would be expected for fall near the center of the histogram of values, suggesting T(yobs) T(yr) that the observed data are highly plausible given the Bayesian posterior parameter estimates. Alternatively, one may compute the proportion of values that are equal to or greater than T(yr) that is, Pr [T(yr) This proportion, known as the T(yobs), T(yobs)]. posterior predictive p value ≥ (ppp), expresses the probability that the predicted values derived from the Bayesian estimates of the model parameters are more extreme than the observed data. For a good-fitting model ppp would be expected to equal approximately 0.5, again indicating that falls in the center of T(yobs) the distribution of predicted test statistics and that the observed sample data are highly T(yr) plausible given the Bayesian estimates of the model parameters (see Muthén & Asparouhov, 2012). A Robust Assessment of Model Fit Given a Small Sample When sample sizes are prohibitively small, any given sample statistic may be T(yobs) biased, which in turn may lead to distorted or incorrect assessments of model fit based on comparisons between and Therefore, rather than comparing with a single T(yr) T(yobs). T(yr) point estimate calculated from the sample data, in the current demonstration analysis, T(yobs) values of were gauged against the sampling distribution of which was obtained T(yr) T(yobs), using bootstrap resampling of the sample data (Efron & Tibshirani, 1993). Briefly, bootstrap resampling entails randomly selecting observations with replacement from a given sample of size until bootstrap samples of size have been drawn using only the observations in the original n B n sample y1, yn. A given test statistic may be computed for each of the bootstrap T(yb) B …, samples (b = 1, The collection of bootstrap test statistics simulates the sampling B). …, distribution of with the expected value estimated as and variance estimated as . The T(yobs) ∙ square root of the estimated variance of the bootstrap sampling distribution, is a robust estimate of the standard error of which quantifies the sampling variability associated with T(yobs),
T(yobs) would be expected for fall near the center of the histogram of T(yr) values, suggesting that the observed data are highly plausible given the Bayesian posterior parameter estimates. Alternatively, one may compute the proportion of T(yr) values that are equal to or greater than T(yobs), that is, Pr [T(yr) ≥ T(yobs)]. This proportion, known as the posterior predictive p value (ppp), expresses the probability that the predicted values derived from the Bayesian estimates of the model parameters are more extreme than the observed data. For a good-fitting model ppp would be expected to equal approximately 0.5, again indicating that T(yobs) falls in the center of the distribution of predicted test statistics T(yr) and that the observed sample data are highly plausible given the Bayesian estimates of the model parameters (see Muthén & Asparouhov, 2012). A Robust Assessment of Model Fit Given a Small Sample When sample sizes are prohibitively small, any given sample statistic T(yobs) may be biased, which in turn may lead to distorted or incorrect assessments of model fit based on comparisons between T(yr) and T(yobs). Therefore, rather than comparing T(yr) with a single point estimate T(yobs) calculated from the sample data, in the current demonstration analysis, values of T(yr) were gauged against the sampling distribution of T(yobs), which was obtained using bootstrap resampling of the sample data (Efron & Tibshirani, 1993). Briefly, bootstrap resampling entails randomly selecting observations with replacement from a given sample of size n until B bootstrap samples of size n have been drawn using only the observations in the original sample y1, …, yn. A given test statistic T(yb) may be computed for each of the B bootstrap samples (b = 1, …, B). The collection of bootstrap test statistics simulates the sampling distribution of T(yobs) with the expected value estimated as and variance estimated as ∙ . The square root of the estimated variance of the bootstrap sampling distribution, is a robust estimate of the standard error of T(yobs), which quantifies the sampling variability associated with
CCP-CCP3-Ozechowski20121361-RR-F1.FINAL.docx