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.^

A hybrid method in combining treatment effects from matched and unmatched studies

In the biomedical studies, the general data structures have been the matched (paired) and unmatched designs. Recently, many researchers are interested in Meta-Analysis to obtain a better understanding from several clinical data of a medical treatment. The hybrid design, which is combined two data structures, may create the fundamental question for statistical methods and the challenges for statistical inferences. The applied methods are depending on the underlying distribution. If the outcomes are normally distributed, we would use the classic paired and two independent sample T-tests on the matched and unmatched cases. If not, we can apply Wilcoxon signed rank and rank sum test on each case. ^ To assess an overall treatment effect on a hybrid design, we can apply the inverse variance weight method used in Meta-Analysis. On the nonparametric case, we can use a test statistic which is combined on two Wilcoxon test statistics. However, these two test statistics are not in same scale. We propose the Hybrid Test Statistic based on the Hodges-Lehmann estimates of the treatment effects, which are medians in the same scale.^ To compare the proposed method, we use the classic meta-analysis T-test statistic on the combined the estimates of the treatment effects from two T-test statistics. Theoretically, the efficiency of two unbiased estimators of a parameter is the ratio of their variances. With the concept of Asymptotic Relative Efficiency (ARE) developed by Pitman, we show ARE of the hybrid test statistic relative to classic meta-analysis T-test statistic using the Hodges-Lemann estimators associated with two test statistics.^ From several simulation studies, we calculate the empirical type I error rate and power of the test statistics. The proposed statistic would provide effective tool to evaluate and understand the treatment effect in various public health studies as well as clinical trials.^

APPLICATIONS OF EPHB4 RECEPTOR SPECIFIC PEPTIDES IN TARGETED CANCER IMAGING AND THERAPY

EphB4 receptors, a member of the largest family of receptor tyrosine kinases, are found over-expressed in a variety of tumors cells including glioma cells as well as angiogenic blood vessels. Noninvasive imaging of EphB4 could potentially increase early detection rates, monitor response to therapy directed against EphB4, and improve patient outcomes. Targeted delivery of EphB4 receptor specific peptide conjugated hollow gold nanoshells (HAuNS) into tumors has great potential in cancer imaging and photothermal therapy. In this study, we developed an EphB4 specific peptide named TNYL-RAW and labeled with radioisotope 64Cu and Cy5.5 dye. We also conjugate this specific peptide with hollow gold nanoshells (HAuNS) to evaluate targeted photothermal therapy of cancers. In vitro, 64Cu-DOTA-TNYL- RAW specifically bind to CT26 and PC-3M cells but not to A549 cells. In vivo, Small-animal PET/CT clearly showed the significant uptake of 64Cu-DOTA-TNYL-RAW in CT26 and PC-3M tumors but not in A549 tumors. Furthermore, µPET/CT and near-infrared optical imaging clearly showed the uptake of the dual labeled TNYL-RAW peptide in both U251 and U87 tumors in the brains of nude mice. In U251 tumors, Cy5.5-labeled peptide can bind to EphB4-expressing tumor blood vessels and tumors cells. But in U87 models, dual labeled peptide only could bind to tumor associated blood vessels. Also, Irradiation of PC-3M and CT-26 cell treated with TNYL-PEG-HAuNS nanopatilces with near-infrared (NIR) laser resulted in selective destruction of these cells in vitro. EphB4 targeted TNYL-PEG-HAuNS showed more photothermal killing effect on CT26 tumor model than PEG-HAuNS did. In summary, tumors with overexpression of EphB4 receptors can be noninvasively visualized by micro PET/CT with 64Cu labeled or dual labeled TNYL-RAW peptide. Targeted delivery of TNYL-RAW conjugated HAuNS into tumors can greatly improve the treatment effect of photothermal therapy. The information acquired with this study should be advantageous in improving diagnostics and future applications in photothermal ablation therapy in clinical.

