Pearson Correlation
Pearson’s r is an integral component in demonstrating and quantifying relationships between variables. A hypothetical situation where Pearson’s r is used is when a researcher wants to investigate the relationship between study duration and the academic performance of a group of students in college. In this situation, the time taken to study is variable Y.
In contrast, the academic achievement during the study period is X. The researcher obtains data on the hours students spend studying every week and their corresponding grades. The magnitude of r reveals the strength of the relationship between Y and X, with values closer to 1 indicating a stronger correlation.
In this case scenario, Pearson’s correlation coefficient (r) can be employed to evaluate the strength and direction of the linear association between study time and students’ grades. The sign of r (+/-) shows the relationship’s direction, whether negative or positive (Warner, 2021). A positive correlation, such as r > 0, indicates that grades also tend to increase as the study duration increases. On the other hand, a negative correlation, such as r < 0, designates that as the amount of time used for studying increases, the scores tend to decrease.
To use Pearson’s r, some assumptions such as normality, linearity, and homoscedasticity must be met. Extensive data screening would encompass examination for violations of these assumptions (Warner, 2021). For statistical tests, a researcher may analyze scatterplots to assess the linearity assumption visually (Flatt & Jacobs, 2019). Additionally, the researcher may employ statistical tests such as Levene’s test for homoscedasticity and the Shapiro-Wilk test for normality to ensure that the data meet these assumptions.
A researcher can utilize the Bonferroni correction to reduce the risk of Type I errors or false positives when reporting different correlations. This correction alters the significance level (alpha) through division using the number of correlations being tested (Warner, 2021). For instance, when a researcher tests various correlations at a significance level of 0.05 and there are five correlations, the new significance level would be 0.05/5, equal to 0.01. The purpose of the correction is to control the overall false positive rate.
Regression Equation
The two forms of the regression equation are raw score and z-score. A raw score is a form that indicates the association between the variables utilizing their unstandardized and original values (Warner, 2021). The regression equation is Y = a + bX, where X is the independent variable, Y is the dependent variable, a is the intercept, and b is the slope coefficient. This form is more useful than the z-score when the variables are measured on a similar scale.
On the other hand, the z-score is a variable that normalizes the variables by changing them into z-scores, with a standard deviation of 1 and a mean of 0. It takes the form of Z(Y) = rZ(X), where Z(X) and Z(Y) are the z scores of the independent and dependent variables, respectively, and r is the correlation coefficient among the variables (Warner, 2021). This form is beneficial when the variables are measured on varying scales.
In bivariate regression, r shows the correlation coefficient between two variables. It determines the direction and strength of the linear association between them. R indicates the numerous correlation coefficient, which finds the direction and strength of the relationship when various independent variables are incorporated into the model (Hanck et al., 2019). B shows the regression coefficient, which designates the alteration in the dependent variable for a unit change in the independent variable if all other independent variables are constant.
The connection between R, r, and B is that B is equivalent to r divided by the standard deviation of X times the standard deviation of Y, and r is the square root of R. This shows that the size of B is dependent on the strength of the relationship among the variables (r) (Warner, 2021)—the variability in the standard deviation of independent variable and the independent variable.
References
Hanck, C., Arnold, M., Gerber, A., & Schmelzer, M. (2019). Introduction to econometrics with R. University of Duisburg-Essen, 1-9. Web.
Flatt, C., & Jacobs, R. L. (2019). Principle assumptions of regression analysis: Testing, techniques, and statistical reporting of imperfect data sets. Advances in Developing Human Resources, 21(4), 484-502. Web.
Warner, R. M. (2021). Applied statistics I: Basic bivariate techniques (3rd ed.). Sage Publications.