Understanding the Consequences of Collinearity for Multilevel Models: The Importance of Disaggregation Across Levels.

In multilevel models, disaggregating predictors into level-specific parts (typically accomplished via centering) benefits parameter estimates and their interpretations. However, the importance of level-specificity has been sparsely addressed in multilevel literature concerning collinearity. In this study, we develop novel insights into the interactivity of centering and collinearity in multilevel models. After integrating the broad literatures on centering and collinearity, we review level-specific and conflated correlations in multilevel data. Next, by deriving formal relationships between predictor collinearity and multilevel model estimates, we demonstrate how the consequences of collinearity change across different centering specifications and identify data characteristics that may exacerbate or mitigate those consequences. We show that when all or some level-1 predictors are uncentered, slope estimates can be greatly biased by collinearity. Disaggregation of all predictors eliminates the possibility that fixed effect estimates will be biased due to collinearity alone; however, under some data conditions, collinearity is associated with biased standard errors and random effect (co)variance estimates. Finally, we illustrate the importance of disaggregation for diagnosing collinearity in multilevel data and provide recommendations for the use of level-specific collinearity diagnostics. Overall, the necessity of disaggregation for identifying and managing collinearity's consequences in multilevel models is clarified in novel ways.

Catching Up on Multilevel Modeling.

This review focuses on the use of multilevel models in psychology and other social sciences. We target readers who are catching up on current best practices and sources of controversy in the specification of multilevel models. We first describe common use cases for clustered, longitudinal, and cross-classified designs, as well as their combinations. Using examples from both clustered and longitudinal designs, we then address issues of centering for observed predictor variables: its use in creating interpretable fixed and random effects of predictors, its relationship to endogeneity problems (correlations between predictors and model error terms), and its translation into multivariate multilevel models (using latent-centering within multilevel structural equation models). Finally, we describe novel extensions-mixed-effects location-scale models-designed for predicting differential amounts of variability.

Association between home and community-based services and depressive symptoms in Chinese older adults: a multilevel analysis.

BACKGROUND: As the phenomenon of ageing continues to intensify, home and community-based services (HCBSs) have been increasingly important in China. However, the association between HCBSs utilization and depressive symptoms in older adults in China is unclear. Consequently, this study aimed to examine the association between HCBSs utilization and depressive symptoms in Chinese older adults. METHODS: This study included 7,787 older adults (≥ 60 years old) who were recruited within the framework of the 2018 China Health and Retirement Longitudinal Study (CHARLS). Depressive symptoms were assessed using the 10-item Center for Epidemiological Studies Depression Scale (CES-D-10). HCBSs utilization was assessed via the question, "What kind of HCBSs were being utilized in their community?". Data were analyzed using binary logistic regression models and generalized hierarchical linear models (GHLM). RESULTS: Of the 7,787 participants, 20.0% (n = 1,556) reported that they utilized HCBSs, and 36.7% (n = 2,859) were evaluated that they had depressive symptoms. After adjusting for individual- and province-level covariates, the HCBSs utilization was found to be associated with depressive symptoms (OR = 1.180, 95% CI: 1.035-1.346, p < 0.05). Additionally, the depressive symptoms were significantly associated with gender, residence, educational level, marital status, number of chronic diseases, self-rated health (SRH), smoking, and provincial Gross Domestic Product (GDP) per capita. CONCLUSIONS: This study found HCBSs utilization might be a protective factor against depressive symptoms in Chinese older adults. It is of utmost significance for the government to provide targeted HCBSs at the community level to address the unmet care needs of older adults, which can reduce the occurrence of negative emotions, consequently contributing to less severe depressive symptoms.

Improving children's fundamental movement skills through a family-based physical activity program: results from the "Active 1 + FUN" randomized controlled trial.

BACKGROUND: Physical activity is related to many positive health outcomes, yet activity levels of many children are low. Researchers have suggested that family-based interventions may improve physical activity behaviors of both children and their parents. In this study, we evaluated the "Active 1 + FUN" program, which was designed based on tenets of self-determination theory. Intervention components included free sporting equipment, ten coach-led workshops and activity sessions, and one booster session. METHODS: We evaluated the intervention program using a randomized controlled trial. One hundred seventy-one families were randomly allocated to either an experimental group or a wait-list control group. Participants were exposed to program contents over a nine-month period, while families in the control did not receive any form of intervention. Measured constructs included moderate-to-vigorous physical activity, co-physical activity behaviors, fundamental movement skills, BMI, and several self-reported questionnaire outcomes. Hierarchical linear modeling was used to compare changes in measured outcomes across the two groups. RESULTS: No significant intervention effects were found for children's and parents' accelerometer-measured moderate-to-vigorous physical activity, or their co-physical activity. However, in terms of children's fundamental movement skills, a significant Time*Group interaction (B = 0.52, 95% CI [0.07, 0.96] for Times 1 to 2; B = 0.24, 95% CI [0.01, 0.48] for Times 1 to 3) in favor of the experimental group was found. CONCLUSIONS: Results suggested that the "Active 1 + FUN" program was effective in improving children's fundamental movement skills. Additional research is needed to examine how family-based initiatives could effectively improve physical activity behaviors too. TRIAL REGISTRATION: ANZCTR, ACTRN12618001524280. Registered 11 September 2018, https://www.anzctr.org.au/Trial/Registration/TrialReview.aspx?id=375660 .

