Market Research Chapter 23: Bivariate Statistical Analysis: Measures Of Association

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Measure of association

Back 1

A general term that refers toa number of bivariate statistical techniques used to measure the strength of a relationship between two variables

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correlation coefficient

Back 2

a statistical measure of the covariation, or association, between two at least interval variables

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covariance

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extent to which two variables are associated systematically with each other

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Negative (inverse) relationship

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Covariation in which the association between variables is in the opposite direciton. As one goes up, the other goes down

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Does covariation in and of itself establish causality?

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No. Think of the example of the covariation between ice cream sales and drownings; or the roosters crow and the rising sun

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Coefficient of Determination (R^2)

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A measure obtained by squaring the correlation coefficient; the proporion of the total variance of a variable accounted for by another value of another value.

Measures that part of the variance of Y that is accounted for by knowing the value of X

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In the example about unemployment and hours worked, r = -.635; therefore r^2=0.403

How much of the variance in unemployment can be explained by the variance in hours worked?

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About 40% of the variance in unemployment can be explained by the variance in hours worked.

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Correlation Matrix

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The standard form for reporting observed correlations among multiple variables. Although any number of variables can be displayed in a correlation matrix, each entry represents the bivariate relationship between a pair of variables.

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What is the procedure for determining statistical significance?

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The procedure for determining statistical significance is the t-test of the significance of a correlation coefficient

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Simple (Bivariate) Linear Regression

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A measure of linear association that investigates straight line relationships between a continuous dependent variable and an independent variable that is usually continuous, but can be a categorical dummy variable

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In simple linear regression, what do the following symbols stand for:
Y
(alpha)
(beta)
X

(Y = (alpha) + (beta)X

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(alpha) represents the Y intercept, or where the line crosses the y-axis

(beta) is the slope coefficient

The slope is the change in Y associated with a change of one unit in X. Slope may also be thought of as rise over run.

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True/False: Beta provides the strength and direction of the relationship between the independent and dependent variable

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True

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True/False: (alpha) Y intercept is a fixed point that is considered a constant

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True

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Standardized regression coefficient

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Estimated coefficient of the strength of relationship between the independent and dependent variables

Expressed on a standardized scale where higher absolute values indicate stronger relationships (between -1 to 1)

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Raw Regression Estimates (b1)

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Raw regression weights have theadvantage of retaining the scale metric (also their key disadvantage). Used if the purpose of the regression analysis is forecasting.

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Standard Regression estimates ((Beta)1)

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Have the advantage of a constant scale. Should be used when the research is testing explanatory hypothesis

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Ordinary Least Squares

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Generates a straight line that minimizes the sum of squared deviations of the actual values from this predicted regression line.

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The equation for the ordinary least squares means that X estimated how?

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The equation menas that the predicted value for any value of X is determined as a funciton of the estimated slope coefficient, plus the estimated intercept coefficient + some error

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Where does the explanatory power of regression lie?

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The explanatory power of regression lies in hypothesis testing

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What two conditions must be satisfied for the outcome of the ypothesis test?

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The regression weight must be in the hypothesized direction. (positivie relationships require a positive coefficient and negative relationships require a negative one)

The t-test associated with the regression weight must be significant

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Multiple Regression Analysis

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An analysis of association in which the effects of two or more independent variables on a single, intervalscaled dependent variable are investigated simultaneously

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Dummy Variable

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The way a dichotomous (two group) independent variable is represnted in regression analysis by assigning 0 to one group and 1 to another

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Partial correlation

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The correlation between two variables after taking into account the fact that they are correlated with other variables too

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R^2 in multiple Regression

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The coefficient of multiple determination in multiple regression indicates the percentage of variation in Y explained by all independent variables

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F-Test

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Tests statistical significance by comparing the variation explained by the regression equation to the residual error variation .

Allows for testing of the relative magnitudes of the sum of squares due to the regression and the error sum of squares