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

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### Purpose of Regression Analysis

### Take Home Lesson

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.

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.
- 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.

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:
- 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.

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 ?
- 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.

Selection of Regression Line

- 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

Calculation of Regression Line

Deviation of Y

Standard

Deviation of X

Correlation of X & Y

Continued

Calculation of Regression LineCalculation of Regression Line

Calculation of Predicted Values and Residuals

Y’ = 1.42 + .0021 X

predicted values. Differencebetween predicted & observed is the residual

Plot of DataSlope showschange in Y

associated to

to change

in one unit of X

Intercept

Calculation of Regression Line Using Standard Deviations

Predicted weight = 811 + 9 Gestation days

Relationship between Weight & Gestation Days

Regression equation: Y` = 811 + 9 X

Intercept

Predicted weight = 811 + 9 Gestation days

Sources of Variation

- 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

Testing Statistical Significance of Regression Coefficient

B. Testing the Regression Coefficient

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

Ha : β≠ 0

Interpretation of Standard Error of Estimate

- 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.

Assumptions

- 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

The Use of Simple Regression

- Answering Research Questions and Testing Hypothesis
- Making Prediction about Some Outcome or Dependent Variable
- Assessing an Instrument Reliability
- Assessing an Instrument Validity

How to conduct Regression Analysis

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