BRM Session 5.pdf

Business Research Methods Session 5 – Regression and Discriminant Analysis Dr. Gopinath Krishnan Assistant Professor Information Systems and Analytics Indian Institute of Management Tiruchirappalli  Correlation The...

Business Research Methods Session 5 – Regression and Discriminant Analysis Dr. Gopinath Krishnan Assistant Professor Information Systems and Analytics Indian Institute of Management Tiruchirappalli  Correlation The product moment correlation, r, summarizes the strength of association between two metric (interval or ratio scaled) variables, say X and Y. It is an index used to determine whether a linear or straight- line relationship exists between X and Y. As it was originally proposed by Karl Pearson, it is also known as the Pearson correlation coefficient. It is also referred to as simple correlation, bivariate correlation, or merely the correlation coefficient. r varies between −1.0 and +1.0. The correlation coefficient between two variables will be the same regardless of their underlying units of measurement.  Correlation From a sample of n observations, X and Y, the product moment correlation, r, can be calculated as: n å i= 1 ( X i - X )(Yi - Y ) r= n n åi= 1 ( Xi - X ) 2 å i= 1 (Yi - Y )2 Division of the numerator and denominator by (n – 1) gives n ( X i - X )(Yi - Y ) å i= 1 n- 1 r= n ( X i - X )2 n (Yi - Y )2 å i= 1 å n - 1 i= 1 n - 1 COVxy = s x sy  Explaining Attitude Toward the City of Residence Respondent Attitude Toward the Duration of Residence Importance Attached to No. City Weather 1 6 10 3 2 9 12 11 3 8 12 4 4 3 4 1 5 10 12 11 6 4 6 1 7 5 8 7 8 2 2 4 9 11 18 8 10 9 9 10 11 10 17 8 12 2 2 5  Decomposition of the Total Variation Explained variation r2 = Total variation SSx = SSy Total variation - Error variation = Total variation SSy - SSerror = SSy r2 measures the proportion of variation in one variable that is explained by the other  Decomposition of the Total Variation When it is computed for a population rather than a sample, the product moment correlation is denoted by ρ, the Greek letter rho. The coefficient r is an estimator of ρ. The statistical significance of the relationship between two variables measured by using r can be conveniently tested. The hypotheses are: Test statistic  Partial Correlation A partial correlation coefficient measures the association between two variables after controlling for, or adjusting for, the effects of one or more additional variables. rxy - (rxz )(ryz ) rxy.z = 1- rxz2 1- ryz2 Partial correlations have an order associated with them. The order indicates how many variables are being adjusted or controlled. The simple correlation coefficient, r, has a zero-order, as it does not control for any additional variables while measuring the association between two variables.  Applications of Partial Correlation How strongly are sales related to advertising expenditures when the effect of price is controlled? Is there an association between market share and size of the sales force after adjusting for the effect of sales promotion? Are consumers’ perceptions of quality related to their perceptions of prices when the effect of brand image is controlled?  Part Correlation Coefficient The part correlation coefficient represents the correlation between Y and X when the linear effects of the other independent variables have been removed from X but not from Y. The part correlation coefficient, ry(x.z) is calculated as follows: rxy - ryz rxz ry ( x. z ) = 1- rxz2 The partial correlation coefficient is generally viewed as more important than the part correlation coefficient. Dependent Variable- Attitude Towards City Independent Variables- Duration of Stay Importance of weather Step-1  The analysis shows strong, moderate, Correlation and weak correlations among attitudes towards a city, duration of stay, and the importance of weather. There's a strong positive correlation (r=.936, p