How can I choose between these two regression models?
R
outputs:
Regression 1
Call:
lm(formula = log.h ~ year)
Residuals:
Min 1Q Median 3Q Max
-1.24004 -0.45221 -0.05301 0.44744 1.42060
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.588e+02 1.565e+01 -10.15 9.47e-13 ***
year 8.809e-02 7.859e-03 11.21 4.65e-14 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.6396 on 41 degrees of freedom
Multiple R-squared: 0.7539, Adjusted R-squared: 0.7479
F-statistic: 125.6 on 1 and 41 DF, p-value: 4.654e-14
Regression 2:
Call:
lm(formula = log.h ~ year + I(year^2) + I(year^3))
Residuals:
Min 1Q Median 3Q Max
-1.4044 -0.3966 0.1141 0.3447 1.2321
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.075e+06 4.663e+05 -2.304 0.0266 *
year 1.622e+03 7.027e+02 2.308 0.0264 *
I(year^2) -8.162e-01 3.529e-01 -2.313 0.0261 *
I(year^3) 1.369e-04 5.909e-05 2.317 0.0259 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.58 on 39 degrees of freedom
Multiple R-squared: 0.8075, Adjusted R-squared: 0.7927
F-statistic: 54.54 on 3 and 39 DF, p-value: 5.093e-14
Thoughts: The point is the first model is good, $R^{2}$ is good, p-values are great, but there are hints of heteroskedasticity of residuals vs predicted values.
The second one has a better adjusted $R^{2}$, a great $p$-value of regression, and the scatterplots have got better (not drastically, but better). On the other hand, the $p$-values of the coefficients are a lot higher than before.
Considering all this, how can I choose between them?
h
is,it doesn't make sense adding square and cubic of time variable without any theoretical justification. $\endgroup$