Index Tracking Using Data-Mining Techniques and Mixed-Binary Linear Programming

Index tracking has become one of the most common strategies in asset management. The index-tracking problem consists of constructing a portfolio that replicates the future performance of an index by including only a subset of the index constituents in the portfolio. Finding the most representative subset is challenging when the number of stocks in the index is large. We introduce a new three-stage approach that at first identifies promising subsets by employing data-mining techniques, then determines the stock weights in the subsets using mixed-binary linear programming, and finally evaluates the subsets based on cross validation. The best subset is returned as the tracking portfolio. Our approach outperforms state-of-the-art methods in terms of out-of-sample performance and running times.

Influence of prosthesis type and retention mechanism on complications with fixed implant-supported prostheses: a systematic review applying multivariate analyses

PURPOSE To identify the influence of fixed prosthesis type on biologic and technical complication rates in the context of screw versus cement retention. Furthermore, a multivariate analysis was conducted to determine which factors, when considered together, influence the complication and failure rates of fixed implant-supported prostheses. MATERIALS AND METHODS Electronic searches of MEDLINE (PubMed), EMBASE, and the Cochrane Library were conducted. Selected inclusion and exclusion criteria were used to limit the search. Data were analyzed statistically with simple and multivariate random-effects Poisson regressions. RESULTS Seventy-three articles qualified for inclusion in the study. Screw-retained prostheses showed a tendency toward and significantly more technical complications than cemented prostheses with single crowns and fixed partial prostheses, respectively. Resin chipping and ceramic veneer chipping had high mean event rates, at 10.04 and 8.95 per 100 years, respectively, for full-arch screwed prostheses. For "all fixed prostheses" (prosthesis type not reported or not known), significantly fewer biologic and technical complications were seen with screw retention. Multivariate analysis revealed a significantly greater incidence of technical complications with cemented prostheses. Full-arch prostheses, cantilevered prostheses, and "all fixed prostheses" had significantly higher complication rates than single crowns. A significantly greater incidence of technical and biologic complications was seen with cemented prostheses. CONCLUSION Screw-retained fixed partial prostheses demonstrated a significantly higher rate of technical complications and screw-retained full-arch prostheses demonstrated a notably high rate of veneer chipping. When "all fixed prostheses" were considered, significantly higher rates of technical and biologic complications were seen for cement-retained prostheses. Multivariate Poisson regression analysis failed to show a significant difference between screw- and cement-retained prostheses with respect to the incidence of failure but demonstrated a higher rate of technical and biologic complications for cement-retained prostheses. The incidence of technical complications was more dependent upon prosthesis and retention type than prosthesis or abutment material.

Accuracy of linear measurements using three imaging modalities: two lateral cephalograms and one 3D model from CBCT data

BACKGROUND The aim of this study was to evaluate the accuracy of linear measurements on three imaging modalities: lateral cephalograms from a cephalometric machine with a 3 m source-to-mid-sagittal-plane distance (SMD), from a machine with 1.5 m SMD and 3D models from cone-beam computed tomography (CBCT) data. METHODS Twenty-one dry human skulls were used. Lateral cephalograms were taken, using two cephalometric devices: one with a 3 m SMD and one with a 1.5 m SMD. CBCT scans were taken by 3D Accuitomo® 170, and 3D surface models were created in Maxilim® software. Thirteen linear measurements were completed twice by two observers with a 4 week interval. Direct physical measurements by a digital calliper were defined as the gold standard. Statistical analysis was performed. RESULTS Nasion-Point A was significantly different from the gold standard in all methods. More statistically significant differences were found on the measurements of the 3 m SMD cephalograms in comparison to the other methods. Intra- and inter-observer agreement based on 3D measurements was slightly better than others. LIMITATIONS Dry human skulls without soft tissues were used. Therefore, the results have to be interpreted with caution, as they do not fully represent clinical conditions. CONCLUSIONS 3D measurements resulted in a better observer agreement. The accuracy of the measurements based on CBCT and 1.5 m SMD cephalogram was better than a 3 m SMD cephalogram. These findings demonstrated the linear measurements accuracy and reliability of 3D measurements based on CBCT data when compared to 2D techniques. Future studies should focus on the implementation of 3D cephalometry in clinical practice.

