt vectors

This group is the additive group of the ring of Witt vectors over the prime field $\mathbb{F}_p$.

Arithmetic functions

Function Value Explanation
order of a profinite group $p^\infty$ for the relevant prime $p$. This is a supernatural number as is the convention for orders of profinite groups.

Powering

The additive group of $p$-adic integers is powered over the set of all primes other than $p$.

Group properties

Abstract group properties

Property Satisfied? Explanation
abelian group Yes
aperiodic group Yes No nonzero element has finite order.

Topological group properties

Here, the topology is from the profinite group structure.

Property Satisfied? Explanation
profinite group Yes By definition
pro-p-group Yes By definition
compact group Yes profinite groups are compact
T0 topological group Yes profinite groups are Hausdorff, hence $T_0$
totally disconnected group Yes profinite groups are totally disconnected
topologically cyclic group Yes $\mathbb{Z}$, the subgroup generated by the element that projects to 1 mod $p^n$ for all $n$, is a dense subgroup. In fact, this group can be thought of as a compactification of $\mathbb{Z}$.
topologically finitely generated group Yes Follows from being topologically cyclic.
connected topological group No It is nontrivial and totally disconnected.

Related notions

Combining all primes

The profinite completion of the integers, denoted $\hat {\mathbb{Z}}$ is a group obtained by taking the profinite completion of the group of integers, which is a residually finite group. Intuitively, this group is obtained by "completing" the group of integers simultaneously at all primes. It turns out that the profinite completion of the integers can be viewed as the external direct product, over all primes $p$, of the additive group of $p$-adic integers.

Dual concept

The additive group of p-adic integers can, in a vague sense, be considered to be constructed using a method dual to the method used to the quasicyclic group. While the $p$-adics are constructed as an inverse limit for surjective maps $\mathbb{Z}/p^n\mathbb{Z} \to \mathbb{Z}/p^{n-1}\mathbb{Z}$, the quasicyclic group is constructed as a direct limit for injective maps $\mathbb{Z}/p^{n-1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$.


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# closed immersion onto an affine scheme - showing affineness

Let $A$ be a ring, $X=\operatorname{Spec}A$ and $f: Z \rightarrow X$ a morphism of schemes such that i) $f$ is a homeomorphism of topological spaces and ii) $f^{\#}:\mathcal{O}_X \rightarrow f_* \mathcal{O}_Z$ is a surjective morphism of sheaves.

Question 1: How can we show that $Z$ is an affine scheme?

I am trying to follow the hint of exercise II.3.11(b) in Hartshorne, but i am getting stuck: i can show that $Z$ can be covered by a finite number of $\operatorname{Spec}B_i$ and each $\operatorname{Spec}B_i$ can be covered by finitely many $D(\phi_i(a_{ij}))$, where $a_{ij} \in A$, and $\phi_i: A \rightarrow B_i$ is the ring homomorphism that corresponds to the restriction of $f$ to $\operatorname{Spec}B_i$. Now, if i am to apply exercise II.2.17(b), i would need to show that each of the $Z_{\phi(a_{ij})}$ is affine, where $\phi: A \rightarrow \Gamma(Z,\mathcal{O}_Z)$ is the ring homomorphism of global sections.

Question 2: Any ideas how we can show that $Z_{\phi(a_{ij})}$ is affine?

Question 3: Alternatively, i have been able to show that each $\operatorname{Spec}B_i$ is isomorphic as a scheme with $\operatorname{Spec}(A/I_i)$, where $I_i$ is the kernel of $\phi_i$. Any ideas how to continue for there?

Take an affine open covering of $Z$ by principal open sets, which will come from principal open sets in $X$. Extend to a covering of $X$ by adding principal open sets. $X$ is quasi-compact since it is affine, so this is a finite covering. Now we have a finite covering of $X$ by principal opens $D(f_i)$ so $f_1, \dots, f_r$ generate the unit ideal of $A$.

Each $Z_{f}$ is a principal open subset of $D(f) \cap Z$, since $Z_f = X_f \cap Z$ and $D(f) = X_f$. $D(f) \cap Z$ was affine, therefore, $Z_f$ is affine.

Ex 2.17b (affineness criterion) applies to show that $Z$ is affine.

As shown, $Z = \text{Spec } B$. By Ex 2.18d now, the map of rings $A \to B$ is surjective. By the Isomorphism Theorem, then we know that $B \cong A / I$.

• Ex 2.17b tells you that the scheme is affine provided that the distinguished open sets are affine and the elements generate the unit ideal. – Future Mar 18 '16 at 19:52
• You can also use a different affineness criterion: if a scheme admits a finite affine covering $(U_i)$ such that $U_i \cap U_j$ also has a finite affine covering, then the scheme is affine. – Future Mar 18 '16 at 20:01
• They are principle open subsets of affine space $D(f_i)$. – Future Mar 18 '16 at 20:03
• For a general scheme $X$, $X_f$ is supposed to be a generalized version of $D(f)$ for affine schemes. – Future Mar 18 '16 at 20:07
• I edited the answer adding both the line of reasoning I explained in above comments and also a new one that tries to incorporate Ex 2.16. I am confident in the reasoning in comments, but I just thought of the Ex 2.16 reasoning so let me know if you feel there is an error in it. – Future Mar 18 '16 at 20:25