New Recommendations on the Use of R-Squared Differences in Multilevel Model Comparisons.

When comparing multilevel models (MLMs) differing in fixed and/or random effects, researchers have had continuing interest in using R-squared differences to communicate effect size and importance of included terms. However, there has been longstanding confusion regarding which R-squared difference measures should be used for which kind of MLM comparisons. Furthermore, several limitations of recent studies on R-squared differences in MLM have led to misleading or incomplete recommendations for practice. These limitations include computing measures that are by definition incapable of detecting a particular type of added term, considering only a subset of the broader class of available R-squared difference measures, and incorrectly defining what a given R-squared difference measure quantifies. The purpose of this paper is to elucidate and resolve these issues. To do so, we define a more general set of total, within-cluster, and between-cluster R-squared difference measures than previously considered in MLM comparisons and give researchers concrete step-by-step procedures for identifying which measure is relevant to which model comparison. We supply simulated and analytic demonstrations of limitations of previous MLM studies on R-squared differences and show how application of our step-by-step procedures and general set of measures overcomes each. Additionally, we provide and illustrate graphical tools and software allowing researchers to automatically compute and visualize our set of measures in an integrated manner. We conclude with recommendations, as well as extensions involving (a) how our framework relates to and can be used to obtain pseudo-R-squareds, and (b) how our framework can accommodate both simultaneous and hierarchical model-building approaches.

Scale effects and countermeasures of cultivated land changes based on hierarchical linear model.

Changes in cultivated land in China are related to food security for nearly 1.4 billion people. Administrative ruling has decomposed the goal of cultivated land protection in China and implemented it from top to bottom, so that cultivated land data have nested attributes. Although related research on cultivated land changes has achieved fruitful results, these studies have neglected the scale effect created by the nested structure of cultivated land data, and it is easy for the policy to lose its effect in scaling. A two-layer linear model of the hierarchical linear model is constructed in this paper based on spatial autocorrelation and scale variance analyses to analyse the different spatial scales for cultivated land changes and reveal the interaction mechanism of the driving factors at different spatial scales. The results show the following: (1) The smaller the spatial scale of the study area, the easier the spatial correlation of the cultivated land quantity distribution. (2) An analysis of the driving factors of cultivated land change in Chongqing finds that 33.80% of the differences are from the functional block scale and 66.20% are from the district or county scale. (3) We found that the spatial scale has a certain impact on the effectiveness of the driving factors of cultivated land changes. Large-scale driving factors will change the correlation between small-scale driving factors and cultivated land change. In the future, the problem of scale effects should be considered in the field of cultivated land management, the effects of different driving forces at various scales should be considered and the scientific decomposition of cultivated land protection tasks according to the spatial scale should be carried out to improve the efficiency of the protection of cultivated land.

Malaria parasite clearance rate regression: an R software package for a Bayesian hierarchical regression model.

BACKGROUND: Emerging resistance to anti-malarial drugs has led malaria researchers to investigate what covariates (parasite and host factors) are associated with resistance. In this regard, investigation of how covariates impact malaria parasites clearance is often performed using a two-stage approach in which the WWARN Parasite Clearance Estimator or PCE is used to estimate parasite clearance rates and then the estimated parasite clearance is regressed on the covariates. However, the recently developed Bayesian Clearance Estimator instead leads to more accurate results for hierarchial regression modelling which motivated the authors to implement the method as an R package, called "bhrcr". METHODS: Given malaria parasite clearance profiles of a set of patients, the "bhrcr" package performs Bayesian hierarchical regression to estimate malaria parasite clearance rates along with the effect of covariates on them in the presence of "lag" and "tail" phases. In particular, the model performs a linear regression of the log clearance rates on covariates to estimate the effects within a Bayesian hierarchical framework. All posterior inferences are obtained by a "Markov Chain Monte Carlo" based sampling scheme which forms the core of the package. RESULTS: The "bhrcr" package can be utilized to study malaria parasite clearance data, and specifically, how covariates affect parasite clearance rates. In addition to estimating the clearance rates and the impact of covariates on them, the "bhrcr" package provides tools to calculate the WWARN PCE estimates of the parasite clearance rates as well. The fitted Bayesian model to the clearance profile of each individual, as well as the WWARN PCE estimates, can also be plotted by this package. CONCLUSIONS: This paper explains the Bayesian Clearance Estimator for malaria researchers including describing the freely available software, thus making these methods accessible and practical for modelling covariates' effects on parasite clearance rates.

