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Correlation Analysis

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Correlation Analysis

X and Y are random variables that are jointly normally distributed and, in addition, that the obtained data consists of a random sample of n independent pairs of observations (X1, Y1), (X2, Y2), . . . . (Xn, Yn) from an underlying bi-variate normal population.

Y = f(X)

any relationships?

Relationships – 3 goalsif any, how strong?

nature or form

Two of the most powerful and versatile approaches for investigating variable relationships are correlation analysis and regression analysis.

Correlation Analysis

- Measures the strength of the relationship between two or more variables
- Correlation
- Measures the degree to which there is an association between two internally scaled variables

Regression Analysis

- Used to understand the nature of the relationship between two or more variables
- A dependent or response variable (Y) is related to one or more independent or predictor variables (Xs)
- Object is to build a regression model relating dependent variable to one or more independent variables
- Model can be used to describe, predict, and control variable of interest on the basis of independent variables

Simple Linear Regression X βo

Yi = βo + β1 xi + εi

Where

- Y
- Dependent variable

- Independent variable

- Model parameter
- Mean value of dependent variable (Y) when the independent variable (X) is zero

Simple Linear Regression (Contd.) εi

- β1
- Model parameter
- Slope that measures change in mean value of dependent variable associated with a one-unit increase in the independent variable

- Error term that describes the effects on Yi of all factors other than value of Xi

Estimating the Model Parameters

- Calculate point estimate bo and b1 of unknown parameter βo and β1
- Obtain random sample and use this information from sample to estimate βo and β1
- Obtain a line of best "fit" for sample data points - least squares line
Yi = bo + b1 xi

Multiple Linear Regression

- A linear combination of predictor factors is used to predict the outcome or response factors
- Involves computation of a multiple linear regression equation
- More than one independent variable is included in a single linear regression model