Re: How to calculate the beta value? What does it mean by a given beta value? .
- From: Bruce Weaver <bweaver@xxxxxxxxxxxx>
- Date: Sat, 01 Oct 2005 21:11:50 -0400
rock31 wrote:
How to calculate the beta value? What does it mean by a given beta value? what is the relationship between the beta value, P-value and alpha?
thanks
I find it helpful to think about this in the context of discrete sample space, like coin flipping experiments. E.g., suppose I have one coin that is known to be fair, and one that is known to be biased towards Tails with the properties described below (under H1). I let you pick one of the coins at random (you don't know which is which), and flip it 11 times.
H0: p = p(Head) = .5, q = p(Tail) = .5 (coin is fair) H1: p = .15, q = .85 (coin is biased towards tails)
Let X = number of Heads in 11 flips.
If you wanted to maintain alpha at .05 or less, you would reject H0 for X = 2 or less, otherwise you would fail to reject H0.
Binomial distributions with N = 11
X p(X|H0) p(X|H1) ----------------------- 0 .0005 .1673 1 .0054 .3248 2 .0269 .2866 Reject H0 ---------------------------------------- 3 .0806 .1517 4 .1611 .0536 5 .2256 .0132 6 .2256 .0023 7 .1611 .0003 Fail to reject H0 8 .0806 .0000 9 .0269 .0000 10 .0054 .0000 11 .0005 .0000 -----------------------
Type I error can only occur if H0 is true, so the probabilities you need to sum to compute alpha are from the distribution that is conditional on H0 being true. Furthermore, Type I error can only occur if you reject H0, so you want the sum of the probabilities in the rejection region (from the distribution that is conditional on H0 being true).
alpha = .0005 + .0054 + .0269 = .0328 < .05
Type II error can only occur if H0 is false, so to calculate beta, you must take the probabilities from the distribution that is conditional on H1 being true. And Type II error can only occur if you fail to reject H0, so you must take them from the region that leads you to not reject H0--i.e., for values of X = 3 or greater in this example.
beta = .1517 + .0536 + ... + .0000 = .2211
Notice that alpha and beta can be computed before you actually do the experiment--before you flip the coin 11 times. But the p-value can only be computed AFTER you've collected the data. Suppose you did flip the coin 11 times, and observed X = 1 Head. The p-value = the probability of the observed outcome, or any more extreme outcome (under the assumption that the null hypothesis is true). So for X = 1 as the observed outcome, p = .0054 + .0005 = .0059. Because p < the prespecified alpha, you would reject H0.
One way to see the relationship between alpha and beta is to change the decision rule. For example, here is the decision rule that minimizes the overall probability of error. (The sum of alpha and beta is lower for this rule than for the one shown above, or for any other rule.)
Decision rule to minimize the overall probability of error.
X p(X|H0) p(X|H1) ----------------------- 0 .0005 .1673 1 .0054 .3248 2 .0269 .2866 Reject H0 3 .0806 .1517 ----------------------------------------- 4 .1611 .0536 5 .2256 .0132 6 .2256 .0023 7 .1611 .0003 Fail to reject H0 8 .0806 .0000 9 .0269 .0000 10 .0054 .0000 11 .0005 .0000 -----------------------
For this rule,
alpha = .0005 + .0054 + .0269 + .0806 = .1134 beta = .1517 + .0536 + ... + .0000 = .0694
Notice that as alpha increased, beta decreased. Notice too that beta (and therefore power) can only be computed when H1 specifies a precise sampling distribution for the test statistic.
-- Bruce Weaver bweaver@xxxxxxxxxxxx www.angelfire.com/wv/bwhomedir .
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