Why would significance of F-value change in linear regression if you change the reference group? For categorical predictors with k levels, it doesnt matter what you choose as your reference group. So why would the F-value significance change in the linear regression if you change the reference group? 
Also is it possible for the F-test to not reject the null hypothesis when it really should? When does this happen?
For reference: 
          The REG Procedure
                                       Model: MODEL1
                      Dependent Variable: HDL HDL cholesterol (mg/dl)

                  Number of Observations Read                       2032
                  Number of Observations Used                       2025
                  Number of Observations with Missing Values           7


                                   Analysis of Variance

                                          Sum of           Mean
      Source                   DF        Squares         Square    F Value    Pr > F

      Model                     4     1380.42472      345.10618       1.89    0.1094
      Error                  2020         368694      182.52193
      Corrected Total        2024         370075


                   Root MSE             13.51007    R-Square     0.0037
                   Dependent Mean       51.47259    Adj R-Sq     0.0018
                   Coeff Var            26.24711











                                       The Reg Procedure
                                       Model: MODEL1
                      Dependent Variable: HDL HDL cholesterol (mg/dl)

                  Number of Observations Read                       2032
                  Number of Observations Used                       2025
                  Number of Observations with Missing Values           7


                                   Analysis of Variance

                                          Sum of           Mean
      Source                   DF        Squares         Square    F Value    Pr > F

      Model                     4     1333.76988      333.44247       1.83    0.1210
      Error                  2020         368741      182.54503
      Corrected Total        2024         370075


                   Root MSE             13.51092    R-Square     0.0036
                   Dependent Mean       51.47259    Adj R-Sq     0.0016
                   Coeff Var            26.24877

The first one is with group 1 as reference group. The second is with group 2 as reference group. Note that the two F values are different.
 A: The model needs a constant.  In its current form,
$$y =\beta x +\varepsilon,$$
with $x$ constrained to be $0$ or $1$, there are two possibilities.  First, let group "A" be coded as $x=0$ (the "reference" group) and group "B" be coded as $x=1$.  For group A the model is
$$y = \beta 0 + \varepsilon = \varepsilon$$
while for group B it is
$$y = \beta + \varepsilon$$.
Thus, it assumes the mean value of $y$ in group A is zero, but it estimates the mean of group B (as $\beta$).  Second, let group A be coded as $x=1$ and group be be coded as $x=0$.  Now the model assumes the mean value of $y$ in group B is zero but estimates the mean of group A.
Neither situation is satisfactory unless one of the group means happens to be close to $0$ (relative to the standard error of the residuals): it has no ability to estimate the mean of the reference group.  The F-tests will differ; whichever reference group has a mean closer to zero ought to yield the smallest p-value.
The solution is to fit
$$y = \alpha + \beta x + \varepsilon;$$
that is, to include a constant in the model.  Now, in group A (with the original coding of $x=0$) the model is
$$y = \alpha + \beta 0 + \varepsilon = \alpha + \varepsilon$$
and in group B it is
$$y = \alpha + \beta 1 + \varepsilon = \alpha + \beta.$$
Whence $\alpha$ estimates the mean of A and $\beta$ estimates the difference in group means.
A pooled t-test will be equivalent to the F-test here: it should yield the same p-value.