Bayesian joint modeling of longitudinal and survival data

The joint modeling of longitudinal and survival data is a new approach to many applications such as HIV, cancer vaccine trials and quality of life studies. There are recent developments of the methodologies with respect to each of the components of the joint model as well as statistical processes that link them together. Among these, second order polynomial random effect models and linear mixed effects models are the most commonly used for the longitudinal trajectory function. In this study, we first relax the parametric constraints for polynomial random effect models by using Dirichlet process priors, then three longitudinal markers rather than only one marker are considered in one joint model. Second, we use a linear mixed effect model for the longitudinal process in a joint model analyzing the three markers. In this research these methods were applied to the Primary Biliary Cirrhosis sequential data, which were collected from a clinical trial of primary biliary cirrhosis (PBC) of the liver. This trial was conducted between 1974 and 1984 at the Mayo Clinic. The effects of three longitudinal markers (1) Total Serum Bilirubin, (2) Serum Albumin and (3) Serum Glutamic-Oxaloacetic transaminase (SGOT) on patients' survival were investigated. Proportion of treatment effect will also be studied using the proposed joint modeling approaches. ^ Based on the results, we conclude that the proposed modeling approaches yield better fit to the data and give less biased parameter estimates for these trajectory functions than previous methods. Model fit is also improved after considering three longitudinal markers instead of one marker only. The results from analysis of proportion of treatment effects from these joint models indicate same conclusion as that from the final model of Fleming and Harrington (1991), which is Bilirubin and Albumin together has stronger impact in predicting patients' survival and as a surrogate endpoints for treatment. ^

A new combined endpoint using U-statistic in analysis in clinical trials

Common endpoints can be divided into two categories. One is dichotomous endpoints which take only fixed values (most of the time two values). The other is continuous endpoints which can be any real number between two specified values. Choices of primary endpoints are critical in clinical trials. If we only use dichotomous endpoints, the power could be underestimated. If only continuous endpoints are chosen, we may not obtain expected sample size due to occurrence of some significant clinical events. Combined endpoints are used in clinical trials to give additional power. However, current combined endpoints or composite endpoints in cardiovascular disease clinical trials or most clinical trials are endpoints that combine either dichotomous endpoints (total mortality + total hospitalization), or continuous endpoints (risk score). Our present work applied U-statistic to combine one dichotomous endpoint and one continuous endpoint, which has three different assessments and to calculate the sample size and test the hypothesis to see if there is any treatment effect. It is especially useful when some patients cannot provide the most precise measurement due to medical contraindication or some personal reasons. Results show that this method has greater power then the analysis using continuous endpoints alone. ^

Are anatomic brain locations important in determining the clinical outcomes in acute ischemic stroke? --- An exploratory analysis of the NINDS Stroke Trial data

Background and purpose. Brain lesions in acute ischemic stroke measured by imaging tools provide important clinical information for diagnosis and final infarct volume has been considered as a potential surrogate marker for clinical outcomes. Strong correlations have been found between lesion volume and clinical outcomes in the NINDS t-PA Stroke Trial but little has been published about lesion location and clinical outcomes. Studies of the National Institute of Neurological Disorders and Stroke (NINDS) t-PA Stroke Trial data found the direction of the t-PA treatment effect on a decrease in CT lesion volume was consistent with the observed clinical effects at 3 months, but measure of t-PA treatment benefits using CT lesion volumes showed a diminished statistical significance, as compared to using clinical scales. ^ Methods. We used the global test to evaluate the hypothesis that lesion locations were strongly associated with clinical outcomes within each treatment group at 3 months after stroke. The anatomic locations of CT scans were used for analysis. We also assessed the effect of t-PA on lesion location using a global statistical test. ^ Results. In the t-PA group, patients with frontal lesions had larger infarct volumes and worse NIHSS score at 3 months after stroke. The clinical status of patients with frontal lesions in t-PA group was less likely to be affected by lesion volume, as compared to those who had no frontal lesions in at 3 months. For patients within the placebo group, both brain stem and internal capsule locations were significantly associated with a lower odd of having favorable outcomes at 3 months. Using a global test we could not detect a significant effect of t-PA treatment on lesion location although differences between two treatment groups in the proportion of lesion findings in each location were found. ^ Conclusions. Frontal, brain stem, and internal capsule locations were significantly related to clinical status at 3 months after stroke onset. We detect no significant t-PA effect on all 9 locations although proportion of lesion findings in differed among locations between the two treatment groups.^

