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ISSN: 2155-6180
Journal of Biometrics & Biostatistics
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Size and Power Properties of Some Test Statistics for Testing the Population Correlation Coefficient ρ

Shipra Banik1 and Golam Kibria BM2*

1Department of Physical Sciences Independent University, Bangladesh Dhaka 1229, Bangladesh

2Department of Mathematics and Statistics Florida, International University Miami, FL 33199, USA

*Corresponding Author:
Golam Kibria BM
Department of Mathematics and Statistics Florida
International University Miami, FL 33199, USA
Tel: +1 305-348-2000
E-mail: [email protected]

Received date: May 31, 2017; Accepted date: June 16, 2017; Published date: June 30, 2017

Citation: Banik S, Golam Kibria BM (2017) Size and Power Properties of Some Test Statistics for Testing the Population Correlation Coefficient ρ. J Biom Biostat 8: 353. doi: 10.4172/2155-6180.1000353

Copyright: © 2017 Banik S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Correlation measures the strength of association between two variables, which plays an important role in various fields, such as Health Science, Economics, Finance, Engineering, Environmental science among others. Several tests for testing the population correlation coefficient are proposed in a literature by various researchers at different time points. This paper evaluates the performance of some of the prominent test statistics for testing the population correlation coefficient based on empirical size and power of the tests. Some bivariate distributions, such as normal, lognormal, gamma and chi-square are considered to compare the performance of the test statistics. We believe that the findings of this paper will make an important contribution to select some good test statistics to find the relationship between two variables.

Keywords

Bivariate distribution; Bootstrapping; Correlation coefficients; Hypothesis tests; Monte Carlo simulation; Power; Size

Introduction

One of the most useful statistical tools for quantifying the relationship between two continuous variables is the coefficient of correlation that developed by Pearson [1] from a related idea introduced by Galton [2]. In statistics, Pearson’s correlation coefficient is used to find the linear relationship between two quantitative variables (say) X and Y. It gives a value between -1 and +1 inclusive, where -1 indicates a perfect negative correlation, 0 is no correlation and +1 indicates a perfect positive correlation between X and Y. Since the population correlation coefficient, ρ is usually unknown, it is necessary to estimate it by estimator, r from the observed data or sample information. Even the sample correlation coefficient (r) is a biased estimator of population correlation coefficient ρ, the biasness disappears with the increase of sample size. When there is a question of estimation, its estimation accuracy and thus the validity through the hypothesis testing is essential. Several researchers considered several confidence intervals for estimating the population correlation coefficient ρ [3]. However, a comparison of several test statistics for testing the population correlation coefficient is limited in literature. In this paper, we have made an attempt to consider several test statistics for testing the population correlation coefficient. Since, a theoretical comparison among the test procedures is not possible, a simulation study will be conducted to compare the performance of the test statistics based on empirical size and power of the test. We believe that the findings of this study will make an important contribution to literature to choose appropriate test statistics for testing the population correlation coefficient for practitioners.

The paper is organized as follows: One proposed and some existing methods for testing the population correlation coefficient are described in section 2. A Monte Carlo simulation study along with results is discussed in section 3. Finally, some concluding remarks are given in section 4.

Methods for Testing the Population Correlation Coefficient

Suppose we are interested to find the linear relationship between two variables X and Y. Then the population correlation coefficient between two variables X and Y is denoted by ρ and is defined by

equation

The corresponding sample correlation coefficient is defined by

equation

It can be shown that −1 ≤ ρ ≤ 1. A value of 1 implies that a linear equation describes the relationship between X and Y perfectly, with all data points lying on a line for which Y increases as X increases. A value of −1 implies that all data points lie on a line for which Y decreases as X increases. A value of 0 implies that there is no linear association between X and Y. Several methods for testing for population correlation coefficient, H0: ρ=0 vs. H1: ρ ≠ 0 are given as follows.

The classical test statistic

Suppose, the sample correlation coefficient, r is a point estimator of ρ. The distribution of r when ρ is zero was for the first time studied by a student [4]. Thus, a common test is that of whether or not a linear relationship exists between two variables X and Y. The test statistic is defined as follows:

equation (1)

where n is the sample size and (n-2) is the degrees of freedom(df). Thus the critical value for this test statistic can be obtained from t-distribution with (n-2) degrees of freedom.

