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 (X 1 , Y 1 ), (X 2 , Y 2 ), . . . . (X n , Y n ) from an underlying bi-variate normal population.

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Correlation Analysis

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Correlation analysis

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 analysis1

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


Simple correlation coefficient

Simple Correlation Coefficient


Regression analysis

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

Simple Linear Regression

Yi = βo + β1 xi + εi

Where

  • Y

    • Dependent variable

  • X

    • Independent variable

  • βo

    • Model parameter

    • Mean value of dependent variable (Y) when the independent variable (X) is zero


  • Simple linear regression contd

    Simple Linear Regression (Contd.)

    • β1

      • Model parameter

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

  • εi

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


  • Estimating the model parameters

    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

    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


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