Diagnostic Accuracy of Full-Body Linear X-Ray Scanning in Multiple Trauma Patients in Comparison to Computed Tomography

Purpose: The purpose of this study was to evaluate the diagnostic accuracy of full-body linear X-ray scanning (LS) in multiple trauma patients in comparison to 128-multislice computed tomography (MSCT). Materials and Methods: 106 multiple trauma patients (female: 33; male: 73) were retrospectively included in this study. All patients underwent LS of the whole body, including extremities, and MSCT covering the neck, thorax, abdomen, and pelvis. The diagnostic accuracy of LS for the detection of fractures of the truncal skeleton and pneumothoraces was evaluated in comparison to MSCT by two observers in consensus. Extremity fractures detected by LS were documented. Results: The overall sensitivity of LS was 49.2 %, the specificity was 93.3 %, the positive predictive value was 91 %, and the negative predictive value was 57.5 %. The overall sensitivity for vertebral fractures was 16.7 %, and the specificity was 100 %. The sensitivity was 48.7 % and the specificity 98.2 % for all other fractures. Pneumothoraces were detected in 12 patients by CT, but not by LS. 40 extremity fractures were detected by LS, of which 4 fractures were dislocated, and 2 were fully covered by MSCT. Conclusion: The diagnostic accuracy of LS is limited in the evaluation of acute trauma of the truncal skeleton. LS allows fast whole-body X-ray imaging, and may be valuable for detecting extremity fractures in trauma patients in addition to MSCT. Key Points: • The overall sensitivity of LS for truncal skeleton injuries in multiple-trauma patients was < 50 %.• The diagnostic reference standard MSCT is the preferred and reliable imaging modality.• LS may be valuable for quick detection of extremity fractures. Citation Format: • Jöres APW., Heverhagen JT, Bonél H et al. Diagnostic Accuracy of Full-Body Linear X-Ray Scanning in Multiple Trauma Patients in Comparison to Computed Tomography. Fortschr Röntgenstr 2016; 188: 163 - 171.

Influence of previous surgery on patient-rated outcome after surgery for degenerative disorders of the lumbar spine.

PURPOSE Few studies have used multivariate models to quantify the effect of multiple previous spine surgeries on patient-oriented outcome after spine surgery. This study sought to quantify the effect of prior spine surgery on 12-month postoperative outcomes in patients undergoing surgery for different degenerative disorders of the lumbar spine. METHODS The study included 4940 patients with lumbar degenerative disease documented in the Spine Tango Registry of EUROSPINE, the Spine Society of Europe, from 2004 to 2015. Preoperatively and 12 months postoperatively, patients completed the multidimensional Core Outcome Measures Index (COMI; 0-10 scale). Patients' medical history and surgical details were recorded using the Spine Tango Surgery 2006 and 2011 forms. Multiple linear regression models were used to investigate the relationship between the number of previous surgeries and the 12-month postoperative COMI score, controlling for the baseline COMI score and other potential confounders. RESULTS In the adjusted model including all cases, the 12-month COMI score showed a 0.37-point worse value [95 % confidence intervals (95 % CI) 0.29-0.45; p < 0.001] for each additional prior spine surgery. In the subgroup of patients with lumbar disc herniation, the corresponding effect was 0.52 points (95 % CI 0.27-0.77; p < 0.001) and in lumbar degenerative spondylolisthesis, 0.40 points (95 % CI 0.17-0.64; p = 0.001). CONCLUSIONS We were able to demonstrate a clear "dose-response" effect for previous surgery: the greater the number of prior spine surgeries, the systematically worse the outcome at 12 months' follow-up. The results of this study can be used when considering or consenting a patient for further surgery, to better inform the patient of the likely outcome and to set realistic expectations.

