Random effects estimation

The process rnust be iterated until convergence and the final estimates are denoted with Plb, bi,LB, and colb- The individual regression parameter can be therefore estimated by replacing the final fixed effects and random effects estimates in the function g so that ... 

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. 

Furthermore, when alternative approaches are applied in computing parameter estimates, the question to be addressed here is Do these other approaches yield similar parameter and random effects estimates and conclusions An example of addressing this second point would be estimating the parameters of a population pharmacokinetic (PPK) model by the standard maximum likelihood approach and then confirming the estimates by either constructing the profile likelihood plot (i.e., mapping the objective function), using the bootstrap (4, 9) to estimate 95% confidence intervals, or the jackknife method (7, 26, 27) and bootstrap to estimate standard errors of the estimate (4, 9). When the relative standard errors are small and alternative approaches produce similar results, then we conclude the model is reliable. 

This turns out to be a rather complex matter because, for any multicentre trial, one can conceive of at least four different sorts of estimate which might be calculated even where the trialist s aim is to allow for differences between centres. Two of these estimates are so-called fixed -effect estimates based on within-centre treatment contrasts, the third is a so-called random effects estimate, and the fourth is one that permits the recovery of between-centre information. (There are various variants on these models and for a fuller account the reader should see Senn (2000).) The nature and properties of these estimators are discussed in further sections of this chapter. It is really only with one of the two fixed-effect estimators that this problem of inefficiency is serious. We shall discuss this point below when comparing type IT and type III approaches to inference. For the moment, suffice it to say that it is not true that trials with unequal numbers of patients per centre are inefficient unless we insist on weighting centres equally (the type III approach). 

What is the difference between fixed- and random-effect estimators ... 

Choices between fixed- or random-effect estimators arise wherever we have data sets with multiple levels within the experimental units for example, patients within trials for a meta-analysis, episodes per patient for a series of n-of-1 trials or patients within centres for a multicentre trial. The issue is an extremely complex one and it is difficult to give hard and fast rules as to which is appropriate. 

On the other hand, we see a clear evidence of heterogeneity among the studies for risk difference since the value of P statistic is 64.5% (95% Cl 6.9%-86.5%), which is large enough to have a substantial impact on the weight. This has the effect of increasing the standard error and the width of the Cl for the overall estimate of racial difference. The Cl based on the random effects model (95% Cl 2.0-7.5 per 1000 subjects) is much wider than that based on the fixed effects model (95% Cl 2.3-S.5 per 1000 subjects). The random effects estimate for risk difference is 2.9 per 1000 subjects, which is also different from the fixed effects estimate of 4.7 per 1000 subjects. The estimate under the random effects model is not even within the 95% Cl under the fixed effects model. 

Wooldridge, Jeffrey M., 2013. Random Effects Estimation. Introductory Econometrics A Modem Approach (Fifth international ed.). Mason, OH South-Western, pp. 474-478. 

Uncertainty expresses the range of possible values that a measurement or result might reasonably be expected to have. Note that this definition of uncertainty is not the same as that for precision. The precision of an analysis, whether reported as a range or a standard deviation, is calculated from experimental data and provides an estimation of indeterminate error affecting measurements. Uncertainty accounts for all errors, both determinate and indeterminate, that might affect our result. Although we always try to correct determinate errors, the correction itself is subject to random effects or indeterminate errors. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

In the PNLS step the current estimates of D and are fixed and the conditional modes of the random effects b and the conditional estimates of the fixed effects p are obtained minimizing the following objective function ... 

As mentioned in Section 4.3.3, bias is the difference between the mean value (x) of a number of test results and an accepted reference value (xo) for the test material. As with all aspects of measurement, there will be an uncertainty associated with any estimate of bias, which will depend on the uncertainty associated with the test results Uj and the uncertainty of the reference value urm> as illustrated in Figure 4.7. Increasing the number of measurements can reduce random effects... 

If the Lukas programme is run with all experimental data, including reasonable estimates for accuracy, it performs its fitting operation by assuming that errors arise from random effects rather than systematic inaccuracies. Systematic errors can be taken into account in at least two ways. Firstly, it is possible to ignore the particular... 

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. 

Robinson GK. 1991. That BLUP is a good thing — the estimation of random effects. Stat Sci 6 32-34. 

The Hausman test was used to test the null hypothesis that the coefficients estimated by the efficient random-effect model are the same as the ones estimated by the consistent fixed-effect model. If this null hypothesis cannot be rejected (insignificant P-value in general, it is larger than 0.05), then the random-effect model is more appropriate. 

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. 

In a Model II ANOVA (random effect model) the result can be decomposed as yij=iu+Aj+eij, where Aj represents a normally distributed variable with mean zero and variance a]j. In this model one is not interested in a specific effect due to a certain level of the factor, but in the general effect of all levels on the variance. That effect is considered to be normally distributed. Since the effects are random it is of no interest to estimate the magnitude of these random effects for any one group, or the differences from group to group. What can be done is to estimate their contribution [Pg.141]

Alternatively, the dummy effect can be taken as the repeatability of the factor effects. Recall that a dummy experiment is one in which the factor is chosen to have no effect on the result (sing the first or second verse of the national anthem as the -1 and +1 levels), and so whatever estimate is made must be due to random effects in an experiment that is free of bias. Each factor effect is the mean of N/2 estimates (here 4), and so a Student s t test can be performed of each estimated factor effect against a null hypothesis of the population mean = 0, with standard deviation the dummy effect. Therefore the t value of the ith effect is ... 

The algorithm used is attributed to J. B. J. Read. For many manipulations on large matrices it is only practical for use with a fairly large computer. The data are arranged in two matrices by sample i and nuclide j one matrix, V, contains the amount of each nuclide in each sample the other matrix, E, contains the variances of these numbers, as estimated from counting statistics, agreement between replicate analyses, and known analytical errors. It is also possible to add an arbitrary term Fik to each variance to account for random effects between samples not considered in the model this is usually done in terms of an additional fractional error. Zeroes are inserted for missing data in cases in which not all nuclides were measured in every sample. 

In order to estimate the random effects model, we need some additional parameter estimates. The group means are y x... 

To estimate the variance components for the random effects model, we also computed the group means regression. The sum of squared residuals from the LSDV estimator is 444,288. The sum of squares from the group means regression is 22382.1. The estimate of a,.2 is 444,288/93 = 4777.29. The estimate of a 2 is 22,382.1/2 - (1/20)4777.29 = 10,952.2. The model is then reestimated by FGLS using these estimates ... 

The F and LM statistics are not useful for comparing the fixed and random effects models. The Hausman statistic can be used. The value appears above. Since the Hausman statistic is small (only 3.14 with two degrees of freedom), we conclude that the GLS estimator is consistent. The statistic would be large if the two estimates were significantly different. Since they are not, we conclude that the evidence favors the random effects model. 

Unbalanced design for random effects. Suppose that the random effects model of Section 13.4 is to be estimated with a panel in which the groups have different numbers of observations. Let 7j be the number of observations in group i. 

Taking all this into consideration, unsaturated designs (f>0) or special designs, which include the influence of interaction effects on linear-effect estimates, are used in practice. An oversaturated design (f<0) was used in Example 2.12 as a random balance method design, but a totally different problem was being solved in that case. 

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