Rising Household Debt and Children's Socioemotional Well-being Trajectories.

Debt is now a substantial aspect of family finances. Yet, research on how household debt is linked with child development has been limited. We use data from the National Longitudinal Survey of Youth 1979 cohort and hierarchical linear models to estimate associations of amounts and types of parental debt (home, education, auto, unsecured/uncollateralized) with child socioemotional well-being. We find that unsecured debt is associated with growth in child behavior problems, whereas this is not the case for other forms of debt. Moreover, the association of unsecured debt with child behavior problems varies by child age and socioeconomic status, with younger children and children from less-advantaged families experiencing larger associations of unsecured debt with greater behavior problems.

Unexpected stasis in a changing world: Lake nutrient and chlorophyll trends since 1990.

The United States (U.S.) has faced major environmental changes in recent decades, including agricultural intensification and urban expansion, as well as changes in atmospheric deposition and climate-all of which may influence eutrophication of freshwaters. However, it is unclear whether or how water quality in lakes across diverse ecological settings has responded to environmental change. We quantified water quality trends in 2913 lakes using nutrient and chlorophyll (Chl) observations from the Lake Multi-Scaled Geospatial and Temporal Database of the Northeast U.S. (LAGOS-NE), a collection of preexisting lake data mostly from state agencies. LAGOS-NE was used to quantify whether lake water quality has changed from 1990 to 2013, and whether lake-specific or regional geophysical factors were related to the observed changes. We modeled change through time using hierarchical linear models for total nitrogen (TN), total phosphorus (TP), stoichiometry (TN:TP), and Chl. Both the slopes (percent change per year) and intercepts (value in 1990) were allowed to vary by lake and region. Across all lakes, TN declined at a rate of 1.1% year-1 , while TP, TN:TP, and Chl did not change. A minority (7%-16%) of individual lakes had changing nutrients, stoichiometry, or Chl. Of those lakes that changed, we found differences in the geospatial variables that were most related to the observed change in the response variables. For example, TN and TN:TP trends were related to region-level drivers associated with atmospheric deposition of N; TP trends were related to both lake and region-level drivers associated with climate and land use; and Chl trends were found in regions with high air temperature at the beginning of the study period. We conclude that despite large environmental change and management efforts over recent decades, water quality of lakes in the Midwest and Northeast U.S. has not overwhelmingly degraded or improved.

Application of Effective Discharge Analysis to Environmental Flow Decision-Making.

Well-informed river management decisions rely on an explicit statement of objectives, repeatable analyses, and a transparent system for assessing trade-offs. These components may then be applied to compare alternative operational regimes for water resource infrastructure (e.g., diversions, locks, and dams). Intra- and inter-annual hydrologic variability further complicates these already complex environmental flow decisions. Effective discharge analysis (developed in studies of geomorphology) is a powerful tool for integrating temporal variability of flow magnitude and associated ecological consequences. Here, we adapt the effectiveness framework to include multiple elements of the natural flow regime (i.e., timing, duration, and rate-of-change) as well as two flow variables. We demonstrate this analytical approach using a case study of environmental flow management based on long-term (60 years) daily discharge records in the Middle Oconee River near Athens, GA, USA. Specifically, we apply an existing model for estimating young-of-year fish recruitment based on flow-dependent metrics to an effective discharge analysis that incorporates hydrologic variability and multiple focal taxa. We then compare three alternative methods of environmental flow provision. Percentage-based withdrawal schemes outcompete other environmental flow methods across all levels of water withdrawal and ecological outcomes.

Tethered lives: A couple-based perspective on the consequences of parenthood for time use, occupation, and wages.

