Estimating Bias and RMSE of Indirect Effects using Rescaled Residual Bootstrap in Mediation Analysis ANWAR FITRIANTO Laboratory of Applied and Computational Statistics, Institute for Mathematical Research University Putra Malaysia 43400 UPM Serdang, Selangor MALAYSIA anwarfitrianto@gmail.com HABSHAH MIDI Laboratory of Applied and Computational Statistics, Institute for Mathematical Research University Putra Malaysia 43400 UPM Serdang, Selangor MALAYSIA habshahmidi@gmail.com Abstract: It is a common practice to estimate the parameters of mediation model by using the Ordinary Least Squares (OLS) method. The construction of T statistics and confidence interval estimates for making inferences on the parameters of a mediation model, particularly the indirect effect, is usually are based on the assumption that the estimates are normally distributed. Nonetheless, in practice many estimates are not normal and have a heavy tailed distribution which may be the results of having outliers in the data. An alternative approach is to use bootstrap method which does not rely on the normality assumption. In this paper, we proposed a new bootstrap procedure of indirect effect in mediation model which is resistant to outliers. The proposed approach was based on residual bootstrap which incorporated rescaled studentized residuals, namely the Rescaled Studentized Residual Bootstrap using Least Squares (ReSRB). The Monte Carlo simulations showed that the ReSRB is more efficient than some existing methods in the presence of outliers. Key-Words: Mediation analysis, outliers, bootstrap, studentized residuals, indirect effect 1 Introduction In order to improve the understanding of the nature of relationship between an independent and dependent variables, a third variable is often suggested to be included in the model. When the third variable is considered as a mediator, it is hypothesized to be linked in a causal chain between the independent and dependent variables [1]. The search for intermediate causal variables is called mediation analysis. Mediation analysis is common in social science research, as they elaborate upon other relationships. In these studies, participants are randomly assigned to receive different experimental conditions, and differences in means in the conditions are either consistent or inconsistent with a mediation theory. For example, cognitive dissonance is a social psychological theory which explains that persons make decisions at least in part to reduce internal discomfort or dissonance. Another example is related to programs to reduce drug use, which often target mediators as resistance skills, hypothesizing that a program that increases resistance skills will decrease drug use [2]. The bootstrap is now a widely used method ([3], [4], [5], [6], [7]). In the simplest form of bootstrapping, called the percentile bootstrap, upper and lower confidence limits are obtained by finding the values of the statistics in the 1000 samples that correspond to the 2.5% and 97.5% percentiles. There are several variations of the bootstrap method that are useful in some situations. Another bootstrap method, called the bias-corrected bootstrap, is important for mediation analysis because of its accuracy for computing confidence intervals for the mediated effect when the mediated effect is nonzero [8]. The method consists of adjusting each bootstrap sample for potential bias in the estimate of the statistic. The bias-corrected bootstrap method removes bias that arises because the true parameter value is not the median of the distribution of the bootstrap estimates. The bias correction is used to obtain a new upper and lower percentile used to adjust the confidence limits in the bootstrap distribution. WSEAS TRANSACTIONS on MATHEMATICS Anwar Fitrianto, Habshah Midi ISSN: 1109-2769 397 Issue 6, Volume 9, June 2010