Methods for combination

Each trial that is to be included in the meta-analysis will provide a measure of treatment effect (difference). For continuous data this could be the mean response on the active treatment minus the mean response in the placebo arm. Alternatively, for binary data the treatment effect could be captured by the difference in the cure rates, for example, or by the odds ratio. For survival data, the hazard ratio would often be the measure of treatment difference, but equally well it could be the difference in the two-year survival rates. 

Assume that we have decided on the best measure for the treatment effect. If this is expressed as a difference, for example, in the means, then there will be an associated standard error measuring the precision of that difference. If the 

If the treatment effect in each of the individual trials is the difference in the mean responses, then d represents the overall, adjusted mean difference. If the treatment effect in the individual trials is the log odds ratio, then d is the overall, adjusted log odds ratio and so on. In the case of overall estimates on the log scale we generally anti-log this final result to give us a measure back on the original scale, for example as an odds ratio. This is similar to the approach we saw in Section 4.4 when we looked at calculating a confidence interval for an odds ratio. 

The methods of the previous subsection give us a combined estimate, d, for the treatment effect. We now need to construct a confidence interval around this estimate. This initially involves obtaining a standard error, se, for d, which is given by  

From this it is easy to obtain a 95 per cent confidence interval for the overall treatment effect as (d — 1.96se, d + 1.96se). 

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Figure 15. Alternative Methods for Combining Volume Fractions of Stationary Phase in GC...

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

Marti, H. F. and R. A. Waller. An Exploratory Comparison of Methods for Combining Failure Rate Data from Different Data Sources. Report No. LA-7556-MS, Los Alamos Scientific Laboratory, 1987. 

A systematic method for combining the velocity and pressure calculations with the previous solutions techniques for composition and temperature starts with known values for all variables and proceeds as follows ... 

Here we discuss a set of traditional methods for combining both the forward and reverse perturbation samples to enhance free energy calculations. 

Table 35-4 ANOVA comparing methods for combined laboratories and operators, all Method B vs. all Method A...

Combined systems The NFPA standards specify acceptable methods for combining the above requirements. 

Newton SS, Dow A, Terwilliger R, Duman R. 2002. A simplified method for combined immunohistochemistry and in-situ hybridization in fresh-frozen, cryocut mouse brain sections. Brain Res Prot 9 214-219. 

Schmid, 1., Uittengobaart, C. H., and Giogi, J. V. (1991) A gentle fixation and permeabilization method for combined cell-snrface and intracellular staining with improved precision in DNA qnantification. Cytometry 12, 279-285. 

An extremely useful addition to the formal method for combining the studies is to represent the data from the individual studies, together with the combination, in a meta-analysis plot. 

The correct statistical methods for combining data in meta-analysis -the summary statistics that are combined must come from independent data sets... 

A full three factorial matrix on the 11 variables in the cure cycle shown in Figure 15.1 would mean 177,147 individual trials. A full two factorial design would still mean 2048 trials. Such a design, however, assumes that all interactions, even between all 11 variables, will be important. DOE provides an ordered means of combining variables to reduce the total number of trials. The assumption made is that high-order interactions (i.e., interactions of three or more variables) are rare and/or insignificant. There are several methods for combining variables by DOE. A detailed discussion of these methods is the subject of another book [9]. 

Yuan E, Haghighi F, White S et al. A single nueleotide polymorphism chip-based method for combined genetic and epigenetic profiling validation in decitabine therapy and tumor/normal comparisons. Cancer Res 2006 66 3443-3351. 

Mossbauer spectroscopy should also be mentioned here as a very promising method for combining the structural and dynamic studies of biomolecular systems. The asymmetry of Mossbauer spectra caused by the anisotropy of vibrations of Mossbauer atoms allowed—for example, to find that the mean square amplitude of vibrations of Fe atoms normal to the plane of porphyrin ring (which are responsible for many important biological functions of hemoproteins) is about five times larger than in the... 

Methods for combined determination of isothermal calorimetric and infrared reaction data... 

Pu JZ, JL Gao, DG Truhlar (2004b) Generalized hybrid orbital (GHO) method for combining ab initio Hartree-Fock wave functions with molecular mechanics. J. Phys. Chem. A 108 (4) 632-650... 

There is no rigorous mathematical method for combining the individual precisions, but the variances of the individual measurements may be combined. The precision, at the 95 percent confidence level, is defined as approximately twice the true standard deviation. Since the latter is the square root of the true variance, this permits evaluation of the precision for the combined operation. 

Linear combination of atomic orbitals (LCAO theory) A method for combining atomic orbitals to approximately compute molecular orbitals. 

Eberle, K. et al.. Device and Method for Combined Purification and Compression of Hydrogen Containing CO and the Use Thereof in Fuel Cell Assemblies, U.S. Patent... 

Constraints of direct methods. The phase angles can be estimated by statistical methods that are based on the concept that the electron density is never negative, and that it consists of isolated, sharp peaks at atomic positions. The statistical methods for combining electron-density waves subject to these conditions are called direct methods. They make it possible to derive phases for a set of structure factors when only information on the magnitudes of F hkl) is available. At present this is the method of choice for small molecules. 

The Decision Forest (DF) is a DT-based consensus modeling method that was conceived as a method for combining heterogeneous yet comparable trees that fully captures the association between molecular structure and biological activity (Figure 6.4) [59], The heterogeneity requirement assures that each tree uniquely contributes to the combined prediction whereas the quality comparability requirement assures that each tree equally contributes to the combined prediction. 

Note that the areas under each curve in Figure 15-4 are the same. The likelihood ratio does not take disease prevalence or any other prior information into account. To arrive at a realistic probability, one must adjust for the best estimate of the probability of disease before obtaining the test result. Bayes theorem provides a method for combining the prior probability information. 

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