The most important thing to do is for you to check if the model makes sense. You have fit a linear model to three continuous predictors, you need to make sure that it makes sense to do so You should look at scatterplots of
y, and adjust the fits of these predictors if needed.
Assuming fitting these predictors linearly is reasonable, fitting the full model with all four predictors is a sensible thing to do.
You have only
25 data points. If you go on a long search through the space of all models (adding and removing variables) you have an extremely high risk of false positives. So, I don't think there is much need to backwards select out variables; if you wish to do so, make sure you use cross validation to make sure doing so improves the fit of the model to unseen data.
The same thing applies to a search for interactions, you have little data, and you are running a large risk of false positives.
If you wish to make inferences using the estimated confidence intervals, you should additionally check a plot of the residuals vs. the fitted values of the model and make sure you do not see any patterns. You're looking to see if they look like they could have been drawn from a normal distribution with constant variance. If this looks reasonably consistent with yor data, then you can make inferences about the
graduation parameter using the linear model
Estimate Std. Error t value Pr(>|t|)
(Intercept) 140.1689 8.3191 16.849 2.77e-13
x1 -1.1428 0.1904 -6.002 7.22e-06
x2 -0.4699 0.1866 -2.518 0.0204
x3yes 2.2259 4.1402 0.538 0.5968
x4 1.2673 1.4922 0.849 0.4058
x3 variable measures graduation, and its parameter lies well within the error of its estimation. So, given that everything above checks out, the data you used to train the model is not inconsistent with the effect of
graduation being indistinguishable from noise.
Thanks so are we really able to judge this just from fitting the full model?
As long as all the caveats are met, I do think the best way to go about this is to fit the full model, and make your inference from that. Like I said, any inference you draw from a model that does variable selection is likely to occurr by chance.
Another way to think about this is, if you go through a variable selection algorithm, the standard errors reported in the model are no longer correct, they are actually much larger than what is reported. To estimate the true standard errors of the parameter estimates under a selection / fitting procedure, you would need to use either nested cross validation or a bootstrap + cross validation. This would drive your data very, very thin, and incur a lot of variance (you are making lots of decisions, each has a chance to be wrong). Your standard errors would be enormous.