Genetic alterations associated with prostate cancer progression

Prostate cancer is the second most commonly diagnosed cancer among men in the United States. In this study, evidence is presented to support the hypothesis that specific chromosomal aberrations (involving one or more chromosomal regions) are associated with prostate cancer progression from organ-confined to locally advanced tumors and that some aberrations seen in high frequency in metastatic tumors may also be present in a subset of primary tumors. To determine the appropriate approach to address this hypothesis, I have established a modified CGH protocol by microdissection and DOP-PCR for use in detecting chromosomal changes in clinical prostate tumor specimens that is more sensitive and accurate than conventional CGH methods. I have successfully performed the improved CGH protocol to screen for genetic changes of 24 organ confined (pT2) and 21 locally advanced (pT3b) clinical prostate cancer specimens without metastases (N0M0). Comparisons of tumors by stage or Gleason scores following contingency table analysis showed that seven regions of the genome differed significantly between pT2 and pT3b tumors or between low and high Gleason tumors suggesting that these regions may be important in local prostate cancer progression. These included losses on 6p21–25, 6q24–27, 8p, 10q25–26, 15q22–26, and 18cen–q12 as well as gain of 3p13–q13. Multivariate analyses showed that loss of 8p (step1) and loss of 6q25–26 (or 6p21–25 or 10q25–26) (step 2) were predictive of pathologic stage or Gleason groups with 80% accuracy. Additional 5–7 steps in the multivariate model increased the predictive value to 91–95%. Comparison of the CGH data from the primary prostate tumors of this study with those obtained from published literature on metastases and recurrent tumors showed that the clinically more aggressive stage pT3b tumors shared more abnormalities in high frequency with metastases and recurrent tumors than less aggressive stage pT2 tumors. Furthermore, loss of 11cen–q22 was shared only between the primary tumors and metastases while gain of Xcen–q13 and loss of 18cen–q12 were in common between primary and recurrent tumors. These analyses suggest that the multistage model of prostate cancer progression is not linear and that some early primary tumors may be predisposed to metastasize or evolve into recurrent tumors due to the presence of specific genetic alterations. ^

Bayesian generalized linear models for meta-analysis of diagnostic tests

With the recognition of the importance of evidence-based medicine, there is an emerging need for methods to systematically synthesize available data. Specifically, methods to provide accurate estimates of test characteristics for diagnostic tests are needed to help physicians make better clinical decisions. To provide more flexible approaches for meta-analysis of diagnostic tests, we developed three Bayesian generalized linear models. Two of these models, a bivariate normal and a binomial model, analyzed pairs of sensitivity and specificity values while incorporating the correlation between these two outcome variables. Noninformative independent uniform priors were used for the variance of sensitivity, specificity and correlation. We also applied an inverse Wishart prior to check the sensitivity of the results. The third model was a multinomial model where the test results were modeled as multinomial random variables. All three models can include specific imaging techniques as covariates in order to compare performance. Vague normal priors were assigned to the coefficients of the covariates. The computations were carried out using the 'Bayesian inference using Gibbs sampling' implementation of Markov chain Monte Carlo techniques. We investigated the properties of the three proposed models through extensive simulation studies. We also applied these models to a previously published meta-analysis dataset on cervical cancer as well as to an unpublished melanoma dataset. In general, our findings show that the point estimates of sensitivity and specificity were consistent among Bayesian and frequentist bivariate normal and binomial models. However, in the simulation studies, the estimates of the correlation coefficient from Bayesian bivariate models are not as good as those obtained from frequentist estimation regardless of which prior distribution was used for the covariance matrix. The Bayesian multinomial model consistently underestimated the sensitivity and specificity regardless of the sample size and correlation coefficient. In conclusion, the Bayesian bivariate binomial model provides the most flexible framework for future applications because of its following strengths: (1) it facilitates direct comparison between different tests; (2) it captures the variability in both sensitivity and specificity simultaneously as well as the intercorrelation between the two; and (3) it can be directly applied to sparse data without ad hoc correction. ^

