Estimation and Nonparametric Tests

Nonparametric methods are important in many statistical applications because they allow us to move beyond traditional approaches that require the specification of valid statistical models. However, establishing the validity of such a model is a complex undertaking. 

Nonparametric methods circumvent this problem by transforming the data into ranks and conditioning on certain quantities derived from the observed distribution of these ranks. The statistics constructed in this way are independent of the distribution of the raw data, which allows for the development of statistical inference procedures that are independent of the underlying model of the data. Furthermore, the loss of statistical efficiency is minimal.
 
This course provides a fairly comprehensive overview of nonparametric methods. It builds upon an introductory course on parametric inferential methods by adapting and expanding the theory behind several advanced concepts, such as conditional tests, the comparative power of tests (measures of effectiveness), and the concept of “effect size.” It emphasizes the practical application of these methodologies by providing an overview of the main R commands and their uses.

To help students understand the limitations of parametric statistical methods; to choose an appropriate nonparametric approach to a given problem based on underlying statistical principles; to implement a solution to their problem using the R software; and to report the conclusions of their analyses and their implications to end users.

An M-level course in inferential statistics

An L-level course in probability
 

 
Recommended prerequisites: A Level L course in descriptive statistics

Comprehensive Continuous Monitoring

1. Definitions of Nonparametric Statistics
   
   2. Two tips for removing dependence on an unknown parameter: conditioning (with applications to contingency tables, chi-square tests, and Fisher-Yates tests) and rank transformation.
   
   3. The Wilcoxon-Mann-Whitney test for two samples: Assumptions and test statistic; exact and asymptotic behavior under H₀, the case of tied data, and test robustness.
    
   4. Variants and extensions of the Wilcoxon-Mann-Whitney test: Point and interval estimates of treatment effects; the case of paired samples; power of the Wilcoxon-Mann-Whitney test i) when the Student’s paradigm holds and ii) when it does not. Concepts of “effect size,” calculation of sample sizes to achieve a target power.
   
   5. Other tests for two samples: Kolmogorov-Smirnov test, Ansari-Bradley test
   
   6. The case of K > 2 samples: Kruskal-Wallis test, Friedman-Tukey test. The problem of multiple comparisons. P-value adjustment: Holm’s method, FDR (false discovery rate) method
   
   7. Independence, correlation, and regression: Pearson’s, Spearman’s, and Kruskal’s correlation coefficients; independence test. Application to regression
   
   Implementation of the main nonparametric tests covered in the course using the R software.

Hours:
Lectures: 15 hours
Tutorials: 15 hours
Lab sessions:
Fieldwork: