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1 Regression Analysis Heibatollah Baghi, and Mastee Badii

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2 Purpose of Regression Analysis Regression analysis procedures have as their primary purpose the development of an equation that can be used for predicting values on some Dependent Variable, Y, given Independent Variables, X, for all members of a population.

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3 Purpose of Linear Relationship One of the most important functions of science is the description of natural phenomenon in terms of ‘functional relationships’ between variables. One of the most important functions of science is the description of natural phenomenon in terms of ‘functional relationships’ between variables. When it was found that the value of a variable Y depends on the value of another variable X so that for every value of X there is a corresponding value of Y, then Y is said to be a ‘function’ of’ X. When it was found that the value of a variable Y depends on the value of another variable X so that for every value of X there is a corresponding value of Y, then Y is said to be a ‘function’ of’ X.

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4 Example of Linear Relationship If one is given a temperature value in the Centigrade Scale ( represented by X), then the corresponding value in the Fahrenheit Scale ( represented by Y), can be calculated by the formula: If one is given a temperature value in the Centigrade Scale ( represented by X), then the corresponding value in the Fahrenheit Scale ( represented by Y), can be calculated by the formula: Y = X Y = X If the Centigrade temperature is 10, the Fahrenheit temperature is calculated to be: If the Centigrade temperature is 10, the Fahrenheit temperature is calculated to be: Y = (10) = = 50 Y = (10) = = 50 Similarly, if the Centigrade temperature is 20, the Fahrenheit temperature must be: Similarly, if the Centigrade temperature is 20, the Fahrenheit temperature must be: Y = (20) = = 68 Y = (20) = = 68 We can plot this relationship on the usual rectangular system of coordinates. We can plot this relationship on the usual rectangular system of coordinates.

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5 Linear Equation Any equation of the following form will generate a straight line Any equation of the following form will generate a straight line Y = a + b X Y = a + b X A straight line is defined by two terms: Slope and Intercept. The slope (b) reflects the angle and direction of regression line. A straight line is defined by two terms: Slope and Intercept. The slope (b) reflects the angle and direction of regression line. The intercept (a) is the point at which regression line intersects the Y axis. The intercept (a) is the point at which regression line intersects the Y axis. Y Intercept Slope of line Dependent variable Independent variable

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6 Regression and Prediction As a university admissions officer, what GPA would you predict for a student who earns a score of 650 on SAT-V ? As a university admissions officer, what GPA would you predict for a student who earns a score of 650 on SAT-V ? If the relationship between X and Y is not perfect, you should attach error to your prediction. If the relationship between X and Y is not perfect, you should attach error to your prediction. Correlation and Regression Correlation and Regression Determining the Line of Best Fit or Regression Line using Least Squares Criterion. Determining the Line of Best Fit or Regression Line using Least Squares Criterion.

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7 Selection of Regression Line Residual or error of prediction = (Y –Y’) Residual or error of prediction = (Y –Y’) Positive or negative Positive or negative Regression line, Y’ = a + bX, is chosen so that the sum of the squared prediction error for all cases, ∑(Y- Y’) 2, is as small as possible Regression line, Y’ = a + bX, is chosen so that the sum of the squared prediction error for all cases, ∑(Y- Y’) 2, is as small as possible

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8 Calculation of Regression Line Calculate sum

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9 Calculation of Regression Line Calculate deviation from average Y

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10 Calculation of Regression Line Calculate deviation from average X

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11 Calculation of Regression Line Calculate product of deviation from X and Y

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12 Calculation of Regression Line

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13 Calculation of Regression Line Standard Deviation of Y Standard Deviation of X Correlation of X & Y

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14 Calculation of Regression Line

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15 Calculation of Regression Line a = 1.42 b =.0021 Y’ = X

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16 Y (GPA)X (SAT)Y’Y-Y’ Sum Average Calculation of Predicted Values and Residuals Y’ = X

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17 Plot of Data

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18 Plot of Data Intercept Slope shows change in Y associated to to change in one unit of X Regression line shows predicted values. Difference between predicted & observed is the residual

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19 Predicted weight = Gestation days Calculation of Regression Line Using Standard Deviations

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20 Intercept Regression equation: Y` = X Relationship between Weight & Gestation Days Predicted weight = Gestation days

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21 Predicting Weight from Gestation Days If a baby’s gestation is… AddIntercept Plus Coefficient TIMES gestation Predicted Weight * (250) * (260) * (270) * (280) * (300) 3511

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22 Sources of Variation The sum of Squares of the Dependent Variable is partitioned into two components: The sum of Squares of the Dependent Variable is partitioned into two components: One due to Regression (Explained) One due to Regression (Explained) One due to Residual (Unexplained) One due to Residual (Unexplained)

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23 Partitioning of Sum of Squares

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24 Testing Statistical Significance of Variance Explained Source of variation SSdf Regression Residual Total3.2011

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25 Testing Statistical Significance of Variance Explained Source of variation SSdfMSRegression Residual Total3.2011

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26 Testing Statistical Significance Source of variation SSdfMSF FFαFFαRegression Residual Total A.Testing the proportion of variance due to regression H 0 : R 2 = 0 Since the F< F α f ail to reject H o H a : R 2 ≠ 0

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27 Testing Statistical Significance of Regression Coefficient B. Testing the Regression Coefficient H 0 : β = 0 Since the p> α Fail to reject H o H a : β ≠ 0

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28 Standard Error of Estimate of Y Regressed on X

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29 Interpretation of Standard Error of Estimate The average amount of error in predicting GPA scores is The average amount of error in predicting GPA scores is The smaller the standard error of estimate, the more accurate the predictions are likely to be. The smaller the standard error of estimate, the more accurate the predictions are likely to be.

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30 Assumptions X and Y are normally distributed X and Y are normally distributed

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31 Assumptions X and Y are normally distributed X and Y are normally distributed The relationship between X and Y is linear and not curved The relationship between X and Y is linear and not curved

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32 Assumptions X and Y are normally distributed X and Y are normally distributed The relationship between X and Y is linear and not curved The relationship between X and Y is linear and not curved The variation of Y at particular values of X is not proportional to X The variation of Y at particular values of X is not proportional to X

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33 Assumptions 1. X and Y are normally distributed 2. The relationship between X and Y is linear and curved 3. The variation of Y at particular values of X is not proportional to X 4. There is negligible error in measurement of X

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34 The Use of Simple Regression Answering Research Questions and Testing Hypothesis Answering Research Questions and Testing Hypothesis Making Prediction about Some Outcome or Dependent Variable Making Prediction about Some Outcome or Dependent Variable Assessing an Instrument Reliability Assessing an Instrument Reliability Assessing an Instrument Validity Assessing an Instrument Validity

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35 Take Home Lesson How to conduct Regression Analysis

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