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Some particular independent variable has low p (high t value) as a linear regressor. That means (if I understand correctly) that I can trust it has some effect on the dependent variable, and then I look to see what effect it has. Normally I would take the coefficient from the regression output and divide it by the mean value of the dependent variable to see the effect. But the r2 of the line is very low. Does that mean I should throw the whole thing out? or how then do I see the effect of the independent variable? To put it another way, if most of the variability is unexplained by the independent variables, then is the effect properly measured by dividing the regression coefficient by the dependent variable's mean, or does one need to account somehow for the r2 when measuring the effect?

Someone has asked me for more information.

R-square    0.494
Adj R-square    0.451
Residual SD 2.107
Sample SD   2.845
N observed  194
N missing   10

        Estimate    Std. Error  t value Pr(>|t|)
0       1,614.974   290.998 5.550   < 0.0001
a       -0.118  0.406   -0.291  0.7717
b       -0.745  0.440   -1.693  0.0923
c       1.161   0.578   2.009   0.0461
d       -2.194  1.104   -1.988  0.0484
e       0.001   0.034   0.031   0.9751
f       -0.009  0.165   -0.057  0.9549
g       -1.289  0.812   -1.588  0.1141
h       0.665   0.648   1.027   0.3058
i       -0.601  0.595   -1.011  0.3132
j       -0.101  0.420   -0.240  0.8109
k       -0.466  0.947   -0.492  0.6233
l       0.057   0.395   0.144   0.8856
m       0.496   0.677   0.732   0.4648
n       -1.009  0.862   -1.170  0.2437
o       -0.037  0.007   -5.478  < 0.0001
kjetil b halvorsen
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Tabitha Gertrude Cappell
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    The coefficients only make sense if the OLS assumptions met. Have you checked? The p-value just tell you if the coefficient is statistically away from zero, nothing more. – SmallChess Mar 26 '17 at 13:15
  • @StudentT, I checked only for multi-collinearity and that the residuals' scatterplot looks independent of the regression equation values (endogeneity). Is there something else to check? – Tabitha Gertrude Cappell Mar 26 '17 at 13:32
  • You need to give us something for us to answer. Give us your summary of your model and your residual plots. Let's start with that. You may also provide a sample data set in R. – SmallChess Mar 26 '17 at 13:33
  • Another way to think about the t-value is as a scale invariant heuristic for effect size. On the other hand, the beta coefficient is not scale invariant (unless standardized *a priori*). In a multiple regression, this metric can be a proxy for the relative importance of the predictors. That the r-square is low is, of course, a statement that most of the variability in the dependent variable remains unexplained by the independent variables. – Mike Hunter Mar 26 '17 at 13:46
  • @DJohnson, I suppose that that is my question: If most of the variability is unexplained by the IVs, then is the effect size properly measured by dividing the regression coefficient by the DV mean, or do I need to account somehow for the *r*^2 when measuring the effect size? – Tabitha Gertrude Cappell Mar 26 '17 at 13:49
  • Did you come up with this effect size metric yourself, creatively, so to speak? Nothing wrong with that, but please explain how dividing the coefficient by the DV mean helps in determining this? Again, the t-value itself is scale invariant. For an excellent literature review of the various heuristics and metrics surrounding effect size and relative variable importance, see Ulrike Groemping's papers, e.g., here ... https://prof.beuth-hochschule.de/fileadmin/user/groemping/downloads/tast_2E2009_2E08199.pdf – Mike Hunter Mar 26 '17 at 13:56
  • @DJohnson, the coefficient is the amount to added to the dependent variable for each increase in the independent variable. However, I wish to know what percentage is added to the dependent variable, so I must divide by a typical value of the dependent variable, and I chose the mean. Is this wrong? Perhaps "effect" is the wrong word, though. – Tabitha Gertrude Cappell Mar 26 '17 at 13:58
  • What you want is an elasticity, a topic dealt with most effectively by economists. The actual calculation is a bit more rigorous, e.g., see Gujarati's *Basic Econometrics* which has a table of elasticities based on the functional form of the model. More easily, if you take the natural log of both dependent and independent variables, that will transform the coefficient into an elasticity where a 1% change in the IV translates into a corresponding percentage change in the DV as a function of the resulting coefficient (the elasticity). – Mike Hunter Mar 26 '17 at 14:17
  • In many fields, 0.49 would not be considered a low R-squared ... – kjetil b halvorsen Dec 15 '20 at 18:38

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