Know your data: Evaluation of normality for continuous variables and tools to deal with non-normal distributions

Rodriguez, Adrian
doi:10.1016/j.medcle.2019.05.011
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          "en" => "<p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Graphs to demonstrate the normality of the data&#58; histograms &#40;A and B&#41;&#44; box plots &#40;C and D&#41; and Q-Q plots &#40;E and F&#41; for the variables &#8216;number of cigarettes&#8217; and &#8216;weight&#8217;&#46;</p>"
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    "textoCompleto" => "<span class="elsevierStyleSections"><p id="par0005" class="elsevierStylePara elsevierViewall">The analysis of continuous quantitative variables is widespread in medical literature&#58; blood or urinary levels&#44; pain scales or anthropometric measurements are some examples of widely used continuous variables&#46; To synthesize the data&#44; researchers can use a measure of central tendency&#44; such as the mean or median&#44; accompanied by measures of dispersion&#44; such as the standard deviation&#44; the interquartile range or the range&#46; However&#44; although many researchers are tempted to use the mean arbitrarily&#44; it is not always the best way to summaries a series of data&#46; If the variable in question does not have a normal &#40;or Gaussian&#41; distribution&#44; the median is usually a better estimate as a measure of central tendency&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">Something similar happens when we have to decide which statistical analysis or test to use to compare a continuous quantitative variable in 2 or more groups&#46; Many researchers tend to use the Student&#39;s t-test or the one-way ANOVA to compare the means between different groups&#46; However&#44; these statistical tests have several assumptions that must be met and&#44; probably&#44; the most important is that the variable is distributed normally&#46; Alternatives to these are the Mann Whitney U test or the Kruskal&#8211;Wallis test&#44; which do not assume any particular distribution of the data&#46; This means that they can always be used&#44; regardless of whether the variable of interest is distributed normally or not&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">This article reviews the different ways of establishing whether a variable follows a normal distribution or not&#44; and the alternatives that we can use in the case of dealing with non-normal distributions&#46;</p><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0005">Normality analysis</span><p id="par0020" class="elsevierStylePara elsevierViewall">When studying the normality of the data&#44; it is convenient to follow a multiple approach in which we evaluate the data using the different tools available&#46; There are specific statistical tests to check if the data follow a normal distribution&#46; The best-known tests are the Shapiro&#8211;Wilk test and the Kolmogorov&#8211;Smirnov test&#46; However&#44; these tests are very sensitive to the sample size&#44; and it is not convenient to rely solely on them to make a decision&#46; However&#44; there are a series of simple graphs that we can represent and that can help us decide if the data follows a normal distribution&#46; These graphs are the histograms&#44; the <span class="elsevierStyleItalic">box plot</span>&#44; and the Q-Q plots&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0010">Graphs</span><p id="par0025" class="elsevierStylePara elsevierViewall">To illustrate how to interpret these graphs&#44; we will take a series of hypothetical data summarized in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#46; This table shows the weight and daily consumption of cigarettes of a hypothetical series of 20 subjects&#46; If we look at the table&#44; we can see that&#44; in the row of cigarette consumption&#44; there are a large number of people &#40;specifically 7 of 20&#41; who are not smokers&#46; This is usually common in surveys on alcohol and tobacco consumption&#44; where there is a high percentage of the population that are not smokers or consume alcohol&#46;</p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><p id="par0030" class="elsevierStylePara elsevierViewall">The first graphs represented in <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a> are the histograms for both variables &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>A and B for cigarette consumption and weight&#44; respectively&#41;&#46; A histogram represents the frequency with which a certain value of the variable appears versus the values of that variable&#46; Thus&#44; for the first case&#44; the X axis shows the number of cigarettes per day and the Y axis shows the frequency with which that value of cigarettes per day appears in our sample&#46; In the second case&#44; the X axis shows the weight of the participants and the Y axis shows the frequency with which these weights appear&#46; These histograms are very useful to clarify if the data of our variable of interest follows a normal distribution&#46; In the first case &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>A&#41; a histogram appears