Estatística e Modelos Probabilísticos - COE241 Aula passada Aula de hoje Goodness of fit: v.a. Regressão linear discreta Goodness of fit: v.a. contínua Kolmogorov- Smirnov Rosa Leão – 2013
Scatter Diagrams A scatter plot scatter plot is a graph that may be used to represent the 2 relationship between two variables. = n − 1 S 2 X n − 1 2 0 Also referred to as a scatter scatter diagram . diagram Rosa Leão – 2013
Dependent and Independent Variables A dependent variable dependent variable is the variable to be predicted or explained in a regression model. This variable is assumed to be functionally related to the independent variable. Rosa Leão – 2013
Dependent and Independent Variables An independent variable independent variable is the variable related to the dependent variable in a regression equation. The independent variable is used in a regression model to estimate the value of the dependent variable. Rosa Leão – 2013
Two Variable Relationships Y X (a) Linear (a) Linear Rosa Leão – 2013
Two Variable Relationships Y X (b) Linear (b) Linear Rosa Leão – 2013
Two Variable Relationship Y X (c) Curvilinear (c) Curvilinear Rosa Leão – 2013
Two Variable Relationships Y X (d) Curvilinear (d) Curvilinear Rosa Leão – 2013
Two Variable Relationships Y X (e) No (e) No Relationship Relationship Rosa Leão – 2013
Correlation The correlation coefficient correlation coefficient is a quantitative measure of the strength of the linear relationship between two variables. The correlation ranges from + 1.0 to - 1.0. A correlation of ± 1.0 indicates a perfect linear relationship, whereas a correlation of 0 indicates no linear relationship. Rosa Leão – 2013
Correlation CORRELATION COEFFICIENT CORRELATION COEFFICIENT ρ= E [( X − E [ X ])( Y − E [ Y ])] σ X σ Y where: ρ = correlation coefficient σ x = standard deviation of X σ y = standard deviation of Y Rosa Leão – 2013
Correlation SAMPLE CORRELATION COEFFICIENT SAMPLE CORRELATION COEFFICIENT ∑ − − ( x x )( y y ) = r ∑ ∑ − − 2 2 [ ( x x ) ][ ( y y ) ] where: r = Sample correlation coefficient n = Sample size x = Value of the independent variable y = Value of the dependent variable Rosa Leão – 2013
Simple Linear Regression Analysis Simple linear regression analysis Simple linear regression analysis analyzes the linear relationship that exists between a dependent variable and a single independent variable. Rosa Leão – 2013
Simple Linear Regression Analysis SIMPLE LINEAR REGRESSION MODEL SIMPLE LINEAR REGRESSION MODEL y = a + bx + ε where: y = Value of the dependent variable x = Value of the independent variable a = Population’s y-intercept b = Slope of the population regression line = Error term, or residual ε Rosa Leão – 2013
Simple Linear Regression Analysis REGRESSION COEFFICIENTS REGRESSION COEFFICIENTS In the simple regression model, there are two coefficients: the intercept and the slope. Rosa Leão – 2013
Simple Linear Regression Analysis The interpretation of the regression regression slope coefficient is that it gives the slope coefficient average change in the dependent variable for a unit increase in the independent variable . The slope coefficient may be positive or negative, depending on the relationship between the two variables. Rosa Leão – 2013
Simple Linear Regression Analysis The least squares criterion least squares criterion is used for determining a regression line that minimizes the sum of squared residuals. Rosa Leão – 2013
Simple Linear Regression Analysis A residual residual is the difference between the actual value of the dependent variable and the value predicted by the regression model. − ˆ y y Rosa Leão – 2013
Simple Linear Regression Analysis = + Y ˆ y 150 60 x Sales in Thousands 390 390 400 300 312 312 20 0 Residual = 312 - 390 = -78 100 X 4 Years with Company Rosa Leão – 2013
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