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 Non- Parametric Test (NP) Test  Applied to a wide variety of situations continuous ordinal scores

 Assumptions of parametric test are violated When are non-parametric tests used?  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 H o :There is no significant difference between the medians of the two samples H a :There is a significant difference between the medians of the two samples Compare the distribution of scores on a quantitative variable obtained from two independent groups. Group A : Group B :

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

snoExposedUnexposed

Deciding Normality

Conclusion???

Large Sample: When n 1,n 2 > 10 (normal approximation) The test statistic is given as where ‘U’ is the smallest sum of ranks between U 1 and U 2

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

Below is the measurement of anxiety before and after two weeks of treatment. SN O BEFORE Rx AFTER Rx Does the treatment have any beneficial effects? Wilcoxon Signed-Rank Test

Result???

Exercise: Wilcoxon signed rank Test Examine whether there is any significant difference in the systolic blood pressure of 16 subjects before and after receiving a standard treatment SnoBeforeAfter

Result???

 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. Advantages of Nonparametric 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 Type of Data Measurement from Normal Population Rank, Score or Measurement from Non-normal Describe one groupMean, SDMedian, Interquartile Range Compare two independent groups Independent sample t-test (unpaired t test) Mann-Whitney test Compare two paired groups Paired t testWilcoxon Signed rank test Quantify relation between two variables Pearson CorrelationSpearman Rank Correlation