If testing of the hypothesis suggests the correlation coefficient (linear) is significant will the regression coefficient say simple (beta 1) will always be significant. Kindly justify with some reference. Thank You
It depends on the variables. First of all, define the variables for testing of hypothesis.
Ashish Kumar Yadav The Pearson product-moment correlation (r) is equal to the standardized regression weight (beta1_standardized) in a simple (bivariate) linear regression, that is, a regression with just a single independent (predictor) variable X :
Y = beta0 + beta1*X + error
Therefore, when the Pearson correlation (r) between Y and X is statistically significant, the regression coefficient beta1 will also be significant. The t statistic and p value for r and beta1 will be the exact same, for example, in SPSS.
However, this is no longer true when you have more than one independent (X) variable in the regression, unless all X variables are uncorrelated with the other X variables (which is atypical in the social sciences).
See
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum Associates Publishers.
Hi Ashish,
The test of the correction between x and y is equivalent to testing the significance of the slope in a simple linear regression. This can be verified using almost any basic statistics textbook. For example, it can be found on P. 207 of
Lomax, RG., (2001). An introduction to statistical concepts for education and behavioral sciences. Lawrence Erlbaum Associations. Mahwah, NJ, USA.
I hope this answered your question.
J. McLean
The correlation coefficient measures the "tightness" of linear relationship between two variables and is bounded between -1 and 1, inclusive. However, The regression slope measures the "steepness" of the linear relationship between two variables and can take any value from −∞ to +∞.
In case of simple regression model,
Yi=α + βXi + εi
estimated by least squares, we know that
β^= cor(Yi, Xi)⋅ [SD(Yi)/SD(Xi)]
Therefore the two only coincide when SD(Yi)=SD(Xi). That is, they only coincide when the two variables are on the same scale, in some sense. The most common way of achieving this is through standardization.
The two, in some sense give you the same information - they each tell you the strength of the linear relationship between Xi and Yi. But, they do each give you distinct information (except, of course, when they are exactly the same):
- The correlation gives you a bounded measurement that can be interpreted independently of the scale of the two variables. The closer the estimated correlation is to ±1, the closer the two are to a perfect linear relationship. The regression slope, in isolation, does not tell you that piece of information.
- The regression slope gives a useful quantity interpreted as the estimated change in the expected value of Yi for a given value of Xi. Specifically, β^ tells you the change in the expected value of Yi corresponding to a 1-unit increase in Xi. This information can not be deduced from the correlation coefficient alone.
The difference between these two statistical measurements is that correlation measures the degree of a relationship between two variables (x and y), whereas regression is how one variable affects another. Moreover, both quantify the direction and strength of the relationship between two numeric variables. When the correlation (r) is negative, the regression slope (b) will be negative. When the correlation is positive, the regression slope will be positive.
It seems you are mixing things. They have nothing to do with each other. The quality of the fit is only related to the correlation coefficient, no matter what the slope is. See Ram Bajpai 's answer.
If the correlation coefficient is positively significant then the R2 value will be high in linear regression eqn.
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