with what is called &#8220;scewed right&#8221;&#46; This occurs when there is a high percentage of the data that have small values of the variable&#46; We could easily deduce that the data is not distributed according to a normal distribution&#46; However&#44; in the case of weight &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>B&#41;&#44; the histogram is shaped like &#8220;bell-curve&#8221;&#44; so it seems that the data does follow a normal distribution&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0035" class="elsevierStylePara elsevierViewall">Then&#44; in <a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>C and D&#44; the <span class="elsevierStyleItalic">box-plot</span> &#40;box and whisker plot&#41; is represented&#46; This is a particularly useful graph&#44; not only to evaluate the distribution of the data but also as a summary of them&#46; This diagram consists of the following parts&#58; a horizontal line with the rectangle &#40;or box&#41;&#44; which represents the median of the 50th percentile&#46; It should be remembered that the median is the value of the data that leaves the bottom 50&#37; of the values below it and the top 50&#37; above it&#46; The 25th percentile is the below the box&#46; That is&#44; the value that leaves 25&#37; of the smallest data below and the remaining 75&#37; above&#46; Therefore&#44; the top of the box is the 75th percentile&#46; Finally&#44; 2 segments &#40;or &#8220;whiskers&#8221;&#41; join&#46; In one case&#44; the 75th percentile with the highest value and&#44; in the other case&#44; the 25th percentile with the lowest value&#46; These graphs are very useful because&#44; with a quick look&#44; we can know the maximum and minimum values of the sample&#44; the value of the median and what are the 25th and 75th percentiles&#46; In addition&#44; in the case at hand&#44; they can also help to clarify whether the data follows an approximate normal distribution&#46; When the <span class="elsevierStyleItalic">box-plot</span> has quite a lot of symmetry&#44; it indicates that the data is distributed normally&#46; In the case of cigarette smoking &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>C&#41;&#44; the <span class="elsevierStyleItalic">box-plot</span> is clearly not symmetrical&#44; since the lowest point and the 25th percentile coincide&#46; However&#44; for the weight variable &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>D&#41; the <span class="elsevierStyleItalic">box-plot</span> is quite symmetrical&#44; which is further proof that the data can be considered normally distributed&#46;</p><p id="par0040" class="elsevierStylePara elsevierViewall">The last type of graph that we can analyse when evaluating the normality of the data is the quantile-quantile graphs&#44; better known as Q-Q plots&#46; These graphs can be generated with most statistical packages &#40;SPSS&#44; R or Stata&#44; etc&#46;&#41;&#46; If the points on the graph are located on the theoretical line&#44; the distribution can be considered as normal&#46; Obviously&#44; it cannot be expected that all the points are exactly on the line&#44; but it is easy to see whether the deviation from normality is great or not&#46; For example&#44; in the case of the consumption of cigarettes &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>E&#41; it can be deduced that the points are very far from being uniformly on the line&#44; while in the case of weight &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Fig&#46; 1</a>F&#41; the points are quite close to the line&#46;</p><p id="par0045" class="elsevierStylePara elsevierViewall">Thus&#44; these 3 graphs allow us to easily conclude that&#44; in our hypothetical data set&#44; cigarette consumption does not meet the requirements to be considered a normally distributed variable&#44; but weight does&#46;</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0015">Specific statistical tests</span><p id="par0050" class="elsevierStylePara elsevierViewall">Although in the vast majority of cases&#44; these 3 graphs can be considered as conclusive&#44; many authors prefer to use specific statistical tests to analyse the normality of the data&#46; These tests are called Shapiro&#8211;Wilk and Kolmogorov&#8211;Smirnov tests&#46; In both cases&#44; the null hypothesis is that the results follow a normal distribution&#46; Thus&#44; assuming the classic 95&#37; confidence level&#44; if p&#8239;&#60;&#8239;0&#46;05&#44; the null hypothesis is rejected&#44; and it is concluded that the distribution is not normal&#46; However&#44; these tests should be complementary to the graphs described above&#44; especially since they are very sensitive to the sample size&#46; If the sample size is small&#44; we can obtain a p&#8239;&#62;&#8239;0&#46;05 and we would conclude that the distribution is normal&#44; when the reality is that the distribution is far from normal&#46; On the other hand&#44; very large sample sizes favour the statistical test being significant &#40;p&#8239;&#60;&#8239;0&#46;05&#41; even for small deviations from normal&#44; and we could erroneously conclude that the distribution is not normal&#44; when in fact it could be considered as