Statistics for Social Studies. Pearson Correlation Test Coursework

Introduction

Pearson correlation test aims to find an association between variables of quantitative data or accept or reject the null hypothesis. Small samples can produce statistical bias or difference between the sample correlation coefficient (r) and the population correlation coefficient (P). Researchers recommend using the Fisher correction formula to minimize this difference.

Pearson application

The aim of scientific research is often to clear up (explain) or foretell (predict) a phenomenon, in other words, to find an association or a correlation between variables. The question do variables change together draws out the aim of explaining, while predicting a factor (dependent) knowing the other factor (independent) considers the second question. Answering the first question is by a correlation test while responding to the second question is by regression analysis (Miller, 1994). The aim of this essay is to spotlight the data that the Pearson formula can be used for, and how to perform hypothesis testing using the statistical application of Pearson (r) (Pearson correlation coefficient)?

Pearson correlation test and coefficient

Pearson correlation test is a parametric test used when there is a need to measure the strength of the association between pairs of variables (quantitative data) without regard to which variable is dependent or independent. In addition, if there is a need to determine the relationship if any; between the variables is a straight line. The variables should be measurement variables (as drug dose and pulse rate) that are kept together in pairs by a nominal variable (as the patient’s name). A precondition to applying the test is that residuals (distances of the data points from the regression line) are normally distributed with constant variance (McDonald, 2008).

The use of the Pearson correlation test is either to test a hypothesis about the cause-effect relationship or to determine whether the two variables are correlated without assuming a cause-effect relationship (McDonald, 2008).

A statistical hypothesis can be in the form of a question to know the outcome of a change. As an example is there a change in the disability associated with a certain disorder, is a certain treatment more effective than another or whether certain abnormal data come from a normal population or not. In this situation, one is testing the null hypothesis. The null hypothesis (H₀) means the expected frequency (proportion of times when it occurs) of an event is not different from that expected (from theoretical reasoning or previous evidence). Alternatively, there can be a research hypothesis different from the null hypothesis as the researcher working hypothesis (researcher’s prediction or expectation). This is called the research hypothesis (H_{1, 2, 3} according to the researcher’s hypotheses in the work) (Sommer, 2006). Hypotheses, by their nature, cannot be proved. They can only be accepted or rejected based on the available statistical evidence (Pipkin, 1986).

Pearson correlation coefficient (r) quantifies the strength of association between the values of two variables. If (r) equals 1; it means that as one variable increases, the other increases exactly linearly (positive correlation). With (r) equals -1; it means that as one variable increases, the other decreases exactly linearly (negative correlation), and a coefficient value of zero tells no relationship (Lane and others, n.d.). In this way, correlation coefficient (r) answers the research alternative hypothesis (H_{1, 2,…}).

Knowing the (r) value, one can calculate the P-value (probability of being wrong in inferring a true association of variables). That is the probability of falsely rejecting the null hypothesis, or committing a Type I error) is calculated. As P-value gets smaller, the probability of variables correlation is greater (the cut-off point is 0.05). This displays the value of the correlation coefficient is expressing the null hypothesis (H₀) (Pipkin, 1986).

The correlation coefficient is termed (r) when drawn from a sample of the population. It is termed (P) when drawn from the population. In other words, its value differs with the sample size and becomes near normal as the sample size (n) increases. Statistical bias because of sample size can reach 0.3 to 0.4 in some cases. Fisher (1915) was the first to suggest a correction formula for (r) which was P = r [1+ (1-r²) / 2n), this was later modified for better correction results to P= [1+(1-r²)/2(n-3) (Zimmerman and other, 2003).

Conclusion

A Pearson correlation test is needed to test a hypothesis about the cause-effect relationship or to determine whether the two variables are correlated without assuming a cause-effect relationship. Variables need to be measurement variables with a nominal variable to keep them in pairs. Data have to be distributed normally with a constant variance. The correlation coefficient (r) answers both the null hypothesis and the research hypothesis. However, its value is affected by the sample size, therefore, using the Fisher correction formula aims to minimize statistical bias, especially in small samples.

References

Lane, D, Lu, J, Peres, C, and Zitek, E. et al (n.d.). Online Statistics: An Interactive Multimedia Course of Study.

McDonald, J H. (2008). Handbook of Biological Statistics. Web.

Miller, L E. (1994). Correlations: Description or Inference? Journal of Agricultural Education, 35(1), 5-7.

Pipkin, F B. (1986). Medical Statistics Made Easy. 2^nd. Edition. Philadelphia: W.B. Saunders.

Sommer, B A. (2006). Null Hypothesis (H0). Web.

Zimmerman, D, W., Zumb B. D., & Williams, R., H. (2003). Bias in estimation and hypothesis testing of correlation. Psicologica, 24, 133-158.