Ratio test

The existence of this situation (for nonporous solids) explains why the ratio test discussed above and exemplified by the data in Table XVII-3 works so well. Essentially, any isotherm fitting data in the multilayer region must contain a parameter that will be found to be proportional to surface area. In fact, this observation explains the success of Ae point B method (as in Fig. XVII-7) and other single-point methods, since for any P/P value in the characteristic isotherm region, the measured n is related to the surface area of the solid by a proportionality constant that is independent of the nature of the solid. 

The vapor/hquid ratio tests measure the amount of vapor formed from a given volume of Hquid at a given temperature at atmospheric pressure. A common measure used in specifying gasoline is the temperature at which the vapor/Hquid ratio is 20 (fV/L=2o ) Although V/L can be measured experimentally, it is a difficult and time consuming test to carry out, and techniques have been developed to calculate it from RVP and D86 values. 

Open-circuit voltage ratio test for slip-ring motors 11/264... 

California bearing ratio tests are used to evaluate subgrades for pavements. These tests may be carried out in the field or in the laboratory. Such tests determine the resistance to penetration of a subgrade soil relative to that of a standard crushed-rock base. 

There are two common methods for comparing results (a) Student s -test and (b) the variance ratio test (F-test). 

Jones MS, Pontzer JF, DelCurto B, Badgett CA, Sather MR. Integrating interactive voice response system and web based systems to support home international normalization ratio testing. Clin Trials 2005 2 S73. 

The interval of convergence for each of the series solutions u and ui may be determined by applying the ratio test. For convergence, the condition... 

Cauchy function 276 Cauchy s ratio test 35-36 central forces 107,132-135 spherical harmonics 134-135 spherical polar coordinates 132-133 chain rule 37, 57, 160 character 153,195,197 orthogonality 197, 204 tables 198-200... 

The most useful test for the convergence of a series is called Cauchy s ratio test It can be summarized as follows for a series defined by Bq. (32). 

Ratio Test. If the absolute ratio of the (n + 1) term divided by the nth term as n becomes unbounded approaches... 

A sequential procedure was further developed by Tong and Crowe (1996) by applying sequential analysis of the principal component test using the sequential probability ratio test (SPRT). Dunia et al. (1996) also used PCA for sensor fault identification via reconstruction. In that paper it was assumed that one sensor had failed and the remaining sensors are used for reconstruction. Furthermore, the transient behavior of a number of sensor faults in various types of residuals is analyzed, and a sensor validity index is suggested, determining the status of each sensor. 

Mendal et al. (1993) compared eight tests of normality to detect a mixture consisting of two normally distributed components with different means but equal variances. Fisher s skewness statistic was preferable when one component comprised less than 15% of the total distribution. When the two components comprised more nearly equal proportions (35-65%) of the total distribution, the Engelman and Hartigan test (1969) was preferable. For other mixing proportions, the maximum likelihood ratio test was best. Thus, the maximum likelihood ratio test appears to perform very well, with only small loss from optimality, even when it is not the best procedure. 

Survival and failure times often follow the exponential distribution. If such a model can be assumed, a more powerful alternative to the Log-Rank Test is the Likelihood Ratio Test. 

Mendell, N.R., Finch, S.J. and Thode, H.C., Jr. (1993). Where is the likelihood ratio test powerful for detecting two component normal mixtures Biometrics 49 907-915. 

There are two goals for significance testing. The first is to estimate ttie number of clusters in the data and the second is to identify the amount of overlap between ttie various clusters. Unfortunately, no completely satisfactory statistical test exists. One is faced with a difficult decision, either to ignore the problem or to make do with available testing methods. The simplest, most straight-forward test is chosen for this study, the sum of squares ratio test. Even though the test method may be flawed, it is necessary to underscore the importance and usefulness of statistical measures of cluster separation. 

The sun of squares ratio test compares two clusters by finding the ratio of the between-clusters sun of squares (B) to the within-clusters sun of squares(W). This is based on the well known sum of squares decomposition. 

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. 

Statisticians (some of whom are craftsmen) like to improve their methodology. Some statisticians feel that a simpler statistical method than the one normally used constitutes an improvement. Quick and dirty methods result one of which can be used as a substitute for the F ratio test. 

F-tost (The Variance Ratio Test) a method for determining, with a given degree of probability, whether the variances of two populations differ significantly from one another ... 

Variance ratio test a test used to determine the difference in variability between two sets of deta. Used in the analysis of variance to compare variation due to a perticular factor with the experimental error. 

Range as estimate of standard deviation, 9 Range ratio test, 13 Range, use in substitute f tests, 19,21... 

There are two different ways of carrying out this test. The first one involves taking a single sample and analysing it by both methods a number of times. The usual procedure is to undertake a number of analyses (preferably not less than 6) for the chosen sample with both methods and calculate the value of the t-statistic. This is then compared with the tabular value for the appropriate degrees of freedom at the selected confidence level. If the calculated value is less than the tabulated t value then the mean values, and hence the methods, are accounted equivalent. This method has the advantage that the number of replicates undertaken for each method does not have to be equal. However, it is not always recognised that for this test to be valid the precision of the two methods should be equal. The method used to compare the precisions of methods is the F-ratio test and is carried out as part of the procedure. 

An F ratio test is hardly needed to decide whether the variances are different A t-test is still applicable but needs to be modified to take into account the variance differences. This is done by calculating the effective number of degrees of freedom using Satterthwaite s method.This is still an area which is controversial and a number of differing approaches and equations have been proposed. 

For the model in Exercise 3, test the hypothesis that X = 0 using a Wald test, a likelihood ratio test, and a Lagrange multiplier test. Note, the restricted model is the Cobb-Douglas, log-linear model. 

Now, to compute the likelihood ratio statistic for a likelihood ratio test of the hypothesis of equal variances, we refer %2 = 401n.58333 - 201n.847071 - 201n.320506 to the chi-squared table. (Under the null hypothesis, the pooled least squares estimator is maximum likelihood.) Thus, %2 = 4.5164, which is roughly equal to the LM statistic and leads once again to rejection of the null hypothesis. 

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