3.1 non parametric test

when you can measure what you are speaking about and express it in numbers, you know something about it but when you cannot measure, when you cannot express it in numbers, your knowledge is of meagre and unsatisfactory kind.”

1. Biostatistics and Research Methodology Ms. Shital S. Patil

2. Biostatistics “when you can measure what you are speaking about and express it in numbers, you know something about it but when you cannot measure, when you cannot express it in numbers, your knowledge is of meagre and unsatisfactory kind.” ....Lord Kelvin

3. Biostatistics Collecting Data, Understanding Data and Numbers. The word is “Statistics” not “ Sadistics ”. Biostatistics

4. 1. Population a group of individuals that we would like to know something about. Biostatistics

5. 2. Parameter a characteristic of the population in which we have a particular interest Examples: The proportion of the population that would respond to a certain drug The association between a risk factor and a disease in a population Biostatistics

6. Consider a clinical trial where study participants are asked to rate their symptom severity following 6 weeks on the assigned treatment. Symptom severity might be measured on a 5 point ordinal scale with response options: Symptoms got much worse, slightly worse, no change, slightly improved, or much improved. Suppose there are a total of n=20 participants in the trial, randomized to an experimental treatment or placebo, and the outcome data are distributed as shown in the figure below.

7. UNIT III

8. Contents Non Parametric tests Wilcoxon Rank Sum Test, Mann-Whitney U test, Kruskal -Wallis test Friedman Test Introduction to Research Need for research, Need for design of Experiments, Experiential Design Technique, P lagiarism Graphs Histogram Pie Chart, Cubic Graph Response surface plot Counter Plot graph Designing the methodology Sample size determination Power of a study, Report writing and presentation of data, Protocol , Cohorts studies, Observational studies, Experimental studies, Designing clinical trial, various phases.

9. Hypothesis Hypothesis is considered as an intelligent guess or prediction, that gives directional to the researcher to answer the research question. Hypothesis or Hypotheses are defined as the formal statement of the tentative or expected prediction or explanation of the relationship between two or more variables in a specified population.

10. Hypothesis A hypothesis is a formal tentative statement of the expected relationship between two or more variables under study. A hypothesis helps to translate the research problem and objective into a clear explanation or prediction of the expected results or outcomes of the study.

11. Classification of Hypothesis Parametric Test Non Parametric Test t-test F-Test Z-test ANOVA Wilcoxon Rank Sum Test, Mann-Whitney U test, Kruskal -Wallis test Friedman Test

12. Non Parametric tests Non Parametric tests: Also known as distribution-free tests because they are based on fewer assumptions (e.g., they do not assume that the outcome is approximately normally distributed). Non parametric statistics refers to a statistical method wherein data is not required to fit a normal distribution. Non parametric statistics uses data that is often ordinal meaning it does not rely on numbers, but rather a ranking or order of sorts.

13. Advantages of Non Parametric Test Non Parametric tests are simple and easy to understand. It will not involve sample complicated theory. No assumptions are made regarding Parent population. This method is only available for Nominal Scale data. This method is easily applicable.

14. Difference between Parametric and Non Parametric Information about the population is completely known about Specific Assumptions are made regarding the population. Null Hypothesis is made on parameters of population distribution. Test statistics is based on the distribution. No information about the population is available No Assumptions are made regarding the population. Null Hypothesis is free from parameters. Test statistics is arbitrary Parametric Non Parametric

15. Difference between Parametric and Non Parametric Parametric test are applicable only for variables No parametric test exist for nominal scale data. Parametric test is powerful, if it exist. No information about the population is available Non parametric test do exist for nominal and ordinal scale data. It is not so powerful. Parametric Non Parametric

16. There are some situations when it is clear that the outcome does not follow a normal distribution. These include situations: when the outcome is an ordinal variable or a rank, when there are definite outliers or when the outcome has clear limits of detection. Non Parametric tests

17. Wilcoxon Rank Sum Test Mann-Whitney U test Kruskal -Wallis test Friedman Test Non Parametric tests

18. The Wilcoxon Rank Sum test is used to test for a difference between two samples. It is the nonparametric counterpart to the two-sample Z or t test. Instead of comparing two population means, we compare two population medians. Wilcoxon Rank Sum Test

19. The problem characteristics of this test are two groups being tested are independent of each other. two groups should have approximately similar distributions. numeric and ordinal data. Wilcoxon Rank Sum Test

20. Wilcoxon Rank Sum Test

21. Step 1: List the data values from both samples in a single list arranged from smallest to largest Step 2: In the next column, assign the numbers 1 to N (where N = n 1 +n 2 ). These are the ranks of the observations. When N is equal to our total sample size, our smallest observation receives a rank of 1, and the largest observation receives a rank of N. If there are ties, assign the average of the ranks the values would receive to each of the tied values. Wilcoxon Rank Sum Test

22. Step 3: The sum of the ranks of the first sample is W, the Wilcoxon Rank-Sum test statistic. If one sample is truly bigger than the other, we’d expect its ranks to be higher than the others. So after we have ranked all of the observations, we sum up the ranks for each of the two samples and we can then compare the two rank sums Wilcoxon Rank-Sum Test

23. Note the following: If there are ties, then we would expect W to be roughly half of [N(N+1)]/2. If there are no ties when the observations are ranked, then we would expect W to be roughly equal to its mean/expected value, µ W = n 1 (N+1)/2. Wilcoxon Rank-Sum Test

