Is Fisher Sharp Null Hypothesis testable
unbiased association measure even under the sharp causal null hypothesis for exposure B.b = Ya = 1
Under the Potential Outcome framework for causal analysis
Suppose that in an application, a negative control outcome is found to be associated with the treatment in view, thus correctly indicating the presence of unobserved confounding. Then, it may seem natural to consider the observed association between the exposure and the control outcome as an estimate of bias due to unmeasured confounding, and one may be tempted to simply correct the confounded estimate of the exposure-outcome association by subtracting the estimated bias. Although this ad hoc bias correction approach may sometimes be appropriate, it often is not. A difficulty with the approach is that it relies on the key assumption that the bias observed for the negative control outcome is somehow equivalent to the bias one would have observed between the exposure and the primary outcome under the null hypothesis of no causal effect of the exposure. A natural prerequisite for this “bias equivalence” assumption is that the outcomes are measured on comparable scales, which would be the case if, for example, the control outcome was a preexposure measure of the outcome process. However, outside of this special case, the assumption may not be appropriate if the outcomes are clearly measured on different scales, such as, for example, if the negative control outcome were dichotomous and the primary outcome were continuous. The assumption would then likely be violated because an additive association of the exposure with the control outcome would a priori be restricted by the binary nature of the outcome, whereas the additive association of the outcome in view with the exposure would not.
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Most statistical tests culminate in a statement regarding the -value, without which reviewers or readers may feel shortchanged. The -value is commonly defined as the probability of obtaining a result (more formally a ) that is at least as extreme as the one observed, assuming that the is true. Here, the specific null hypothesis will depend on the nature of the experiment. In general, the null hypothesis is the statistical equivalent of the “innocent until proven guilty” convention of the judicial system. For example, we may be testing a mutant that we suspect changes the ratio of male-to-hermaphrodite cross-progeny following mating. In this case, the null hypothesis is that the mutant does not differ from wild type, where the sex ratio is established to be 1:1. More directly, the null hypothesis is that the sex ratio in mutants is 1:1. Furthermore, the complement of the null hypothesis, known as the or , would be that the sex ratio in mutants is different than that in wild type or is something other than 1:1. For this experiment, showing that the ratio in mutants is different than 1:1 would constitute a finding of interest. Here, use of the term “significantly” is short-hand for a particular technical meaning, namely that the result is , which in turn implies only that the observed difference appears to be real and is not due only to random chance in the sample(s). . Moreover, the term significant is not an ideal one, but because of long-standing convention, we are stuck with it. Statistically or statistically may in fact be better terms.
A Definition of Causal Effect | Causality | Statistics
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Entire books are devoted to the statistical method known as . This section will contain only three paragraphs. This is in part because of the view of some statisticians that ANOVA techniques are somewhat dated or at least redundant with other methods such as (see ). In addition, a casual perusal of the worm literature will uncover relatively scant use of this method. Traditionally, an ANOVA answers the following question: are any of the mean values within a dataset likely to be derived from populations that are truly different? Correspondingly, the null hypothesis for an ANOVA is that all of the samples are derived from populations, whose means are identical and that any difference in their means are due to chance sampling. Thus, an ANOVA will implicitly compare all possible pairwise combinations of samples to each other in its search for differences. Notably, in the case of a positive finding, an ANOVA will not directly indicate which of the populations are different from each other. An ANOVA tells us only that at least one sample is likely to be derived from a population that is different from at least one other population.
Interestingly, there is considerable debate, even among statisticians, regarding the appropriate use of one- versus two-tailed -tests. Some argue that because in reality no two population means are ever identical, that all tests should be one tailed, as one mean must in fact be larger (or smaller) than the other (). Put another way, the null hypothesis of a two-tailed test is always a false premise. Others encourage standard use of the two-tailed test largely on the basis of its being more conservative. Namely, the -value will always be higher, and therefore fewer false-positive results will be reported. In addition, two-tailed tests impose no preconceived bias as to the direction of the change, which in some cases could be arbitrary or based on a misconception. A universally held rule is that one should never make the choice of a one-tailed -test after determining which direction is suggested by your data In other words, if you are hoping to see a difference and your two-tailed -value is 0.06, don't then decide that you really intended to do a one-tailed test to reduce the -value to 0.03. Alternatively, if you were hoping for no significant difference, choosing the one-tailed test that happens to give you the highest -value is an equally unacceptable practice.
