# 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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- May 7, 2008 at 1:00 pm #50023
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?

Kris0May 7, 2008 at 2:46 pm #171812

Robert ButlerParticipant@rbutler**Include @rbutler in your post and this person will**

be notified via email.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.0May 12, 2008 at 5:15 am #171897

BulusuParticipant@ramachandrudu**Include @ramachandrudu in your post and this person will**

be notified via email.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.0May 12, 2008 at 11:57 am #171902

Eric C. LindParticipant@Eric-C.-Lind**Include @Eric-C.-Lind in your post and this person will**

be notified via email.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,

Eric~

0May 12, 2008 at 1:40 pm #171905Kris,

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.0May 12, 2008 at 2:00 pm #171906Kris, 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.

0May 14, 2008 at 10:04 pm #171977

Chiranjiv NagiParticipant@Chiranjiv-Nagi**Include @Chiranjiv-Nagi in your post and this person will**

be notified via email.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 Y0 - AuthorPosts

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