Normality Tests-2012

Endocrinology Metabolism International Journal of www.EndoMetabol.comKOWSAR Int J Endocrinol Metab. 2012;10(2):486-489. DOI: 10.5812/ijem.3505 Normality Tests for Statistical Analysis: A Guide for Non-Statisticians Asghar Ghasemi 1, Saleh Zahediasl 1* 1 Endocrine Research Center, Research Institute for Endocrine Sciences, Shahid Beheshti University of Medical Sciences, Tehran, IR Iran A R T I C L E I N F O A B S T R A C T Article history: Received: 21 Nov 2011 Revised: 21 Jan 2012 Accepted: 28 Jan 2012 Keywords: Normality Statistical Analysis Article type: Statistical Comment Please cite this paper as: Ghasemi A, Zahediasl S. Normality Tests for Statistical Analysis: A Guide for Non-Statisticians. Int J Endocrinol Metab. 2012;10(2):486-9. DOI: 10.5812/ijem.3505 Implication for health policy/practice/research/medical education: Data presented in this article could help for the selection of appropriate statistical analyses based on the distribution of data. 1. Background Statistical errors are common in scientific literature, and about 50% of the published articles have at least one error (1). Many of the statistical procedures including correlation, regression, t tests, and analysis of variance, namely parametric tests, are based on the assumption that the data follows a normal distribution or a Gaussian distribution (after Johann Karl Gauss, 1777–1855); that is, it is assumed that the populations from which the samples are taken are normally distributed (2-5). The assumption of normality is especially critical when constructing ref- erence intervals for variables (6). Normality and other as- sumptions should be taken seriously, for when these as- sumptions do not hold, it is impossible to draw accurate Statistical errors are common in scientific literature and about 50% of the published ar- ticles have at least one error. The assumption of normality needs to be checked for many statistical procedures, namely parametric tests, because their validity depends on it. The aim of this commentary is to overview checking for normality in statistical analysis us- ing SPSS. and reliable conclusions about reality (2, 7). With large enough sample sizes (> 30 or 40), the viola- tion of the normality assumption should not cause ma- jor problems (4); this implies that we can use parametric procedures even when the data are not normally distrib- uted (8). If we have samples consisting of hundreds of observations, we can ignore the distribution of the data (3). According to the central limit theorem, (a) if the sam- ple data are approximately normal then the sampling distribution too will be normal; (b) in large samples (> 30 or 40), the sampling distribution tends to be normal, regardless of the shape of the data (2, 8); and (c) means of random samples from any distribution will themselves have normal distribution (3). Although true normality is considered to be a myth (8), we can look for normality vi- sually by using normal plots (2, 3) or by significance tests, that is, comparing the sample distribution to a normal one (2, 3). It is important to ascertain whether data show a serious deviation from normality (8). The purpose of this report is to overview the procedures for checking normality in statistical analysis using SPSS. Copyright c 2012 Kowsar Corp. All rights reserved. * Corresponding author: Saleh Zahediasl, Endocrine Research Center, Research Institute for Endocrine Sciences, Shahid Beheshti University of Medical Sciences, P.O. Box: 19395-4763, Tehran, IR Iran. Tel: +98-2122409309, Fax: +98-2122402463, E-mail: zahedi@endocrine.ac.ir DOI: 10.5812/ijem.3505 Copyright c 2012 Kowsar Corp. All rights reserved. 487Int J Endocrinol Metab. 2012;10(2) Ghasemi A et al.Normality Tests for Non-Statisticians 2. Visual Methods Visual inspection of the distribution may be used for assessing normality, although this approach is usually unreliable and does not guarantee that the distribution is normal (2, 3, 7). However, when data are presented visu- ally, readers of an article can judge the distribution as- sumption by themselves (9). The frequency distribution (histogram), stem-and-leaf plot, boxplot, P-P plot (prob- ability-probability plot), and Q-Q plot (quantile-quantile plot) are used for checking normality visually (2). The frequency distribution that plots the observed values against their frequency, provides both a visual judgment about whether the distribution