Solution Manual Of Introduction To Statistical Theory Part 1
Magdalen Dano
It can be tempting to jump prematurely into a statistical analysis when undertaking a systematic review. The production of a diamond at the bottom of a plot is an exciting moment for many authors, but results of meta-analyses can be very misleading if suitable attention has not been given to formulating the review question; specifying eligibility criteria; identifying and selecting studies; collecting appropriate data; considering risk of bias; planning intervention comparisons; and deciding what data would be meaningful to analyse. Review authors should consult the chapters that precede this one before a meta-analysis is undertaken.
An important step in a systematic review is the thoughtful consideration of whether it is appropriate to combine the numerical results of all, or perhaps some, of the studies. Such a meta-analysis yields an overall statistic (together with its confidence interval) that summarizes the effectiveness of an experimental intervention compared with a comparator intervention. Potential advantages of meta-analyses include the following:
solution manual of introduction to statistical theory part 1
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Of course, the use of statistical synthesis methods does not guarantee that the results of a review are valid, any more than it does for a primary study. Moreover, like any tool, statistical methods can be misused.
This chapter describes the principles and methods used to carry out a meta-analysis for a comparison of two interventions for the main types of data encountered. The use of network meta-analysis to compare more than two interventions is addressed in Chapter 11. Formulae for most of the methods described are provided in the RevMan Web Knowledge Base under Statistical Algorithms and calculations used in Review Manager (documentation.cochrane.org/revman-kb/statistical-methods-210600101.html), and a longer discussion of many of the issues is available (Deeks et al 2001).
Figure 10.2.a Example of a forest plot from a review of interventions to promote ownership of smoke alarms (DiGuiseppi and Higgins 2001). Reproduced with permission of John Wiley & Sons
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.
The inverse-variance method is so named because the weight given to each study is chosen to be the inverse of the variance of the effect estimate (i.e. 1 over the square of its standard error). Thus, larger studies, which have smaller standard errors, are given more weight than smaller studies, which have larger standard errors. This choice of weights minimizes the imprecision (uncertainty) of the pooled effect estimate.
A variation on the inverse-variance method is to incorporate an assumption that the different studies are estimating different, yet related, intervention effects (Higgins et al 2009). This produces a random-effects meta-analysis, and the simplest version is known as the DerSimonian and Laird method (DerSimonian and Laird 1986). Random-effects meta-analysis is discussed in detail in Section 10.10.4.
Most meta-analysis programs perform inverse-variance meta-analyses. Usually the user provides summary data from each intervention arm of each study, such as a 22 table when the outcome is dichotomous (see Chapter 6, Section 6.4), or means, standard deviations and sample sizes for each group when the outcome is continuous (see Chapter 6, Section 6.5). This avoids the need for the author to calculate effect estimates, and allows the use of methods targeted specifically at different types of data (see Sections 10.4 and 10.5).
There are four widely used methods of meta-analysis for dichotomous outcomes, three fixed-effect methods (Mantel-Haenszel, Peto and inverse variance) and one random-effects method (DerSimonian and Laird inverse variance). All of these methods are available as analysis options in RevMan. The Peto method can only combine odds ratios, whilst the other three methods can combine odds ratios, risk ratios or risk differences. Formulae for all of the meta-analysis methods are available elsewhere (Deeks et al 2001).
When data are sparse, either in terms of event risks being low or study size being small, the estimates of the standard errors of the effect estimates that are used in the inverse-variance methods may be poor. Mantel-Haenszel methods are fixed-effect meta-analysis methods using a different weighting scheme that depends on which effect measure (e.g. risk ratio, odds ratio, risk difference) is being used (Mantel and Haenszel 1959, Greenland and Robins 1985). They have been shown to have better statistical properties when there are few events. As this is a common situation in Cochrane Reviews, the Mantel-Haenszel method is generally preferable to the inverse variance method in fixed-effect meta-analyses. In other situations the two methods give similar estimates.
