Regression and Steepness of Slope

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This topic contains 6 replies, has 7 voices, and was last updated by  Chiranjiv Nagi 11 years, 5 months ago.

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    I have a statistics question.  More specifically a question on Regression.  I understand how the correlation coefficient is calculated, but my question is about steepness of slope.  How does correlation coefficient reflect the steepness of slope? 


    Robert Butler

    For a simple linear regression Y = b0 +b1*X
    the slope b1 is a scaled version of the correlation coefficient Rxy.
    The correlation coefficient is a measure of the linear association between X and Y while b1 measures the size in the change of Y due to a unit change in X.
    The actual relationship is b1 = (Sy/Sx)*Rxy
    where Sy is the measure of the variation in Y and Sx is the measure of the variation in X.
      In more general regression problems the regression coefficients are also related to the correlation coefficient but in a more complicated manner.  See pp.45 Applied Regression Analysis 2nd edition Draper and Smith for more details.



    Regression Coefficient is the rate at which the dependent variable(y) increases per unit increase in the independent variable(x).
    Correlation coefficient measures the closeness of the observed values of the dependent variables to the expected, whatever be the gradient.
    If b=0, r also will be 0, a state of no relationsip between y and x.


    Eric C. Lind

    If you remember back to your algebra days, the slope of any line is represented by the “m” in the y=mx+b formula.
    Basically all the “m” means is that there are either more x’s for every y, more y’s for every x, or an equal number of y’s and x’s. If you have more x’s than y’s then the line is very steep. If you have more y’s than x’s then the line isn’t as steep.
    How corellation comes in is that is establishes whether a relationship exists between two variables. It doesn’t say that one causes the other however. For example, if I have a 46% corellation, then as one variable moves I should expect to see a move in the other variable as a proportion of the 1st.
    Here’s an easy way to think about corellation. Does crime cause church or does church cause crime?
    Regression however decribes a cause and effect. If we hypothesize that crime causes church, then my statistical result should be significant and account for a good portion of the variance.
    Hope that helps,



    In a nutshell,
    r = Sxy/(Sxx*Syy)^(1/2)
    b = Sxy/Sxx
    Sxy is a measure of the covariance between x and y.  (Covariance per se is Sxy/(n-1).
    r is just a normalization of Sxy, to get a measure of co-variation between -1 and +1.  b is the ratio of Sxy to the variation in x: how much variation in y is due to x.  r and b simply differ in the denominator.



    Kris, good question, more subtle than some realize, Robert and BC have provided the basic math.  Remember that if the slope is not statistically signficantly different from zero, then there is not a significant correlation between the x’s and y’s.  The algebra provided by BC and Robert make that clear, as b –> 0, r^2 –> 0 too.


    Chiranjiv Nagi

    The most interesting (and largely overlooked) parameter in a linear model is usually the slope. If the slope is zero, the line is flat, so there’s no relationship between the variables.
    The Gradient of the slope signifies the strength of variable with Y

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