1. Lecture 6 Correlation and Regression STAT 3120 Statistical Methods I

2. STAT3120 – Correlation and Linear Regression Dependent Variable Independent (predictor) Variable Statistical Test Comments QuantitativeCategoricalT-TEST (one, two or paired sample) Determines if categorical variable (factor) affects dependent variable; typically used for experimental or planned change studies Quantitative Correlation /Regression Analysis Test establishes a regression model; used to explain, predict or control dependent variable Categorical Chi-SquareTests if variables are statistically independent (i.e. are they related or not?)

3. STAT3120 - Correlation  Correlation coefficients assess strength of linear relationship between two quantitative variables. The correlation measure ranges from -1 to +1. A negative correlation means that X and Y are inversely related. A positive correlation means that X and Y are directly related. zero correlation means that X and Y are not linearly related. A correlation of +1 indicates X and Y are directly related and that all the points fall on the same straight line. A correlation of -1 indicates X and Y are inversely related and that all the points fall on the same straight line  Plot Scatter Diagram of Each Predictor variable and Dependent Variable Look of Departures from Linearity Look for extreme data points (Outliers)  Examine Partial Correlation Can’t determine causality, but isolate confounding variables

4. STAT3120 - Correlation For example, lets take two variables and evaluate their correlation…open the stats98 dataset in Excel… What would you expect the correlation of the Verbal SAT scores and the Math SAT scores to be? Why? What would you expect the correlation of the Math SAT scores and the percent taking the test to be? Why?

5. STAT3120 - Correlation What would you expect the correlation of the Verbal SAT scores and the Math SAT scores to be? Why?

6. STAT3120 - Correlation What would you expect the correlation of the Math SAT scores and the Percent of HS students that took the test? Why?

7. STAT3120 - Correlation Lets pull up the 2000 Florida Vote Count in Excel…

8. STAT3120 - Correlation Lets pull up the UCDAVIS2 dataset in Excel…plot Ideal Height versus Actual Height…what would you expect the correlation value to be? Can you explain someone’s Ideal Height using their Actual Height?

9. STAT3120 – Regression

10. STAT3120 - Regression From the previous slide, the “regression line” has been imposed onto the relationship between ideal height and height. The equation of this line takes the general form of y=mx+b, where: Y is the dependent variable (ideal height) M is the slope of the line X is the independent variable (actual height) B is the Y-intercept. When we discussion regression models, we transform this equation to be: Y = b o + b 1 x 1 + …b n x n Where b o is the y-intercept and b 1 is the slope of the line. The “slope” is also the effect of a one unit change of x on y.

11. STAT3120 - Regression From the previous slide, the model equation is presented in the form of the equation of a line: y=.8174x +14.271. From this, we would say: 1.For every 1 inch of change in someone’s actual height, there is a.8174 inch change in their ideal height. 2.Everyone “starts” with 14.271 inches. 3.If someone has an actual height of 68 inches, their ideal height is 69.85 inches. That R2 value of.7372 is interpreted as “73.72% of the change in ideal height can be explained by a linear model with actual height as the only predictor”.

12. STAT3120 - Regression Lets do this in SAS. After you import the data, the code to run a correlation looks like this: Proc Corr data=jlp.ucdavis2; Var Idealht Height; Run; The output looks like this:

13. STAT3120 - Regression The SAS Code to develop a regression model on the data looks like this: Proc Reg data=jlp.ucdavis2; Model idealht = height/p r; output out=preds p=pred r=resid; run; In this code, the regression model is developed using the “model” statement. Here, the dependent variable of interest is set first. The independent variable(s) then follow after the = sign. The p and r options after the / will produce the predictions and the residuals respectively. The output statement will create a new (temporary) dataset called “preds” that will contain the predictions and the residuals – so that we can examine them.

14. STAT3120 - Regression Here is some of the associated output: This information tells us about the performance of the model This information tells us about the Equation of the model and the impact of the predictor(s).