Bayesian estimation of multilevel item response model with missing data

This study proposed a novel statistical method that modeled the multiple outcomes and missing data process jointly using item response theory. This method follows the "intent-to-treat" principle in clinical trials and accounts for the correlation between outcomes and missing data process. This method may provide a good solution to chronic mental disorder study. ^ The simulation study demonstrated that if the true model is the proposed model with moderate or strong correlation, ignoring the within correlation may lead to overestimate of the treatment effect and result in more type I error than specified level. Even if the within correlation is small, the performance of proposed model is as good as naïve response model. Thus, the proposed model is robust for different correlation settings if the data is generated by the proposed model.^

A simulation study of the effects of small center enrollment in multi -center clinical trials

Multi-center clinical trials are very common in the development of new drugs and devices. One concern in such trials, is the effect of individual investigational sites enrolling small numbers of patients on the overall result. Can the presence of small centers cause an ineffective treatment to appear effective when treatment-by-center interaction is not statistically significant?^ In this research, simulations are used to study the effect that centers enrolling few patients may have on the analysis of clinical trial data. A multi-center clinical trial with 20 sites is simulated to investigate the effect of a new treatment in comparison to a placebo treatment. Twelve of these 20 investigational sites are considered small, each enrolling less than four patients per treatment group. Three clinical trials are simulated with sample sizes of 100, 170 and 300. The simulated data is generated with various characteristics, one in which treatment should be considered effective and another where treatment is not effective. Qualitative interactions are also produced within the small sites to further investigate the effect of small centers under various conditions.^ Standard analysis of variance methods and the "sometimes-pool" testing procedure are applied to the simulated data. One model investigates treatment and center effect and treatment-by-center interaction. Another model investigates treatment effect alone. These analyses are used to determine the power to detect treatment-by-center interactions, and the probability of type I error.^ We find it is difficult to detect treatment-by-center interactions when only a few investigational sites enrolling a limited number of patients participate in the interaction. However, we find no increased risk of type I error in these situations. In a pooled analysis, when the treatment is not effective, the probability of finding a significant treatment effect in the absence of significant treatment-by-center interaction is well within standard limits of type I error. ^

MODIFICATION AND IMPLEMENTATION OF THE RPC HETEROGENEOUS THORAX PHANTOM FOR VERIFICATION OF PROTON THERAPY TREATMENT PROCEDURES

The Radiological Physics Center (RPC) provides heterogeneous phantoms that are used to evaluate radiation treatment procedures as part of a comprehensive quality assurance program for institutions participating in clinical trials. It was hypothesized that the existing RPC heterogeneous thorax phantom can be modified to assess lung tumor proton beam therapy procedures involving patient simulation, treatment planning, and treatment delivery, and could confirm agreement between the measured dose and calculated dose within 5%/3mm with a reproducibility of 5%. The Hounsfield Units (HU) for lung equivalent materials (balsa wood and cork) was measured using a CT scanner. The relative linear stopping power (RLSP) of these materials was measured. The linear energy transfer (LET) of Gafchromic EBT2 film was analyzed utilizing parallel and perpendicular orientations in a water tank and compared to ion chamber readings. Both parallel and perpendicular orientations displayed a quenching effect underperforming the ion chamber, with the parallel orientation showing an average 31 % difference and the perpendicular showing an average of 15% difference. Two treatment plans were created that delivered the prescribed dose to the target volume, while achieving low entrance doses. Both treatment plans were designed using smeared compensators and expanded apertures, as would be utilized for a patient in the clinic. Plan 1a contained two beams that were set to orthogonal angles and a zero degree couch kick. Plan 1b utilized two beams set to 10 and 80 degrees with a 15 degree couch kick. EBT2 film and TLD were inserted and the phantom was irradiated 3 times for each plan. Both plans passed the criteria for the TLD measurements where the TLD values were within 7% of the dose calculated by Eclipse. Utilizing the 5%/3mm criteria, the 3 trial average of overall pass rate was 71% for Plan 1a. The 3 trial average for the overall pass rate was 76% for Plan 1b. The trials were then analyzed using RPC conventional lung treatment guidelines set forth by the RTOG: 5%/5mm, and an overall pass rate of 85%. Utilizing these criteria, only Plan 1b passed for all 3 trials, with an average overall pass rate of 89%.

