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Minimum Sample Sizes for Likelihood Ratio Tests: a Simulation Study

Mst. Moushumi Pervin*
Department of Statistics, Jagannath University, Dhaka, Bangladesh

Md. Siddiqur Rahman
Department of Statistics, Jagannath University, Dhaka, Bangladesh

Shahanaj Parvin
Department of Statistics, Jagannath University, Dhaka, Bangladesh

Salma Akter
Department of Statistics, Jagannath University, Dhaka, Bangladesh

Abstract/ Executive Summary:

The necessity of sample size determination for various research purposes arises due to cost, time or availability of data, and also to offer sufficient statistical power. In this paper, we have tried to determine the minimum sample sizes for which −2 log λ (X)converges to chi-square distribution with 1 degree of freedom for several discrete and continuous distributions. We have applied a simulation study on 100000 datasets each of size to calculate −2 log λ (X) for the considered distributions i.e. Bernoulli, Poisson, Geometric, Exponential and Normal. In order to determine the minimum sample sizes, we have drawn Quantile-Quantile and Probability-Probability plots of−2 log λ (X) values against the theoretical quantiles and histogram of the generated datasets are plotted with the density curve of chi-square distribution, and finally, sample sizes are determined by careful scrutinization of these three figures. From our study, we have found that sample sizes should be at least 200 for any value of > 0(probability) in Bernoulli distribution so that −2 log λ (X)∼1 2. For Poisson and Geometric distributions, the minimum sample sizes are approximately 40 and 80 respectively. Exponential and Normal requires roughly about only 10 samples so that−2 log λ (X) follows chi-square distribution with 1 degree of freedom.

Keywords: Sample size determination, Chi-square distribution, Quantile-Quantile plot, Probability-Probability, Histogram, Bernoulli distribution

Location: In Bangladesh

Start Date:

End Date:

Program Area: Resource Development and Management

Non-commodity: Value addition

Objectives:

In an estimation study, the main objective of determining exact sample size is to estimate the parameter with a desired level of precision and confidence. To do a research more precisely a researcher should select a large sample size because as the minimal sample size increases the margin of error decreases that is precision increases. Similarly, level of confidence increases with an increased sample size (Binuet al., 2014).

Methodology:

There is a vast literature review for determining sample size in various social, biological, medical etc. fields. Borderer formula was used to calculate sample size for estimating sensitivity and specificity but not for testing (Kim, 2009). Simel et al. used sample size based on desired likelihood ratios (LR) confidence interval but not calculate sample size for a wide range of marginal errors around LR (Simelet al., 1991). A review of sample size formula had also been provided for various diagnostic test but the practical tables for calculating sample sizes were not provided (Obuchowski, 1998). Several methods had been developed for sample size calculation in diagnostic medicine because their methods were not used frequently due to the lack of available software and it was complex to use for the clinician (Fosgate,2009; Li and Fine, 2004; Steinberg et al., 2009; Kumar and Indrayan, 2002). A model-based approach was used to predict the sample size for classifying microarray data (Dobbin et al., 2008). Normal approximations are sometimes used for calculating sample size (Lachin, 1981). Sample size calculations are often based even for data that are not Gaussian and which are analyzed using generalized linear models (GLMs) (Wong et al., 2010; Watson-Jones et al., 2008; Kessler et al., 2009; Holland et al., 2005; Kaul et al., 2004). Methods have been proposed for sample size determination of logistic (Whittemore, 1981; Hsieh et al., 1998; Vaeth and Skovlund, 2004; Alam et al., 2010) or Poisson (Signorini, 1991) models or both (Shieh, 2001) or for the negative binomial (Zhu and Lakkis, 2014) and for generalized linear models (Self and Mauritsen, 1988; Shieh, 2000). The maximum likelihood method is a powerful technique and it also provides a powerful framework for addressing the aspects of a phylogenetic analysis. Maximum likelihood appears to be both efficient and robust in terms of the estimation of relationship (Kuhner and Felsenstein, 1994; Tateno et al., 1994; Huelsenbeck, 1995a, 1995b). However, likelihood provides a natural way of testing hypothesis and this hypothesis is done through the likelihood ratio test (Edwards, 1972).

Source of Information: Jn U J Sci., Vol 07, No. I, Dec. 2020, pp. 63?75

Article Link (Online Journal):

Funding Source:

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Total Budget (Tk.) :

Achievement/Findings:

In this paper, the main concern was to determine the sample size so that −2logλ(X) converges to1 2distribution. In real fields, we are not sure about the exact or approximate sample size which raises the necessity of sample size determination. Sample size should be as small as possible because of cost and feasibility. In this study, 100000 datasets each of size are generated from several important distributions such as Bernoulli, Poisson, Geometric, Exponential and Normal with unknown variance, and Y = −2logλ(X) are calculated for each of 100000 datasets. We drew quantile-quantile (Q-Q) plot, probability-probability (PP) plot of Y = −2logλ(X), and then we plot histograms of generated datasets with the density curve of chi-square distribution for the considered distribution to find the minimum sample sizes. Sample sizes should be at least 200 for any value of > 0(probability) in Bernoulli distribution so that −2 log λ (X)∼1 2. For Poisson distribution, minimum sample size should be approximately 40 and for geometric distribution, it should be roughly 80. But we can see from our analysis that we need only 10 size of samples for exponential and normal distributions with unknown variance.

Knowledge Management: Journal