I am running some linear regression models in R using the
lm function and calling the
summary() function on my linear model gives a nice summary of the degrees of freedom, F-statistic, sum of squares, etc.
Call: lm(formula = y ~ a + b + c + d, data = my_data) Residuals: Min 1Q Median 3Q Max -0.0313491 -0.0034180 -0.0000105 0.0030041 0.0306064 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.004e-04 3.396e-04 0.362 0.717 a -2.739e-02 0.893e-02 -1.447 0.149 b 5.019 5.127e-01 19.249 <2e-16 *** c 7.090 2.954e-01 31.171 <2e-16 *** d 4.241 2.760e-01 22.978 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.006236 on 354 degrees of freedom (1 observation deleted due to missingness) Multiple R-squared: 0.881, Adjusted R-squared: 0.8797
Now, this gives p-values / significance tests based on the null hypothesis that: $$H_0: \mu_1 = \mu_2 =\mu_3 =\mu_4 = 0$$
Therefore, since each p-value is statistically significant at $\alpha=0.05$, we can reject the null hypothesis and conclude there is a linear relationship between each variable and the response variable.
Now I am interested in determining if one variable is statistically different from another variable, i.e. that it my model can be simply explained by just $b$ or just $b$ and $c$, etc. (I guess this is the same as determining if a two variables are co-linear but I am uncertain.)
What tools can I use clarify this and find the most significant model?