Same SE for all coefficients of a linear model I am trying to find shellfish densities for a field-based aquaculture program and wanted to understand the impact of reef structure and site on the response.  This experiment only has a sample size of 36. I looked at the distribution and log-transformed it for normality before running a linear model:
model<-lm(data$logged ~ data$reef + data$site)

My output looks like this:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  10.2593     0.1599  64.179  < 2e-16 ***
data$reef2   -1.2147     0.1958  -6.204 1.06e-06 ***
data$reef3   -1.4936     0.1958  -7.629 2.61e-08 ***
data$reef4   -2.6405     0.1958 -13.487 9.03e-14 ***
data$reef5   -1.0867     0.1958  -5.550 6.18e-06 ***
data$reef6   -0.8922     0.1958  -4.557 9.30e-05 ***
data$siteB   -0.6314     0.1384  -4.561 9.21e-05 ***
data$siteC   0.2567     0.1384   1.854   0.0743 .  

Residual standard error: 0.3391 on 28 degrees of freedom
Multiple R-squared:  0.8943,    Adjusted R-squared:  0.8679 
F-statistic: 33.85 on 7 and 28 DF,  p-value: 4.937e-12

What I don't understand is why my SE value are the same for all of my coefficients. Can someone help me understand how to fix this issue?
Thanks.

The (usual) equation for the variance of a regression parameter (not the intercept) $\hat\beta_j$ is as follows.
$$
\widehat{\text{var}}\left(
\hat\beta_j
\right)
=
\dfrac{
s^2
}{
(n-1)s^2_{X_j}
}\times\dfrac{1}{
1-R^2_j
}
$$
$s^2$ is the residual variance.
$n$ is the sample size
$s^2_{X_j}$ is the variance of the feature $j$
$R^2_j$ is the variance-inflation factor (VIF) for feature $j$, which is calculated by running a regression of feature $j$ on the other features and calculating the $R^2$ of that regression.
To get your observed result, one of the following must have occurred.

*

*The change in $s^2_{X_j}$ perfectly offset the change in the VIF for every variable


*All $s^2_{X_j}$ and all $R^2_j$ are equal.
(Perhaps there could be a combination of the two where some changes in $R^2_j$ are offset by changes in $s^2_{X_j}$ and the rest of the features have the same $s^2_{X_j}$ and $R^2_j$.)
Possibility #1 requires a lot of coincidences. On the other hand, for a designed experiment where all of the features have equal variance and are independent of each other, possibility #2 will occur.
And since not all of your slope coefficients are equal, apply the above logic to the groups of coefficients that do have equal coefficients.
The equation for $
\widehat{\text{var}}\left(
\hat\beta_j
\right)
$ is available on the VIF Wikipedia article, which has references to other material.
