Simulation Studies. We conducted simulation studies to evaluate the impact on ASE bias and mapping quality when using the three competing mapping strategies: using the universal ref- erence genome which is the status quo, using the masked universal reference genome which is introduced by Xxxxxx et al. 2009; and using the diploid personalized ref- erence genome which we propose. In order to represent the diversity of human population and investigate its impact on the results, we selected three individuals from the HapMap panel, one Caucasian from CEPH (NA12865), one African from YRI (NA19238) and one Asian from CHB (NA18621). For each individual, we down- loaded the individual’s genotype information (2007-03 version) from the International HapMap Project. As Xxxxxx et al. did in their study, we randomly inserted sequenc- ing errors on reads generated. We tested three different sequencing error rates: 0, 0.01 and 0.05. When simulating reads, we choose the sequencing read length to be 35 bp and 100 bp, and then randomly sample DNA fragments across the whole diploid reference genome except chromosome X and Y. We only keep reads that cover at least one heterozygous SNP. For each of the three sequencing error rates, 2 million reads were generated. Either reference or alternative allele was selected with equal proba- bility thus assume balanced allele specific expression. To create the masked reference genome, all SNPs identified from the 214 individuals in the International HapMap Project (genotype information obtained from the 2007-03 version) are masked. In order to increase the precision with more mapped reads, we consider SNPs located in both exons and introns.
Simulation Studies. We conducte simulation studies to evaluate performance of the proposed methods compared with existing methods for many different scenarios. To generate the networks and observations on nodes, we follow the simulation settings in Li and Li (2008), Zhe et al. (2013) and Xxx et al. (2013). See Figure
Simulation Studies. Results from the state space model suggest that rewards that help players explore the game become less attractive after purchase and currency rewards will decrease players’ purchase intention. Also, I have found empirical evidence of points pressure for all three types of rewards and level-ups. In this section I seek to gauge the impact of some adjustments of the rewards structure according to the findings on players’ play and purchase decisions. Adjustments in the rewards structures I propose are easy to implement for the developer and I will show how those changes influence player conversion and engagement. The results from the play equation showed that rental rewards and currency rewards become not very appealing to pay players. So, in the first simulation study I want to use customer information (whether players have purchased the character pack) to customize the reward structure. Specifically, for players who own the character pack several rental rewards are replaced by progress accelerators since results from play equation results suggest that rewards that help players progress stays relevant while rewards that help players become less influential. The impact of replacing 1/2/3 rental rewards by progress accelerator rewards on players’ play activity are compared to the case with the original rewards structure and the results are displayed in Table 10. By changing one rental reward to an accelerator reward after players’ purchase, the number of game sessions played daily by character pack owners increased by about 4.95%. This number goes up to 11.54% when I replace 3 rental rewards instead. The model predicts that by simply adjusting the reward type based on player investment status the developer can have a decent boost in customer engagement for the group of committed players. Results of the purchase equation imply that currency rewards work against players’ character pack purchase intentions. However, the results of the play equation show that progress towards currency rewards can motivate free users more than accelerators. In the second simulation study I try to gauge the impact of replacing several currency rewards with accelerator rewards. The impact of this change on player purchase and play are shown in Table 11. When I replace one currency reward by an accelerator the decrease in sessions played daily is only 1.45% while the increase in number of character pack purchases is about 4%. If two currency rewards are replaced, the percentage decreas...
Simulation Studies. In this section, the 1D simulation analysis is conducted to compare our newly pro- posed voxel-wise univariate heritability estimation methods with those frequently used methods in terms of prediction accuracy, validity, sensitivity, and the over- all computation time for different variance component settings. The ROC-based simulation studies generate 2D imaging data for power evaluation and comparison between the voxel- and cluster-wise heritability inference approaches with various simulation settings.