such&#46; Therefore&#44; these tests must be used with great discretion&#44; and the aim should be to use them as complementary tests to the normality graphs&#44; which offer more and better information&#46; By applying the Shapiro&#8211;Wilk test to our set of hypothetical data we see that&#44; for weight&#44; p&#8239;&#61;&#8239;0&#46;84 and for the consumption of cigarettes p&#8239;&#61;&#8239;0&#46;004&#46; Thus&#44; again&#44; in this first case we would retain the null hypothesis normality data&#44; while in the case of the cigarette consumption variable we would discard the null hypothesis and conclude that the data is not distributed normally&#46;</p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0020">Description of the data</span><p id="par0055" class="elsevierStylePara elsevierViewall">Once we have seen how to evaluate the normality of the data&#44; we must consider how to report that data in an article or communication&#44; and what types of statistical tests to apply in each case&#46; When the data follow a normal distribution&#44; the most common occurrence is to report the mean as a measure of central tendency of said data&#44; and the standard deviation &#40;or standard error in some cases&#41; as a measure of dispersion&#46; On the other hand&#44; the confidence intervals of the mean can be reported&#44; generally at 95&#37;&#46; For example&#44; in the case of our hypothetical weight variable&#44; the mean &#40;standard deviation&#41; is 70 &#40;4&#41; kg&#44; and the 95&#37; confidence interval for the average is &#40;68&#8211;72&#41;&#46;</p><p id="par0060" class="elsevierStylePara elsevierViewall">On the other hand&#44; in the case of cigarette consumption&#44; the mean would not be a good measure of central tendency&#44; since the distribution is far from normal&#46; Thus&#44; the alternative is to use measures of central tendency that do not assume any distribution of the data&#46; Without a doubt&#44; the most used is the median&#46; As a measure of dispersion&#44; the interquartile path is generally used &#40;which is the difference between the 25th and 75th percentiles&#41;&#44; although the percentiles or the range &#40;minimum and maximum values&#41; can also be reported&#46; In our hypothetical data set&#44; the median cigarette consumption is 5&#46; The 25th and 75th percentiles are&#44; respectively&#44; 0 and 11 cigarettes&#44; so the interquartile range is 11&#46; The range is 0&#8211;30&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025">Data analysis</span><p id="par0065" class="elsevierStylePara elsevierViewall">When analysing data&#44; it is important to keep in mind that there is nothing inherently wrong in the fact that the data does not follow a normal distribution&#46; There are simply variables that are not distributed according to this model&#46; Luckily&#44; we have tools so that the data ends up well expressed and summarised in both cases&#46; In addition&#44; it should be borne in mind that&#44; whenever possible&#44; the mean is usually used&#46; But this does not mean that using the median in the case of normal distributions is incorrect&#59; the median can always be used&#44; since it does not presuppose a specific distribution of the data&#46; However&#44; there is a clear consensus among statisticians that&#44; whenever possible&#44; it is better to use the mean as a measure of central tendency and the so-called parametric tests for hypothesis contrasts&#46;</p><p id="par0070" class="elsevierStylePara elsevierViewall">Once again&#44; the statistical tests will depend on the distribution of the data&#46;<a class="elsevierStyleCrossRef" href="#bib0005"><span class="elsevierStyleSup">1</span></a> This article aims to be an introduction to the study of the normality of the data&#44; so not all statistical tests will be explained in detail&#46; However&#44; it is important to have an overview&#44; to know when a certain statistical test can be applied and to take into account the assumptions to be analysed&#46; To illustrate&#44; I will focus on the tests to compare a quantitative variable between 2 or more groups&#46;</p><p id="par0075" class="elsevierStylePara elsevierViewall">The so-called &#8220;parametric&#8221; alternatives estimate a parameter of a given distribution&#44; so it is assumed that the dependent variable of interest follows that distribution &#40;in our case&#44; the normal distribution&#41;&#46; Thus&#44; before proceeding to apply this type of statistical test&#44; we must check the normality of the data&#46; This check is only critical when we have small sample sizes&#44; as explained below&#46; For example&#44; when we want to compare the means between 2 or more independent groups&#44; we would use the Student&#8217;s t-test or the one-way analysis of variance &#40;one-way ANOVA&#41;&#44; respectively&#46; However&#44; when we have small sample sizes and the data does not follow a normal distribution&#44; non-parametric methods can be used&#46; Specifically&#44; in this case&#44; we would be talking about the Mann&#8211;Whitney U test &#40;sometimes known as &#8220;Wilcoxon Rank-Sum test&#8221;&#41; or the Kruskal&#8211;Wallis test&#44; which would be the