Fisher’s large sample test statistic

Since the sampling distribution of Pearson's r is not normally distributed, Pearson's r is converted to Fisher's z and the test statistic for testing H0: ρ=0 vs. H1: ρ ≠ 0 is computed using Fisher's [5] transformation and is given as follows:

equation (2)

Where equation and z0 is the value of z under the null hypothesis where equation

The distribution of tz has a standard normal distribution.

Gorsuch and Lehmann test statistics

To improve the performances of the classical statistic, Gorsuch and Lehmann [6] modified the classical statistic and the Fisher statistic and proposed the following four statistics for testing H0: ρ=0 vs. H1: ρ ≠ 0 based on different standard errors of r:

Modified classical statistics:

equation (3)

equation (4)

Where equation the critical value of tGL1 is assumed to be 2 (details see Gorsuch and Lehmann [6]) and the distribution of tGL2 follows t distribution with (n-1) df.

Modified Fisher statistics:

equation (5)

equation (6)

where the critical value of tGL3 is assumed to be 2 (details see Gorsuch and Lehmann [6]) and tGL4 has a t-distribution with (n-1) degrees of freedom.

Proposed test statistic

We know that equation , where b1 is estimator of β1 for the model yt01xt+et, β0 is the constant, β1 is the regression coefficient of y on x and et ~ N (0, σ2). The test statistic for testing H0: ρ=0 vs. H1: ρ ≠ 0 is given by

equation

The distribution of tSK has t-distribution with (n-2) df.

Parametric bootstrap test statistic

Let equation where ith random samples are denoted by x(i)and y(i) for i =1,2, …, B and B is the number of bootstrap samples [7]. The test statistic for testing H0: ρ=0 vs. H1: ρ ≠ 0 is given by

equation (8)

Where critical values of the above statistic is the equation which is the (α/2)th sample quintiles of tpboot.

Parametric bootstrap Fisher z test statistic

The test statistic for testing H0: ρ=0 vs. H1: ρ ≠ 0is computed using Fisher’s z [8] transformation and is given as follows:

equation (9)

Where critical values of the above statistic is the equation which is the (α/2)th sample quintiles of tFboot

Parametric bootstrap version of proposed test statistic

The test statistic for testing H0: ρ=0 vs. H1: ρ ≠ 0is computed follows:

equation (10)

Where critical values of the above statistic is the equation , which is the (α/2)th sample quintiles of tSKboot

Bootstrap bias corrected acceleration test statistic

This method is introduced by Efron and Tibshirani [9]. The test statistic for testing H0: ρ=0 vs. H1: ρ ≠ 0is the t-statistic defined in eqn. (1) and the critical value is calculated by equation , where r() is the (100α)% percentile of the distribution of equation is the standard normal cumulative distribution function, bias correction equation is the inverse function of cumulative distribution function of the Z distribution, acceleration factor equation , r is the correlation between x and y and ri is the correlation between x and y of (n-1) observations without the ith observation.

Simulation Study

The main goal of this paper is to evaluate the performance of test statistics for testing population correlation coefficient based on size and power properties, discussed in section 2. Since a theoretical comparison among the tests is not possible, a simulation study has been conducted in this section.

Simulation design

MATLAB (2015) programming language was used to run simulations and to make necessary tables. The most common level of significance α=0.05 is considered and assumed random sample sizes n=10, 30, 50, 80 and 100 and ρ1=-0.5, -0.9, 0.3, 0.8 and 0.99. We have considered 2500 replications for our simulation experiments and 1500 bootstrap samples for each selected random samples sizes. Random samples produced from the following population distributions:

(a) Bivariate normal with μ1=15, μ2=20 and σ1=10, σ2=4

(b) Bivariate log normal with μ1=15, μ2=20 and σ1=10, σ2=4

(c) Bivariate gamma with shape parameters 1 and mean parameters 2

(d) Bivariate chi-square with dfs 1 and 3

Results Discussion

Table 1 presents estimated sizes of the selected test statistics for selected values of n and ρ, when random samples are generated from the bivariate normal distribution. For a visual expression, simulation results are presented graphically in Figure 1.