Analysis of prognostic factors for the development of metastases and survival in renal cell carcinoma patients

Introduction and objective. A number of prognostic factors have been reported for predicting survival in patients with renal cell carcinoma. Yet few studies have analyzed the effects of those factors at different stages of the disease process. In this study, different stages of disease progression starting from nephrectomy to metastasis, from metastasis to death, and from evaluation to death were evaluated. ^ Methods. In this retrospective follow-up study, records of 97 deceased renal cell carcinoma (RCC) patients were reviewed between September 2006 to October 2006. Patients with TNM Stage IV disease before nephrectomy or with cancer diagnoses other than RCC were excluded leaving 64 records for analysis. Patient TNM staging, Furhman Grade, age, tumor size, tumor volume, histology and patient gender were analyzed in relation to time to metastases. Time from nephrectomy to metastasis, TNM staging, Furhman Grade, age, tumor size, tumor volume, histology and patient gender were tested for significance in relation to time from metastases to death. Finally, analysis of laboratory values at time of evaluation, Eastern Cooperative Oncology Group performance status (ECOG), UCLA Integrated Staging System (UISS), time from nephrectomy to metastasis, TNM staging, Furhman Grade, age, tumor size, tumor volume, histology and patient gender were tested for significance in relation to time from evaluation to death. Linear regression and Cox Proportional Hazard (univariate and multivariate) was used for testing significance. Kaplan-Meier Log-Rank test was used to detect any significance between groups at various endpoints. ^ Results. Compared to negative lymph nodes at time of nephrectomy, a single positive lymph node had significantly shorter time to metastasis (p<0.0001). Compared to other histological types, clear cell histology had significant metastasis free survival (p=0.003). Clear cell histology compared to other types (p=0.0002 univariate, p=0.038 multivariate) and time to metastasis with log conversion (p=0.028) significantly affected time from metastasis to death. A greater than one year and greater than two year metastasis free interval, compared to patients that had metastasis before one and two years, had statistically significant survival benefit (p=0.004 and p=0.0318). Time from evaluation to death was affected by greater than one year metastasis free interval (p=0.0459), alcohol consumption (p=0.044), LDH (p=0.006), ECOG performance status (p<0.001), and hemoglobin level (p=0.0092). The UISS risk stratified the patient population in a statistically significant manner for survival (p=0.001). No other factors were found to be significant. ^ Conclusion. Clear cell histology is predictive for both time to metastasis and metastasis to death. Nodal status at time of nephrectomy may predict risk of metastasis. The time interval to metastasis significantly predicts time from metastasis to death and time from evaluation to death. ECOG performance status, and hemoglobin levels predicts survival outcome at evaluation. Finally, UISS appropriately stratifies risk in our population. ^

A multivariate frailty model for disease recurrences and survival

A multivariate frailty hazard model is developed for joint-modeling of three correlated time-to-event outcomes: (1) local recurrence, (2) distant recurrence, and (3) overall survival. The term frailty is introduced to model population heterogeneity. The dependence is modeled by conditioning on a shared frailty that is included in the three hazard functions. Independent variables can be included in the model as covariates. The Markov chain Monte Carlo methods are used to estimate the posterior distributions of model parameters. The algorithm used in present application is the hybrid Metropolis-Hastings algorithm, which simultaneously updates all parameters with evaluations of gradient of log posterior density. The performance of this approach is examined based on simulation studies using Exponential and Weibull distributions. We apply the proposed methods to a study of patients with soft tissue sarcoma, which motivated this research. Our results indicate that patients with chemotherapy had better overall survival with hazard ratio of 0.242 (95% CI: 0.094 - 0.564) and lower risk of distant recurrence with hazard ratio of 0.636 (95% CI: 0.487 - 0.860), but not significantly better in local recurrence with hazard ratio of 0.799 (95% CI: 0.575 - 1.054). The advantages and limitations of the proposed models, and future research directions are discussed. ^

