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# Regression Analysis - PowerPoint PPT Presentation

Regression Analysis. Heibatollah Baghi, and Mastee Badii. Purpose of Regression Analysis.

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### Regression Analysis

Heibatollah Baghi, and

Mastee Badii

### 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.

• 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.

• 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 = 32 + 1.8 X

• If the Centigrade temperature is 10, the Fahrenheit temperature is calculated to be:

• Y = 32 + 1.8 (10) = 32 + 18 = 50

• Similarly, if the Centigrade temperature is 20, the Fahrenheit temperature must be:

• Y = 32 + 1.8 (20) = 32 + 36 = 68

• We can plot this relationship on the usual rectangular system of coordinates.

Dependent variable

Independent variable

Slope of line

Y Intercept

Linear Equation

• Any equation of the following form will generate a straight line

• 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.

• The intercept (a) is the point at which regression line intersects the Y axis.

• 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.

• Correlation and Regression

• Determining the Line of Best Fit or Regression Line using Least Squares Criterion.

• Residual or error of prediction = (Y –Y’)

• 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

Calculate

sum

Calculation of Regression Line

Calculate

deviation

from average Y

Calculation of Regression Line

Calculate

deviation

from average X

Calculation of Regression Line

Calculate

product of deviation from X and Y

Calculation of Regression Line

Deviation of Y

Standard

Deviation of X

Correlation of X & Y

Continued

Calculation of Regression Line

Calculation of Regression Line

Calculation of Regression Line

a = 1.42

b = .0021

Y’ = 1.42 + .0021 X

Y’ = 1.42 + .0021 X

predicted values. Differencebetween predicted & observed is the residual

Plot of Data

Slope showschange in Y

associated to

to change

in one unit of X

Intercept

Predicted weight = 811 + 9 Gestation days

Regression equation: Y` = 811 + 9 X

Intercept

Predicted weight = 811 + 9 Gestation days

• The sum of Squares of the Dependent Variable is partitioned into two components:

• One due to Regression (Explained)

• One due to Residual (Unexplained)

Testing Statistical Significance of Variance Explained

Testing Statistical Significance

• Testing the proportion of variance due to regression

• H0 : R2 = 0 Since the F< Fα fail to reject Ho

• Ha : R2≠ 0

B. Testing the Regression Coefficient

H0 : β = 0 Since the p> αFail to reject Ho

Ha : β≠ 0

• The average amount of error in predicting GPA scores is 0.49.

• The smaller the standard error of estimate, the more accurate the predictions are likely to be.

• X and Y are normally distributed

Assumptions

• X and Y are normally distributed

• The relationship between X and Y is linear and not curved

Assumptions

• X and Y are normally distributed

• 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

Assumptions

• X and Y are normally distributed

• The relationship between X and Y is linear and curved

• The variation of Y at particular values of X is not proportional to X

• There is negligible error in measurement of X

• Answering Research Questions and Testing Hypothesis

• Making Prediction about Some Outcome or Dependent Variable

• Assessing an Instrument Reliability

• Assessing an Instrument Validity

### Take Home Lesson

How to conduct Regression Analysis