Test statistic for correlation vs for slope coefficients

[SpareTime]

Why is the formula for the test statistic for correlation so much different than for others?

The test stat for the correlation coefficient is t = r(n-2)^1/2 / (1-r^2)^1/2

While later in example 7 we see that t = bhat - b1 / standarderror of bhat. This one makes sense to me because it represents how many standard errors away from the value we’re testing it against. But what in the heck is that first formula? Why isn’t it the same as the second?

What’s the difference between a correlation coefficient and a slope coefficient?

It’s only linear regression and I’m already having quant issues…ugh.

[tickersu]

@SpareTime

Why is the formula for the test statistic for correlation so much different than for others?

The test stat for the correlation coefficient is t = r(n-2)^1/2 / (1-r^2)^1/2

It’s actually the exact same thing as the formula below… if R is the hypothesized correlation (true) and r is the sample then we can see —> t-stat = (r-R) / (standard error for r) what the CFAI books lack is a definition of the standard error for r. It’s [(1-r^2)/(n-2)]^(0.5) …

So, if we’re testing a null of R=0 then (r-0)/ (s.e. of r) = (r-0)/ [(1-r^2)/(n-2)]****^(0.5)

This is the same as t-stat = [r * (n-2)^0.5]/ ([1-r^2]^0.5)

While later in example 7 we see that t = bhat - b1 / standarderror of bhat. This one makes sense to me because it represents how many standard errors away from the value we’re testing it against. But what in the heck is that first formula? Why isn’t it the same as the second?

What’s the difference between a correlation coefficient and a slope coefficient? Sorry, missed this at first. The correlation coefficient is a bounded indication as to the strength (how closely do they move together) and direction of the linear relationship between the two variables. The slope coefficient is a (partial) effect of how the DV changes in response to (1 unit) changes in the explanatory variable (ceteris paribus). Mathematically, the slope is the covariance of y and xi, standardized by the total variation in xi. The correlation is the covariance of xi and y divided by the product of the standard deviations of x and y. Correlation is contained within a slope calculation, but the slope is not contained within the correlation.

If the correlation of xi and y is +1, then they always move together in the same direction. If it is -1, then the always move together in opposite directions**. In between -1 and +1, there can be some “breaks” in the co-movement “pattern”. If you look at a scatterplot and see data points tightly clustered in a direction, the linear relation is strong. However, if the points are loosely scattered around, but in a general direction, the correlation will not be so strong. If the correlation is 0, there is no linear association. This could mean a perfect (or less than perfect) non-linear relation, or just a random scattering of the data**.

To the slope, though, we aren’t looking at how closely the variables co-move. Rather, we are looking at how sharply (and in what direction) the DV changes with respect to changes in the regressor (steepness of the line, covariance standardized by variance of regressor).

Hope this helped!

It’s only linear regression and I’m already having quant issues…ugh.

[S2000magician]

@SpareTime

What’s the difference between a correlation coefficient and a slope coefficient?

I wrote an article on beta that illustrates the difference between a correlation coefficient (ρ) and a slope coefficient (β): http://financialexamhelp123.com/beta/

It doesn’t specifically explain why the test statistics are so different (mainly because it’s a Level I article, not a Level II article), but it does illustrate that there’s a significant difference between correlation and slope, so it’s not surprising that there would also be a significant difference in their test statistics.

[S2000magician]

@tickersu

If the correlation is 0, there is no linear association. This could mean a perfect (or less than perfect) non-linear relation, or just a random scattering of the data.

Or it could mean that the independent variable is constant (which means that there is a (perfect) linear relation: one whose slope is zero).

[tickersu]

@S2000magician

@tickersu

If the correlation is 0, there is no linear association. This could mean a perfect (or less than perfect) non-linear relation, or just a random scattering of the data.

Or it could mean that the independent variable is constant (which means that there is a (perfect) linear relation: one whose slope is zero).

Thanks for the catch!

[tickersu]

@S2000magician

@SpareTime

What’s the difference between a correlation coefficient and a slope coefficient?

I wrote an article on beta that illustrates the difference between a correlation coefficient (ρ) and a slope coefficient (β): http://financialexamhelp123.com/beta/

It doesn’t specifically explain why the test statistics are so different (mainly because it’s a Level I article, not a Level II article), but it does illustrate that there’s a significant difference between correlation and slope, so it’s not surprising that there would also be a significant difference in their test statistics.

I’m only seeing a real difference in the standard errors (which differ for different estimators) used in the test statistics. Other than that, they’re the same: (observed value - hypothesized)/(s.e. of the estimator)

Is there something else I’m missing?

[jounin83]

Hi Sparetime,

Just to notify…

EOC no 21 is one of the application this formula.

to determine wheter sample corr Stellar vs CPIENG is SIGNIFICANT, the curriculum using the formula :

t = r(n-2)^1/2 / (1-r^2)^1/2

result : -2.3017

for ONE indep variable the t value should be the SAME (in this case slope for CPIENG)

[SpareTime]

Thanks so much everyone. I think I have it down now.

One thing that really frustrates me is when a formula is given with no explanation. When they give me a totally different way to calculate the t-test, it throws my thinking way out of wack. Now that I understand it’s really the same thing, it’s just that the standard error is calculated differently and this certain formula is showing that explicitly, I understand. I wish they would have thrown it in there - it would take all of a couple sentences.

And now I realize that I was mixing up the strength of the relationship with the change in the DV for 1 unit change in the IV. I could have a very large slope with a weak correlation, or vice versa.

Thanks again!

[S2000magician]

You’re quite welcome.

[tickersu]

@SpareTime

Thanks so much everyone. I think I have it down now.

One thing that really frustrates me is when a formula is given with no explanation. When they give me a totally different way to calculate the t-test, it throws my thinking way out of wack. Now that I understand it’s really the same thing, it’s just that the standard error is calculated differently and this certain formula is showing that explicitly, I understand. I wish they would have thrown it in there - it would take all of a couple sentences.

And now I realize that I was mixing up the strength of the relationship with the change in the DV for 1 unit change in the IV. I could have a very large slope with a weak correlation, or vice versa.

Thanks again!

Glad to help!