Correlation and Regression
Correlation What type of relationship exists between the two variables and is the correlation significant? x y Cigarettes smoked per day Score on SAT Height Hours of Training Explanatory (Independent) Variable Response (Dependent) Variable A quantitative relationship between two interval or ratio level variables Number of Accidents Shoe SizeHeight Lung Capacity Grade Point Average IQ
Correlation measures and describes the strength and direction of the relationship Bivariate techniques requires two variable scores from the same individuals (dependent and independent variables) Multivariate when more than two independent variables (e.g effect of advertising and prices on sales) Variables must be ratio or interval scale
Negative Correlation–as x increases, y decreases x = hours of training (horizontal axis) y = number of accidents (vertical axis) Scatter Plots and Types of Correlation Hours of Training Accidents
Positive Correlation–as x increases, y increases x = SAT score y = GPA GPA Scatter Plots and Types of Correlation Math SAT
No linear correlation x = height y = IQ Scatter Plots and Types of Correlation Height IQ
Strong, negative relationship but non-linear! Scatter Plots and Types of Correlation
Correlation Coefficient A measure of the strength and direction of a linear relationship between two variables The range of r is from –1 to 1. If r is close to 1 there is a strong positive correlation. If r is close to –1 there is a strong negative correlation. If r is close to 0 there is no linear correlation. –1 0 1
Outliers..... Outliers are dangerous Here we have a spurious correlation of r=0.68 without IBM, r=0.48 without IBM & GE, r=0.21
x y Absences Final Grade Application Final Grade X Absences
xy x 2 y2y2 Computation of r x y
r is the correlation coefficient for the sample. The correlation coefficient for the population is (rho). The sampling distribution for r is a t-distribution with n – 2 d.f. Standardized test statistic For a two tail test for significance: Hypothesis Test for Significance (The correlation is not significant) (The correlation is significant)
A t-distribution with 5 degrees of freedom Test of Significance The correlation between the number of times absent and a final grade r = – There were seven pairs of data.Test the significance of this correlation. Use = Write the null and alternative hypothesis. 2. State the level of significance. 3. Identify the sampling distribution. (The correlation is not significant) (The correlation is significant) = 0.01
t –4.032 Rejection Regions Critical Values ± t 0 4. Find the critical value. 5. Find the rejection region. 6. Find the test statistic. df\p
t 0 –4.032 t = –9.811 falls in the rejection region. Reject the null hypothesis. There is a significant negative correlation between the number of times absent and final grades. 7. Make your decision. 8. Interpret your decision.
The equation of a line may be written as y = mx + b where m is the slope of the line and b is the y- intercept. The line of regression is: The slope m is: The y-intercept is: Regression indicates the degree to which the variation in one variable X, is related to or can be explained by the variation in another variable Y Once you know there is a significant linear correlation, you can write an equation describing the relationship between the x and y variables. This equation is called the line of regression or least squares line. The Line of Regression
Ad $ = a residual (xi,yi)(xi,yi) = a data point revenue = a point on the line with the same x-value Best fitting straight line
Calculate m and b. Write the equation of the line of regression with x = number of absences and y = final grade. The line of regression is:= –3.924x xy x 2 y2y2 x y
Absences Final Grade m = –3.924 and b = The line of regression is: Note that the point = (8.143, ) is on the line. The Line of Regression
The regression line can be used to predict values of y for values of x falling within the range of the data. The regression equation for number of times absent and final grade is: Use this equation to predict the expected grade for a student with (a) 3 absences(b) 12 absences (a) (b) Predicting y Values = –3.924(3) = = –3.924(12) = = –3.924x
The correlation coefficient of number of times absent and final grade is r = – The coefficient of determination is r 2 = (–0.975) 2 = Interpretation: About 95% of the variation in final grades can be explained by the number of times a student is absent. The other 5% is unexplained and can be due to sampling error or other variables such as intelligence, amount of time studied, etc. Strength of the Association The coefficient of determination, r 2, measures the strength of the association and is the ratio of explained variation in y to the total variation in y.