Tests n=10 n=30 n=50 n=80 n=100
t 0.0476 0.0520 0.0480 0.0588 0.0560
Fisher 0.0488 0.0548 0.0484 0.0604 0.0560
GL1 0.0920 0.0608 0.0528 0.0592 0.0552
GL2 0.0592 0.0552 0.0512 0.0608 0.0576
GL3 0.0440 0.0512 0.0452 0.0560 0.0520
GL4 0.0248 0.0476 0.0436 0.0564 0.0532
SK 0.0460 0.0360 0.0356 0.0496 0.0468
tBoot 0.0300 0.0332 0.0336 0.0400 0.0392
FBoot 0.0424 0.0524 0.0552 0.0560 0.0500
SKBoot 0.1328 0.1020 0.0860 0.0624 0.0568
BCABoot 0.0540 0.0558 0.0520 0.0480 0.0412

Table 1: Estimated sizes at the 5% level of significance for bivariate normal data.

biometrics-biostatistics-bivariate-normal-distribution

Figure 1: Estimated sizes for various values of n in case of the bivariate normal distribution.

We can see from Figure 1 and Table 1 is that for all sample sizes, all proposed test statistics except GL1 and SK Boot have empirical sizes close to the 5% nominal level.

We have presented estimated sizes when data are generated from the bivariate lognormal distribution in Table 2 and depicted results for visual inspection in Figure 2. From Table 2 and Figure 2, we observe that for moderate to large sample sizes, tBoot, FBoot, SKBoot and BCABoot have sizes close to the nominal level, while rest of the tests achieve nominal level only when sample sizes are large.

Tests n=10 n=30 n=50 n=80 n=100
t 0.0896 0.0608 0.0496 0.0436 0.0312
Fisher 0.0904 0.0608 0.0504 0.0436 0.0312
GL1 0.1028 0.0620 0.0508 0.0436 0.0312
GL2 0.0960 0.0608 0.0508 0.0436 0.0316
GL3 0.0876 0.0604 0.0488 0.0436 0.0312
GL4 0.0792 0.0600 0.0488 0.0436 0.0312
SK 0.0896 0.0608 0.0496 0.0436 0.0312
tBoot 0.0464 0.0312 0,0252 0.0260 0.0204
FBoot 0.0228 0.0356 0.0240 0.0320 0.0200
SKBoot 0.0812 0.0436 0.0296 0.0340 0.0208
BCABoot 0.0576 0.0336 0.0360 0.0368 0.0340

Table 2: Estimated sizes at the 5% level of significance for bivariate lognormal data.

biometrics-biostatistics-bivariate-lognormal-distribution

Figure 2: Estimated sizes for various values of n in case of the bivariate lognormal distribution.

In Tables 3 and 4, we have reported estimated sizes when data generated from the bivariate gamma and bivariate chi-square distribution respectively. We find that all tests have correct sizes expect GL1 and SKboot. GL1 test has small sizes than the nominal level and SKboot has higher sizes than the nominal level (Figures 3 and 4).

Tests n=10 n=30 n=50 n=80 n=100
t 0.0520 0.0500 0.0536 0.0468 0.0420
Fisher 0.0524 0.0504 0.0540 0.0472 0.0420
GL1 0.0924 0.0588 0.0564 0.0472 0.0412
GL2 0.0652 0.0532 0.0560 0.0484 0.0440
GL3 0.0500 0.0484 0.0492 0.0440 0.0396
GL4 0.0288 0.0432 0.0480 0.0452 0.0408
SK 0.0520 0.0500 0.0536 0.0468 0.0420
tBoot 0.0152 0.0276 0.0184 0.0252 0.0228
FBoot 0.0348 0.0368 0.0508 0.0436 0.0348
SKBoot 0.0852 0.1804 0.2292 0.1048 0.1552
BCABoot 0.0412 0.0376 0.0436 0.0384 0.0380

Table 3: Estimated sizes at the 5% level of significance for the bivariate gamma data.

Tests n=10 n=30 n=50 n=80 n=100
t 0.0456 0.0452 0.0560 0.0544 0.0468
Fisher 0.0472 0.0456 0.0568 0.0548 0.0468
GL1 0.0912 0.0504 0.0596 0.0548 0.0468
GL2 0.0608 0.0480 0.0592 0.0568 0.0476
GL3 0.0420 0.0428 0.0520 0.0500 0.0448
GL4 0.0220 0.0388 0.0516 0.0516 0.0460
SK 0.0456 0.0452 0.0560 0.0544 0.0468
tBoot 0.0064 0.0008 0.0228 0.0312 0.0224
FBoot 0.0348 0.0432 0.0400 0.0408 0.0424
SKBoot 0.2108 0.1828 0.2000 0.1744 0.1700
BCABoot 0.0364 0.0344 0.0500 0.0488 0.0392 BCABoot

Table 4: Estimated sizes at the 5% level of significance for the bivariate chi-square data.

biometrics-biostatistics-bivariate-gamma-distribution

Figure 3: Estimated sizes for various values of n in case of the bivariate gamma distribution.

biometrics-biostatistics-bivariate-chi-distribution

Figure 4: Estimated sizes for various values of n in case of the bivariate chi square distribution.