Inoculum effect with cefazolin among clinical isolates of methicillin-susceptible Staphylococcus aureus: frequency and possible cause of cefazolin treatment failure.

Methicillin (meticillin)-susceptible Staphylococcus aureus (MSSA) strains producing large amounts of type A beta-lactamase (Bla) have been associated with cefazolin failures, but the frequency and impact of these strains have not been well studied. Here we examined 98 MSSA clinical isolates and found that 26% produced type A Bla, 15% type B, 46% type C, and none type D and that 13% lacked blaZ. The cefazolin MIC(90) was 2 microg/ml for a standard inoculum and 32 microg/ml for a high inoculum, with 19% of isolates displaying a pronounced inoculum effect (MICs of >or=16 microg/ml with 10(7) CFU/ml) (9 type A and 10 type C Bla producers). At the high inoculum, type A producers displayed higher cefazolin MICs than type B or C producers, while type B and C producers displayed higher cefamandole MICs. Among isolates from hemodialysis patients with MSSA bacteremia, three from the six patients who experienced cefazolin failure showed a cefazolin inoculum effect, while none from the six patients successfully treated with cefazolin showed an inoculum effect, suggesting an association between these strains and cefazolin failure (P = 0.09 by Fisher's exact test). In summary, 19% of MSSA clinical isolates showed a pronounced inoculum effect with cefazolin, a phenomenon that could explain the cases of cefazolin failure previously reported for hemodialysis patients with MSSA bacteremia. These results suggest that for serious MSSA infections, the presence of a significant inoculum effect with cefazolin could be associated with clinical failure in patients treated with this cephalosporin, particularly when it is used at low doses.

Effect of VEGF treatment on the blood-spinal cord barrier permeability in experimental spinal cord injury: dynamic contrast-enhanced magnetic resonance imaging.

Compromised blood-spinal cord barrier (BSCB) is a factor in the outcome following traumatic spinal cord injury (SCI). Vascular endothelial growth factor (VEGF) is a potent stimulator of angiogenesis and vascular permeability. The role of VEGF in SCI is controversial. Relatively little is known about the spatial and temporal changes in the BSCB permeability following administration of VEGF in experimental SCI. Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) studies were performed to noninvasively follow spatial and temporal changes in the BSCB permeability following acute administration of VEGF in experimental SCI over a post-injury period of 56 days. The DCE-MRI data was analyzed using a two-compartment pharmacokinetic model. Animals were assessed for open field locomotion using the Basso-Beattie-Bresnahan score. These studies demonstrate that the BSCB permeability was greater at all time points in the VEGF-treated animals compared to saline controls, most significantly in the epicenter region of injury. Although a significant temporal reduction in the BSCB permeability was observed in the VEGF-treated animals, BSCB permeability remained elevated even during the chronic phase. VEGF treatment resulted in earlier improvement in locomotor ability during the chronic phase of SCI. This study suggests a beneficial role of acutely administered VEGF in hastening neurobehavioral recovery after SCI.