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Cohn, M. A, Fredrickson, B. L., Brown, S. L., Mikels, J., & Conway, A. M. (2009). Happiness unpacked: Positive emotions increase life satisfaction by building resilience. 361-8. doi:10.1037/a0015952 Happiness-a composite of life satisfaction, coping resources, and positive emotions-predicts desirable life outcomes in many domains. The broaden-and-build theory suggests that this is because positive emotions help people build lasting resources. To test this hypothesis, the authors measured emotions daily for 1 month in a sample of students (N = 86) and assessed life satisfaction and trait resilience at the beginning and end of the month. Positive emotions predicted increases in both resilience and life satisfaction. Negative emotions had weak or null effects and did not interfere with the benefits of positive emotions. Positive emotions also mediated the relation between baseline and final resilience, but life satisfaction did not. This suggests that it is in-the-moment positive emotions, and not more general positive evaluations of one's life, that form the link between happiness and desirable life outcomes. Change in resilience mediated the relation between positive emotions and increased life satisfaction, suggesting that happy people become more satisfied not simply because they feel better but because they develop resources for living well.
and verifying whether 0 falls in the above interval as would be consistent with the null hypothesis of no confounding, where and are consistent estimates of the asymptotic variance of and respectively (). Note that, although under the null hypothesis of no unobserved confounding, converges to a positive number with increasing sample size, it can be negative in the observed finite sample or if the null hypothesis is false, in which case its square root is not a real number. In such cases, it is recommended to instead use the nonparametric bootstrap approach to estimate the variance of The above 95% confidence intervals were (−0.36, 1.268) for Y, indicating no statistically significant evidence of bias due to unobserved confounding for the crude association between consumption of contaminated fish and level of mercury in the blood; and (−1.40, 6.28) for Y*, indicating no statistically significant evidence of bias due to unobserved confounding for the crude association between consumption of large quantities of fish contaminated with methylmercury and percent chromosome abnormality. In closing, we should note that the foregoing analysis and its conclusions may dismiss unobserved confounding by certain, but not all, hidden variables. Assumption 1 may not be entirely credible if, say, an ingredient other than methylmercury in contaminated fish caused the chromosomal abnormalities, or if lack of eating meat by fish consumers were the culprit. This is because the unobserved confounder may no longer be shared between the outcome and the negative control outcome, so that the negative control outcome would have no power to detect unobserved confounding, let alone correct for it. The analysis should be interpreted with caution, particularly because no additional covariates C were available for adjustment, which would have helped to make the identifying assumption more credible.
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Getting back to -values, let's imagine that in an experiment with mutants, 40% of cross-progeny are observed to be males, whereas 60% are hermaphrodites. A statistical significance test then informs us that for this experiment, = 0.25. We interpret this to mean that even if there was no actual difference between the mutant and wild type with respect to their sex ratios, we would still expect to see deviations as great, or greater than, a 6:4 ratio in 25% of our experiments. Put another way, if we were to replicate this experiment 100 times, random chance would lead to ratios at least as extreme as 6:4 in 25 of those experiments. Of course, you may well wonder how it is possible to extrapolate from one experiment to make conclusions about what (approximately) the next 99 experiments will look like. (Short answer: There is well-established statistical theory behind this extrapolation that is similar in nature to our discussion on the SEM.) In any case, a large -value, such as 0.25, is a red flag and leaves us unconvinced of a difference. It is, however, possible that a true difference exists but that our experiment failed to detect it (because of a small sample size, for instance). In contrast, suppose we found a sex ratio of 6:4, but with a corresponding -value of 0.001 (this experiment likely had a much larger sample size than did the first). In this case, the likelihood that pure chance has conspired to produce a deviation from the 1:1 ratio as great or greater than 6:4 is very small, 1 in 1,000 to be exact. Because this is very unlikely, we would conclude that the null hypothesis is not supported and that mutants really do differ in their sex ratio from wild type. Such a finding would therefore be described as statistically significant on the basis of the associated low -value.
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The Central Limit Theorem having come to our rescue, we can now set aside the caveat that the populations shown in are non-normal and proceed with our analysis. From we can see that the center of the theoretical distribution (black line) is 11.29, which is the actual difference we observed in our experiment. Furthermore, we can see that on either side of this center point, there is a decreasing likelihood that substantially higher or lower values will be observed. The vertical blue lines show the positions of one and two SDs from the apex of the curve, which in this case could also be referred to as SEDMs. As with other SDs, roughly 95% of the area under the curve is contained within two SDs. This means that in 95 out of 100 experiments, we would expect to obtain differences of means that were between “8.5” and “14.0” fluorescence units. In fact, this statement amounts to a 95% CI for the difference between the means, which is a useful measure and amenable to straightforward interpretation. Moreover, because the 95% CI of the difference in means does not include zero, this implies that the -value for the difference must be less than 0.05 (i.e., that the null hypothesis of no difference in means is not true). Conversely, had the 95% CI included zero, then we would already know that the -value will not support conclusions of a difference based on the conventional cutoff (assuming application of the two-tailed -test; see below).
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