Insignificant slope with statistically significant r value

Question

I collected data of 60 countries to identify whether there is a relationship between average income per person and average life expectancy. However, I am having some trouble interpreting my lin-reg and r value.

y = 0.000437x + 67.68
r = 0.814

Does this inconsistency have to do with the nature of my data? Considering that that the data for income is in the 10-30 thousands, while the data for life expectancy is all below 100, would that influence the reliability of the persons r correlation test? If so, how?

Also, I have checked and double checked both calculations a number of times, on the calculator, manually, on excel and through an online calculator and I still get the same answers.

If someone could help me interpret this, that would be very helpful.

Answer 1

First, please make sure you're not using "insignificant" to describe 0.000437 being very small. The word "significant" is associated with hypothesis test and p-value and if you mix this up with the magnitude of the regression coefficient, people will be very confused.

Here is a formula that can explain your problem:

Given a regression mode $y = \beta_0 + \beta_1 x$,

$\beta_1 = \rho \frac{s_y}{s_x}$

Where $\rho$ is the Pearson's correlation coefficient, $s_y$ is the standard deviation of $y$, and $s_x$ is the standard deviation of $x$.

You can see that, more than often, your $\beta_1$ will not be exactly equal to $\rho$. The only time it will happen is when both your dependent and independent variables have the same standard deviation. So, it's very understandable that you can have a very low $\beta_1$ while a very high $\rho$.

Second, the magnitude of $\beta_1$ can be easily manipulated. For example, if you recode 1000 to 1, and change the unit from "dollar" to "thousand dollar," the $\beta_1$ will change. Instead of comparing the $\beta_1$ to your correlation coefficient, think if your $\beta_1$ makes sense. For example, why would you expect the $\beta_1$ to be close to 0.8? If some country increases its national income by 1000 dollars, would it make sense that life expectancy increases by 800 years? It can't be. In fact, your 0.000437 is much more realistic.

Answer 2

Based on the formula or definition of Pearson r correlation coefficient. http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient

So the scale of the variables won't matter since it is divided by Std. For test of significance, we need to check the assumptions that we make. The simplest checking method in this case is plot the data to visually check whether they have certain linear relationship.