Simulation Studies. We evaluate the proposed methods in Monte Carlo simulations of 500 datasets. Each Monte Carlo dataset contains 500 observations with patient characteristics, principal stratum assignment, treatment information, and out- come data generated as described in the following sections. In all simulation scenarios, principal stratum, treatment assignment, and survival status must be determined prior to outcome generation, as Yi can only exist for those indi- viduals with Si = 1. Therefore, while Yi is generated for all observations with Gi = LL, it can only be generated for observations with Gi = LD and Zi = 1 or Gi = DL and Zi = 0, and it cannot be generated for observations wtih Gi = DD at all. In each of the 500 Monte Carlo datasets, four variables are generated to represent patient characteristics D. D1, D2, D4 are generated from Uniform distributions of varying ranges (Unif [1, 1], Unif [−2, 2], and Unif [0, 1] respec- tively). D3 is generated for each individual using a Bernoulli distribution with p = 0.5. Overlapping subsets of these covariates are used in the models that generate principal strata assignment (D2, D3, D4) and the treatment assign- ment (D1, D2, D3) for each observation. Principal strata assignment, Gi is generated from a discrete distribution with probabilities generated from a multinomial logit model, as presented in equation (2.1) in Section 2.2.3. The parameters of this model, αg, are se- lected such that LD is the reference group and the average values of the probabilities of principal strata in simulation population have the preferred relationship πLL > πLD > πDL > πDD. Specifically the parameter values are αLL = [0.85, 1.0, −0.5, 0.5], αDL = [−0.55, 1.5, −0.5, 0.25], and αDD = [−1.2, −0.8, −0.5, −0.5]. Treatment assignment, Zi, is generated from a Bernoulli distribution with probability pZ,i. A logistic regression model is used to xx- xxxxxxx pZ,i, with parameters β = [0.1, 1.5, −1.0, −0.5], selected to achieve an average probability that is slightly greater than 0.5 for the simulation popula- tion. g
Simulation Studies. Simulation studies will be used to compare the performance of the proposed diagnostic measures in detecting the true number of components for the latent trajectory model. One-, two-, and three-class model solutions were compared using each of the possible criteria. In several instances, models that overfit the data were found to lead to numerical problems or divergent solutions. This is not surprising since cross-sectional finite mixture models often encounter identifiability and numerical issues when too many latent classes are assumed[34]. These numerical issues arise because the regularity conditions required for the asymptotic theory of maximum likelihood to apply are sometimes violated for small data sets, mixtures with small component weights, and overfitting mixtures with too many components[30]. Thus, starting values that led to a GEE that could not be successfully estimated without error were excluded from consideration in the following simulation studies. In addition, divergent solutions were not considered. A divergent solution was deemed to be any solution for which the absolute value of one or more of the parameter estimates associated with the polytomous logistic regression model or the GEE for one of the binary feature variables exceeded 10. Finally, any root that had not converged in 100 iterations was excluded. After the aforementioned roots were excluded from consideration, weak identifiability was assessed [30]. Specifically, when considering a C-component mixture model, any class for which the mixing proportion, πg(g = 1, . . . , C), was less than 0.01 was deemed an empty class and treated as a C − 1 class solution. Similarly, if the maximum absolute difference between elements of βg and βg′ for g ̸= g′(g, g′ = 1, . . . , C) was less than 0.01, the two classes were deemed to be equivalent and the root was treated as a C − 1 class solution.
Simulation Studies. Extensive simulations are conducted to evaluate the performance of the two proposed methods MICE-XXXX and MICE-IURR in comparison with the standard MICE and several other existing methods under general missing data patterns. For MICE-XXXX and MICE-IURR, we consider three regularization methods, namely, lasso, EN and Alasso. We summarize the simulation results over 200 Monte Carlo (MC) data sets. Following Shah et al. (2014), when applying MI methods, we generate 10 imputed data sets for subsequent analysis, which is our primary goal. To benchmark the bias and loss of efficiency in parameter estimation, two additional approaches that do not involve imputations are also included: a gold standard (GS) method that uses the underlying complete data before missing data are generated, and a complete-case analysis (CC) method that uses only complete-cases for which all the variables are observed(Little and Xxxxx, 2002). The setup of the simulations is similar to what was used in Zhao and Long (2013). Specifically, the sample size is fixed at n = 100 and each simulated data set includes Y , the fully observed outcome variable, and Z = (Z1, . . . , Zp), the set of predictors and auxiliary variables. We consider settings with p = 200 and p = 1000. We consider Z1, Z2, and Z3 having missing values, which follow a general missing data pattern. We first generate (Z4, . . . , Zp) from a multivariate normal distribution with mean (0, . . . , 0)p−4 and a first order autoregressive covariance matrix with autocorrelation ρ varying as 0, 0.1, 0.5, and 0.9. Given (Z4, . . . , Zp), variables Z1, Z2, and Z3 are generated independently from a normal distribution N (1 + ZSα, 4), where S represents the true active set with a cardinality of q. We further consider settings where q = 4 and 20, and α = (1, . . . , 1)j4 for q = 4; α = (0.2, . . . , 0.2)j20 for q = 20. For q = 4 and 20, the corresponding true active set ZS = {Z4, Z5, Z50, Z51} and {Z4, . . . , Z13, Z50, . . . , Z59}. Given Z, the outcome variable Y is generated from Y = β0 + β1Z1 + β2Z2 + β3Z3 + β4Z4 + β5Z5 + s, where βi = 1, and s ∼ N (0, 6) is random noise and independent of Zi. Missing values are created in Z1, Z2, and Z3 using the following logit models for the corresponding missing indicators, x0, x0, and x0, xxxxx(Xx(x0 = 1)) = −1−Z4+2Z5 −Y , logit(Pr(δ2 = 1)) = −1−Z4+2Z51 −Y , and logit(Pr(δ3 = 1)) = −1 − Z50 + 2Z51 − Y , resulting in approximately 40% of observations having missing values. We compare our proposed ...
Simulation Studies. We examine in this section the performance of the proposed methods. Suppose our vertically partitioned data X consist p = 6 independent variables from K = 3 institutions. We assume each institution possesses two independent variables and has access to the outcome variable Y . We are interested in a multiple linear regression: Y = Xθ + s site1 site2 site3 = ( , Xx1s, X˛2, ¸X3x,sX˛4, X¸ 5x,sX˛6) × (θ0, x0, x0, x0, x0, x0, x0)X + s, when X1 has missing values. We generate X2, ..., X6 independently from the uniform distribution on (-1, 1). We
