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# Checking for normality (Normal distribution)

A collection of methods on how to check data for normal distribution

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### Checking for normality (Normal distribution)

1. 1. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Checking for Normality (Normal Distribution) Dr. S. A. Rizwan, M.D., Public Health Specialist, Saudi Board of Preventive Medicine, Riyadh, Kingdom of Saudi Arabia. Nov 2019 1
2. 2. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course When is non-normality a problem? • Non-normality can be a problem when the sample size is small. • Highly skewed data create problems. • Highly leptokurtic data are problematic, but not as much as skewed data. • Normality becomes a serious concern when there is “activity” in the tails of the data set. – Outliers are a problem. – “Clumps” of data in the tails are worse Nov 2019 2
3. 3. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Ways to check for normality 1. Thumb rules – Mean & Range & SD – Skewness and kurtosis – Compare mean, median and mode – Trimmed mean – Outliers 2. Graphs – Histogram with theoretical normal curve – QQ plot – Box plot and outlier detection – Stem and leaf plot 3. Formal statistical tests – W/S test – Jarque-Bera test – Shapiro-Wilks test – Kolmogorov-Smirnov test – D’Agostino test – Grubbs and Dixon test (for outliers) 4. Comparing non-parametric and parametric test results Nov 2019 3
4. 4. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course 1. THUMB RULES Nov 2019 4
5. 5. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Thumb rules • SD less than half of mean -> may be normally distributed • Calculate the observed mean minus the lowest possible value (or the highest possible value minus the observed mean), and divide this by the standard deviation. – A ratio less than 2 suggests skew (Altman 1996). If the ratio is less than 1 there is strong evidence of a skewed distribution. Nov 2019 5
6. 6. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Thumb rules • Skewness – <-1 or >1, highly skewed. – -1 to -0.5 or 0.5 to 1, moderately skewed. – -0.5 to 0.5, approximately symmetric • Kurtosis – For normally distributed data the value is 3, theoretically – Statistical packages adjust this to zero – Platykursis < 0 and leptokursis > 0 – Extremely non-normal distributions may have high positive or negative kurtosis values, while nearly normal distributions close to 0 – Kurtosis is positive if the tails are “heavier” and negative if the tails are “lighter” Nov 2019 6
7. 7. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Skewness and kurtosis Nov 2019 7
8. 8. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Thumb rules • In a normal distribution, mean=median=mode • 5% Trimmed Mean – lower and upper 5% of values deleted – If the value of the 5% trimmed mean is very different from the mean, this indicates outliers • Examine the lowest and highest 5 extreme values to look for outliers Nov 2019 8
9. 9. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course 2. GRAPHS Nov 2019 9
10. 10. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course NORMAL CURVE For each mean and standard deviation combination a theoretical normal distribution can be determined This distribution is based on the proportions shown Nov 2019 10
11. 11. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course This theoretical normal distribution can then be compared to the actual distribution of the data. • Are the actual data statistically different than the computed normal curve? Theoretical normal distribution calculated from amean of 66.51 anda standard deviation of 18.265. Theactual data distribution that hasa mean of 66.51 and a standard deviation of 18.265. Nov 2019 11
12. 12. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Comparison of histograms Nov 2019 12
13. 13. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Q-Q plots • Q-Q plots display the observed values against normally distributed data (represented by the diagonal line). • Normally distributed data fall along theline. Nov 2019 13
14. 14. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Example Q-Q plots • The Q-Q and detrended Q-Q plots show systematic deviations from normality • Notice that the overall shape of the detrended plot is parabolic (U-shaped). • Also notice that the deviations from normality are relatively large: the y-axis of the detrended normal q-q plot indicates that the deviations range in magnitude from -0.2 to 1.2. Nov 2019 14
15. 15. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Beware! • Graphical methods are typically not very useful when the sample size is small. • These data do not ‘look’ normal, but they are not statistically different than normal. Nov 2019 15
