# How do you interpret a low coefficient yet statistically significant with a high R-squared?

Consider the data below. (Dependent Variable: Human Development Index (HDI))

           Coefficient   Std. Error   t-ratio   p-value
const        -0.260523      0.129316     2.0146    0.05693  *
LITRATE       0.0031374    0.000965695   3.2488    0.00384  ***
LIFEXP        0.0051052    0.00268771    1.8995    0.07133  *
GINI          0.00598775   0.00175813    3.4057    0.00266  ***

R-squared           0.826974
S.D. dependent var  0.053285
S.E. of regression  0.023695
Sum squared resid   0.011791
Durbin-Watson       0.618921


How do you interpret these results? Given the independent variables are significant yet it result to a small coefficient, but the R-squared is high?

• This is hard to answer without some context about what these variables are, etc., but I can say that the values of the coefficients are entirely scale dependent so it's not clear that these coefficients are large, or small, or what. For example, if typical values of LITRATE were on the order of $10^6$, then the coefficient values actually seem pretty large. – Macro Aug 28 '13 at 17:41
• Isn't HDI computed via measures of education, health, and economic equality as inputs? If so, it seems like you're just confirming that HDI tracks with the same variables that HDI purports to measure -- that is, using a thermometer to "predict" temperature. – Sycorax says Reinstate Monica Aug 28 '13 at 17:42

$R^2$ is easily misinterpreted (see this excellent answer: is-r2-useful-or-dangerous). Note that $R^2$ is a function of several things: the marginal variance of $Y$, the residual variance, and both the slope of the relationships between your $X$s & $Y$ and how spread out your $X$s are. Consider the figure below:

The underlying data generating process is the same: the data were generated with the same slope and intercept, and with the same residual variance. Only the amount by which the $X$ values are spread out differs. However, the $R^2$ changes as we move from left to right. The three observed values are: $0.02$, $0.10$, and $0.24$. If I were to spread the values far enough apart, I could get the $R^2$ to asymptotically approach $1.0$.

Thus, how small the coefficients are is not sufficient to determine how large $R^2$ can be.

Further notes: @Martyn's candidate solution is popularly held, however, it won't necessarily answer your question, unfortunately. You will simply lose information about how far your $X$ values are spread out in terms of the original units. In addition, I don't know about the nature of the Human Development Index, but if @DJE is correct, the whole issue would be misconceived.

• +1. But I upvoted Martyn's answer, too: you don't lose any information about the distribution of $X$. (There isn't any in the output in the first place.) And I think that other answer comes closer to the heart of the matter, which is that unstandardized coefficients scale with the units of measurement. – whuber Jan 8 '14 at 23:53
• @whuber, I'm not sure we're saying anything different, although maybe I had phrased it poorly. I edited the footnote to make it clearer. I hadn't meant "distribution" in terms of (say) normal vs uniform, but only max(x)-min(x) is 1 km or 1 cm. – gung - Reinstate Monica Jan 9 '14 at 0:07

In order to get an idea of scale you may want to produce standardised coefficients by converting each of your predictor variables to a z-score. They will be easier to compare and will all be on the same scale of standard deviations.

Others have suggested standardized coefficients as a solution, which is one good approach, but they did not explain why you are seeing what you are seeing.

Remember that the interpretation of the slope is: the average amount of increase in y for a 1 unit increase in x (holding the other variables constant). This depends on the units for both x and y.

Imagine a plant that on average grows 1 foot (additional height) each year (on average). So if you measure a bunch of plants at different times and fit a regression model with y being the height in feet and x is time in years then you will see a coefficient of about 1.

But what if we measure time in months instead of years (but keep height in feet), then the slope will be about 1/12. On the other hand if we keep x measured in years, but measure height in inches instead of feet then the slope will be about 12. Go even further and measure time (x) in seconds and height (y) in miles and you will see a slope that is extremely small (about 1/(365*24*60*60*5280)). If you measure time in centuries and height in millimeters then the slope will be huge.

But in all of these cases (assuming no rounding errors) the relationship between the 2 variables will be exactly the same, things like R-squared and the significance level will not change (the units all cancel out in the calculations).

The reason using standardized coefficients helps is because it changes the units to standard deviations, puts them on a more comparable scale. Another approach is to just change your units to something that may be more meaningful or easier to understand. Or just accept that the scales/units are such that you would expect a small coefficient.

I would also suggest that you look for the standardized coefficients, not the coefficients themselves. They are also called (Beta) in statistics software packages.

• Thank you, Niousha. By including "also" you are indicating your answer somehow goes beyond @Martyn's suggestion last August, but you seem to be saying exactly the same thing. How does your answer differ? – whuber Jan 8 '14 at 20:04