Please write an answer for the following discussion. Please follow the instructions at the bottom .
Correlation and Regression Questions
D6.8.1 Scatterplots and Regression lines
First and foremost, we graph scatterplots and compute regression lines later on for two or several sets of data for the sole purpose of determining the presence of a substantial correlation between them. Scatter plots are akin to line graphs except that scatter plots do not connect individual dots immediately, but they express a trend with the help of a regression line, a mathematical expression describing the relationship (NC State University, n.d.). In a visual inspection of scatter plots, one can distinguish the strength of the relationship between a few variables under study by looking at how close the points are to a line. Overall patterns and deviations are apparently and likely suggested from the scatter plots (Illowsky and Dean, 2019). On top of these, if one variable is proven to be a predictor of another variable, “linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data” (Yale University, 1998).
D6.8.2 Refer to Output 8.2, Correlation Coefficient, R2, and Pearson and Spearman correlations
D.6.8.2(a) Correlation Coefficient
Shortly after creating a scatter plot, a linear equation usually follows, but the strength of the correlation between variables is not clear and categorical since that would be based only on a visual inspection of the scatter plot. The correlation coefficient, r, mathematically tells how strong this relationship is. This number is mostly known as the Pearson product-moment correlation coefficient after Karl Pearson. As r approaches +1, the stronger the positive correlation between the variables. On the other hand, the near it is to -1, the stronger the negative correlation (Calkins, 2005).
D6.8.2 (b) R2
While the correlation coefficient tells the strength of the relationship between several variables, R2, or the coefficient of determination, tells what percentage of the variation in the dependent variable can be explained by the independent variables (Department of Statistics: Eberly College of Science, 2018). For instance, if the correlation coefficient of the two variables is 0.951, then the coefficient of determination is 90.4%. Simply tells, 90.4% of the variation in the dependent variable can be explained by the independent variable.
In output 8.2, the obtained Pearson correlation is 0.34 for N=75. With that, the coefficient of determination, or R2, is 0.1156. It means that the model’s independent variable can explain the 11.56% variation in the dependent variable.
D6.8.2 (c) Pearson and Spearman correlations: Correlation size and significance level
Pearson measures the strength of the relationship between an independent variable and the independent variable(s), with the following criteria: “interval or ratio level; linearly related; bivariate and normally distributed” (Stats Tutor, n.d.). The absence of these assumptions leads to deferment to Spearman’s rank correlation, where a monotonic relationship is a priority. A monotonic function is a function where the independent variable increases but there is an un-uniform movement with the dependent variable. Also, to carry out Spearman, one needs to rank the data. And more importantly, Pearson’s correlation requires normality, unlike Spearman’s correlation.
Referring to output 8.2, the Pearson correlation is 0.34 almost the same as the Spearman rho, 0.32. In terms of significance, expressed as p-value, Pearson (p-value = 0.003) and Spearman (p-value = 0.006) are near to each other. If we use significance level of 0.05, both of these numbers suggest that the model’s independent variable is a significant predictor of the dependent variable.
D6.8.2 (d) Condition to use Pearson and Spearman
Recall that the Pearson correlation gauges the linear relationship between two or more continuous variables, whereas the Spearman correlation evaluates the monotonic relationship based on the ranked values rather than raw data. Hence, the decision to use either of these two lies in the type of data. For the interval scale, Pearson is the most appropriate, while Spearman is for the ordinal scale. Note too that normality favors Pearson so it is useful to create boxplots or calculate the skewness of each dataset. Again, if the scatter plots do not seem to show a linear relationship, then it is appropriate to use Spearman. This is one of the reasons why scatter plots must precede any correlation tests.
Going back to output 8.2, since both correlations are almost equal, apply the skewness by creating a boxplot. Here, we can see that the mother’s education is skewed; therefore, cases of this type must use Spearman rho correlation.
D6.8.5 Coefficients of predictors
The standardized regression weights or coefficients indicate how much the dependent variable will increase or decrease if the coefficient is positive or negative, respectively, provided that the independent variable is a proven significant predictor of the dependent variable based on their p-values, when this particular variable increases by one unit (Princeton University, n.d.).
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