Why do parameter estimates change when a different reference is set in generalized linear models? Generating a binomial regression with a response as Survival and predictors Time, Life.Stage and Trial, output in r provides comparisons of each factor level for Life.Stage  "B" , "C" and "D" and compares to the intercept estimated using "A" as a baseline. 
Eg: 
mod=glm((cbind(Alive, Dead))~Time+Life.Stage+Trial, data=, family=binomial(link="logit"))
Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)      4.8883     0.5361   9.118  < 2e-16 ***
Time            -1.5748     0.1886  -8.352  < 2e-16 ***
Life.StageD     10.9599     1.5906   6.891 5.56e-12 ***
Life.StageC      7.9772     1.3710   5.818 5.94e-09 ***
Life.StageB      5.2570     0.9619   5.465 4.63e-08 ***
factor(Trial)3   0.1628     0.4426   0.368    0.713    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 507.691  on 42  degrees of freedom
Residual deviance:  67.457  on 37  degrees of freedom
AIC: 104.44

Number of Fisher Scoring iterations: 7

Why, when I change the reference level to "B" or any other level, do the parameter estimates change?
I've referred to a similar question with an anova, however the explanation is a bit beyond my understanding. 
The ultimate goal of my analysis is to determine whether Survival at each Life. Stage is different than every other Life.Stage.
 A: Mathematically: 
Because the model matrix is different. The coefficients are calculated from the model matrix. 
Specifically, Let $X$ to be the model matrix. We calculate $\beta$ by doing (assuming linear regression squared loss, BTW this is not a glm specific problem but more general). If $X$ changes, and $y$ is the same, the $\beta$ will change.
$$
\text{minimize} ~~\|X\beta-y\|^2
$$
Here is a simplified example: linear regression on mtcars, with cyl as categorical predictor. You can observe after relevel, the model matrix has been changed.
> mtcars$cyl=factor(mtcars$cyl)

> head(model.matrix(mpg~cyl,mtcars))
                  (Intercept) cyl6 cyl8
Mazda RX4                   1    1    0
Mazda RX4 Wag               1    1    0
Datsun 710                  1    0    0
Hornet 4 Drive              1    1    0
Hornet Sportabout           1    0    1
Valiant                     1    1    0

> mtcars$cyl=relevel(mtcars$cyl,2)

> head(model.matrix(mpg~cyl,mtcars))
                  (Intercept) cyl4 cyl8
Mazda RX4                   1    0    0
Mazda RX4 Wag               1    0    0
Datsun 710                  1    1    0
Hornet 4 Drive              1    0    0
Hornet Sportabout           1    0    1
Valiant                     1    0    0

Intuitively: 
In addition, the coefficient change after relevel is also very intuitive. Because we are comparing with different base level. In the example above, without relevel we are comparing a car with 4 cylinder. After,  we compare with a 6 cylinder car. If we change the reference, intuitively how much it impacts to MPG will be changed. 
For example, I have a car with 8 cyl, if the coefficient describes how it compare to 4 cyl car, the coefficient will be different when we want to compare a 6 cyl car.
