Statistical Power of Hypothesis Testing Using Parametric and Nonparametric Methods

Statistical Power of Hypothesis Testing Using Parametric and Nonparametric Methods

Chapter One

Aim and Objectives of the Study

The study/research is aimed at investigating the Statistical Power of Hypothesis Testing using Parametric and Nonparametric Method with a view to achieving the following objectives to compare.

The decision of parametric test with non-parametric test when normal or exponential distribution is used for simulation.
The power of parametric with non-parametric approaches in test of hypotheses using small and large sample size.
The consistency of the two approaches in hypothesis testing.

CHAPTER TWO

LITERATURE REVIEW

Introduction

This chapter present review of literature on parametric and nonparametric tests by different scholars in attempt to present the most powerful among the two classes of statistical test methodologies.

The Power of a Test

The power of a statistical test is the probability that it will correctly lead to the rejection of a false null hypothesis Greene, (2000). The statistical power is the ability of a test to detect an effect, if the effect actually exists Green, (2000). Cohen (1988) says, it is the probability that it will result in the conclusion that the phenomenon exists.

Siegel (1956), Runyon (1977) and Goon and Gupta (2003) stated that a particular test (A) may be more powerful than its counterpart test (B) when their sample sizes are equal but its counterpart (B) may be as powerful as its efficiency (A) if its sample size is increased.

Goon and Gupta (2003) stated that the power of a statistical hypothesis test depends on the following factors:

The power depends on the population standard deviation: The smaller the population standard deviation, the greater the power.
The power depends on the sample size used: The larger the sample, the greater the power.
The power depends on the level of significance of the test: The smaller the level of significance, the smaller the power.

Goon and Gupta (2003), consistency is one such criterion which is particularly useful in non-parametric methods. A test which is consistent for a subclass of alternatives is said to be specially sensitive to that type of alternatives and can be recommended for use in detecting differences of the type mentioned in the alternatives. They defined consistency as “The sequence of tests corresponding to {W_n} is consistent if for every

value of θ lying in (Θ – Θ₀), the power, P_θ (W_n), tends to 1 as n tends to ∞. The ideaB behind the concept of consistency can be viewed from considering the behaviour of a test for increasing n. Naturally one would like that the power should increase with the number of random variables taken into account. In cases X₁, X₂, …, X_n form a random sample, for instance, our test should have, for large sample sizes, a high probability of rejecting a false hypothesis that is a high power.

Goon and Gupta (2003) stated that another useful criterion is the concept of asymptotic relative efficiency (ARE). In point estimation, the efficiency of two unbiased estimators for a parameter is defined as the inverse ratio of their variances. In the case of power efficiency, Pitman (1948) defined it as:

Let T_n and T_n*, n = 1, 2, …, be two sequences of test statistics of the same null hypothesis H₀ at the same significance level α. Let the distributions be indexed by a real parameter γ, so that γ = 0 gives a distribution in H₀ and other γ’s correspond to distributions native hypothesis. Considering a sequence of alternatives γ_i, If, for the same power with respect to the same alternative γ_i, the test T_n requires n_i observations and the test T_n* requires

CHAPTER THREE

RESEARCH METHODOLOGY

Introduction

In this section, the different test statistics for the parametric and non parametric approaches were explained in detail. The data used for the research is from the test on an improved designed threshing machine for sorghum obtainable from Division of Agric college (DAC) Zaria. The speed rate for the machine blowing fan is at three levels after threshing the sorghum the percentage of mechanical grain damage (MGD) was estimated. Two parametric tests and their nonparametric tests equivalents; the independent t-test and one-way ANOVA their nonparametric equivalents are Wilcoxon Rank-Sum and Kruskal-Wallis test respectively.

The data is simulated fifty times using normal and exponential distribution. The equality sample number of the case n = 10, n=30 and n = 45 were used for the study. The comparison is done using different significant level = 0.05, 0.01 and 0.1. The simulated data is used for the comparison, the simulated data was generated using SPSS 20 and the various tests were done using SPSS 20. Also a bar chart is used to represent the power of the test.

CHAPTER FOUR

PRESENTATION OF RESULTS AND DISCUSION

Introduction

This chapter presents the result and discussion of the analysis conducted for the research study.

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATION

Introduction

In this chapter, we present the summary, conclusion and recommendations based on the results obtained in chapter four as an overview of the study.