Statistical and methodological challenges for disaster preparedness and medical needs assessment in Rio Grande Valley of Texas

In recent years, disaster preparedness through assessment of medical and special needs persons (MSNP) has taken a center place in public eye in effect of frequent natural disasters such as hurricanes, storm surge or tsunami due to climate change and increased human activity on our planet. Statistical methods complex survey design and analysis have equally gained significance as a consequence. However, there exist many challenges still, to infer such assessments over the target population for policy level advocacy and implementation. ^ Objective. This study discusses the use of some of the statistical methods for disaster preparedness and medical needs assessment to facilitate local and state governments for its policy level decision making and logistic support to avoid any loss of life and property in future calamities. ^ Methods. In order to obtain precise and unbiased estimates for Medical Special Needs Persons (MSNP) and disaster preparedness for evacuation in Rio Grande Valley (RGV) of Texas, a stratified and cluster-randomized multi-stage sampling design was implemented. US School of Public Health, Brownsville surveyed 3088 households in three counties namely Cameron, Hidalgo, and Willacy. Multiple statistical methods were implemented and estimates were obtained taking into count probability of selection and clustering effects. Statistical methods for data analysis discussed were Multivariate Linear Regression (MLR), Survey Linear Regression (Svy-Reg), Generalized Estimation Equation (GEE) and Multilevel Mixed Models (MLM) all with and without sampling weights. ^ Results. Estimated population for RGV was 1,146,796. There were 51.5% female, 90% Hispanic, 73% married, 56% unemployed and 37% with their personal transport. 40% people attained education up to elementary school, another 42% reaching high school and only 18% went to college. Median household income is less than $15,000/year. MSNP estimated to be 44,196 (3.98%) [95% CI: 39,029; 51,123]. All statistical models are in concordance with MSNP estimates ranging from 44,000 to 48,000. MSNP estimates for statistical methods are: MLR (47,707; 95% CI: 42,462; 52,999), MLR with weights (45,882; 95% CI: 39,792; 51,972), Bootstrap Regression (47,730; 95% CI: 41,629; 53,785), GEE (47,649; 95% CI: 41,629; 53,670), GEE with weights (45,076; 95% CI: 39,029; 51,123), Svy-Reg (44,196; 95% CI: 40,004; 48,390) and MLM (46,513; 95% CI: 39,869; 53,157). ^ Conclusion. RGV is a flood zone, most susceptible to hurricanes and other natural disasters. People in the region are mostly Hispanic, under-educated with least income levels in the U.S. In case of any disaster people in large are incapacitated with only 37% have their personal transport to take care of MSNP. Local and state government’s intervention in terms of planning, preparation and support for evacuation is necessary in any such disaster to avoid loss of precious human life. ^ Key words: Complex Surveys, statistical methods, multilevel models, cluster randomized, sampling weights, raking, survey regression, generalized estimation equations (GEE), random effects, Intracluster correlation coefficient (ICC).^

THE USE OF MULTIVARIATE ANALYSIS OF COVARIANCE IN EVALUATING THE COMPARABILITY OF LABORATORY DETERMINATIONS IN COOPERATIVE STUDIES

The role of clinical chemistry has traditionally been to evaluate acutely ill or hospitalized patients. Traditional statistical methods have serious drawbacks in that they use univariate techniques. To demonstrate alternative methodology, a multivariate analysis of covariance model was developed and applied to the data from the Cooperative Study of Sickle Cell Disease.^ The purpose of developing the model for the laboratory data from the CSSCD was to evaluate the comparability of the results from the different clinics. Several variables were incorporated into the model in order to control for possible differences among the clinics that might confound any real laboratory differences.^ Differences for LDH, alkaline phosphatase and SGOT were identified which will necessitate adjustments by clinic whenever these data are used. In addition, aberrant clinic values for LDH, creatinine and BUN were also identified.^ The use of any statistical technique including multivariate analysis without thoughtful consideration may lead to spurious conclusions that may not be corrected for some time, if ever. However, the advantages of multivariate analysis far outweigh its potential problems. If its use increases as it should, the applicability to the analysis of laboratory data in prospective patient monitoring, quality control programs, and interpretation of data from cooperative studies could well have a major impact on the health and well being of a large number of individuals. ^

