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Is there any way to interpret the standard error of dummy variables created to represent a categorical variable in a linear regresion model? I saw a great mathematical explanation here, but I am still having trouble with the "real-life" interpretation, if any.

As an example, let's say that this is the result of a case study:

> summary(results)$coefficients

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  31.35750    2.32037  13.514 1.55e-14 ***
sexwoman     -0.53100    1.77624  -0.299   0.7670 

Representing the average sallary of employees of a company (in thousands) based on their sex. How would we interpret this standard error?

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1 Answer 1

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The underlying math is that the intercept ($\beta_0$) indicates the average salary of men, in \$1000; the parameter $\beta_1$ indicates the difference between the average woman's and average man's salary. So we could say something like "on average, women make \$530 less than men (with a standard error of \$1,776)". The $\pm 2 \mbox{SE}$ confidence intervals on the difference range from women's salaries being \$2,306 less to \$1,246 more than men's.

$\pm 2 \textrm{SE}$ is a commonly used shortcut; it is a little wider than the Normal-based 95% confidence interval ($\pm 1.96 \textrm{SE}$) and is a good approximation when your residual degrees of freedom (number of observations minus the number of model parameters) is moderate. In particular, the SE multiplier (which is can be computed in R via qt(0.975, df) for a given df) is:

  • 1.96 (actually 1.9599) as df $\to \infty$
  • 2.008 for df = 50
  • 2.086 for df = 20
  • 2.228 for df = 10
  • 2.571 for df = 5.

and so on. The SE multiplier that I back-calculated from your comments is approximately 2.6, so my guess is that you had approximately 5 residual df (the residual df are displayed in the summary() output, but you haven't shown us ...)

Interpreting dummy variables associated with factors can get considerably more complicated when different contrasts, and interactions among predictors, are present, but your case is simple.

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  • $\begingroup$ I see. And this confidence interval is not related to the 95% confidence interval that we get from confint(results), correct? $\endgroup$ Commented Jun 7, 2022 at 20:38
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    $\begingroup$ No, it's approximately the same as what you get from confint(results). Rather than use $\pm 2 \textrm{SE}$, confint will use a multiplier based on a t distribution with the residual degrees of freedom (number of observations - 2 in this case) , which will be only approximately equal to 2. $\endgroup$
    – Ben Bolker
    Commented Jun 7, 2022 at 20:43
  • $\begingroup$ This does not really agree with my data, as >confint(results) results in the following: 2.5 % 97.5 % (Intercept) 26.625080 36.0899220 sexwoman -4.153664 3.0916739<br> However, this may be due to the fact that I had some more variebles in this case study that i did not want to bother you with. Thank you very much for your help! $\endgroup$ Commented Jun 7, 2022 at 20:50
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    $\begingroup$ Just read the edit. So helpful. Thank you very much! $\endgroup$ Commented Jun 7, 2022 at 20:58

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