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Non Parametric Tests
- 1. Dr. Zarrin Ansari SecondYear Resident Pharmacology Dept. LTMMC & GH, Sion June 28, 2021 1
- 2. 2 Table of Content When to Choose a Statistical Tests OR When NOT to Choose? Types of Statistical Tests (Parametric vs. Non-Parametric Tests) Parameters to check when Choosing a Statistical Test: - Distribution of Data - Type of data/Variable - Types of Analysis (What’s the hypothesis) - No of groups or data-sets - Data Group Design Snapshot of all statistical test and “How” to Choose using above parameters Explanation using Examples: - Mann Whitney U Test - Wilcoxon Sign Rank Test - Spearman’s co-relation - Chi-Square Test
- 3. 3 When to Choosea Statistical Tests? Or When NOT to Choose a Statistical Tests?
- 4. Parametric Test Non-Parametrictest Normal Distribution Skewed Distribution Based on assumptions Less rigid & fewer assumptions Quantitative Data Qualitative Data More Statistical power Less Statistical Power No outliers Outliers present Mainly compares means +/- SDs Mainly compares % and proportions Jain V. (2019) Review of preventive & social medicine: Including biostatistics. 11th ed. 4 Types of Statistical Test
- 5. 5 Sample Size Types ofdata No of groups/ data sets Distribution of data Data Group design How to Choose a Statistical Test? Parameters to Check R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
- 6. Normal Non-Normal Dichotomous Distribution of Data All bodily parameters follow Normal Distribution exception; Ab titers Measure of ranks/scores viz. Apgar, Scoring of liver fibrosis, Pain in visual analogue scale follow Non- normal Distribution Variables measured as counts e.g. no of people having headache etc. are binomial measurements. These for Non-normal distribution If Mean < 2 times SD then the distribution is Non- Normal Distribution Mean=Median=Mode Normal Distribution 6
- 7. Left sided Skewing Mean< Median < Mode Right sided Skewing Mean > Median > Mode Non-Normal Distribution Normal Distribution No Skewing Mean = Median = Mode Dakhale GN, Hiware SK, Shinde AT, Mahatme MS. Basic biostatistics for post-graduate students. Indian journal of pharmacology. 2012 Jul;44(4):435. 7
- 8. Types of Data/Variable Categorical/Non-numerical/ Qualitative Numerical/ Quantitative Ordinal (Order/Hierarchy) E.g. TumorStaging Nominal (any Order/Hierarchy) E.g. Gender/Race etc. Discrete (Whole numbers) E.g. No of patients Continuous (Can take Decimals) E.g. Weight 8 R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
- 9. Types of Analysis (Whatwe want to do?) (What’s the Hypothesis) Comparison Between Mean, Median & Proportions of one or > one group Co-relation Analysis Find relationship between two variable Regression Analysis: Predicting one variable with another R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8 9
- 10. No of Groups &Data sets One group (Two or > Two Data sets) Two groups (Two data sets) > Two groups/ > 2 data sets 10 R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
- 11. UNPAIRED / INDEPENDANT PAIRED / MATCHED •2GROUPS OR > 2 GROUPS OF TOTALLY DIFFERENT SET OF SUBJECTS INDEPENDENT OF EACH OTHER • ONE GROUP BEFORE OR AFTER INTERVENTION Data Group Design 11 R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
- 12. C1 C2 C3 ComparisonAssociation of two variables Regression analysis of two variables Of 2 data sets Of > 2 data sets Paired Unpaired Paired Unpaired C1a C1b C1c C1d R1 Normally distributed continuous data (Summarized as means) Paired t test Unpaired t test Repeated Measures ANOVA (Two way ANOVA) One-way ANOVA Pearson correlation Linear regression R2 Scores, ranks & non- normally distributed continuous data (Summarized as Median) Wilcoxon singed rank test Mann Whitney U Test Friedman test Kruskal- Wallis test Spearman’s rank correlation Non- parametric regression R3 Dischotomous Data (Summarized as proportions) Mc Nemar’s test Fischer’s test or Chi square test Cochrane Q test Chi square test Contingency coefficient Logistic regression How to Choose a Statistical Test? 12 R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
- 13. Example 1 Drug X(New anti-asthmatic) given in one group of patients (n=5). Placebo given in second group (n=5). Measure number of asthma attack/weak Steps: 1. Aim: To evaluate anti-asthmatic effect of new drug 2. Hypothesis: (Null) The new drug is NOT effective anti-asthmatic. 3. No of groups/data: 2 groups 4. Data Group design: Unpaired/Independant 5. Distribution: Non-normal (Small sample size) 6. Type of analysis: Comparison Table: Mann Whitney U Test 13
- 14. C1 C2 C3 ComparisonAssociation of two variables Regression analysis of two variables Of 2 data sets Of > 2 data sets Paired Unpaired Paired Unpaired C1a C1b C1c C1d R1 Normally distributed continuous data (Summarized as means) Paired t test Unpaired t test Repeated Measures ANOVA (Two way ANOVA) One-way ANOVA Pearson correlation Linear regression R2 Scores, ranks & non- normally distributed continuous data (Summarized as Median) Wilcoxon singed rank test Mann Whitney U Test Friedman test Kruskal- Wallis test Spearman’s rank correlation Non- parametric regression R3 Dischotomous Data (Summarized as proportions) Mc Nemar’s test Fischer’s test or Chi square test Cochrane Q test Chi square test Contingency coefficient Logistic regression How to Choose a Statistical Test? 14 R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
- 15. Consider a PhaseII clinical trial designed to investigate the effectiveness of a new drug to reduce symptoms of asthma in children. A total of n=10 participants are randomized to receive either the new drug or a placebo. Participants are asked to record the number of episodes of shortness of breath over a 1 week period following receipt of the assigned treatment. https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/bs704_nonparametric4.html#:~:text=The%20test%20statistic%20for%20the,our%20example%2C%20U%3D3. 15
