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One Piece Of Paper: The Simple Approach To Powe...

There is a more interesting use for trap handlers that comes up when computing products such as that could potentially overflow. One solution is to use logarithms, and compute exp instead. The problem with this approach is that it is less accurate, and that it costs more than the simple expression , even if there is no overflow. There is another solution using trap handlers called over/underflow counting that avoids both of these problems [Sterbenz 1974].

One Piece of Paper: The Simple Approach to Powe...


An interesting example of error analysis using formulas (19), (20), and (21) occurs in the quadratic formula . The section Cancellation, explained how rewriting the equation will eliminate the potential cancellation caused by the operation. But there is another potential cancellation that can occur when computing d = b2 - 4ac. This one cannot be eliminated by a simple rearrangement of the formula. Roughly speaking, when b2 4ac, rounding error can contaminate up to half the digits in the roots computed with the quadratic formula. Here is an informal proof (another approach to estimating the error in the quadratic formula appears in Kahan [1972]).

The section Optimizers, mentioned the problem of accurately computing very long sums. The simplest approach to improving accuracy is to double the precision. To get a rough estimate of how much doubling the precision improves the accuracy of a sum, let s1 = x1, s2 = s1 x2..., si = si - 1 xi. Then si = (1 + i) (si - 1 + xi), where i , and ignoring second order terms in i gives

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

A very common and simple version of the meta-analysis procedure is commonly referred to as the inverse-variance method. This approach is implemented in its most basic form in RevMan, and is used behind the scenes in many meta-analyses of both dichotomous and continuous data.

There are statistical approaches available that will re-express odds ratios as SMDs (and vice versa), allowing dichotomous and continuous data to be combined (Anzures-Cabrera et al 2011). A simple approach is as follows. Based on an assumption that the underlying continuous measurements in each intervention group follow a logistic distribution (which is a symmetrical distribution similar in shape to the normal distribution, but with more data in the distributional tails), and that the variability of the outcomes is the same in both experimental and comparator participants, the odds ratios can be re-expressed as a SMD according to the following simple formula (Chinn 2000):

A formal statistical approach should be used to examine differences among subgroups (see MECIR Box 10.11.a). A simple significance test to investigate differences between two or more subgroups can be performed (Borenstein and Higgins 2013). This procedure consists of undertaking a standard test for heterogeneity across subgroup results rather than across individual study results. When the meta-analysis uses a fixed-effect inverse-variance weighted average approach, the method is exactly equivalent to the test described by Deeks and colleagues (Deeks et al 2001). An I2 statistic is also computed for subgroup differences. This describes the percentage of the variability in effect estimates from the different subgroups that is due to genuine subgroup differences rather than sampling error (chance). Note that these methods for examining subgroup differences should be used only when the data in the subgroups are independent (i.e. they should not be used if the same study participants contribute to more than one of the subgroups in the forest plot).

Review authors may undertake sensitivity analyses to assess the potential impact of missing outcome data, based on assumptions about the relationship between missingness in the outcome and its true value. Several methods are available (Akl et al 2015). For dichotomous outcomes, Higgins and colleagues propose a strategy involving different assumptions about how the risk of the event among the missing participants differs from the risk of the event among the observed participants, taking account of uncertainty introduced by the assumptions (Higgins et al 2008a). Akl and colleagues propose a suite of simple imputation methods, including a similar approach to that of Higgins and colleagues based on relative risks of the event in missing versus observed participants. Similar ideas can be applied to continuous outcome data (Ebrahim et al 2013, Ebrahim et al 2014). Particular care is required to avoid double counting events, since it can be unclear whether reported numbers of events in trial reports apply to the full randomized sample or only to those who did not drop out (Akl et al 2016).

You don't always have to import full detailed data. The Power Query Editor makes it easy to pre-aggregate data during import. Technically, it's possible to import exactly the aggregate data you need for each visual. While DirectQuery is the simplest approach to large data, importing aggregate data might offer a solution if the underlying data source is too slow for DirectQuery. 041b061a72

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