Variance-Covariance matrix interpretation

Assume we have a linear model Model1 and vcov(Model1) gives the following matrix:

             (Intercept)    latitude  sea.distance   altitude
(Intercept)    28.898100 -23.6439000  -34.1523000  0.50790600
latitude      -23.643900  19.7032500   28.4602500 -0.42471450
sea.distance  -34.152300  28.4602500   42.4714500 -0.62612550
altitude        0.507906  -0.4247145   -0.6261255  0.00928242


For this example, what does this matrix actually display? What assumptions we can safely make for our model and it’s independent variables?

This matrix displays estimates of the variance and covariance between the regression coefficients. In particular, for your design matrix $\mathbf{X}$, and an estimate of the variance, $\widehat{\sigma}^2$, your displayed matrix is $\widehat{\sigma}^2(\mathbf{X}’\mathbf{X})^{-1}$.

The diagonal entries are the variance of the regression coefficients and the off-diagonals are the covariance between the corresponding regression coefficients.

As far as assumptions go, apply the cov2cor() function to your variance-covariance matrix. This function will convert the given matrix to a correlation matrix. You wil get estimates of the correlations between the regression coefficients. Hint: for this matrix, each of the correlations will have large magnitudes.

To say something about the model in particular, we need point estimates of the regression coefficients to say anything further.