STAT106 LEC1

• Nonparametric methods


  Statistical techniques that deal with ordinal data
• Nonparametric methods


  • Alternative techniques to parametric tests when assumptions like normality and homogeneity of variances are not met
• Parametric statistical inference


  Methods for estimating and testing hypotheses about population parameters based on the assumption of normal distribution
• Nonparametric statistical inference


  Methods that make little to no assumptions about the population distribution
• Questions addressed by nonparametric tests

  • What if the population is not normal?
  • What if the scale of measurement is nominal or ordinal?
  • What if the interest is not about population parameters?
• Distribution-free tests


  Techniques of statistical inference that make little to no assumptions about the population distribution
• Nonparametric tests


  Tests for hypotheses that are not statements about parameter values
• Advantages of nonparametric tests

  • Quick and easy to apply
  • Can test hypotheses not addressed by parametric tests
  • Exact sampling distribution can often be determined
  • Less problem with violated assumptions
  • May be less expensive and time-consuming to collect data
• Disadvantages of nonparametric tests

  • May be used inappropriately when parametric tests are more appropriate
  • Arithmetic can be tedious for large samples without a computer
• When to use nonparametric tests

  • Hypothesis does not involve a population parameter
  • Data is measured on a nominal or ordinal scale
  • Assumptions for parametric tests are violated
  • Quick results are needed without a computer
• One-Sample Wilcoxon Signed-Rank Test

  Nonparametric alternative to One-Sample T-Test when data cannot be assumed to be normally distributed
• Assumptions for One-Sample Wilcoxon Signed-Rank Test

  • Random sample from population with unknown median
  • Continuous variable
  • Symmetric population
  • Interval or higher scale of measurement
  • Independent observations
• Steps to calculate One-Sample Wilcoxon Signed-Rank Test statistic

  1. Compute Di = Xi - M0 for each observation

  2. Rank the |Di| and assign sign of Di to rank

  3. Determine W+ (sum of positive ranks) and W- (sum of negative ranks)

  4. Test statistic is smaller of W+ or W- for two-tailed test, W+ for one-tailed test > M0, W- for one-tailed test < M0

  
• Use p-value method for hypothesis testing instead of comparing to tables
• value


  ≤ α, otherwise do not reject
• Test the hypothesis at the 0.05 level of significance that this particular lamp operates, on the average, 1.8 hours before requiring a recharge
• Hypothesis testing


  1. Ho: M=1.8 hours

  2. Ha: M≠1.8 hours

  3. α = 0.05

  4. Test statistic W=13.0 with p-value=0.152

  5. Decision: Cannot reject Ho

  6. Conclusion: There is no sufficient evidence that the average number of hours the lamps operate is not 1.8 hours before requiring a recharge at 5% level of significance
• Mann-Whitney U test


  • Assumptions:
  • Random samples from two populations
  • Independent samples
  • Ordinal measurement scale
  • Continuous random variable
  • Distributions differ only in location

  
• Mann-Whitney U test


  1. Ho: MX = MY

  2. Ha: MX < MY or MX > MY

  3. Test statistic U = S - n1(n1+1)/2

  4. Decision rule: Reject Ho if p-value ≤ α
• Self-concept scale data


  • Normal subjects ranks
  • Psychiatric patients ranks
• Hypothesis testing for Mann-Whitney U test


  1. Ho: MX = MY

  2. Ha: MX ≠ MY

  3. α = 0.05

  4. Test statistic U=39.5 with p-value=0.003

  5. Decision: Reject Ho

  6. Conclusion: Median scores for psychiatric patients significantly differs from normal subjects
• Wilcoxon Signed-rank Test


  • Assumptions:
  • Random sample
  • Continuous variable
  • Symmetric population
  • Ordinal measurement scale
  • Independent pairs
• Wilcoxon Signed-rank Test


  1. Ho: M = Mo

  2. Ha: M < Mo or M > Mo

  3. Test statistic W = W+ or W-

  4. Decision rule: Reject Ho if p-value ≤ α
• Reduction in forced vital capacity (FVC) data


  • Placebo
  • Drug
• Hypothesis testing for Wilcoxon Signed-rank Test

  1. Ho: M = 0

  2. Ha: M ≠ 0

  3. α = 0.05

  4. Test statistic W=1.0 with p-value<0.001

  5. Decision: Cannot reject Ho

  6. Conclusion: Average difference in reduction of FVC significantly differs from 0
• Kruskal-Wallis Test


  • Assumptions:
  • k random samples
  • Independent observations
  • Continuous variable
  • Ordinal measurement scale
  • Populations differ only in location

  
• Kruskal-Wallis Test


  1. Ho: Medians are the same

  2. Ha: Medians are not all equal

  3. Compute test statistic H

  4. Decision rule: Reject Ho if p-value ≤ α

  
• Growth rates of rats on 3 diets

  • Diet I
  • Diet II
  • Diet III
• Hypothesis testing for Kruskal-Wallis Test

  1. Ho: M1 = M2 = M3

  2. Ha: Not all medians are the same

  3. α = 0.10

  4. Test statistic H=5.18 with p-value=0.075

  5. Decision: Reject Ho

  6. Conclusion: Median growth rates significantly differ across the three diets

  
• Is there sufficient evidence to conclude that there is a differential effect among the treatments at 0.10 level of significance.
• We are interested whether the three diets have the same average growth rates. Let M1, M2, and M3, be the median growth rates of rats in the Diet I, Diet II, and Diet III, respectively. Performing Kruskal-Wallis using a software, we have the following table:
• Ho: M1= M2= M3
• Ha: Not all medians are the same.
• α = 0.10
• Based on the table above, the test statistic H=5.18 with p-value=0.075.

  
• Decision Rule: Reject Ho if p − value ≤ 0.10.
• We reject Ho because the p-value is less than 0.10.
• At 0.10 level of significance, there is sufficient evidence to say that median growth rates significantly differ across the three diets.
• The course pack provided to you in any form is intended only for your use in connection with the course that you are enrolled in. It is not for distribution or sale. Permission should be obtained from your instructor for any use other than for what it is intended.
• At the end of this unit, the student will

  • Perform correlation analysis
  • Build linear models using simple or multiple linear regression analysis
  • Perform diagnostic checking on the adopted linear model
  • Perform remedial measures if necessary
  • Interpret results of linear regression analysis