# Re: [S] graphics device functions for S-PLUS

Erin Hodgess (hodgess@uhdux2.dt.uh.edu)
Thu, 12 Feb 1998 15:43:11 -0600

Dear Frank:

A. The only stupid question is the one that you don't ask.

B. Re: Correlation Coefficient. By rejecting the null of rho=0,
you are saying that for this particular sample, the sample r is
significantly different from zero. When you fail to reject, you
are saying that the sample r is not significantly different from zero.
That does not mean that the sample r is exactly = to zero, but it is
not sig. diff. from zero. Also, (and you mentioned this too), the
correlation coefficient measures a linear relationship only.
For instance, if you select points from a circle, calculate
the sample r and do the rho=0 test, you will almost surely
fail to reject. That doesn't mean that there is no relationship
between the points; i.e., it just means that there is no LINEAR
relationship between points.

C. Re: Linear Regression. It depends. By that, I mean you can consider
simple linear regression(one x variable, one y variable), vs. multiple
linear regression(one y variable, several x variables). In the case
of simple linear regression, when you test the beta1 = 0, you
are, in a sense, testing for model adequacy. If you reject beta1=0, you
are saying that the sample beta1 is sign. diff from zero, which means
that the regression line has a non-zero slope. If that line has a non-zero
slope, then you can efficiently use that model for explanation/prediction
purposes. If the line has a zero slope, then the best estimate for
prediction/explanation purposes is sample mean(y).

In terms of multiple linear regression, when you test some betai = 0,
you are testing for significance of that individual variable. If you
fail to reject on a particular variable, you might as well remove that
varible from the model, since it doesn't really help in your analysis.

I hope this is useful!

Sincerely,
Erin Hodgess
Assistant Professor
Dept. of Computer and Mathematical Sciences
University of Houston -- Downtown
e-mail: hodgess@uhdux2.dt.uh.edu

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I have two simple (maybe stupid?) questions about hypothesis tests, for example in correlation and regression.

1.
The significance test in correlation analysis asks if there is a significant correlation in the population, i.e., you test the null hypothesis rho = 0. If you reject the null, you conclude that there is significant correlation. However, the correlation coefficient itself indicates whether there is a correlation (if zero, there is no correlation).

If the conclusion of the hypothesis test is that the null should not be rejected, you conclude there is no correlation. But what is the difference between this conclusion and having a zero correlation coefficient in a significant correlation? Do they mean the same thing?

The question is whether failure to reject the null amounts to the same conclusion, a conclusion of no relationship?

2.
The question for regression is basically the same: does an insignificant regression imply that that there is no relationhip (of the kind specified in the regression, for example, linear) between the variables?

Of course, I understand that for correlation or linear regression, the conclusion is that there is no *linear* relationship.
Frank O'Hare
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