The effect of shoulder variation on IMRT and SmartArc for Head and Neck Cancer

Purpose: First, to determine an average and maximum displacement of the shoulder relative to isocenter over the course of treatment. Second, to establish the dosimetric effect of shoulder displacements relative to correct isocenter alignment on the dose delivered to the target and the surrounding structures for head and neck cancer patients. Method and Materials: The frequency of shoulder shifts of various magnitudes relative to isocenter was assessed for 4 patients using image registration software. The location of the center of the right and left humeral head relative to isocenter (usually C2) was found daily from CT on rails scans, and was compared to the location of the humeral heads relative to isocenter on the initial simulation CT. Three Baseline head and neck IMRT and SmartArc plans were generated in Pinnacle based on simulation CTs. The CT datasets (external contour and boney structures) were then modified to represent shifts of the shoulder (relative to isocenter) between 3 mm and 15 mm in the SI, AP, and LR directions. The initial plans were recalculated on the image sets with shifted shoulders. Results: On average, shoulder variation was 2-5 mm in each direction, although displacements of over 1 cm in the inferior and posterior directions occurred. Shoulder shifts induced perturbations in the dose distribution, although generally only for large shifts. Most substantially, large, superior shifts resulted in coverage loss by the 95% isodose line for targets in the lower neck. Inferior shifts elevated the dose to the brachial plexus by 0.6-4.1 Gy. SmartArc plans showed similar loss of target coverage as IMRT plans. Conclusions: The position of the shoulder can have an impact on target coverage and critical structure dose. Shoulder position may need to be considered for setup of head and neck patients depending on target location.

An evaluation of sleep hypnotic and stimulant effect on human performance using a computerized psychological test battery

This study evaluated the administration-time-dependent effects of a stimulant (Dexedrine 5-mg), a sleep-inducer (Halcion 0.25-mg) and placebo (control) on human performance. The investigation was conducted on 12 diurnally active (0700-2300) male adults (23-38 yrs) using a double-blind, randomized sixway-crossover three-treatment, two-timepoint (0830 vs 2030) design. Performance tests were conducted hourly during sleepless 13-hour studies using a computer generated, controlled and scored multi-task cognitive performance assessment battery (PAB) developed at the Walter Reed Army Institute of Research. Specific tests were Simple and Choice Reaction Time, Serial Addition/Subtraction, Spatial Orientation, Logical Reasoning, Time Estimation, Response Timing and the Stanford Sleepiness Scale. The major index of performance was "Throughput", a combined measure of speed and accuracy.^ For the Placebo condition, Single and Group Cosinor Analysis documented circadian rhythms in cognitive performance for the majority of tests, both for individuals and for the group. Performance was best around 1830-2030 and most variable around 0530-0700 when sleepiness was greatest (0300).^ Morning Dexedrine dosing marginally enhanced performance an average of 3% with reference to the corresponding in time control level. Dexedrine AM also increased alertness by 10% over the AM control. Dexedrine PM failed to improve performance with reference to the corresponding PM control baseline. With regard to AM and PM Dexedrine administrations, AM performance was 6% better with subjects 25% more alert.^ Morning Halcion administration caused a 7% performance decrement and 16% increase in sleepiness and a 13% decrement and 10% increase in sleepiness when administered in the evening compared to corresponding in time control data. Performance was 9% worse and sleepiness 24% greater after evening versus morning Halcion administration.^ These results suggest that for evening Halcion dosing, the overnight sleep deprivation occurring in coincidence with the nadir in performance due to circadian rhythmicity together with the CNS depressant effects combine to produce performance degradation. For Dexedrine, morning administration resulted in only marginal performance enhancement; Dexedrine in the evening was less effective, suggesting the 5-mg dose level may be too low to counteract the partial sleep deprivation and nocturnal nadir in performance. ^

The heart challenge: The effect of regular aerobic exercise on body fat distribution in third and fourth grade school children: A randomized clinical trial