Detecting genetic and nutritional lung cancer risk factors related to folate metabolism using Bayesian generalized linear models

Complex diseases, such as cancer, are caused by various genetic and environmental factors, and their interactions. Joint analysis of these factors and their interactions would increase the power to detect risk factors but is statistically. Bayesian generalized linear models using student-t prior distributions on coefficients, is a novel method to simultaneously analyze genetic factors, environmental factors, and interactions. I performed simulation studies using three different disease models and demonstrated that the variable selection performance of Bayesian generalized linear models is comparable to that of Bayesian stochastic search variable selection, an improved method for variable selection when compared to standard methods. I further evaluated the variable selection performance of Bayesian generalized linear models using different numbers of candidate covariates and different sample sizes, and provided a guideline for required sample size to achieve a high power of variable selection using Bayesian generalize linear models, considering different scales of number of candidate covariates. ^ Polymorphisms in folate metabolism genes and nutritional factors have been previously associated with lung cancer risk. In this study, I simultaneously analyzed 115 tag SNPs in folate metabolism genes, 14 nutritional factors, and all possible genetic-nutritional interactions from 1239 lung cancer cases and 1692 controls using Bayesian generalized linear models stratified by never, former, and current smoking status. SNPs in MTRR were significantly associated with lung cancer risk across never, former, and current smokers. In never smokers, three SNPs in TYMS and three gene-nutrient interactions, including an interaction between SHMT1 and vitamin B12, an interaction between MTRR and total fat intake, and an interaction between MTR and alcohol use, were also identified as associated with lung cancer risk. These lung cancer risk factors are worthy of further investigation.^

New methods for quantification and analysis of quantitative real-time polymerase chain reaction data

Quantitative real-time polymerase chain reaction (qPCR) is a sensitive gene quantitation method that has been widely used in the biological and biomedical fields. The currently used methods for PCR data analysis, including the threshold cycle (CT) method, linear and non-linear model fitting methods, all require subtracting background fluorescence. However, the removal of background fluorescence is usually inaccurate, and therefore can distort results. Here, we propose a new method, the taking-difference linear regression method, to overcome this limitation. Briefly, for each two consecutive PCR cycles, we subtracted the fluorescence in the former cycle from that in the later cycle, transforming the n cycle raw data into n-1 cycle data. Then linear regression was applied to the natural logarithm of the transformed data. Finally, amplification efficiencies and the initial DNA molecular numbers were calculated for each PCR run. To evaluate this new method, we compared it in terms of accuracy and precision with the original linear regression method with three background corrections, being the mean of cycles 1-3, the mean of cycles 3-7, and the minimum. Three criteria, including threshold identification, max R2, and max slope, were employed to search for target data points. Considering that PCR data are time series data, we also applied linear mixed models. Collectively, when the threshold identification criterion was applied and when the linear mixed model was adopted, the taking-difference linear regression method was superior as it gave an accurate estimation of initial DNA amount and a reasonable estimation of PCR amplification efficiencies. When the criteria of max R2 and max slope were used, the original linear regression method gave an accurate estimation of initial DNA amount. Overall, the taking-difference linear regression method avoids the error in subtracting an unknown background and thus it is theoretically more accurate and reliable. This method is easy to perform and the taking-difference strategy can be extended to all current methods for qPCR data analysis.^

Application of the general linear model to assess the effect of missing data on bone marrow mononuclear cell therapy with infusion timing and follow-up MRI