Effect measures for dichotomous data are described in Chapter 6, Section 6.4.1. The effect of an intervention can be expressed as either a relative or an absolute effect. The risk ratio (relative risk) and odds ratio are relative measures, while the risk difference and number needed to treat for an additional beneficial outcome are absolute measures. A further complication is that there are, in fact, two risk ratios. We can calculate the risk ratio of an event occurring or the risk ratio of no event occurring. These give different summary results in a meta-analysis, sometimes dramatically so.
The selection of a summary statistic for use in meta-analysis depends on balancing three criteria (Deeks 2002). First, we desire a summary statistic that gives values that are similar for all the studies in the meta-analysis and subdivisions of the population to which the interventions will be applied. The more consistent the summary statistic, the greater is the justification for expressing the intervention effect as a single summary number. Second, the summary statistic must have the mathematical properties required to perform a valid meta-analysis. Third, the summary statistic would ideally be easily understood and applied by those using the review. The summary intervention effect should be presented in a way that helps readers to interpret and apply the results appropriately. Among effect measures for dichotomous data, no single measure is uniformly best, so the choice inevitably involves a compromise.
Consistency Empirical evidence suggests that relative effect measures are, on average, more consistent than absolute measures (Engels et al 2000, Deeks 2002, Rcker et al 2009). For this reason, it is wise to avoid performing meta-analyses of risk differences, unless there is a clear reason to suspect that risk differences will be consistent in a particular clinical situation. On average there is little difference between the odds ratio and risk ratio in terms of consistency (Deeks 2002). When the study aims to reduce the incidence of an adverse event, there is empirical evidence that risk ratios of the adverse event are more consistent than risk ratios of the non-event (Deeks 2002). Selecting an effect measure based on what is the most consistent in a particular situation is not a generally recommended strategy, since it may lead to a selection that spuriously maximizes the precision of a meta-analysis estimate.
Mathematical properties The most important mathematical criterion is the availability of a reliable variance estimate. The number needed to treat for an additional beneficial outcome does not have a simple variance estimator and cannot easily be used directly in meta-analysis, although it can be computed from the meta-analysis result afterwards (see Chapter 15, Section 15.4.2). There is no consensus regarding the importance of two other often-cited mathematical properties: the fact that the behaviour of the odds ratio and the risk difference do not rely on which of the two outcome states is coded as the event, and the odds ratio being the only statistic which is unbounded (see Chapter 6, Section 6.4.1).
Ease of interpretation The odds ratio is the hardest summary statistic to understand and to apply in practice, and many practising clinicians report difficulties in using them. There are many published examples where authors have misinterpreted odds ratios from meta-analyses as risk ratios. Although odds ratios can be re-expressed for interpretation (as discussed here), there must be some concern that routine presentation of the results of systematic reviews as odds ratios will lead to frequent over-estimation of the benefits and harms of interventions when the results are applied in clinical practice. Absolute measures of effect are thought to be more easily interpreted by clinicians than relative effects (Sinclair and Bracken 1994), and allow trade-offs to be made between likely benefits and likely harms of interventions. However, they are less likely to be generalizable.
It is generally recommended that meta-analyses are undertaken using risk ratios (taking care to make a sensible choice over which category of outcome is classified as the event) or odds ratios. This is because it seems important to avoid using summary statistics for which there is empirical evidence that they are unlikely to give consistent estimates of intervention effects (the risk difference), and it is impossible to use statistics for which meta-analysis cannot be performed (the number needed to treat for an additional beneficial outcome). It may be wise to plan to undertake a sensitivity analysis to investigate whether choice of summary statistic (and selection of the event category) is critical to the conclusions of the meta-analysis (see Section 10.14).
For rare outcomes, meta-analysis may be the only way to obtain reliable evidence of the effects of healthcare interventions. Individual studies are usually under-powered to detect differences in rare outcomes, but a meta-analysis of many studies may have adequate power to investigate whether interventions do have an impact on the incidence of the rare event. However, many methods of meta-analysis are based on large sample approximations, and are unsuitable when events are rare. Thus authors must take care when selecting a method of meta-analysis (Efthimiou 2018).
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