is bell shaped and in- sights about gaps in the data and outliers outlying values (10). The stem-and-leaf plot is a method similar to the his- togram, although it retains information about the actual data values (8). The P-P plot plots the cumulative prob- ability of a variable against the cumulative probability of a particular distribution (e.g., normal distribution). After data are ranked and sorted, the corresponding z-score is calculated for each rank as follows: . This is the expected value that the score should have in a normal distribution. The scores are then themselves converted to z-scores. The actual z-scores are plotted against the ex- pected z-scores. If the data are normally distributed, the result would be a straight diagonal line (2). A Q-Q plot is very similar to the P-P plot except that it plots the quan- tiles (values that split a data set into equal portions) of the data set instead of every individual score in the data. Moreover, the Q-Q plots are easier to interpret in case of large sample sizes (2). The boxplot shows the median as a horizontal line inside the box and the interquartile range (range between the 25th to 75th percentiles) as the length of the box. The whiskers (line extending from the top and bottom of the box) represent the minimum and maximum values when they are within 1.5 times the in- terquartile range from either end of the box (10). Scores greater than 1.5 times the interquartile range are out of the boxplot and are considered as outliers, and those greater than 3 times the interquartile range are extreme outliers. A boxplot that is symmetric with the median line at approximately the center of the box and with sym- metric whiskers that are slightly longer than the subsec- tions of the center box suggests that the data may have come from a normal distribution (8). 3. Normality Tests The normality tests are supplementary to the graphi- cal assessment of normality (8). The main tests for the assessment of normality are Kolmogorov-Smirnov (K-S) test (7), Lilliefors corrected K-S test (7, 10), Shapiro-Wilk test (7, 10), Anderson-Darling test (7), Cramer-von Mises test (7), D’Agostino skewness test (7), Anscombe-Glynn kurtosis test (7), D’Agostino-Pearson omnibus test (7), and the Jarque-Bera test (7). Among these, K-S is a much used test (11) and the K-S and Shapiro-Wilk tests can be conducted in the SPSS Explore procedure (Analyze → De- scriptive Statistics → Explore → Plots → Normality plots with tests) (8). The tests mentioned above compare the scores in the sample to a normally distributed set of scores with the same mean and standard deviation; the null hypothesis is that “sample distribution is normal.” If the test is signif- icant, the distribution is non-normal. For small sample sizes, normality tests have little power to reject the null hypothesis and therefore small samples most often pass normality tests (7). For large sample sizes, significant re- sults would be derived even in the case of a small devia- tion from normality (2, 7), although this small deviation will not affect the results of a parametric test (7). The K-S test is an empirical distribution function (EDF) in which the theoretical cumulative distribution function of the test distribution is contrasted with the EDF of the data (7). A limitation of the K-S test is its high sensitivity to extreme values; the Lilliefors correction renders this test less conservative (10). It has been reported that the K-S test has low power and it should not be seriously consid- ered for testing normality (11). Moreover, it is not recom- mended when parameters are estimated from the data, regardless of sample size (12). The Shapiro-Wilk test is based on the correlation be- tween the data and the corresponding normal scores (10) and provides better power than the K-S test even after the Lilliefors correction (12). Power is the most frequent mea- sure of the value of a test for normality—the ability to detect whether a sample comes from a non-normal dis- tribution (11). Some researchers recommend the Shapiro- Wilk test as the best choice for testing the normality of data (11). 