The Statistical Imagination Chapter 15. Correlation and Regression Part 2: Hypothesis Testing and Aspects of a Relationship

When to Test a Hypothesis Using Correlation and Regression 1)There is one representative sample from a single population 2)There are two interval/ratio or interval-like ordinal variables 3)There are no restrictions on sample size, but generally, the larger the n, the better 4)A scatterplot of the coordinates of the two variables fits a linear pattern

Test Preparation Before proceeding with the hypothesis test, check the scatterplot for a linear pattern Calculate the Pearson’s r correlation coefficient and the regression coefficient, b Compute the means of X and Y and use them and b to compute a Specify the regression equation, insert values of X, solve for Ý, and plot the line on the scatterplot Provide a conceptual diagram

Features of the Hypothesis Test Step 1. Stat. H: ρ = 0 That is, there is no relationship between X and Y The Greek letter rho (ρ) is the correlation coefficient obtained if Pearson’s correlation coefficient were computed for the population A ρ of zero asserts that there is no correlation in the population and that the regression line has no slope

Features of the Hypothesis Test (Continued) Step 2. The sampling distribution is the t- distribution with df = n - 2 When the Stat. H is true, sample Pearson’s r’s will center around zero This test does not require a direct calculation of a standard error

Features of the Hypothesis Test (Continued) Step 4. The test effect is the value of Pearson’s r The test statistic is t r The p-value is estimated from the t-distribution table, Statistical Table C in Appendix B

Four Aspects of a Relationship With correlation and regression analysis, because both variables are of interval/ratio level, the analysis is mathematically rich All four aspects of a relationship apply

Existence of a Relationship Test the Stat. H that ρ = 0, that there is no relationship between X and Y If the Stat. H is rejected, a relationship exists

Direction of a Relationship Direction is indicated by the sign of r and b, and by observing the slope of the pattern of coordinates in a scatterplot A positive relationship is revealed with an upward slope, and r and b will be positive A negative relationship is revealed with a downward slope, and r and b will be negative

Strength of a Relationship Strength is determined by the proportion of the total variation in Y explained by X This proportion is quickly obtained by squaring Pearson’s r correlation coefficient

Nature of a Relationship 1)Interpret the regression coefficient, b, the slope of the regression line. State the effect on Y of a one-unit change in X 2)Provide best estimates using the regression line equation. Insert chosen values of X, compute Ý ’s and interpret them in everyday language

Careful Interpretation of Findings A correlation applies to a population, not to an individual E.g., predictions of Y for a value of X provide the best estimate of the mean of Y for all subjects with that X-score A statistical relationship may exist but not mean much. It is important to distinguish statistical significance (i.e., the existence of a relationship) from practical significance (i.e., the strength of the relationship

Spurious Correlation A spurious correlation is one that is conceptually false, nonsensical, or theoretically meaningless E.g., in the 1990s there is a positive correlation between the amount of carbon dioxide released into the atmosphere and the level of the Dow Jones stock index