- 16. https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/bs704_nonparametric4.html#:~:text=The%20test%20statistic%20for%20the,our%20example%2C%20U%3D3. The first stepis to assign ranks and to do so we order the data from smallest to largest. This is done on the combined or total sample (i.e., pooling the data from the two treatment groups (n=10)), and assigning ranks from 1 to 10. R1= 37 R2= 18 16
- 17. The test statisticfor the Mann Whitney U Test is denoted U and is the smaller of U1 and U2, defined below. Ucalculated=3 Smaller values of U support the research hypothesis, and larger values of U support the null hypothesis. U tabular = 2 https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/bs704_nonparametric4.html#:~:text=The%20test%20statistic%20for%20the,our%20example%2C%20U%3D3. 17
- 18. Decision rule isto reject H0 if Ucalc < U tab Ucalc = 3 > Utab =2 H0 is NOT rejected The difference between the two arms is not statistically different https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/bs704_nonparametric4.html#:~:text=The%20test%20statistic%20for%20the,our%20example%2C%20U%3D3. 18
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- 20. Example 2 Drug Xgiven in one group of autistic children (n=8). The number of repetitive behavior was observed over period of three hours before and after the administration of the new drug for a period of one weak Steps: 1. Aim: To evaluate anti-autistic effect of new drug 2. Hypothesis: (Null) The new drug is NOT effective in treating autism 3. No of groups/data: one group but two data sets 4. Group Data design: Paired 5. Distribution: Non-normal (Small sample size) 6. Type of analysis: Comparison Table: Wilcoxon singed rank test 20
- 21. C1 C2 C3 ComparisonAssociation of two variables Regression analysis of two variables Of 2 data sets Of > 2 data sets Paired Unpaired Paired Unpaired C1a C1b C1c C1d R1 Normally distributed continuous data (Summarized as means) Paired t test Unpaired t test Repeated Measures ANOVA (Two way ANOVA) One-way ANOVA Pearson correlation Linear regression R2 Scores, ranks & non- normally distributed continuous data (Summarized as Median) Wilcoxon singed rank test Mann Whitney U Test Friedman test Kruskal- Wallis test Spearman’s rank correlation Non- parametric regression R3 Dischotomous Data (Summarized as proportions) Mc Nemar’s test Fischer’s test or Chi square test Cochrane Q test Chi square test Contingency coefficient Logistic regression How to Choose a Statistical Test? 21 R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
- 22. W+ (sum ofthe positive ranks) =32 W- (sum of the negative ranks) =4 The test statistic is W (cal) = 4. https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/BS704_Nonparametric6.html 22
- 23. If W (calculated)=/< W (tabulated), then reject H0 If W (calculated) > W (tabulated), then DO NOT reject H0 W (calculated) = 4 W (tabulated) = 6 Thus, null hypothesis is rejected ….. https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/BS704_Nonparametric6.html 23
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- 25. Example 3 Does thenumber of symptoms in a patient predict their willingness to take medication? Steps: 1. Aim: To evaluate co-relation between number of symptoms the patients have to their willingness to take medicine 2. Hypothesis: (Null) There is NO correlation between the two variable 3. No of groups/data: one groups 4. Group Data design: NA 5. Distribution: Non-normal (ordinal data) 6. Type of analysis: Correlation Table: Spearman rank correlation analysis 25
- 26. C1 C2 C3 ComparisonAssociation of two variables Regression analysis of two variables Of 2 data sets Of > 2 data sets Paired Unpaired Paired Unpaired C1a C1b C1c C1d R1 Normally distributed continuous data (Summarized as means) Paired t test Unpaired t test Repeated Measures ANOVA (Two way ANOVA) One-way ANOVA Pearson correlation Linear regression R2 Scores, ranks & non- normally distributed continuous data (Summarized as Median) Wilcoxon singed rank test Mann Whitney U Test Friedman test Kruskal- Wallis test Spearman’s rank correlation Non- parametric regression R3 Dischotomous Data (Summarized as proportions) Mc Nemar’s test Fischer’s test or Chi square test Cochrane Q test Chi square test Contingency coefficient Logistic regression How to Choose a Statistical Test? 26 R Raveendran et al. (2014) A Practical Approach to PG Dissertation…..a handbook of research methodology for postgraduate students. Second Edition. Chapter 8
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- 29. Chi-Square Test • Usedto test significance of association between 2 or more qualitative characteristics • It checks for patterns and relationships in categorical variable • Is used to compare proportions in 2 or more groups • Is used for Non-normal distribution • Applications of Chi-square test - Test of Independence - Test of Goodness of Fit Jain V. Review of preventive & social medicine: Including biostatistics. 11th ed. New Delhi, India: Jaypee Brothers Medical; 2019. 30
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- 31. To see relationshipbetween Smoking and Cancer H0 = There is NO relationship between Smoking and Cancer H1 = Smoking is associated with Cancer 32
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- 33. 34 Some Perks inStatistical Concepts: • Uniform Distribution • Degree of Freedom • Kurtosis • Normality testing
- 34. 35 Methods to Checkfor Normality of Data Visual Methods: Histogram, Box-plots, Q-Q plots Normality Tests Kolmogorov-Smirnov (K-S) test Lilliefors corrected K-S test Shapiro-Wilk test Anderson-Darling test Cramer-von Mises test Kurtosis Skweness Ghasemi et al. Normality Tests for Statistical Analysis: A Guide for Non-Statisticians. Int J Endocrinol Metab. 2012;10(2):486-489.
