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Ok for simplicity let’s assume this is a 1-sample test. So there is a certain “default” mean, say mu, and we are concerned with whether the mean of some random

Ok for simplicity let’s assume this is a 1-sample test. So there is a certain “default” mean, say mu, and we are concerned with whether the mean of some random variable on whatever population we are sampling from is within, say, epsilon of this default value. For every number in [mu – epsilon, mu + epsilon] we can get a p-value giving the probability that we would have observed the data if this was the true mean. In order to get a probability that we would have observed the data given that the mean was somewhere in this interval, we need some prior distribution on the means in this interval, which we don’t have. However, we can just take the sup of all the p-values for each mean in this range to get an upper bound. (I think this is similar to how confidence intervals work also but take that with a grain of salt)

nonbel 8y ago
Looking closer at this you are describing one model but testing different values of one of the model parameters.
- tprice7 8y ago
  “Looking closer at this you are describing one model but testing different values of one of the model parameters.” I am inferring from context that this is probably supposed to be a criticism but it doesn’t make much sense to me. Of course we have to consider different values of the mean, the whole point is to get a p-value corresponding to a range of potential different means. But anyways, I do think that an explanation based on the t_0 statistic I defined in my other post is better. 1. We can define a statistic t_0 that is the infimum of the absolute value of all t-statistics for every candidate mean in the interval. 2. Suppose the mean is some value mu’ in the interval. Whenever the t_0 statistic is at least as big as its observed value, the t-statistic corresponding to mu’ is also at least as big in absolute value as the observed value of t_0, by definition. 3. So we can give an upper bound for the probability of the former event by the probability of the latter event. But the probility of the latter event is the same no matter what mu’ is, and can be computed using Student’s T distribution. 4. Therefore, we have an upper bound for the probability of t_0 attaining a value at least as big as the observed value, assuming that the mean is somewhere in the specified interval (plus the other standard assumptions). This is the p-value. Do you agree with those assertions? If not, specifically where is the problem?
  - tprice7 8y ago
    “Looking closer at this you are describing one model but testing different values of one of the model parameters.” Furthermore, this is even done with the usual 1-sample t-test because the variance can be anything.

nonbel 8y ago

Looking closer at this you are describing one model but testing different values of one of the model parameters.

tprice7 8y ago
“Looking closer at this you are describing one model but testing different values of one of the model parameters.” I am inferring from context that this is probably supposed to be a criticism but it doesn’t make much sense to me. Of course we have to consider different values of the mean, the whole point is to get a p-value corresponding to a range of potential different means. But anyways, I do think that an explanation based on the t_0 statistic I defined in my other post is better. 1. We can define a statistic t_0 that is the infimum of the absolute value of all t-statistics for every candidate mean in the interval. 2. Suppose the mean is some value mu’ in the interval. Whenever the t_0 statistic is at least as big as its observed value, the t-statistic corresponding to mu’ is also at least as big in absolute value as the observed value of t_0, by definition. 3. So we can give an upper bound for the probability of the former event by the probability of the latter event. But the probility of the latter event is the same no matter what mu’ is, and can be computed using Student’s T distribution. 4. Therefore, we have an upper bound for the probability of t_0 attaining a value at least as big as the observed value, assuming that the mean is somewhere in the specified interval (plus the other standard assumptions). This is the p-value. Do you agree with those assertions? If not, specifically where is the problem?
- tprice7 8y ago
  “Looking closer at this you are describing one model but testing different values of one of the model parameters.” Furthermore, this is even done with the usual 1-sample t-test because the variance can be anything.