Summary and Conclusion

We recall that the main objectives of this research work is to assess the parametric test strength with its nonparametric equivalents. The comparison of the class of test were subject to three cases which are the sample size, n < 30, n=30 and n > 30. From, the analysis performed for Two Independent Samples and for more than Two Independent Samples under the normal distribution it was observed that the small sample sizes as higher powers and under the exponential distribution the large sample sizes have higher powers.

As stated by Siegel (1956), Runyon (1977) and Goon and Gupta (2003) that a particular test (A) may be more powerful than its counterpart test (B) when their n‟s are equal, but its counterpart (B) may be as powerful as its efficiency (A) if its sample size is increased. This result could be largely influenced by outliers in the data set and as it is well known the parametric test make use of the mean which is highly affected by outliers compared to the non-parametric test that make use of the median, which is highly resistant to outliers Masoud and Rahim, (2010) likewise, the mean square error, MSE for the ANOVA F-test can be unsecured, Sergio and Joan (2001) stated that like variance, MSE has the disadvantage of heavily weighting outliers. They said it is because of the squaring of each term, which effectively weights large errors more heavily than small ones. This property, undesirable in many applications, has led researchers to use alternatives such as the mean absolute error, or those based on the median.

However, when the samples are continuous, but the experiment is not sure of the assumption of normality and equality of variance, the nonparametric is to be considered. Hence, it can be concluded from this research that the parametric and non-parametric test are reliable when their assumptions are met and their efficiency can be greatly determined by their sample size therefore, where the sample size is not small, any of the two approaches will perform well for the purpose of such comparison irrespective of the distribution.

Recommendation

In this research study we present the following recommendation when trying to decide on which test to apply in any study

We recommend that the assumption criteria must be met by fully exploring the various test to ascertain the criteria present and these assumptions are independency, normality, constant variance and linearity. If these conditions are met, then the parametric test is preferred to the nonparametric test.
Test for the presence of outliers should be done and decision on whether to delete the data point(s) or not and if it is actually an important data point(s) that have to be included then care should be taken on using the parametric test or better still test that make use of the median (nonparametric test) should be used.
If the data collected is in interval or ratio scale the parametric test should be applied except the underlying conditions are not met even after data transformation because it could lead to loss of in-depth analytical information but if the data is ordinal or nominal all temptations towards the parametric test should be abolished.
Also, when samples are continuous but the experiment is not sure whether the assumption of normality and homogeneity of variance are valid, the nonparametric is to be considered.
We recommend this research for further open criticism and improvement to researchers in the field of statistics and related fields.

Contribution to Knowledge

The old perception of the nonparametric being less powerful in all cases has been proven to be otherwise because this referral work has shown that nonparametric can be as powerful as parametric given sufficient sample size.

The work has also shown that as the sample size increases, the power also increases for exponential distribution regardless of the level of significance. While for the normal distribution, the level of significance influences the relationship between the sample size and the power of test.

Recommendation for Further Studies.

For further studies it is recommended that other parametric and nonparametric distributions should be compared using small and large samples size under different distributions and also for three or more independent groups.

REFERENCES

Abebe, A. (2009): Introduction to Design and Analysis of Experiment with the SAS System(Stat 7010 Lecture Notes).Department of Discrete and Statistical Sciences, Auburn University, Auburn.Auburn University Press.
Andrew, M. J, Nigel, R. and Silvana, R (2011): A Comparison of Parametric and Nonparametric, University of York, York. Ac.
Atman, D. G. (1991): Practical Statistics for Medical Research London; Chapman and Hall (monograph).
Atman, D. G. (1948): Parametric and Non-parametric Methods, ed Curt Hinrichs.
Bradley, E. (1996): Empirical Bayes Methods for combining likelihoods J. Amer statistics Association.
Breheny, P. (2012): Connection between Parametric and Nonparametric Theory, The Oxford handbook Applied Nonparametric and Semi-parametric Econometrics course. Hero Inc.
Bridge, P. D. and Sawilowsky, S. S.(1999): Increasing Physicians Awareness of the Impact of Statistics on Research Outcomes: Comparative Power of the t-test and Wilcoxon Rank-Sum test, Journal of Clinical Epidemiology.
Briggs, A. H, Wonderling, D. E, and Mooney, C. Z(1997): Pulling Cost-Effectiveness Analysis up by its boot steps, A Nonparametric Approach to Confidence Interval Estimation. Health Econ.
Cambell, G. K(2006): An Introduction to Nonparametric Statistics for Health Scientists. University of Albert Health Journal, Volume 3. Issue 1.20 reviews

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