16. 16. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Stem and leaf plot • Frequency – This is the frequency of the leaves • Stem – It is the number in the 10s place of the value of the variable • Leaf – It is the number in the 1s place of the value of the variable. – The number of leaves tells you how many of these numbers is in the variable Nov 2019 16
17. 17. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Parts of a boxplot a) Maximum score unless there are values more than 1.5 times the interquartile range above Q3, in which, it is the third quartile plus 1.5 times the interquartile range b) Third quartile (Q3) or 75th percentile. c) Median (q2) or 50th percentile. d) First quartile (q1) or 25th percentile. e) Minimum score unless there are values less than 1.5 times the interquartile range below Q1, in which case, it is the first quartile minus 1.5 times the interquartile range. Nov 2019 17
18. 18. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Comparison of box plots Nov 2019 18
19. 19. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Example of outliers in boxplot Nov 2019 19
20. 20. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course 3. FORMAL STATISTICAL TESTS Nov 2019 20
21. 21. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Background of tests • Statistical tests for normality are more precise since actual probabilities are calculated. • Tests for normality calculate the probability that the sample was drawn from a normal population. • The hypotheses used are: – Ho: The sample data are not significantly different than a normal population. – Ha: The sample data are significantly different than a normal population. Nov 2019 21
22. 22. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Background of tests • Typically, we are interested in finding a difference between groups. When we are, we ‘look’ for small probabilities. • If the probability of finding an event is rare (less than 5%) and we actually find it, that is of interest. • When testing normality, we are not ‘looking’ for a difference. • In effect, we want our data set to be NO DIFFERENT than normal. We want to accept the null hypothesis. • So when testing for normality: – Probabilities > 0.05 mean the data are normal. – Probabilities < 0.05 mean the data are NOT normal. Nov 2019 22
23. 23. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course W/S test • A fairly simple test that requires only the sample standard deviation and the data range. • Should not be confused with the Shapiro-Wilk test. • Based on the q statistic, which is the ‘studentized’ (meaning t distribution) range, or the range expressed in standard deviation units. Tests kurtosis. whereq istheteststatistic,w istherangeof thedataands is the standarddeviation. s q  w Nov 2019 23
24. 24. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Range constant, SD changes Range changes, SD constant Nov 2019 24
25. 25. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Village Population Density Aranza 4.13 Corupo 4.53 SanLorenzo 4.69 Cheranatzicurin 4.76 Nahuatzen 4.77 Pomacuaran 4.96 Sevina 4.97 Arantepacua 5.00 Cocucho 5.04 Charapan 5.10 Comachuen 5.25 Pichataro 5.36 Quinceo 5.94 Nurio 6.06 Turicuaro 6.19 Urapicho 6.30 Capacuaro 7.73 Standard deviation (s)=0.866 Range(w) = 3.6 n =17 0.866 3.6 s qCritical Range  3.06 to 4.31  4.16q  q  w The W/S test uses a critical range. IF the calculated value falls WITHIN the range, then accept Ho. IF the calculated value falls outside the range then reject Ho. Since 3.06 < q=4.16 < 4.31, then we accept Ho. W/S test Nov 2019 25
26. 26. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Table for W/S test • Since we have a critical range, it is difficult to determine a probability range for our results. • Therefore we simply state our alpha level. • The sample data set is not significantly different than normal (W/S4.16, p > 0.05). Nov 2019 26
27. 27. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Jarque–Bera Test • A goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. • Where x is each observation, n is the sample size, s is the standard deviation, k3 is skewness, and k4 is kurtosis. Nov 2019 27
28. 28. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Example dataset for JB test Village Population Density Mean Deviates Mean Deviates3 Mean Deviates4 Aranza 4.13 -1.21 -1.771561 2.14358881 Corupo 4.53 -0.81 -0.531441 0.43046721 SanLorenzo 4.69 -0.65 -0.274625 0.17850625 Cheranatzicurin 4.76 -0.58 -0.195112 0.11316496 Nahuatzen 4.77 -0.57 -0.185193 0.10556001 Pomacuaran 4.96 -0.38 -0.054872 0.02085136 Sevina 4.97 -0.37 -0.050653 0.01874161 Arantepacua 5.00 -0.34 -0.039304 0.01336336 Cocucho 5.04 -0.30 -0.027000 0.00810000 Charapan 5.10 -0.24 -0.013824 0.00331776 Comachuen 5.25 -0.09 -0.000729 0.00006561 Pichataro 5.36 0.02 0.000008 0.00000016 Quinceo 5.94 0.60 0.216000 0.12960000 Nurio 6.06 0.72 0.373248 0.26873856 Turicuaro 6.19 0.85 0.614125 0.52200625 Urapicho 6.30 0.96 0.884736 0.84934656 Capacuaro 7.73 2.39 13.651919 32.62808641 12.595722 37.433505 x  5.34 s  0.87 Nov 2019 28