24. Wilcoxon Rank-Sum Test

25. Commonly portrayed as the non-parametric substitute for Student's t-test when samples are not normally distributed. To compute the Mann Whitney U test: Rank the scores in both groups (together) from highest to lowest. Sum the ranks of the scores for each group. The sum of ranks for each group are used to make the statistical comparison. 2. Mann- Whitney (U test)

26. Null hypothesis states that there is no difference in the scores of the populations from which samples were drawn. The Mann- Whitney (U test) is sensitive to both the central tendency of the scores and the distribution of the scores. The Mann- Whitney (U test) statistic is smaller of U 1 and U 2. 2. Mann- Whitney (U test)

27. U 1 = n 1 n 2 + [n 1 (n 1 +1)/2] - R 1 U 2 = n 1 n 2 + [n 2 (n 2 +1)/2] - R 2 Where, n 1 = No. of observations in group 1 n 2 = No. of observations in group 2 R 1 = Sum of ranks assigned to group 1 R 2 = Sum of ranks assigned to group 1 2. Mann- Whitney (U test)

28. Null Hypothesis : There is no difference in scores of the two groups (i.e. the sum of ranks for group 1 is no different than the sum of ranks for group 2). Alternative Hypothesis: There is a difference between the scores of the two groups (i.e. the sum of ranks for group 1 is significantly different from the sum of ranks for group 2) 2. Mann- Whitney (U test)

29. 3. Kruskal -Wallis test The Kruskal –Wallis one-way analysis of variance by ranks is a non- parametric method for testing whether samples originate from the same distribution. It is also called Kruskal -Wallis H test. Kruskal -Wallis was presented by : William Kruskal and W. Allen Wallis. The Kruskal -Wallis test is the nonparametric test equivalent to the one-way ANOVA, and an extension of the Mann-Whitney U test to allow the comparison of more than two independent groups.

30. Following; The continuous distributions for the test variable are exactly the same (except their medians) for the different populations. The cases represent random samples from the populations, and the scores on the test variable are independent of each other. The chi-square statistic for the Kruskal - Wallis test is only approximate and becomes more accurate with larger sample sizes. 3. Kruskal -Wallis test

31. HYPOTHESIS: Ho= All population has the same median yield. H1 = Not all median yield are equal. Non-parametric tests hypothesize about the median instead of the mean (as parametric tests do). We order the scores that we have from lowest to highest, ignoring the group that the scores come from, and then we assign the lowest score a rank of 1, the next highest a rank of 2 and so on. We take the responses from all groups and rank them; then we sum up the ranks for each group . 3. Kruskal -Wallis test Methodology

32. The test statistic H is calculated: 3. Kruskal -Wallis test

33. Chi-squared distribution with K-1 degrees of freedom when Ho is true. R is the assumed value of sum of ranks, for i= 1,2,….k. N be the observation in the ith sample. 3. Kruskal -Wallis test

34. Critical Region: H > Χ2 α, reject Ho at the α-level of significance, otherwise fail to reject Ho. 3. Kruskal -Wallis test

35. Friedman Test Friedman test is a non parametric statistical method developed by Dr. Milton Friedman

36. Friedman Test The Friedman test is a non-parametric alternative to ANOVA with repeated measures. It is used to test for differences between groups when the dependent variable being measured is ordinal. The Friedman test tests the Null hypothesis of identical populations for dependent data. The test is similar to the Kruskal -Wallis Test. It uses only the rank information of the data.

37. Assumptions 1. The r blocks are independent so that the measurements in one block have no influence on the measurements in any other block. 2. The underlying random variable of interest is continuous (to avoid ties). 3. The observed data constitute at least an ordinal scale of measurement within each of the r blocks. 4. There is no interaction between the m blocks and the k treatment levels. 5. The c populations have the same variability. 6. The c populations have the same shape. Friedman Test

38. Steps involved in testing 1) Formulation of hypothesis 2) Significance level 3) Test statistics 4) Calculations 5) Critical region 6) Conclusion Friedman Test

39. 1) Formulation of hypothesis we check the equality of means of different treatments as in ANOVA, The hypothesis will be stated as: Ho= M1=M2=……=Mk H1= not all medians are equal Friedman Test

40. 2) Level of significance: It is selected as given if not given 0.05 is taken. 3) Test statistics: Where, R 2 .j is the square of the rank total for group j (j = 1, 2, . . . , c) m is the number of independent blocks k is the number of groups or treatment levels Friedman Test

41. 4) Calculations: Start with n rows and k columns. Rank order the entries of each row independently of the other rows. Sum the ranks for each column. Sum the squared column totals. Using test statistic calculate the value of Q. Friedman Test

42. 5) Critical region: Reject H if Q ≥ critical value at α= 5% If the values of k and/or n exceed those given in tables, the significance of Q may be looked up in chi-squared (χ2) distribution tables with k-1 degrees of freedom. 6) Conclusion : If the value of Q is less than the critical value then we’ll not reject H 0. If the value of Q is greater than the critical value then we’ll reject H0. Friedman Test

43. Introduction to Research 1.1 MEANING OF RESEARCH 1.2 NEED AND OBJECTIVES OF RESEARCH 1.3 CHARACTERISTICS OF RESEARCH 1.4 CRITERIA OF A GOOD RESEARCH 1.5 QUALITIES OF GOOD RESEARCH 1.6 RESEARCH MOTIVATIONS 1.7 TYPES OF RESEARCH 1.8 PROBLEMS IN RESEARCH 1.9 RESEARCH APPROACHES 1.10 RESEARCH PROCESS 1.11 LITERATURE REVIEW 1.12 HYPOTHESIS 1.13 CRITERIA OF GOOD RESEARCH 1.14 PROBLEMS ENCOUNTERED BY RESEARCHERS