non-parametric alternatives to the Student&#39;s t test and the one-way ANOVA&#44; respectively&#46; The primary difference is that the non-parametric tests do not assume any type of data distribution to estimate a specific parameter and&#44; therefore&#44; they can be used for any data set&#46;</p><p id="par0080" class="elsevierStylePara elsevierViewall">Although it is true that non-parametric tests are relatively known and used in medical literature&#44; some authors advise against their use&#44; while others use them for any type of analysis&#46; In the opinion of the author&#44; non-parametric tests are useful in some cases&#44; but parametric tests are normally preferable&#44; provided that due assumptions are met&#44; since they provide confidence intervals in addition to a p-value&#44; and they can be generalised to be used in more complex statistical analyses&#46;</p></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0030">Alternatives to non-parametric tests</span><p id="par0085" class="elsevierStylePara elsevierViewall">One of the alternatives that we can use when the distribution of the data is not normal&#44; and when we do not want to resort to non-parametric tests&#44; is to perform a data transformation&#46;<a class="elsevierStyleCrossRef" href="#bib0010"><span class="elsevierStyleSup">2</span></a> In many cases&#44; if we perform a mathematical operation on the data &#40;for example&#44; a decimal logarithm or a square root&#41;&#44; the transformed data does follow a normal distribution and&#44; therefore&#44; we could use parametric tests&#46; Although some authors have defended its use&#44; the truth is that its use is not widespread in medical literature&#46; The reason is that the statistics must be carried out on the transformed data&#46; Therefore&#44; the reported confidence intervals will also be based on the transformed data and their interpretation may result in errors or be somewhat complex&#46;</p><p id="par0090" class="elsevierStylePara elsevierViewall">Throughout this article I have repeated several times that the assumption of normality in parametric tests is only essential when the sample size is small&#46; Although there is no established cut-off point&#44; there is a fairly broad consensus that&#44; when sample sizes are greater than 80&#8211;100&#44; the premise of normal distribution loses importance&#44; in a way&#44; when choosing between a parametric test or a non-parametric one&#46;<a class="elsevierStyleCrossRef" href="#bib0015"><span class="elsevierStyleSup">3</span></a> This is because of the so-called central limit theorem&#46;<a class="elsevierStyleCrossRef" href="#bib0020"><span class="elsevierStyleSup">4</span></a> Without going into detail&#44; this theorem establishes that&#44; although the distribution of the data is not normal&#44; the distribution of the means is normal&#46; This theorem allows parametric tests to be applied when the sample size is high&#44; even when the variable of interest is not distributed normally&#46;</p></span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">Conclusion</span><p id="par0095" class="elsevierStylePara elsevierViewall">When reporting a measure of central tendency for a certain quantitative variable or when choosing a certain statistical test&#44; in many cases it is important to examine the normality of the data&#46; For this&#44; we have different tools&#44; such as graphs and statistical tests&#44; which can be very useful for this purpose&#46; The contrasting statistical hypothesis testing for normal distribution should be interpreted with caution&#44; and always as a complementary test to the graphic description of the data&#46;</p></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">Financing</span><p id="par0100" class="elsevierStylePara elsevierViewall">The author confirms that there has been no financial support for the preparation of this article&#46;</p></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Conflict of interests</span><p id="par0105" class="elsevierStylePara elsevierViewall">The author confirms that there is no conflict of interests&#46;</p></span><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0050">Thanks</span><p id="par0110" class="elsevierStylePara elsevierViewall">The author thanks the Alfonso Mart&#237;n Escudero Foundation for the financing of the postdoctoral fellowship he currently enjoys&#46;</p></span></span>"
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        "nota" => "<p class="elsevierStyleNotepara" id="npar0005">Please cite this article as&#58; Rodriguez A&#46; Conoce tus datos&#58; evaluaci&#243;n de la normalidad de variables continuas y estrategias para tratar con distribuciones no normales&#46; Med Clin &#40;Barc&#41;&#46; 2020&#59;154&#58;142&#8211;145&#46;</p>"
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          "en" => "<p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Graphs to demonstrate the normality of the data&#58; histograms &#40;A and B&#41;&#44; box plots &#40;C and D&#41; and Q-Q plots &#40;E and F&#41; for the variables &#8216;number of cigarettes&#8217; and &#8216;weight&#8217;&#46;</p>"
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