Simple Linear Regression and Correlation by Asst. Prof. Dr. Min Aung

When SLR? Study a relationship between two variables Paired-Samples or matched data Interval or ratio level measurement

Independent and dependent variables You want to guess or estimate or compute the values of the dependent variable. In estimating, you will use the values of the independent variable.

Predictor and Predicted variables Predictor = independent variable. Predicted variable = dependent variable.

Scatter Diagram X-axis = independent variable. Y-axis = dependent variable. Each pair of data A point (x, y) X Y 2 3 (2, 3)

X Purpose of Drawing Scatter Diagram Is there a linear relationship between the two variables X and Y? Linear relationship = Scatter points (roughly at least) form the shape of a straight line. Y X Y Linear relationshipNo linear relationship

Measuring Strength of Linear Relationship Pearsons coefficient of correlation r Formula (2) (Not used in exam. Just for knowledge) Calculator Work For Casio 350MS Switch the calculator on. 1.Set calculator in LR (Linear Regression) mode: Press Mode. Press 3 for Reg (Regression). Press 1 for Linear. Check n. (Checking whether there are old data): Press Shift 1, next 3, and then =.

Calculator Work for r 3.Enter Data in Pairs: x-value, y-value M+ 4.Check n again: see step 2 above. 5.Press shift 2, then move by arrow to the right, press 3 for r, and then press =. Now you see the value of r.

Interpretation of r (Direct linear relationship) 1.If r is 1 or – 1, then all scatter points are on a straight line. 2.If r is 1, all points are on a straight line with a positive slope. 3.If r is -1, all points are on a straight line with a negative slope. 4.If a straight line has a positive slope, it rises up to the right. 5.If a straight line has a positive slope, if x increases, then y increases for the points (x, y) on it. (small x, small y) (large x, large y) 6.In this situation, we say that the two variables X and Y are directly or positively correlated.

Interpretation of r (Inverse linear relationship) 1.If r is -1, all points are on a straight line with a negative slope. 2.If a straight line has a negative slope, if x increases, then y decreases for the points (x, y) on it. (small x, large y) (large x, small y) 6.In this situation, we say that the two variables X and Y are inversely or negatively correlated.

Interpretation of r (strength) 1.If r is not exactly 1 or – 1, but it is.9 or -.9, then the points are around a straight line. They are close to a straight-line shape. 2.If r is.8 or -.8, then the points are close to a straight-line shape, but not so well as in case of.9 or Thus, the closer r is to 1 or – 1, the closer are the points to a straight-line shape. 4.Thus, the closer r is to 0, the farther are the points from a straight-line shape. 5.In r-values, 0.9 are stronger than 0.8, and 0.8 are weaker than 0.9.

Interpretation of r (strength) Values of r 0 No linear relationship 0.5 Weak linear relationship Weak linear relationship 1 Strong Perfect -1 Strong Perfect

Testing Linear Relationship 1.Pearson invented a formula to measure the strength and direction of a linear relationship between two variables. 2.The number given by his formula is called correlation coefficient. We call it Pearsons coefficient of correlation. 3.We write r for this value in a sample, and we write for this value in a population. 4.Testing whether the correlation is significant is scientific guessing whether there should be a correlation, in the population, between the two variables under consideration.

Null and Alternate Hypothesis 1.Test correlation: H 0 : = 0 and H a : 0 2.Test direct correlation: H 0 : 0 and H a : > 0 3.Test inverse correlation: H 0 : 0 and H a : < 0 4.Test positive correlation: H 0 : 0 and H a : > 0 5.Test inverse correlation: H 0 : 0 and H a : < 0

Three types of test 1.H 0 : = 0 and H a : 0 Two-tailed test 2.H 0 : 0 and H a : < 0 Left-tailed test 3.H 0 : 0 and H a : > 0 Right-tailed test

Critical value 1.Read t table. 2.Degrees of freedom (Df) = n n = number of pairs of data 4.Right-tailed test Positive sign 5.Left-tailed test Negative sign 6.Two-tailed test Both positive and negative sign

Test Statistic 1.Test statistic = Strength of evidence supporting alternate hypothesis H a 2.Original test statistic to test is r. 3.Convert r to t by Formula (10). 4.Learn to compute t by your calculator correctly.

Rejection region 1 For a two tailed-test, the rejection region is on the right of positive critical value and on the left of negative critical value. Real number line for t values 0Positive Critical Value Negative Critical Value Total area = Level of significance = Probability = α Rejection region T curve

Rejection region 2 For a left-tailed test, the rejection region is on the left of (negative) critical value. Real number line for t values 0 (Negative) Critical Value α = Area = Level of significance = Probability Rejection region t curve

Rejection region 3 For a right-tailed test, the rejection region is on the right of the (positive) critical value. Real number line for t values 0 (Positive) Critical Value Area = Level of significance = Probability = α Rejection region t curve

Decision Rule If the test statistic (TS) is in the rejection region, then reject H 0. Reject H 0 = H 0 is false, and hence H a is true. Fail to reject H 0 = H 0 is true, and hence H a is false.

Conclusion Conclusion = Decision Decision is the last step of statistical procedure. Conclusion is the report to the one who asked the original question.