3 years ago

First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

This is illustrated in

This is illustrated in Figure 1, which shows histograms of 20,000 simulatedX 1 values for each of the four base distributions, with φ 1,2 = 0.75 and areliability of 0.90 (these values are part of the “mild” parameter configurationused in some simulations that come later in the paper).Figure 1: Simulated values of X 1Normal Base DistributionPareto Base DistributionRelative Frequency0.0 0.1 0.2 0.3 0.4 0.5 0.6Relative Frequency0.0 0.1 0.2 0.3 0.4 0.5 0.6−15 −10 −5 0 5 10 15X1−15 −10 −5 0 5 10 15X1T Base DistributionUniform Base DistributionRelative Frequency0.0 0.1 0.2 0.3 0.4 0.5 0.6Relative Frequency0.0 0.1 0.2 0.3 0.4 0.5 0.6−15 −10 −5 0 5 10 15X1−15 −10 −5 0 5 10 15X1As intended, the t base distribution yields heavy-tailed symmetric distributions,the Pareto yields heavy-tailed nonsymmetric distributions, and theuniform yields light-tailed distributions. One thing that is not apparent inFigure 1 is the high outliers generated by the Pareto base distribution, andthe high and low outliers generated by the t base distribution. For the Paretobase distribution, simulated values of X 1 range from -1.19 to 14.53; for thet, they range from -14.44 to 12.00. Of course the true variance of X 1 is thesame regardless of the base distribution; in this case it is 10/9 ≈ 1.11, andthe sample variances of the simulated data are all close to this value.14

1.2.2 ResultsAgain, this is a complete factorial experiment with 5 × 5 × 3 × 5 × 5 ×4 = 7, 500 treatment combinations. Within each treatment combination,we independently generated 10,000 random sets of data, yielding 75 millionsimulated data sets in all. For each one, we ignored measurement error, fittedModel (4) and tested H 0 : β 2 = 0 with the usual “extra sum of squares” F -test. The proportion of simulated data sets for which the null hypothesis wasrejected at α = 0.05 is a Monte Carlo estimate of the Type I error rate.Considerations of space do not permit us to reproduce the entire set of resultshere. Instead, we give excerpts that tell the main part of the story, referringthe reader to the rest. On the Web, the full set of results is available in the form ofa 6-dimensional table with 7,500 cells, and also in the form of a data filewith 7,500 lines, suitable as input data for further analysis. Complete sourcecode for the special-purpose fortran programs we wrote is also available fordownload, along with other supporting materials.Table 1 shows the results when all the variables are normally distributedand the reliabilities of both independent variables equal 0.90; that is, only10% of the variance of the independent variables arises from measurementerror. In the social and behavioral sciences, a reliability of 0.90 would beconsidered impressively high, and one might think there was little to worryabout.Table 1 shows that except when the latent independent variables ξ 1 andξ 2 are uncorrelated, applying ordinary least squares regression to the correspondingobservable variables X 1 and X 2 results in a substantial inflation ofthe Type I error rate. As one would predict from Expression 5 with θ 1,2 = 0,the problem becomes more severe as ξ 1 and ξ 2 become more strongly related,as ξ 1 and Y become more strongly related, and as the sample size increases.We view these Type I error rates as shockingly high, even for fairly moderatesample sizes and modest relationships among variables.This pattern of results holds for all four base distributions, and for alltwenty-five combinations of reliabilities of the independent variables. In addition,the Type I error rates increased with decreasing reliability of X 1 ,and decreased with decreasing reliability of X 2 , the variable being tested.The distribution of the error terms and independent variables did not mattermuch, though average Type I error rates were slightly lower when the basedistribution was the skewed and heavy-tailed Pareto. These trends are sum-15

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