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Non – Parametric Test Dr.L.Jeyaseelan Dept. of Biostatistics Christian Medical College Vellore, India Introduction No rigid assumptions about the distribution of the populations - “Distribution-free tests” Answers the same sort of questions as the parametric test – for each Parametric tests (PT) there is an alternative NonParametric Test (NP) Test Applied to a wide variety of situations continuous ordinal scores When are non-parametric tests used? Assumptions of parametric test are violated Non-normal or skewed Variance very high in relative to mean Data is on an ordinal scale Very few observations Mann-Whitney U Test Most powerful of the NP test – Wilcoxon Rank Sum test Alternative to the parametric independent t-test To test whether two independent groups have been drawn from the same population Assumptions: Two sample are selected independently and at random from their respective population Variable of interest is continuous Measurement scale is at least ordinal. Mann-Whitney U Test Compare the distribution of scores on a quantitative variable obtained from two independent groups. Group A : 2 Group B : 8 4 2 6 4 8 8 4 10 12 11 Ho: There is no significant difference between the medians of the two samples Ha: There is a significant difference between the medians of the two samples Man Whitney U test output Mann-Whitney U Test (contd.) A researcher designed an experiment to assess the effects of prolonged inhalation of cadmium oxide. 15 lab animals served as experimental subjects while 10 similar animals served as controls. Variable of interest was Hb level following the experiment. Conclude that there is no difference between exposed and unexposed animals in their Hb level sno Exposed Unexposed 1 13.1 17.4 2 15.6 16.2 3 17.3 17.1 4 16.4 17.5 5 14.3 15 6 15.5 16 7 14.9 16.9 8 15.6 15.1 9 14.1 16.1 10 15.3 17.2 11 15.7 12 16.7 13 13.7 14 15.6 15 14 Deciding Normality Conclusion??? Large Sample: When n1,n2 > 10 (normal approximation) The test statistic is given as U E (U ) Z var(U ) n1n2 U 2 Z (n1 n2 1) n1n2 12 where ‘U’ is the smallest sum of ranks between U1 and U2 Wilcoxon Signed-Rank Test An alternative to the parametric paired t-test Used to compare 2 samples from populations are not independent eg., measure a variable in each subject before and after an intervention Assumptions Samples must be paired Pairs are randomly selected from the larger population Probability distribution from which the sample of paired differences drawn is continuous Wilcoxon Signed-Rank Test Below is the measurement of anxiety before and after two weeks of treatment. SN O BEFORE Rx AFTER Rx 1 130 120 2 170 163 3 125 120 4 170 135 5 130 143 6 130 136 7 145 144 8 160 120 Does the treatment have any beneficial effects? Result??? Exercise: Wilcoxon signed rank Test Sno Before After 1 142 143 2 148 146 3 144 147 4 142 138 5 140 136 6 144 139 7 146 141 8 150 145 subjects before and after receiving a standard 9 149 143 10 treatment 142 136 11 149 145 12 143 140 13 145 143 14 146 142 15 143 140 16 146 141 Examine whether there is any significant difference in the systolic blood pressure of 16 Result??? Advantages of Nonparametric Methods Nonnumeric data such as Taste of food: bad, good, great and excellent, Smoking habit: light, moderate and heavy etc. Involves simpler computations than the corresponding parametric methods and are therefore easier to understand. Since the inference is based on ranks, Nonparametric methods are quick for small samples and less subject to measurement error than parametric methods. Disadvantages of NP Methods No parameters to describe and it becomes more difficult to make quantitative statements about the actual difference between populations Tend to waste information because it deals with ranks and discard the actual values Less powerful where parametric test is applicable. Selecting a Statistical Test Measurement from Normal Population Rank, Score or Measurement from Non-normal Describe one group Mean, SD Median, Interquartile Range Compare two independent groups Independent sample t-test (unpaired t test) Mann-Whitney test Compare two paired groups Paired t test Wilcoxon Signed rank test Pearson Correlation Spearman Rank Correlation Type of Data Quantify relation between two variables