29. 29. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Jarque–Bera Test Nov 2019 29
30. 30. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Jarque–Bera Test • The Jarque-Bera statistic can be compared to the χ2 distribution (table) with 2 degrees of freedom (df or v) to determine the critical value at an alpha level of 0.05. • The critical χ2 value is 5.991. Our calculated Jarque-Bera statistic is 4.12 which falls between 0.5 and 0.1, which is greater than the critical value. • Therefore we accept Ho that there is no difference between our distribution and a normal distribution (Jarque-Bera χ2, 0.5 > p > 0.1). Nov 2019 30
31. 31. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Non-Normally Distributed Data a. Lilliefors SignificanceCorrection Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. Average PM10 .142 72 .001 .841 72 .000 KS and Shapiro-Wilk test • Remember that LARGE probabilities denote normally distributed data NormallyDistributedData *. Thisisalowerboundofthetruesignificance. a.LillieforsSignificanceCorrection Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. AsthmaCases .069 72 .200* .988 72 .721 Nov 2019 31
32. 32. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course NormallyDistributedData *. Thisisalowerboundofthetruesignificance. a.LillieforsSignificanceCorrection In SPSS output above the probabilities are greater than 0.05 (the typical alpha level), so we accept Ho that these data are not different from normal. Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. AsthmaCases .069 72 .200* .988 72 .721 KS and Shapiro-Wilk test Nov 2019 32
33. 33. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Non-Normally Distributed Data a. Lilliefors SignificanceCorrection Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. Average PM10 .142 72 .001 .841 72 .000 • In the SPSS output above the probabilities are less than 0.05 (the typical alpha level), so we reject Ho that these data are significantly different from normal. • Important: As the sample size increases, normality parameters becomes MORE restrictive and it becomes harder to declare that the data are normally distributed. • So for very large data sets, normality testing becomes less important. KS and Shapiro-Wilk test Nov 2019 33
34. 34. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course D’Agostino Test • A very powerful test for departures from normality. • Based on the D statistic, which gives an upper and lower critical value. • where D is the test statistic, SS is the sum of squares of the data and n is the sample size, and i is the order or rank of observation X. The df for this test is n (sample size). • First the data are ordered from smallest to largest or largest to smallest. Nov 2019 34
35. 35. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course 0.26050 4.53 (39) 4.69 ...(17 9) 7.73 • Df = n = 17 • If the calculated value falls within the critical range, accept Ho. • Since 0.2587 < D = 0.26050 < 0.2860; accept Ho. • The sample data set is not significantly different than normal (D0.26050, p > 0.05). 212 1.46410 Village Population Density i Mean Deviates2 Aranza 4.13 1 1.46410 (4.13  5.34)2 1. Corupo 4.53 2 0.65610 SanLorenzo 4.69 3 0.42250 n 1 17 1 Cheranatzicurin 4.76 4 0.33640   9 Nahuatzen 4.77 5 0.32490 2 2 Pomacuaran 4.96 6 0.14440 T (i 9)X1 Sevina 4.97 7 0.13690 T  (19) 4.13  (2 9) Arantepacua 5.00 8 0.11560 Cocucho 5.04 9 0.09000 T  63.23 Charapan 5.10 10 0.05760 Comachuen 5.25 11 0.00810 Pichataro 5.36 12 0.00040 D  63.23  Quinceo 5.94 13 0.36000 (173 )(11.9916) Nurio 6.06 14 0.51840 D  0.2587,0.286 Turicuaro 6.19 15 0.72250 Critical Urapicho 6.30 16 0.92160 Capacuaro 7.73 17 5.71210 Mean =5.34 SS=11.9916 Nov 2019 35
36. 36. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Usethe nextlower nonthe tableif your samplesizeisNOT listed. Nov 2019 36
37. 37. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Normality Test Statistic Calculated Value Probability Results W/S q 4.16 >0.05 Normal Jarque-Bera χ2 4.15 0.5 >p >0.1 Normal Kolmogorov-Smirnov D 0.2007 0.067 Normal Shapiro-Wilk W 0.8827 0.035 Non-normal D’Agostino D 0.2605 >0.05 Normal Comparative performance of the tests • Different normality tests produce vastly different probabilities. • This is due to where in the distribution (central, tails) or what moment (skewness, kurtosis) they are examining. Nov 2019 37
