Stats: Relationship between Alpha and Beta My question has to do with the relationship between alpha and beta and their definitions in statistics.
alpha = type I error rate = significance level under consideration that the NULL hypothesis is correct
Beta = type II error rate
If alpha is lowered (specificity increases as alpha = 1- specificity), beta increases (sensitivity/power decreases as beta = 1 - sensitivity/power)
How does a change in alpha affects beta ?
Is there a linear relationship or not ?
Do the ratio alpha/beta always the same, in other words the ratio specificity/sensitivity is always the same ?
If yes, it means that by using a bonferroni correction we're just shifting to lower sensitivity and higher specificity but we're not changing sensitivity/specificity ratio. Is that correct to say so ?
Update (Case-specific question):
For a given experimental design, we run 5 Linear Models on the data. We have a True Positive Rate (sensitvity/power) at 0.8 and a True Negative Rate (specificity) at 0.7. (Let's imagine we know what should be positive and what should not.). If we now correct the significance level using Bonferroni to 0.05 / 5 = 0.01. Can we numerically estimate the resulting True Positive Rate (sensitivity/power) and True Negiative Rate (Specificity) ?
Thanks a lot for your help. 
 A: For others in the future: 
In Sample Size estimation, the Ztotal is calculated by adding the Z corresponding to alpha and Z corresponding to power (1-beta). So mathematically, if sample size is kept constant, increasing Z for alpha means you decrease the Z for power by the SAME amount e.g., increasing Zalpha from 0.05 to 0.1 decreases Zpower by 0.05.
The difference is the Z for alpha is two-tailed while the Z for beta is 1-tailed. So, while the Z value changes by the same amount, but the probability % that this Z value corresponds to does not change by the same amount.
Example: 
5% alpha (95% confidence) with 80% power (20% beta) gives the same sample size as
20% alpha (80% confidence) with 93.6% power (6.4% beta) rather than the 95% power we would have if the relationship were 1:1.
A: There is no general relation between alpha and beta.
It all depends on your test, take the simple exemple: 
(Wikipedia)
In colloquial usage type I error can be thought of as "convicting an innocent person" and type II error "letting a guilty person go free".
A jury can be severe: no type II error, some type I
A jury can be "kind": no type I but some type II
A jury can be normal: some type I and some type II
A jury can be perfect: no error
In practice there is two antagonist effect: 
When the quality of the test goes up, type I and type II error decrease until some point.
When a jury improves, he tends to give better judgment over both innocent and guilty people.
After some point the underlying problem appears in the building of the test. Type I or II are more important for the one who runs the test. With the jury exemple, type I errors are more important and so the law process is build to avoid type I. If there is any doubt the person is free.  Intuitively this lead to a growth in type II error.
Concerning Bonferroni:
(Wikipedia again)
Bonferroni correction controls the probability of false positives only. The correction ordinarily comes at the cost of increasing the probability of producing false negatives, and consequently reducing statistical power. When testing a large number of hypotheses, this can result in large critical values.
