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| visits | member for | 1 year, 10 months |
| seen | May 27 at 16:09 | |
| stats | profile views | 20 |
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Aug 2 |
awarded | Yearling |
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Sep 7 |
comment |
Chi-square test for checking if values are close to zero? @whuber: I made no such assumptions. As your own example illustrates, it is not true that "large differences in the variance of sampling distributions" can explain a high SD/mean ratio (without having even higher unexplained ratios in the observations for the high variance components). In that example of two regressions, an overall SD/mean ratio of 10 was derived from a ratio of 22000 for the higher-variance regression. This phenomenon and its generalizations are calculations and theorems, not "assumptions". |
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Sep 7 |
comment |
Chi-square test for checking if values are close to zero? @whuber: Notice, as in your example, that in order for such things to occur, a conspiracy (that is, an actual phenomenon in the data generation) has to occur to suppress the high-variance terms from contributing to the mean. So one still can say something, and it will likely be interesting in relation to the problem of interest. It will not be as simple as the canonical case of i.i.d or nearly iid gradients. But you still have to explain how a mean that "could" have been larger is ten times smaller than the SD. Numerical accidents can be ruled out by taking subsamples. |
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Sep 7 |
comment |
Chi-square test for checking if values are close to zero? @whuber: re "that might or might not be consistent with the true mean being zero" -- how could it be inconsistent with the mean being zero? It is consistent with the mean being zero, and also consistent with the mean being nonzero but the observations of the mean having SD higher than the value of the mean. As was pointed out in the answer. Are you saying there are relevant classes of models where a small observed ratio (mean/SD) disfavors the hypothesis that the true mean is zero? |
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Sep 7 |
comment |
Chi-square test for checking if values are close to zero? The OP called 1/10000 "small" and knows what units and meaning it has in his problem. Here the mean and SD have the same units and their ratio is dimensionless. It is not necessary to assume a single common sampling distribution, only similar variances (and if that is not the case, a small ratio of mean to SD still imposes strong constraints). More information is needed but as e.g. the Chebyshev calculation indicates one can say something, just a lot less than with full information. |
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Sep 7 |
answered | Chi-square test for checking if values are close to zero? |
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Aug 29 |
revised |
Meaning of p-values in regression added 52 characters in body |
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Aug 29 |
revised |
Meaning of p-values in regression clarify re omission of constant term |
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Aug 29 |
awarded | Commentator |
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Aug 29 |
comment |
Meaning of p-values in regression @Andy W, why should it be clear a priori that the hypothesis tests for the individual parameters do not say anything about significance of the overall model? (It is not assumed in OP's question or my comments, by the way, that the model significance can be quantified only by F-tests, and even in that case there are examples where F is equivalent to a t-test, so why not contemplate the possibility of a more complicated F being computable or estimable from a suite of t-tests?). |
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Aug 29 |
comment |
Meaning of p-values in regression @Andy, it seems to me that link #2 does not address the question in any apparent way. Having high p-values for the regressors and low p-value for the overall model, does not indicate whether the former "can ... be combined into a p-value for the whole model". Maybe under some strong assumptions on what "combine" can mean, such as using a formula that extends continuously to the limit where some regressor p-values are zero, or something more than that, plus the ability to produce unboundedly extreme examples of the type seen in link #2. But all this is well beyond link #2 contents. |
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Aug 29 |
comment |
Meaning of p-values in regression @Andy W: none of the F-test links pertain to item 3 of the question, which was whether one can determine the model p-value from coefficient p-values (or, interpreted more broadly, whether there is some other relation between the two types of p-value). |
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Aug 29 |
comment |
Meaning of p-values in regression The first comment with link to [p-values] tag was removed by moderator or otherwise by the time I saw the migrated thread. I deleted my comment (still up at the original) about stat.SE since the context was gone and, although accurate in my opinion, the comment could cause disputes if posted here. Both are still visible at the math.SE original posting. I don't remember if there were other comments there that got lost in the shuffle. |
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Aug 29 |
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Meaning of p-values in regression @cardinal: which part of which last comment involves possible confusion? If re: (3) -- the hyp.tests are simply different for alpha=beta=0 versus alpha=0 so there is no a priori ordering of their p-values. However, a test of, say "-1 < alpha < +1" would always have higher p-value (for any given value of the estimator) than a test of "alpha=0" since presumably the same estimator would be used and interpreted the same way in both tests, but the set of models in the first test is broader than in the second. |
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Aug 29 |
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Meaning of p-values in regression re (3) and the relative size of different p-values, I've deleted the original comments but would be interested to know if any valid inequalities do hold (thanks again for the correction). Although a narrower null model such as "alpha = beta = 0" does have higher p-values than a less specific one such as "alpha=0" in tests with the same estimator and alternative hypothesis, they are not the same in the case of linear regression. |
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Aug 29 |
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Meaning of p-values in regression @cardinal: it seems more accurate to say that a p-value is associated to a hypothesis test. The parameters appear in the null hypothesis of the test and the pair (observed value of estimator, alternative hypothesis) then determine a p-value. The null hypotheses should be described using parameters, such as α=0 rather than estimators a=0 as was [carelessly] done in the original answer, now edited (thanks for pointing out the error). However, the supposedly confused or missing distinction "the estimators are bivariate normal, not the parameters" was stated explicitly in the answer. |
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Aug 29 |
revised |
Meaning of p-values in regression some corrections in light of comments |
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Aug 29 |
comment |
Meaning of p-values in regression (Some comments referenced above were lost in migration. They are visible at the original math.SE posting, linked below next to the words "migrated from...") |
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Aug 29 |
awarded | Editor |
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Aug 29 |
revised |
Meaning of p-values in regression added 18 characters in body |