In Table 5, we have presented the estimated powers when data are generated from the bivariate normal distribution for various sample sizes and various values of ρ. We observed that for small sample size n=10 (Figure 5), Fisher, G0L3, GL4, FBoot and SKBoot have good powers as compare to other test statistics. For sample sizes 50 or above, (Figure 6 for n=50) we found that all test statistics have good powers except for ρ=0.3. We noted that for weak positive correlation, SKboot has highest power as compare to rest of the test statistics.

  t Fisher GL1 GL2 GL3 GL4 SK tBoot FBoot SKBoot BCABoot
pho=-0.5           n=10          
0.2360 0.3300 0.3680 0.2860 0.3140 0.2200 0.2360 0.1440 0.2900 0.5800 0.2760
pho=-0.9 0.6000 0.9740 0.7780 0.6820 0.9740 0.9640 0.6000 0.4740 0.9840 0.9700 0.7400
pho=0.3 0.0940 0.1040 0.1680 0.1080 0.1020 0.0660 0.0940 0.0420 0.0940 0.3280 0.1080
pho=0.8 0.4680 0.8120 0.6260 0.5300 0.8000 0.7220 0.4680 0.3500 0.8460 0.8760 0.5600
pho=0.99 0.6620 1.0000 0.8520 0.7620 1.0000 1.0000 0.6620 0.4860 1.0000 0.9420 0.6260
pho=-0.5           n=30          
0.7720 0.8420 0.7940 0.7760 0.8320 0.8260 0.7720 0.6660 0.8260 0.8800 0.7320
pho=-0.9 0.9080 0.9080 0.9620 0.9200 0.9080 0.9580 0.9080 0.7560 0.9560 0.9580 0.9080
pho=0.3 0.2480 0.2800 0.2720 0.2520 0.2640 0.2480 0.2480 0.1740 0.3180 0.5380 0.2140
pho=0.8 0.8440 0.9300 0.8600 0.8500 0.9180 0.9120 0.8440 0.6820 0.8940 0.9300 0.8400
pho=0.99 0.9012 0.9745 0.9340 0.9310 0.9235 0.9205 0.9012 0.7930 0.9012 0.9745 0.8923
pho=-0.5           n=50          
0.9500 0.9740 0.9520 0.9520 0.9660 0.9660 0.9500 0.9240 0.9440 0.9740 0.9340
pho=-0.9 0.9920 0.9980 0.9920 0.9920 0.9980 0.9980 0.9920 0.9800 0.9960 0.9980 0.9920
pho=0.3 0.4420 0.4780 0.4500 0.4460 0.4600 . 0.4560 0.4420 0.3720 0.5220 0.6960 0.4260
pho=0.8 0.9660 0.9900 0.9740 0.9720 0.9900 0.9900 0.9660 0.9420 0.9920 0.9980 0.9540
pho=0.99 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
pho=-0.5           n=80          
0.9940 0.9980 0.9960 0.9960 0.9980 0.9980 0.9940 0.9860 0.9980 0.9980 0.9940
pho=-0.9 0.9960 1.0000 0.9960 0.9960 1.0000 1.0000 0.9960 0.9940 1.0000 1.0000 0.9940
pho=0.3 0.5480 0.5860 0.5540 0.5560 0.5740 0.5740 0.5480 0.4420 0.5920 0.7780 0.4780
pho=0.8 0.9900 0.9980 0.9900 0.9920 0.9960 0.9960 0.9960 0.9900 0.9780 1.0000 0.9840
pho=0.99 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
pho=-0.5           n=100          
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
pho=-0.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
pho=0.3 0.7200 0.7620 0.7200 0.7260 0.7440 0.7540 0.7200 0.5820 0.7020 0.8480 0.6860
pho=0.8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
pho=0.99 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Table 5: Powers at the 5% level of significance for bivariate normal data.

biometrics-biostatistics-bivariate-normal-distribution

Figure 5: Estimated powers for n=10 and various values of ρ for bivariate normal distribution.

biometrics-biostatistics-bivariate-normal-distribution

Figure 6: Estimated powers for n=50 and various values of ρ in case of bivariate normal distribution.

