Hypothesis testing ratio

To compute the power for a hypothesis test based on standard deviation, we would have to read off the corresponding probability points from a chi-square table for 95% confidences on both alpha and beta, the square root of the ratio of 2(0.95, v) and 2(0.05, v) (v = the degrees of freedom, close enough to n for now) is the ratio of standard deviations that can be distinguished at that level of power. Similarly to the case of the means, v would also be related to the square of that ratio, but x2 would still have to be read from tables (or computed numerically). As an example, for 35 samples, the precision of the instrument could not be tested to be better than... 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

The analysis of the mean response and the s/n ratio can be performed employing the usual ANOVA and/or hypothesis tests to detect which factors or interactions have statistical significance. Taguchi proposed a conceptual approach based on the graphical display of the effects (they are called factor plots or marginal means followed by a qualitative evaluation. This provides objective information and a test for the significance of each design factor on the two observed responses mean and s/n ratio. 

The null hypothesis tested with the F-ratio is a general hypothesis stating that the true coefficients are all zero (note that b, is not included). The / "-ratio has an F-distribution with df= m and [Pg.126]

The previous two examples (3.23 and 3.24) implement the hypothesis-testing procedure for two straightforward statistical tests. Hypothesis testing may also be extended to the difference between the means of two samples or the ratio of two variances. The relevant test statistics for these calculations were presented in Section 3.3 of this chapter. The NIST/SFMATECH handbook... 

As Talias (2007) has pointed out, there is an interesting analogy between the Pearson Index and the Neyman-Pearson lemma. (The Pearsons in question are different. Alan Pearson is the author of the Pearson index and Egon Pearson, 1895-1980, was the son of Karl Pearson, 1857-1936 and the collaborator of Jerzey Neyman, 1894-1981, in developing hypothesis testing.) Both are relevant to optimizing a function subject to a constraint. In the case of the Pearson index this is profit subject to total cost, and for the Neyman-Pearson lemma it is power subject to the constraint of an overall type I error rate. In both cases a ratio plays a key role. For the Neyman-Pearson lemma this is the likelihood ratio and for the Pearson index the index itself is a ratio of expected profit to expected cost. 

Interval estimates and hypothesis tests for the parameters can be performed, in principle, by considering the as)miptotic normal distribution of the maximum likelihood estimates and the asymptotic chi-squared distribution of the likelihood ratio statistics, respectively (Lawless, 1982). 

The question remains if there is a significant difference in the ability of the different probability distribution functions to describe the distillation data. It is generally not recommended to apply null hypothesis testing to information-theoretic ranking data to determine if the best model is significantly better than any of the lower ranked models (Burnham and Anderson, 1998). Model selection is best achieved through inspection of evidence ratios and residuals. A summary of the AIC and evidence ratios of the best 10 ranked functions are presented in Table 12.24. It can be... 

Hence, for a given P, G and Ah, the hypothesis will be correct if two tubes of different length, but with the same Ljd ratio, give the same burn-out flux. Barnett (B3) has tested Eq. (2), and Fig. 21 gives one of his test results, showing that the hypothesis is not generally valid. Thus, any burn-out theory or correlation which combines L and d only as a ratio cannot generally be correct. 

Intriguing to hypothesize [12] that a single monolayer will exhibit less disorder (larger ratio) than the 2 monolayer sample. Tests of this hypothesis are In progress. 

Van der Voet [21] advocates the use of a randomization test (cf. Section 12.3) to choose among different models. Under the hypothesis of equivalent prediction performance of two models, A and B, the errors obtained with these two models come from one and the same distribution. It is then allowed to exchange the observed errors, and c,b, for the ith sample that are associated with the two models. In the randomization test this is actually done in half of the cases. For each object i the two residuals are swapped or not, each with a probability 0.5. Thus, for all objects in the calibration set about half will retain the original residuals, for the other half they are exchanged. One now computes the error sum of squares for each of the two sets of residuals, and from that the ratio F = SSE/JSSE. Repeating the process some 100-2(K) times yields a distribution of such F-ratios, which serves as a reference distribution for the actually observed F-ratio. When for instance the observed ratio lies in the extreme higher tail of the simulated distribution one may... 

The observation that the transition state volumes in many Diels-Alder reactions are product-like, has been regarded as an indication of a concerted mechanism. In order to test this hypothesis and to gain further insight into the often more complex mechanism of Diels-Alder reactions, the effect of pressure on competing [4 + 2] and [2 + 2] or [4 + 4] cycloadditions has been investigated. In competitive reactions the difference between the activation volumes, and hence the transition state volumes, is derived directly from the pressure dependence of the product ratio, [4 + 2]/[2 + 2]p = [4 + 2]/[2 + 2]p=i exp —< AF (p — 1)/RT. All [2 + 2] or [4 + 4] cycloadditions listed in Tables 3 and 4 doubtlessly occur in two steps via diradical intermediates and can therefore be used as internal standards of activation volumes expected for stepwise processes. Thus, a relatively simple measurement of the pressure dependence of the product ratio can give important information about the mechanism of Diels-Alder reactions. 

The primary pathway of Hg to aquatic systems is considered to be atmospheric transport and subsequent deposition, which has caused an accumulation of Hg in watershed soils (Mason, et al., 1994). Recent work has shown that watershed disturbance such as clear cutting results in the increased export of Hg from the watershed to aquatic systems (Porvari, et al., 2003) and others have found agricultural and urbanized land use to be important factors influencing Hg export (Fitzgibbon, et al., 2008 Mason Sullivan, 1998). Thus the recovery of aquatic systems from anthropogenic Hg may depend on watershed characteristics which will differ among watersheds. Therefore we compared time to recovery to watershed attributes (e.g., %urban, susceptibility to erosion, watershed to lake area ratio, etc...) to test the hypothesis that the rate of recovery from Hg enrichment is influenced by watershed controlled pathways and stressors. 

In order to validate the hypothesis mentioned above, the Ni retention capacity of the Lac Tio waste rock was estimated using a batch sorption test performed on a fresh (C1) and weathered (C4) sample, followed by a 3-step Sequential Extraction Procedure (or SEP). The batch sorption test was done using a 10 mg/L Ni solution with an initial pH of 6, an ionic force adjusted to 0.05 M with NaN03 and with a liquid/solid ratio of 25. Some of the batch sorption results are presented in Figure 3. 

A test of the hypothesis that the linear trends in Figures 10-12 are due to mixing comes from comparisons with oxygen isotopes. It has been conventional wisdom that Mg does not correlate in detail with O excesses in primitive chondrite components, including CAIs. However, combining the new highly precise MC-ICPMS data with isotope ratio data for the... 

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