D6.8.1 Why would we graph scatterplots and regression lines?

            Relations in data are described and analyzed by fitting a regression line to the data. The graphical equivalent is a line fit to a scatterplot, typically via the least squares fit (Gillan, 2020). The most important reason is to check for violations in the assumptions of correlation and regression. Both the Pearson correlation and regression statistic assume a linear relationship. Viewing the scatterplot lines allows the researcher to check to see if there are marked violations of linearity (Morgan, 2020). In regression, there may be better fitting lines such as a quadratic (one bend) or cubic (two bends) that would explain the data more accurately. Graphing the data also allows you to easily see outliers (Morgan, 2020).

D6.8.2 In Output 8.2, (a) What do the correlation coefficients tell us?

Correlation coefficients are used to measure the strength of the relationship between two variables. Pearson correlation is the one most used in statistics. This measures the strength and direction of a linear relationship between two variables. Values always range between -1 (strong negative relationship) and +1 (strong positive relationship). Values at or close to zero imply a weak or no linear relationship. Correlation coefficient values less than +0.8 or greater than -0.8 are not considered significant (Schober, 2018).

D6.8.2 (b) What is r2 for the Pearson correlation? What does it mean?

In output 8.2, the Pearson correlation is reported to be r(73) = .34, p= .003. The 73 is what they call the degrees of freedom which is (N-2). In this output the N was equal to 75.  The Pearson correlation is one of the most used statistic. The Pearson correlation measures the strength and direction of liner relationship between variables. R2 equals the square of the Pearson correlation coefficient between the observed and modeled (predicted) data values of the dependent variable ( Gardner & Neufeld, 2013). In our example, in output 8.2 r= .34 so R2 = .1156  if R-squared value < 0.3 this value is generally considered a None or Very weak effect size,  if R-squared value 0.3 < r < 0.5 this value is generally considered a weak or low effect size, if R-squared value r > 0.7 this value is generally considered strong effect size (Schober, 2018).

D6.8.2 (c) Compare the Pearson and Spearman correlations on both correlation size and significance level.

The Pearson Correlation coefficient is .34; the significance level (Sig.) or p is .003 and the number of participants with both variables (math achievement and mother’s education) is 75 (Morgan, 2020). The correlation value for Spearman’s rho or Rs (.32) is slightly different from r, but usually, as in this case, it has a similar significance level (p = .006) (Morgan,2020).

D6.8.2 (d) When should you use which type in this case?

In output 8.2, this shows the different types of correlations, we have the Pearson and Spearman’s rho. The Pearson is used when you have two variables that are normal/scale, and the Spearman rho is used when one or both variables are ordinal or if either or both variable is not normally distributed (Morgan,2020). Pearson can only be used if the data is not skewed. In the example, in output 8.2, we can see that mother’s education was markedly skewed, Show the more appropriate choice would be to use the Spearman’s rho correlation.

D6.8.5 In Output 8.5, what do the standardized regression weights or coefficients tell you about the ability of the predictors to predict the dependent variable?

Standardized coefficients are better to use for predicting the dependent variable from the predictor scores. The standardized coefficients allow you to compare the amount that each variable contributes to predicting the outcome (like math achievement in Output 8.5) (Morgan,2020).

References

Gillan, D. J. (2020). Fitting Regression Lines to Scatterplots: The Role of Perceptual Heuristics. Proceedings of the Human Factors and Ergonomics Society Annual Meeting, 64(1), 1650–1654.

Schober, P. (2018). Correlation Coefficients. Anesthesia & Analgesia., 126(5), 1763–1768.

Gardner, R. C., & Neufeld, R. W. J. (2013). What the Correlation Coefficient Really Tells Us About the Individual. Canadian Journal of Behavioral Science, 45(4), 313-319.

Morgan, G. A., Barrett, K. C., Leech, N. L., & Gloeckner, G. W. (2020). IBM SPSS for Introductory Statistics: Use and Interpretation (6th ed.). New York: Routledge.