With most clinical trials, missing data presents a statistical problem in evaluating a treatment's efficacy. There are many methods commonly used to assess missing data; however, these methods leave room for bias to enter the study. This thesis was a secondary analysis on data taken from TIME, a phase 2 randomized clinical trial conducted to evaluate the safety and effect of the administration timing of bone marrow mononuclear cells (BMMNC) for subjects with acute myocardial infarction (AMI).^ We evaluated the effect of missing data by comparing the variance inflation factor (VIF) of the effect of therapy between all subjects and only subjects with complete data. Through the general linear model, an unbiased solution was made for the VIF of the treatment's efficacy using the weighted least squares method to incorporate missing data. Two groups were identified from the TIME data: 1) all subjects and 2) subjects with complete data (baseline and follow-up measurements). After the general solution was found for the VIF, it was migrated Excel 2010 to evaluate data from TIME. The resulting numerical value from the two groups was compared to assess the effect of missing data.^ The VIF values from the TIME study were considerably less in the group with missing data. By design, we varied the correlation factor in order to evaluate the VIFs of both groups. As the correlation factor increased, the VIF values increased at a faster rate in the group with only complete data. Furthermore, while varying the correlation factor, the number of subjects with missing data was also varied to see how missing data affects the VIF. When subjects with only baseline data was increased, we saw a significant rate increase in VIF values in the group with only complete data while the group with missing data saw a steady and consistent increase in the VIF. The same was seen when we varied the group with follow-up only data. This essentially showed that the VIFs steadily increased when missing data is not ignored. When missing data is ignored as with our comparison group, the VIF values sharply increase as correlation increases.^

Robust effect sizes and their confidence intervals for group difference between trajectories in hierarchical linear growth model

Hierarchical linear growth model (HLGM), as a flexible and powerful analytic method, has played an increased important role in psychology, public health and medical sciences in recent decades. Mostly, researchers who conduct HLGM are interested in the treatment effect on individual trajectories, which can be indicated by the cross-level interaction effects. However, the statistical hypothesis test for the effect of cross-level interaction in HLGM only show us whether there is a significant group difference in the average rate of change, rate of acceleration or higher polynomial effect; it fails to convey information about the magnitude of the difference between the group trajectories at specific time point. Thus, reporting and interpreting effect sizes have been increased emphases in HLGM in recent years, due to the limitations and increased criticisms for statistical hypothesis testing. However, most researchers fail to report these model-implied effect sizes for group trajectories comparison and their corresponding confidence intervals in HLGM analysis, since lack of appropriate and standard functions to estimate effect sizes associated with the model-implied difference between grouping trajectories in HLGM, and also lack of computing packages in the popular statistical software to automatically calculate them. ^ The present project is the first to establish the appropriate computing functions to assess the standard difference between grouping trajectories in HLGM. We proposed the two functions to estimate effect sizes on model-based grouping trajectories difference at specific time, we also suggested the robust effect sizes to reduce the bias of estimated effect sizes. Then, we applied the proposed functions to estimate the population effect sizes (d ) and robust effect sizes (du) on the cross-level interaction in HLGM by using the three simulated datasets, and also we compared the three methods of constructing confidence intervals around d and du recommended the best one for application. At the end, we constructed 95% confidence intervals with the suitable method for the effect sizes what we obtained with the three simulated datasets. ^ The effect sizes between grouping trajectories for the three simulated longitudinal datasets indicated that even though the statistical hypothesis test shows no significant difference between grouping trajectories, effect sizes between these grouping trajectories can still be large at some time points. Therefore, effect sizes between grouping trajectories in HLGM analysis provide us additional and meaningful information to assess group effect on individual trajectories. In addition, we also compared the three methods to construct 95% confident intervals around corresponding effect sizes in this project, which handled with the uncertainty of effect sizes to population parameter. We suggested the noncentral t-distribution based method when the assumptions held, and the bootstrap bias-corrected and accelerated method when the assumptions are not met.^