- 35. 36 Thank you foryour Attention
Editor's Notes
- #5 More statistical power means more likely to detect a difference when truly exists…
- #8 Uniform Distribution….
- #10 Co-relation analysis: E.g. We measure BP & body weight in one group. We measure whether BP increases with increase in weight and vice-a-versa Regression Analysis: If a patient comes to my OPD weighing 50 Kg, what would be his BP. Predicting variables is called regression analysis.
- #17 We call the placebo group 1 and the new drug group 2 (assignment of groups 1 and 2 is arbitrary). We let R1 denote the sum of the ranks in group 1 (i.e., R1=37), and R2 denote the sum of the ranks in group 2 (i.e., R2=18). If the null hypothesis is true (i.e., if the two populations are equal), we expect R1 and R2 to be similar. In this example, the lower values (lower ranks) are clustered in the new drug group (group 2), while the higher values (higher ranks) are clustered in the placebo group (group 1). This is suggestive, but is the observed difference in the sums of the ranks simply due to chance? To answer this we will compute a test statistic to summarize the sample information and look up the corresponding value in a probability distribution.
- #19 In every test, we must determine whether the observed U supports the null or research hypothesis. This is done following the same approach used in parametric testing. Specifically, we determine a critical value of U such that if the observed value of U is less than or equal to the critical value, we reject H0 in favor of H1 and if the observed value of U exceeds the critical value we do not reject H0. The critical value of U can be found in the table below. To determine the appropriate critical value we need sample sizes (for Example: n1=n2=5) and our two-sided level of significance (α=0.05). For Example 1 the critical value is 2, and the decision rule is to reject H0 if U < 2. We do not reject H0 because 3 > 2. We do not have statistically significant evidence at α =0.05, to show that the two populations of numbers of episodes of shortness of breath are not equal. However, in this example, the failure to reach statistical significance may be due to low power. The sample data suggest a difference, but the sample sizes are too small to conclude that there is a statistically significant difference.
- #24 The test statistic for the Wilcoxon Signed Rank Test is W, defined as the smaller of W+ (sum of the positive ranks) and W- (sum of the negative ranks). If the null hypothesis is true, we expect to see similar numbers of lower and higher ranks that are both positive and negative (i.e., W+ and W- would be similar). If the research hypothesis is true we expect to see more higher and positive ranks (in this example, more children with substantial improvement in repetitive behavior after treatment as compared to before, i.e., W+ much larger than W-). In this example, W+ = 32 and W- = 4. Recall that the sum of the ranks (ignoring the signs) will always equal n(n+1)/2. As a check on our assignment of ranks, we have n(n+1)/2 = 8(9)/2 = 36 which is equal to 32+4. The test statistic is W = 4. Next we must determine whether the observed test statistic W supports the null or research hypothesis. This is done following the same approach used in parametric testing. Specifically, we determine a critical value of W such that if the observed value of W is less than or equal to the critical value, we reject H0 in favor of H1, and if the observed value of W exceeds the critical value, we do not reject H0.
- #26 Compared to the Pearson correlation coefficient, the Spearman correlation does not require continuous-level data (interval or ratio), because it uses ranks instead of assumptions about the distributions of the two variables. This allows us to analyze the association between variables of ordinal measurement levels. Moreover, the Spearman correlation does not assume that the variables are normally distributed. A Spearman correlation analysis can therefore be used in many cases in which the assumptions of the Pearson correlation (continuous-level variables, linearity, heteroscedasticity, and normality) are not met.
- #28 Whereas correlation explains the strength of the relationship between an independent and dependent variable, R-squared explains to what extent the variance of one variable explains the variance of the second variable.
- #33 Expected values= row total multiplied by column total divided by All total