The pattern of body fat distribution known as "centralized", and characterized by a predominance of subcutaneous fat on the trunk and a "pot belly", has been associated with an increased risk of chronic disease. These patterns of fat distribution, as well as the lifestyle habit variables associated with adult fatness and chronic morbidity clearly begin to develop during childhood, indicating the need for intervention and primary prevention of obesity, particularly the centralized form, during childhood or adolescence. The purpose of this study was to determine whether regular aerobic exercise could beneficially alter the distribution of body fat in 8 and 9 year old children. One hundred and eighty-eight participants were randomized into either a regular aerobic exercise treatment group or a standard physical education program control group. A variety of aerobic activities was used for intervention 5 days per week during physical education class for a period of 12 weeks. Fat distribution was measured by a number of the most commonly used indices, including ratios of body circumferences and skinfolds and indices derived from a principal components analysis. Change over time in average pulse rate was used to determine if intervention actually occurred. Approximately 10% of the students were remeasured, allowing the calculation of intra- and interexaminer measurement reliability estimates for all indices.^ This study group was comparable to the U.S. population, though the study children were slightly larger for certain measures. No effect of the exercise intervention was found. The most likely explanation for this was inadequacy of the intervention, as indicated by the lack of any change in average pulse rate with treatment. The results of the measurement reliability analysis are reported and indicate that body circumference ratios are more precise than skinfold ratios, particularly when multiple observers are used. Reliability estimates for the principal component indices were also high.^ It remains unclear whether the distribution of body fat can be altered with exercise. It is likely that this issue will remain undecided until one highly reliable, valid, and sensitive measure of fat distribution can be found. ^

Treatment modalities of female Operation Iraqi Freedom/Operation Enduring Freedom Veterans with posttraumatic stress disorder: A systematic review of the literature

Objective. To review professional literature regarding treatment modalities of post-traumatic stress disorder (PTSD) amongst female Operation Iraqi Freedom (OIF) and Operation Enduring Freedom (OEF) veterans, to assess the efficacy of these treatment options, and to summarize implications of the findings from this literature. Design. Systematic review of published literature. Data sources. Medline, Pubmed, Psycinfo. Review Methods. Articles selected for the literature review pertain to the treatment options of female OIF or OEF veterans who have a diagnosis of PTSD. In addition, other relevant articles, such as articles that discuss the prevalence of the problem, access to care, and similar treatment modalities for PTSD in other war settings, were selected for background information for the review. Results. The search strategy identified 1,305 potential journal articles, taken from thorough searches in Medline, Pubmed, and Psycinfo. These articles were then imported into Refworks. Following final screening, there were 18 articles included in the systematic review and 28 articles used as background information. The remaining articles were excluded following screening of abstract and/or full text of articles. Treatment modalities presented in these trials include: Exposure Therapy (average of 68% reduction in PTSD symptoms), Imagery Rehearsal Therapy (23% reduction), Body-Oriented Therapy (57% reduction), Electroconvulsive Therapy (35% reduction), Holographic Reprocessing (47% reduction), a self-defense training program (13% reduction), Cognitive Behavioral Therapy (65% reduction) and a variety of pharmacotherapies (antipsychotics at 81% reduction, sympatholytic drug at 100% reduction). Outcomes of the studies included in this systematic review were measured by using personal assessment of whether there was a reduction in symptoms of PTSD, based on the results in each study. Conclusion. Overall, all of the treatment modalities investigated in the systematic review proved to be somewhat effective in relieving the burden of symptoms of PTSD amongst female veterans of OIF/OEF. In addition to pharmacotherapy, which had the highest reduction in PTSD symptoms, both the Exposure Therapy and the Cognitive Behavioral Therapy techniques proved to have the most positive results. As all of the therapies had a positive effect on this population, to some degree, a study needs to be done in the future to compare and contrast the efficacy of each therapy intervention when applied to a standardized population.^