Random-effects models/analysis

With regard to relevant statistical methodologies, it is possible to dehne 2 situations, which can be termed a meta-analysis context and a shrinkage estimation context. Similar statistical models, in particular random-effects models, may be applicable in both situations. However, the results of such a model will be used somewhat differently. 

Methods of statistical meta-analysis may be useful for combining information across studies. There are 2 principal varieties of meta-analytic estimation (Normand 1995). In a hxed-effects analysis the observed variation among estimates is attributable to the statistical error associated with the individual estimates. An important step is to compute a weighted average of unbiased estimates, where the weight for an estimate is computed by means of its standard error estimate. In a random-effects analysis one allows for additional variation, beyond statistical error, making use of a htted random-effects model. 

There is a growing literature that addresses the transferability of a study s pooled results to subgroups. Approaches include evaluation of the homogeneity of different centers and countries results use of random effects models to borrow information from the pooled results when deriving center-specific or country-specific estimates direct statistical inference by use of net monetary benefit regression and use of decision analysis. 

The fixed effects model considers the studies that have been combined as the totality of all the studies conducted. An alternative approach considers the collection of studies included in the meta-analysis as a random selection of the studies that have been conducted or a random selection of those that could have been conducted. This results in a slightly changed methodology, termed the random effects model The mathematics for the two models is a little different and the reader is referred to Fleiss (1993), for example, for further details. The net effect, however, of using a random effects model is to produce a slightly more conservative analysis with wider confidence intervals. 

In the panel data models estimated in Example 21.5.1, neither the logit nor the probit model provides a framework for applying a Hausman test to determine whether fixed or random effects is preferred. Explain. (Hint Unlike our application in the linear model, the incidental parameters problem persists here.) Look at the two cases. Neither case has an estimator which is consistent in both cases. In both cases, the unconditional fixed effects effects estimator is inconsistent, so the rest of the analysis falls apart. This is the incidental parameters problem at work. Note that the fixed effects estimator is inconsistent because in both models, the estimator of the constant terms is a function of 1/T. Certainly in both cases, if the fixed effects model is appropriate, then the random effects estimator is inconsistent, whereas if the random effects model is appropriate, the maximum likelihood random effects estimator is both consistent and efficient. Thus, in this instance, the random effects satisfies the requirements of the test. In fact, there does exist a consistent estimator for the logit model with fixed effects - see the text. However, this estimator must be based on a restricted sample observations with the sum of the ys equal to zero or T muust be discarded, so the mechanics of the Hausman test are problematic. This does not fall into the template of computations for the Hausman test. 

In addition it is now time to think about the two assumption models, or types of analysis of variance. ANOVA type 1 assumes that all levels of the factors are included in the analysis and are fixed (fixed effect model). Then the analysis is essentially interested in comparing mean values, i.e. to test the significance of an effect. ANOVA type 2 assumes that the included levels of the factors are selected at random from the distribution of levels (random effect model). Here the final aim is to estimate the variance components, i.e. the variance fractions with respect to total variance caused by the samples taken or the measurements made. In that case one is well advised to ensure balanced designs, i.e. equally occupied cells in the above scheme, because only then is the estimation process straightforward. 

A statistical methodology that is particularly relevant where experimentation is meant to identify important unregulated sources of variation in a response is that of variance component estimation, based on so-called ANalysis Of VAriance (ANOVA) calculations and random effects models. As an example of what is possible, consider the data of Table 5.6 Shown here are copper content measurements for some bronze castings. Two copper content determinations were made on each of two physical specimens cut from each of 11 different castings. 

The main statistical issue is the choice between fixed effects and random effects models. Fleiss describes and discusses the two approaches in detail. Peto argues for the former as being assumption-free, as it is based just on the studies being considered at the time of analysis. This assumes that the same true statement effect underlies the apparent effect seen in each trial, study to study variation being due to sampling error. On the other... 

Even for simpler approaches to the estimation of costs, not based on survival analysis, it can be difficult to calculate an individual cost for every patient. For example, if one only uses patients still in the trial to calculate costs and there are progressive drop-outs, the mean cost between visits will be based on progressively fewer patients. If one then uses these means to estimate naively what the average patient would consume in resources at each period in order to cumulate these, then one has a total cost which is not based on individual total costs and for which a measure of variability cannot be directly computed. On the other hand, if one were to use only those patients who provided cost information for the whole period - the completers - one would be throwing away information. Ideally some sort of random effect model with suitable assumptions would be used but it is idle to pretend that this is easy and currently within the grasp of most investigators. 

In this experiment, the three DIETs may be the only ones of concern to the experimenter. In this case it is called a fixed-effect model, and the conclusion drawn is applicable only to these specific three options. However, situations may arise in which the experimenter is seeking for a conclusion apphcable to aU possible DIET options and yet he or she can only handle three options. Then, three of the available options should be drawn from aU possible at random. This is called a random-effect model. In addition to the difference in the scope of conclusion, different types of effect models also imply a somewhat different approach of data analysis. In what is to follow, only the fixed-effect model is considered. 

We carry out a Bayesian analysis of the historical data in Table 2.2. Under the random effects model, the posterior distribution of (Yo/ 0o/ 4o) based on the... 

In discussion of meta-analysis, there is often much attention given to the random effects model versus the fixed effects model. Random effects models assume that the true treatment effects of the individual trials represent a random sample from some population. The random effects model estimates the population mean of the treatment effects and accounts for the variation in the observed effects. It is sometimes stated that the fixed effects model assumes that the individual trial effects are constant. However, this is not a necessary assumption. An alternative view is that the fixed effects model estimates the mean of the true treatment effects of the trials in the meta-analysis. Senn (2000) discussed the analogy with center effects in multicenter trials. In safety, random effects models may be problematic because of the need to estimate between-trial effects with sparse data. Additionally, the random effects model is less statistically powerful than the fixed effects model, albeit the hypotheses are different. In the fixed effects model, the variance estimate should account for trial effect differences either through stratification, conditioning, or modeling of fixed effects. 

This section presents the two most widely used statistical methods for meta-analysis, namely, the fixed effecfs model and the random effects model. In addition, we want to emphasize that an analysis based on crude pooling of adverse event numbers across different studies to compare treatment groups should be avoided, as the analysis is vulnerable to the mischief of Simpson s paradox (Chuang-Stein and Beltangady, 2011). 

Pooled estimate of relative risk imder the fixed effect model (Figure 16.2a) found that black patients had a relative risk of angioedema of 3.0 (95% C3 2.5-S.7) compared with nonblack patients. The pooled estimate and the Cl from the random effect model were almost equal to those from the fixed effects model because the P statistics did not suggest noticeable heterogeneity among the studies. Meta-analysis using odds ratio provided similar results as risk ratio because the proportion of patients with angioedema was very low in all studies. 

Schwartz (1994) did a meta-analysis on three longitudinal and four cross-sectional studies relating PbB to full scale IQ changes in schoolchildren. The methodology involved inverse variance weighting in a random effects model. Schwartz reported that an increase of PbB of 10—20 ng/dl produced a decline of 2.6 (rounding) IQ points for all seven original data sets. The effect... 

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