38. 38. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Normality tests & sample sizes • Notice that as the sample size increases, the probabilities decrease. • In other words, it gets harder to meet the normality assumption as the sample size increases since even small departures from normality are detected. Nov 2019 38
39. 39. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Which normality test to use? • W/S: – Simple, but effective. – Not available in SPSS. • Jarque-Bera: – Tests for skewness and kurtosis very effective. – Not available in SPSS. • D’Agostino: – Powerful omnibus (skewness, kurtosis, centrality) test. – Not available in SPSS • Kolmogorov-Smirnov: – Not sensitive to problems in the tails. – For datasets>50. • Shapiro-Wilks: – Doesn't work well if several values in the data set are the same. – Works best for <50 datasets, but can be used with larger data sets. – Probably the better test in SPSS Nov 2019 39
40. 40. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Obs 15 7 6 6 5 5 5 4 4 3 Ho:Thesuspectedoutlier isnot different than the sampledistribution. Ha:Thesuspectedoutlier isdifferent than the sampledistribution. • Thecritical value for ann =10from Grubbsmodified t table (G table)at anα =0.05is2.18. • Since2.671>2.18, reject Ho. • Thesuspected outlier isfrom asignificantly different sample population (GMax,2.671,p <0.01). Grubbs test for outlier Nov 2019 40
41. 41. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Grubbs test for outlier Nov 2019 41
42. 42. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course • df = n • where xn is the suspected outlier, xn-1 is the next ranked observation, and x1 is the last ranked observation. • Ho: The suspected outlier is not different than the sample distribution. • Ha: The suspected outlier is different than the sample distribution. • Thecritical value for ann =10from VermaandQuiroz-Ruiz expanded Dixon table atan α =0.05is0.4122.Since0.6667> 0.4122,reject Ho. • Thesuspected outlier isfrom asignificantly different sample population (Q0.6667, p <0.005). Obs 15 7 6 6 5 5 5 4 4 3 Dixon test for outlier Nov 2019 42
43. 43. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Dixon test for outlier Nov 2019 43
44. 44. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Requirements for outlier tests • The data are from a normal distribution • There are not multiple outliers (3+) • The data are sorted with the suspected outlier first. • If 2 observations are suspected as being outliers and both lie on the same side of the mean, this test can be performed again after removing the first outlier from the data set. • Caution must be used when removing outliers. Only remove outliers if you suspect the value was caused by an error of some sort, or if you have evidence that the value truly belongs to a different population. • In small sample size, extreme caution should be used when removing any data. Nov 2019 44
45. 45. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Tests ofNormality a. Lilliefors Significance Correction Kolmogorov-Smirnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. Age .110 1048 .000 .931 1048 .000 Tests of Normality a. Lilliefors Significance Correction Kolmogorov-Sm irnov a Shapiro-Wilk Statistic df Sig. Statistic df Sig. TOTAL_VALU .283 149 .000 .463 149 .000 Tests ofNormality *. This is a lower bound of the true significance. a. Lilliefors Significance Correction Kolmogorov-Smirnova Shapiro-Wilk Statistic df Sig. Statistic df Sig. Z100 .071 100 .200* .985 100 .333 A combination of approaches (SPSS) Nov 2019 45
46. 46. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course 4. COMPARE PARAMETRIC AND NON PARAMETRIC TESTS Nov 2019 46
47. 47. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Comparing tests • It is advisable to first check for normality or your data distribution • If it is normally distributed, then use parametric tests • However, it is better to also do non-parametric tests, because they do not make modeling assumptions, and if you get similar results, report your approach in the methods section • If you get dissimilar results, then also report both to ensure transparency, but give proper justification of which you think is better and why • Beware! By using multiple tests, the alfa error will be inflated so use this strategy only when necessary Nov 2019 47
48. 48. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Take home messages • All methods may not agree with each other • If you perform a normality test, do not ignore the results. • If the data are not normal, use non-parametric tests. • If the data are normal, use parametric tests. • If you have groups of data, you MUST test each group for normality • Decision must be taken on a case by case basis Nov 2019 48
49. 49. Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course THANK YOU Kindly email your queries to sarizwan1986@outlook.com Nov 2019 49