**Comparison of Some Parametric and Non-Parametric Statistical Methods**

**Chapter One**

**Purpose of the**** ****Study**

- To find out if there exists any relationship between indexes of human development reports using both parametric and nonparametric
- To equally test for independence using both multivariate parametric method and nonparametric method to see if it can produce the same
- To find out similarities/differences between the two statistical methods based on the result of the analysis used in this
- To analyze statistically, multivariate data by using parametric and nonparametric tests for independence

**CHAPTER TWO**

**LITERATURE REVIEW**

**INTRODUCTION**

Multivariate analysis deals with the observation of more than one variable where there is some inherent interdependence between variables. There is a wide variety of multivariate techniques. The choice of the most appropriate method depends on the type of data, the problem, and the sort of objectives that are envisaged for analysis. The review in this chapter extends from the existing literature by providing both multivariate parametric and nonparametric tests for independence.

**Multinormality**** ****Theory**

Multivariate analysis lays too much interest on the assumption that all random vectors come from multivariate normal distribution. By definition, the probability density function of a normal variable with mean m and variance s^{2} is given by

f (x) = (2ps^{2}) exp – ½ (x-m)(s^{2})^{-1}(x-m)

Then the extension to the p-variate is

*f** *(*x*) = (2*p** *) 2 å

– 1

2 exp-

1 (*x** *– *m** *)1 -1 (*x** *– *m** *)

The reasons for its (normal distribution) preference in the multivariate case are among others. (Hollander M and Wolfe DA, 1973)

- The multivariate normal distribution is entirely defined by its first and second
- The multivariate distribution is an easy generalization of its univariate counterpart, and the multivariate analysis runs almost parallel to the corresponding analysis based on univariate
- Linear functions of a multinormal vector are themselves univariate normal.
- In the case of normal variables, zero correlation implies independence and pairwise independence implies total independence.
- Equiprobability contours of the multivariate distribution are simple ellipses, which by a suitable change of coordinates can be made into a circle.
- When the original data is not multinormal, one can often appeal to central limit theorems which prove that certain functions such as the sample mean are normal for large

**Parametric versus non-parametric statistics in**** ****the analysis of randomized trials with non-normally distributed**** ****data.**

The following ideas are the contributions and conclusions gotten from **Andrew J Vickers, **Wolfowitz, (1942) Siegel & Castellan, (1988) & Dr Matthew Ellis (2002). It has generally been argued that parametric statistics should not be applied to data with non-normal distributions. Empirical research has demonstrated that Mann-Whitney (Wilcoxon) generally has greater power than the *t*-test unless data are sampled from the normal. In the case of randomized trials, we are typically interested in how an endpoint, such as blood pressure or pain, changes following treatment. Such trials should be analyzed using ANCOVA, rather than a *t*-test. The objectives of this study were:

**CHAPTER THREE**

**MATERIALS USED FOR THE STUDY**

The data set used in this work, listed as Appendix A, consists of eight (8) indicator variables selected from 38 African Countries (HDR 2005).

These indicators are

- HDI Human Development Index
- EI Education Index
- GDP/Index Gross Domestic Product Index
- LEI Life Expectancy Index
- HPI-I Human Poverty Index values(%)
- PNUIWS Population not Using Improved Water Sources
- AIR Adult Illiteracy Rate
- PBNS Probability at birth of not surviving to age 40(%).

In the indicators table, countries and areas are ranked in descending order by their human development index, value, or by their human poverty index. The indicators present both both similarities or differences. The group of human development Index includes indicators such as LEI, EI, GDP, HDI and the second group includes HPI, PNUIWS, AIR, PBNS.

The human development index (HDI): This measures ‘the average achievements in a country in three basic dimensions of human development,once the dimension indices have been calculated, then the HDI is a simple average of the three dimensions indices. HDI = ⅓ (Life expectancy Index LEI + Education Index El’ + Gross Domestic Product Index ‘GDP’)

**CHAPTER FOUR**

** ANALYSIS**

**Hoteling’s-T**^{2}**-Test**

**Hoteling T**^{2}** test of independence**

**CHAPTER FIVE**

**SUMMARY AND**** ****CONCLUSION**

**Summary**

In this research work multivariate parametric and non parametric tests for independence were performed using multivariate data collected from World Bank on Human Development Report. The null hypothesis (Ho: variables are independent) is rejected for both the selected parametric (Hotelling’s T^{2}, Wilks Lambda, Canonical correlation analysis) and non parametric (Friedman, Kendall W. Wilcoxon and Cochran Q) tests.

**Conclusion**

Multivariate parametric and non parametric are two statistical methods of inference. Multivariate parametric methods depend upon the assumption of a specific distributional form for example an approximate multivariate normal distribution. And data for these method will be interval or ratio scales. On the other hand the multivariate non parametric method is referred to as distribution free method using ordinal or normal data and even interval and ratio data when the distributional assumption is unspecified. The analysis of the data from human development report using both methods yields the results. From the parametric test of independence, Hotelling’s T^{2},

Wilks Lambda and Canonical correlation analysis the null hypothesis that is variable are independent and is rejected in favour of the alternative. From the nonparametric test, Wilcoxon signed-ranks, Friedman, Kendall and Cochran, the null hypothesis is also rejected at the same levels of significant so both methods yield the same result. But a multivariate parametric method for example, Canonical correlation analysis have some times an advantage over nonparametric method. That is, if the null hypothesis is rejected, it shows which of the components hypothesis led to the rejection of the null hypothesis. As well as knowing that the null hypothesis should be rejected, one could enquire which specific linear combination led to its rejection. This work has equally shown us the similarities and differences between parametric and non parametric method in analyzing specific data.

**Recommendation**

The essence of this work is to expose the relationship between component of human development reports. The result shows that the education index and poverty index are closely related. Hence, assistance need to be given to Countries with high poverty indices in terms of their educational institutions.

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