4. Testing Normality Using SPSS We consider two examples from previously published data: serum magnesium levels in 12–16 year old girls (with normal distribution, n = 30) (13) and serum thy- roid stimulating hormone (TSH) levels in adult control subjects (with non-normal distribution, n = 24) (14). SPSS provides the K-S (with Lilliefors correction) and the Sha- piro-Wilk normality tests and recommends these tests only for a sample size of less than 50 (8). In Figure, both frequency distributions and P-P plots show that serum magnesium data follow a normal dis- tribution while serum TSH levels do not. Results of K-S with Lilliefors correction and Shapiro-Wilk normality tests for serum magnesium and TSH levels are shown in Table. It is clear that for serum magnesium concentra- tions, both tests have a p-value greater than 0.05, which indicates normal distribution of data, while for serum TSH concentrations, data are not normally distributed as both p values are less than 0.05. Lack of symmetry (skewness) and pointiness (kurtosis) are two main ways in which a distribution can deviate from normal. The values for these parameters should be zero in a normal 488 Int J Endocrinol Metab. 2012;10(2) Ghasemi A et al. Normality Tests for Non-Statisticians Figure. Histograms (Left) and P-P Plots (Right) for Serum Magnesium and TSH Levels N o . M ea n ± S D a M ea n ± S EM a Sk ew n es s SE Sk ew n es s Z Sk ew n es s K u rt o si s SE K u rt o si s Z K u rt o si s K -S a W it h L il li ef o rs C o rr ec ti o n T es t Sh ap ir o -W il k T es t St at is ti cs D f a P va lu e St at is ti cs D fa P- va lu e Se ru m m ag ne si um , m g/ dL 30 2. 0 8 ± 0 .17 5 2. 0 8 ± 0 .0 3 0 .7 45 0 .4 27 1.7 4 0 .5 67 0 .8 33 0 .6 81 0 .13 7 30 0 .15 6 0 .9 55 30 0 .2 36 Se ru m T SH a , m U /L 24 1.6 7 ± 1.5 3 1.6 7 ± 0 .3 1 1.5 94 0 .4 72 3. 38 1.4 0 1 0 .9 18 1.5 2 0 .2 30 24 0 .0 0 2 0 .7 50 24 < 0 .0 0 1 Ta b le . S ke w n es s, k u rt os is , a n d N or m al it y Te st s fo r Se ru m M ag n es iu m a n d T SH L ev el s Pr ov id ed b y SP SS a A b b re vi at io n s: D f, D eg re e of fr ee d om ; K -S , K ol m og or ov -S m ir n ov ; S D , S ta n d ar d d ev ia ti on ; S EM , S ta n d ar d e rr or o f m ea n ; T SH , T h yr oi d s ti m u la ti n g h or m on e 489Int J Endocrinol Metab. 2012;10(2) Ghasemi A et al.Normality Tests for Non-Statisticians distribution. These values can be converted to a z-score as follows: Z Skewness= Skewness-0 SE Skewness and Z Kurtosis= Kurtosis-0 SE Kurtosis . An absolute value of the score greater than 1.96 or lesser than -1.96 is significant at P < 0.05, while greater than 2.58 or lesser than -2.58 is significant at P < 0.01, and greater than 3.29 or lesser than -3.29 is significant at P < 0.001. In small samples, values greater or lesser than 1.96 are sufficient to establish normality of the data. However, in large samples (200 or more) with small standard errors, this criterion should be changed to ± 2.58 and in very large samples no criterion should be applied (that is, significance tests of skewness and kur- tosis should not be used) (2). Results presented in Table indicate that parametric statistics should be used for serum magnesium data and non-parametric statistics should be used for serum TSH data. 5. Conclusions According to the available literature, assessing the nor- mality assumption should be taken into account for us- ing parametric statistical tests. It seems that the most popular test for normality, that is, the K-S test, should no longer be used owing to its low power. It is preferable that normality be assessed both visually and through normal- ity tests, of which the Shapiro-Wilk test, provided by the SPSS software, is highly recommended. The normality as- sumption also needs to be considered for validation of data presented in the literature as it shows whether cor- rect statistical tests have been used. Acknowledgments The authors thank Ms. N. Shiva for critical editing of the manuscript for English grammar and syntax and Dr. F. Hosseinpanah for statistical comments. Financial Disclosure None declared. Funding/Support None declared. References 1. Curran-Everett D, Benos DJ. Guidelines for reporting statistics in journals published by the American Physiological Society. Am J Physiol Endocrinol Metab. 2004;287(2):E189-91. 2. Field A. Discovering statistics using SPSS. 3 ed. London: SAGE publi- cations Ltd; 2009. p. 822. 3. Altman DG, Bland JM. Statistics notes: the normal distribution. Bmj. 1995;310(6975):298. 4. Pallant J. SPSS survival manual, a step by step guide to data analysis us- ing SPSS for windows. 3 ed. Sydney: McGraw Hill; 2007. p. 179-200. 5. Driscoll P, Lecky F, Crosby M. 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