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I want to compare the regression coefficients for two variables from the same sample.

My models may look like this: $$ \hat{Y}_1 = \alpha + \beta_1 * V_1 + \beta_2 * V_2\\ \hat{Y}_2 = \alpha + \beta_3 * V_3 + \beta_4 * V_4 $$

I want to compare both $\beta_1$ to $\beta_2$ as well as $\beta_1$ to $\beta_3$.

As far as I understand, both should be possible with a paired t-test. I know that I could also look at the confidence intervals, however it would be way easier to conduct a power analysis for the paired t-test.

Is it possible to do this with a paired t-test? How would I conduct the respective statistic? Would the power be the same regardless of whether I use a paired t-test or look at the confidence intervals of the coefficients?

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  • $\begingroup$ A paired t-test requires that you have a sample of pairs that are independent of each other. As far as I can see, you only have one estimate for every $\beta$, so the paired t-test seems the wrong tool for the job. $\endgroup$ – Lewian Oct 26 '20 at 10:44
  • $\begingroup$ In regard to the comparisons I would like to make, I would have one pair (e.g., $\beta_1$ and $\beta_2$) for each person. I am quite sure that this would have to be treated as dependent samples. $\endgroup$ – Max J. Oct 26 '20 at 10:47
  • $\begingroup$ Correction: I have one pair (e.g., $V_1$ and $V_2$) per person. I only have one $\beta_1\$ for my whole sample. $\endgroup$ – Max J. Oct 26 '20 at 11:34
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I've found out that for my original answer below I misunderstood the setting. Here is something regarding testing the equality of two coefficients $\beta_1$ and $\beta_2$ in a regression. You will need to use something that is called "general linear hypothesis" in linear regression for testing $\beta_1-\beta_2=0$. I will not look this up for you because my time is limited and I had answered originally thinking that I could say something without doing some reading to remind myself.

Regarding $\beta_1$ and $\beta_3$ from two different regressions, it is important to model the dependence between the regressions, which is probably best done running a multivariate regression with outputs $Y_1$ and $Y_2$ and all four variables $V_1$, $V_2$, $V_3$, $V_4$. The again a linear hypothesis regarding $\beta_1-\beta_3$ can be tested.

The paired t-test doesn't address this problem because there is no sample of independent pairs here.

The following answer was written based on wrong understanding of the problem. I leave it here in case anybody has the problem addressed here:

Response in comment says: "I would have one pair (e.g., $\beta_1$ and $\beta_2$) for each person". Assuming that the persons don't influence each other and the $\beta$ are computed separately for each person not involving other persons' data, yes, you could take the $\beta_1,\beta_2$ for each person as a pair to be analysed by a paired t-test (have a look at your data whether there are outliers or strong skewness in the differences between the $\beta$ though). I'm also here assuming that the null hypothesis that you want to test is $\beta_1=\beta_2$. The model then implies that there are random coefficients $\beta_{1i}, \beta_{2i}$ for each person $i$ with a fixed expected value of $\beta_{1i}-\beta_{2i}$, zero under the null hypothesis. This may make sense, although I obviously don't know the details of what you're doing.

Note that this involves a dependence (namely between $\beta_1$ and $\beta_2$ of the same person) and also an independence assumption (between different persons).

I'm not sure what exactly you mean by "looking at confidence intervals of coefficients" - you mean for every single person? This will give you twice as many confidence intervals as you have persons (as long as you're only looking at $\beta_1$ and $\beta_2$) - I'm not quite sure how you'd interpret that. Also these confidence intervals will not take dependence between the different $\beta$-coefficients into account.

"How would I conduct the respective statistic?" You'd just run it, with the $\beta_{1i}, \beta_{2i}$ for each person $i$ as paired values (which actually amounts to running a one-sample t-test on $\beta_{1i}-\beta_{2i}$).

As long as there is no specific knowledge on how your two regressions within a person depend on each other (or not), the same holds for comparing $\beta_1$ and $\beta_3$ etc.

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  • $\begingroup$ Thank you for the reply. Your description of my problem seems accurate. If I run my regression analysis, I will only obtain one estimate (including a SD value) for each $\beta$ instead of individual $\beta_i$ for each person. So how would I then compute a t-statistic? $\endgroup$ – Max J. Oct 26 '20 at 11:09
  • $\begingroup$ As I wrote before, if you have separate $\beta$-values for each person (which you have if my description is accurate), do it as described. If you only have one $\beta_1,\beta_2$ etc. estimate overall, the paired t-test cannot be used. So I'm confused about what your situation actually is because on one hand you confirm my description but on the other hand you write as if it is something else. Also in your first reply you had written that you have $\beta$-values for each person!? $\endgroup$ – Lewian Oct 26 '20 at 11:26
  • $\begingroup$ I am sorry if that was misleading. My first reply was incorrect. I do have one pair per person, however these are pairs of the variables themselves (e.g., $V_1$ and $V_2$). When running my regression, I end up with one $\beta$ per variable for my whole sample. As all the $\beta$-values describe the influence of variables from the same sample, I would still think that there is a dependency between two betas of the same person. That's why I was thinking about a paired t-test. $\endgroup$ – Max J. Oct 26 '20 at 11:32
  • $\begingroup$ I'm not sure what you mean be "two betas for the same person" if your betas are not defined individually for the persons. Maybe my updated answer can help. $\endgroup$ – Lewian Oct 26 '20 at 11:44
  • $\begingroup$ Thank you for your help! I am quite sure that there is a simple solution to this. As I mentioned in my question, it should basically suffice to check the confidence intervals of the respective beta estimates. I am still wondering if there is a way to do this using a t-test, for the sake of being able to easily conduct a power-analysis. $\endgroup$ – Max J. Oct 26 '20 at 12:10

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