Which is the strongest prediction in linear regression?

Question

Assume a linear regression with metric predictors: y ~ x1 + x2 + x3

Assume all x are significant predictors.

Now I want to find out if predictors differ from each other, that is, if one predictor is a stronger predictor of y than the others, and on top of it, if it is significantly stronger.

Results:

X | Unstd. B | Std. Beta | t value | p value | CI lower | CI upper

1--- .140----- .170 ------ 9.806 --- .000 --- .112 --- .168 ------

2--- .022----- .035 ------ 2.252 --- .024 --- .003 --- .041 ------

3--- .256----- .152 ------ 9.898 --- .000 --- .210 --- .302 ------

Row 1 = x1, row 2 = x2, row 3 = x3. From the p values x2 is the weakest predictor, and also has the lowest std beta weight. But is it significantly weaker than the other predictors? Which of the other two is stronger?

EDIT:

So that we can understand this question, please tell us what it means for predictors to be "strong" or "weak."

I am predicting impairment/disability by different symptoms of a disorder, and want to find out whether one symptom is associated with more impairment than the other symptoms, that is, whether it explains more variance of impairment than the others, although all symptoms are significant predictors. It's about the degree of prediction, not about significance.

Answer 1

I will decompose your question into two parts.

Which is the strongest predictor?

I assume the following form

$Y=\beta_1 X_1+\beta_2 X_{2}+\beta_3 X_{3}$

which does not include a bias. Now after fitting the $\beta$'s we want to find out the strength of each predictor. We will define this as

$S_1=\beta_1\sum_{i=1}^N x_{1,i}$

for the strength of $X_1$ where the lowercase symbols are the realized values present in your dataset. $N$ is the number of data points in your dataset. The same holds true for $S_2$ and $S_3$. The relative strength of predictor $X_1$ can then be defined as $\frac{S_1}{\sum_{j=1}^3 S_j}$.

Is one predictor significantly stronger/weaker than the others?

I don't believe there is an easy answer to this questions since it's hard to decide what significantly stronger/weaker means in this context. My advice is to seperate significance and strength and discuss them seperately.

I hope this helps you.