# Interpreting non-significant regression coefficients

Out of seven, six of the independent variables (predictors) are not significant ($$p>0.05$$), but their correlation values are small to moderate. Moreover, the $$p$$-value of the regression itself is significant ($$p<0.005$$; Table 2).

I understand in a partial-least squares analysis or SEM, the weights (standardized coefficients in Table 1) are considered rather than the correlation coefficient $$r$$ (Table 4). I am trying to achieve the same using multiple regression analysis.

1. Should I ignore the var that are non-significant in the coefficient table?

2. Do we account for significance or non-signficance from the corresponding 1-tailed sig in Table 4 (correlations) for each variable or should we consider the 2-tailed sig in Table 1 (coefficients)?

I am planning to investigate how each variable in a framework are related to each other (directly and indirectly) using multiple regression. Kindly advise.

• I understand that the path coefficients are the beta weights (standardized coefficients) in the coefficient table, between the independent var and the dependent var. Do we actually make use of the values of Pearson Correlation in a Path Analysis run through Multiple Regression? – Vyas Sep 26 '18 at 3:14
• Please include the output you are referring to. – Frans Rodenburg Sep 26 '18 at 11:30

I'll try to address your questions in order, but I don't think this is the right approach, so you may also skip to the third quote.

Should I ignore the variables that are non-significant in the coefficient table?

Non-significant results are also results and you should definitely include them in the results. However, you should not focus too much on what the implications of their estimated coefficients might be. Namely, their large standard errors (or similarly: high $$p$$-values) suggest that you might as well have observed an effect this large if the true effect were zero.

Do we account for significance or non-signficance from the corresponding 1-tailed sig in Table 4 (correlations) for each variable or should we consider the 2-tailed sig in Table 1 (coefficients)?

Table 1 shows the estimated coefficients of your explanatory variables. While bearing in mind that no causal relationship has been demonstrated, you can interpret significance here as: Does a unit change in this explanatory variable correspond to a significant change in the response variable?

Table 4 appears to show the correlations of your fixed effects. Whether or not your explanatory variables strongly correlate to each other says more about whether you might have potential problems with estimating this model, rather then their effect on the outcome. Significance here could mean that you have collinearity issues, but there are better ways to diagnose collinearity.

Now as to why I don't think you can best answer your research objective with multiple regression:

I am planning to investigate how each variable in a framework are related to each other

Unless you have a variable that can clearly be considered the outcome of the others, and you have some idea of which interactions to test for, I don't think multiple regression is the way to go here. Using multiple regression, you would have to regress all variables on all other variables and interpret a multitude of output tables. You are almost guaranteed to find spurious correlations and I doubt any $$p$$-values would be significant after correcting for multiple testing.

If you really want to use multiple regression, I suggest you forget about significance and instead construct a set of confidence intervals using the reported standard errors in table 1. You should clearly state that the goal is exploration and then you can propose which variables might correlate with which. A future study could then try to confirm/refute these findings.

Instead, you might be interested in graphical models: In brief, you can find the partial correlations between variables by standardizing the precision matrix (the inverse of the covariance matrix).$$^1$$ Using a form of regularization (e.g. LASSO), you can shrink the smallest partial correlations to zero, such that variables with zero partial correlation can be considered conditionally independent. The remaining non-zero partial correlations can then form an undirected graph, which gives you a single, intuitive representation of which variables 'interact' with one another.

$$^1$$ This also has the interpretation of regressing all variables on each other, but with a single resulting network to interpret the results with.

I don't know of any SPSS implementations, but you can download $$\textsf{R}$$ for free and use the glasso package (or try rags2ridges for a ridge regularization approach).

• Friedman, Jerome and Hastie, Trevor and Tibshirani, Robert (2008): "Sparse inverse covariance estimation with the graphical lasso"
• Thank you for the elaboration. Given my limited knowledge to conduct anlayses beyond Multiple Regression, I believe it may be a wiser approach to explore the relationships among the Independent Var. By ignoring p-value > 0.05, do you imply to focus only on the beta standardized coefficient from Table 1? How do we construct a set of confidence intervals using the reported standard errors in table 1? Please provide an example. (2) How complicated is it to run a PLS and analyze the output in SPSS which I have access to? Can we use more than one dependent var in PLS? Thanks again – Vyas Sep 27 '18 at 2:18
• @Vyas You're welcome! I don't use SPSS, but you could calculate $t$-based confidence intervals as $\hat{\beta} \pm t_{1-\frac{\alpha}{2}, \text{df}} \cdot \text{SE}$. If you have enough degrees of freedom, this is almost the same as a normal-based confidence interval: $\text{Estimate} \pm 1.96 \cdot \text{standard error}$. – Frans Rodenburg Sep 27 '18 at 2:31
• Also note that if you have new questions, you should post them separately, the comments are only for clarification. – Frans Rodenburg Sep 27 '18 at 2:32
• Hi Frans, are you referring to the confidence intervals generated (coefficients first table as shown above) relating to each variable? Looking at each confidence interval value, except for BX_Adv_HQ variable, I note that the confidence intervals include zero, meaning that we can confidently state that there is no relation between X and Y. Is this how I can interpret the multiple regression results using confidence interval? Thank you – Vyas Sep 27 '18 at 13:16
• Almost. You can state that there is no significant relationship between variables which CI includes the null-hypothesis ($\text{H}_0:\, \beta_j = 0$). However, you cannot confidently conclude that there is no relationship at all, because that isn't what the confidence interval shows. In fact, the true $\beta$ could be any value on (or even outside) the confidence interval. Have a look here about inference about the null-hypothesis: stats.stackexchange.com/q/85903/176202. If you want to demonstrate that a variable has no effect, you would need a test of equivalence. – Frans Rodenburg Sep 28 '18 at 1:14