Regresi dan Korelasi Pertemuan 10 Matakuliah : D0722 - Statistika dan Aplikasinya Tahun : 2010 Regresi dan Korelasi Pertemuan 10

Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : menghubungkan dua variabel dalam analisis regresi dan korelasi linier sederhana dapat menunjukkan hubungan antara variabel berdasarkan hasil uji hipotesis

The Simple Linear Regression Model The population simple linear regression model: Y= 0 + 1 X +  Nonrandom or Random Systematic Component Component where Y is the dependent variable, the variable we wish to explain or predict X is the independent variable, also called the predictor variable  is the error term, the only random component in the model, and thus, the only source of randomness in Y. 0 is the intercept of the systematic component of the regression relationship. 1 is the slope of the systematic component. The conditional mean of Y:

Picturing the Simple Linear Regression Model X Y E[Y]=0 + 1 X Xi } 1 = Slope 1 0 = Intercept Yi { Error: i Regression Plot The simple linear regression model gives an exact linear relationship between the expected or average value of Y, the dependent variable, and X, the independent or predictor variable: E[Yi]=0 + 1 Xi Actual observed values of Y differ from the expected value by an unexplained or random error: Yi = E[Yi] + i = 0 + 1 Xi + i

10-3 Estimation: The Method of Least Squares Estimation of a simple linear regression relationship involves finding estimated or predicted values of the intercept and slope of the linear regression line. The estimated regression equation: Y = b0 + b1X + e where b0 estimates the intercept of the population regression line, 0 ; b1 estimates the slope of the population regression line, 1; and e stands for the observed errors - the residuals from fitting the estimated regression line b0 + b1X to a set of n points.

Fitting a Regression Line Y Y Data Three errors from the least squares regression line X X Y Errors from the least squares regression line are minimized Three errors from a fitted line X X

Errors in Regression Y . { X Xi

Least Squares Regression b0 SSE b1 Least squares b0 Least squares b1 At this point SSE is minimized with respect to b0 and b1

Sums of Squares, Cross Products, and Least Squares Estimators

Error Variance and the Standard Errors of Regression Estimators X Y Square and sum all regression errors to find SSE.

Standard Errors of Estimates in Regression

Confidence Intervals for the Regression Parameters Length = 1 Height = Slope Least-squares point estimate: b1=1.25533 Upper 95% bound on slope: 1.35820 Lower 95% bound: 1.15246 (not a possible value of the regression slope at 95%)

Correlation The correlation between two random variables, X and Y, is a measure of the degree of linear association between the two variables. The population correlation, denoted by, can take on any value from -1 to 1.    indicates a perfect negative linear relationship -1 <  < 0 indicates a negative linear relationship    indicates no linear relationship 0 <  < 1 indicates a positive linear relationship    indicates a perfect positive linear relationship The absolute value of  indicates the strength or exactness of the relationship.

Illustrations of Correlation Y X  = 0 Y X  = 1 Y X  = -1 Y X  = -.8 Y X  = 0 Y X  = .8

Covariance and Correlation Example 10 - 1: = r SS XY X Y 51402852. 4 40947557. 84 66855898 52321943 29 9824 ( )( ) . *Note: If  < 0, b1 < 0 If  = 0, b1 = 0 If  > 0, b1 >0

Hypothesis Tests for the Correlation Coefficient H0:  = 0 (No linear relationship) H1:   0 (Some linear relationship) Test Statistic:

Hypothesis Tests about the Regression Relationship X Constant Y Unsystematic Variation Nonlinear Relationship A hypothes is test fo r the exis tence of a linear re lationship between X and Y: H 1 Test stati stic for t he existen ce of a li near relat ionship be tween X an d Y: ( - ) where is the le ast squares es timate of the regres sion slope and ) is the s tandard er ror of . When the null hypot hesis is t rue, the stati stic has a distribu tion with degrees o f freedom. : b 2 = ¹ t n s

Hypothesis Tests for the Regression Slope

How Good is the Regression? The coefficient of determination, r2, is a descriptive measure of the strength of the regression relationship, a measure of how well the regression line fits the data. . { Y X } Total Deviation Explained Deviation Unexplained Deviation Percentage of total variation explained by the regression.

The Coefficient of Determination Y Y Y X X X SST SST SST S E r2=0 SSE r2=0.50 SSE SSR r2=0.90 SSR 5 4 3 2 1 7 6 M i l e s D o a r

Analysis of Variance and an F Test of the Regression Model

Use of the Regression Model for Prediction Point Prediction A single-valued estimate of Y for a given value of X obtained by inserting the value of X in the estimated regression equation. Prediction Interval For a value of Y given a value of X Variation in regression line estimate Variation of points around regression line For an average value of Y given a value of X

Prediction Interval for a Value of Y

Prediction Interval for the Average Value of Y

RINGKASAN Regresi : Bentuk hubungan anatara variabel bebas dengan variabel tak bebas Korelasi: Keeratan dan arah hubungan antara dua variabel Uji hipotesis parameter regresi Uji hipotesis korelasi