Figure 8 presents estimated powers when data are generated from the bivariate lognormal distribution for n=80. We observed that all tests have good powers except bootstrap versions of t, Fisher, SK and BCA for ρ=0.3.

biometrics-biostatistics-bivariate-lognormal-distribution

Figure 7: Estimated powers for n=10 and various values of ρ in case of bivariate lognormal distribution.

biometrics-biostatistics-bivariate-lognormal-distribution

Figure 8: Estimated powers for n=80 and various values of ρ in case of bivariate lognormal distribution.

we have tabulated estimated power of various test statistics when data are generated from bivariate gamma and bivariate chi-square distributions (see Figure 9 (n =10) and Figure 10 (n=30) for better understanding). It is observed that power properties of the selected tests are similar when data generated from bivariate normal distribution or lognormal.

biometrics-biostatistics-bivariate-gamma-distribution

Figure 9: Estimated powers for n=10 and various values of ρ for bivariate gamma distribution.

biometrics-biostatistics-bivariate-chi-square-distribution

Figure 10: Estimated powers for n=30 and various values of ρ in case of bivariate chi-square distribution.

In Figures 11 and 12, we have plotted estimated powers for various values of n for ρ=-0.5 and ρ=0.8 to check effects of n on the selected tests. It is observed from these graphs that as n increases powers are also increases for all selected tests. We noted that for small sample sizes, Fisher, GL2, Fboot and SKboot tests are more powerful than the other considered tests. It is also noted that our proposed bootstrap version SKboot is more powerful than the other considered tests.

biometrics-biostatistics-bivariate-normal-distribution

Figure 11: Estimated powers for various values of n and ρ = -0.5 in case of bivariate normal distribution.

biometrics-biostatistics-bivariate-normal-distribution

Figure 12: Estimated powers for various values of n and ρ = 0.8 in case of bivariate normal distribution.

In Figures 13 and 14, we have plotted estimated powers for selected values of n and two selected values of ρ. Here also we observed same patterns like Figures 11 and 12 as sample size increases, estimated powers also increases. As compare to the Figure 11, we noted very low powers n=10 when data generated from the bivariate lognormal distribution. We noted that tboot and BCAboot tests have very low power compared to the other tests.

biometrics-biostatistics-bivariate-lognormal-distribution

Figure 13: Estimated powers for various values of n and ρ = -0.5 and bivariate lognormal distribution.

biometrics-biostatistics-bivariate-lognormal-distribution

Figure 14: Estimated powers for various values of n and ρ = 0.8 and bivariate lognormal distribution.

Figures 15-18 present estimated powers for various values of n and two selected values of ρ when data are generated from the bivariate gamma distribution and bivariate chi-square distribution respectively. Similar interpretation can be drawn from these figures, as we observed when data are generated from the bivariate normal distribution.

biometrics-biostatistics-bivariate-gamma-distribution

Figure 15: Estimated powers for various values of n and ρ = -0.5 in case of bivariate gamma distribution.

biometrics-biostatistics-bivariate-gamma-distribution

Figure 16: Estimated powers for various values of n and ρ = 0.8 in case of bivariate gamma distribution.

biometrics-biostatistics-bivariate-chi-square-distribution

Figure 17: Estimated powers for various values of n and ρ = -0.5 in case of bivariate chi-square distribution.

biometrics-biostatistics-bivariate-chi-square-distribution

Figure 18: Estimated powers for various values of n and ρ = 0.8 in case of bivariate chi-square distribution.

Conclusion

In this paper, we study the performance of several methods for testing the population correlation coefficient by means of a simulation study. Data were generated randomly from several bivariate distributions, namely, bivariate normal, bivariate lognormal, bivariate gamma and bivariate chi-square with a range of sample sizes. Overall, we found that test statistics, t, Fisher, GL2, GL3, GL4, SK and FBoot have sizes close to the 5% nominal level. Fisher, GL3, GL4, FBoot and SKBoot have good powers as compare to other test statistics. It appears from the simulation study is that the test statistics, Fisher, GL3, GL4, FBoot and SKBoot can be recommended for practitioners because these test statistics have good sizes and powers compare to the rest of selected test statistics.

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