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I ran a linear regression model using four regressors. The four regressors all share the same scale / unit.

Here is the regression table:

Linear Regression Table

I realize there is multicollinearity between Ethnic and Ac Age, I even tested the correlation between them (Pearson r = 0.81 with p-value<0.001).

Here is my question, I want to make sure that I am interpreting the table correctly:

1- Is it safe to say that while we know that Ethnic and Ac Age are correlated, Ethnic is "stronger", since:

  • in R3 Ac Age lost its significance but Ethnic remained strong

  • Since the scales on the regressors are the same, it makes their coefficients comparable in this instance. And since Ethnic has a much higher coefficient value, it means it has a stronger effect on Y?

  • in R2 when testing Ac Age alone, R-squared became really low (0.27).

  • between R6 and R8, R-squared did not change, so Ac Age had no effect?

2- Is it safe to say that Field and Aff. have no effect on my dependent variable (Y), since they are always insignificant (p-value<0.1, significance being <0.05)

3- That while in R2 and R7 Ac Age seem to be highly significant (p<0.01), the fact that R-squared is small makes the significance not matter?

4- Does this mean that Ac Age's results are now negligible? and shouldn't be studied further, as Ethnic and Ac Age have multicollinearity?

I'm not quite sure how to explain this in correct scientific terminology, so any help in analyzing this table is thanked.

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  • $\begingroup$ What do the columns of the table mean? What are R1, etc.? $\endgroup$ – Kodiologist Sep 24 '17 at 20:17
  • $\begingroup$ oh I thought this was standard... each column is a different run of the linear regression, but each time I use a different set of regressors... $\endgroup$ – BKS Sep 25 '17 at 5:41
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    $\begingroup$ Numbering the models "R#" and then presenting the r-squared without superscript ($R2$ instead of the proper $R^2$) feel very confusing. $\endgroup$ – Penguin_Knight Sep 25 '17 at 17:22
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  1. It's certainly true that the sample correlation of Ethnic with the dependent variable (DV) is stronger than that of Ac Age with the DV (r = .88 vs. .52). Likewise, in the regression model with all four predictors (R8), Ethnic's coefficient is much larger than Ac Age's. How these relationships would change in any conceivable regression model with other variables or interaction terms is not something one can tell in advance. Also, since the sample is relatively small, these estimates might not be very accurate. You can look at confidence intervals to get a sense of how accurate the estimates are.

  2. No. Failure to reject a null hypothesis is not evidence for the null hypothesis. In fact, the null hypothesis is almost always false.

  3. Significance (i.e., whether $p < α$) and model fit (e.g., $R^2$) are ultimately apples and oranges. Whether something matters depends on things like what the variables represent and the purpose of the analysis. But, significance rarely matters anyway, because you should've already known the null hypothesis was false before collecting any data.

  4. That, too, depends on substantive issues like the purpose of the analysis. One way to use a set of collinear variables without throwing any out is to combine them, as by conducting a principal-components analysis on them and using the first component. Since your variables are on the same scale, simply averaging them might also make sense.

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