Prior research on parenthood effects has typically used single-sex models and estimated average effects. By contrast, we estimate population-level variability in partners' changes in housework hours, paid work hours, occupation traits, and wages after becoming parents, and we explore whether one partner's adjustment offsets or supplements the other's. We find tradeoffs between spouses on paid work adjustments to parenthood, but complementarity in adjustments to housework hours, occupation traits, and wages. The effect of parenthood on wives' behaviors is larger and more variable than on husbands' behaviors in every domain. The modest variation between husbands in work responses to parenthood explains little of the variation in the motherhood penalty, while variation in wives' own behaviors plays a larger role. We refer to this pattern as tethered autonomy: variation across American couples in work responses to parenthood is shaped primarily by variation in wives' adjustments, while husbands' work acts largely as a fixed point.

Bayesian hierarchical regression on clearance rates in the presence of "lag" and "tail" phases with an application to malaria parasites.

We present a principled technique for estimating the effect of covariates on malaria parasite clearance rates in the presence of "lag" and "tail" phases through the use of a Bayesian hierarchical linear model. The hierarchical approach enables us to appropriately incorporate the uncertainty in both estimating clearance rates in patients and assessing the potential impact of covariates on these rates into the posterior intervals generated for the parameters associated with each covariate. Furthermore, it permits us to incorporate information about individuals for whom there exists only one observation time before censoring, which alleviates a systematic bias affecting inference when these individuals are excluded. We use a changepoint model to account for both lag and tail phases, and hence base our estimation of the parasite clearance rate only on observations within the decay phase. The Bayesian approach allows us to treat the delineation between lag, decay, and tail phases within an individual's clearance profile as themselves being random variables, thus taking into account the additional uncertainty of boundaries between phases. We compare our method to existing methodology used in the antimalarial research community through a simulation study and show that it possesses desirable frequentist properties for conducting inference. We use our methodology to measure the impact of several covariates on Plasmodium falciparum clearance rate data collected in 2009 and 2010. Though our method was developed with this application in mind, it can be easily applied to any biological system exhibiting these hindrances to estimation.

Multicollinearity in hierarchical linear models.

This study investigates an ill-posed problem (multicollinearity) in Hierarchical Linear Models from both the data and the model perspectives. We propose an intuitive, effective approach to diagnosing the presence of multicollinearity and its remedies in this class of models. A simulation study demonstrates the impacts of multicollinearity on coefficient estimates, associated standard errors, and variance components at various levels of multicollinearity for finite sample sizes typical in social science studies. We further investigate the role multicollinearity plays at each level for estimation of coefficient parameters in terms of shrinkage. Based on these analyses, we recommend a top-down method for assessing multicollinearity in HLMs that first examines the contextual predictors (Level-2 in a two-level model) and then the individual predictors (Level-1) and uses the results for data collection, research problem redefinition, model re-specification, variable selection and estimation of a final model.

Weight change trajectory in women with breast cancer receiving chemotherapy and the effect of different regimens.

AIMS AND OBJECTIVES: To investigate the trajectory of weight change in Taiwanese women with breast cancer after starting chemotherapy and the impact of chemotherapy regimens on weight change while controlling for age, menopausal status, body mass index, lymph node involvement and changes in habits of dietary fat intake and exercise. BACKGROUND: Weight gain after adjuvant chemotherapy in women with breast cancer has negative impact on health outcomes. DESIGN: Longitudinal, clinical observational study. METHODS: Weights were repeatedly measured in 147 women with breast cancer stages I-III. Hierarchical linear modelling was used to analyse these longitudinal data. RESULTS: The overall pattern of weight change was a cubic form beginning with a mean of 56·9 kg before chemotherapy. It gradually increased to 59·4 kg at 8·5 months after the first chemotherapy followed by a decrease to 58·5 kg at 21·5 months. During the last 2·5 months, weight increased slightly and never returned to the initial level. After controlling for confounders, steeper weight change was observed among women receiving cyclophosphamide, methotrexate and fluorouracil. The highest weight gain in the cyclophosphamide, methotrexate and fluorouracil group was 2·9 kg (5%) vs. 0·9 kg (1%) in the anthracycline-based group. CONCLUSION: The trajectory of body weight change within two years after chemotherapy shows a trend of gradual ascent, followed by a small decline and a slight increase in the last 2·5 months. The chemotherapy regimen can predict the trend after controlling for other confounders; women on cyclophosphamide, methotrexate and fluorouracil have a steeper weight change. RELEVANCE TO CLINICAL PRACTICE: Nurses can inform women with breast cancer about the expected changes in body weight after chemotherapy to reduce their uncertainty. Future studies on effective interventions to minimise chemotherapy-induced weight gain are needed.

Reducing Attenuation Bias in Regression Analyses Involving Rating Scale Data via Psychometric Modeling.

Many studies in fields such as psychology and educational sciences obtain information about attributes of subjects through observational studies, in which raters score subjects using multiple-item rating scales. Error variance due to measurement effects, such as items and raters, attenuate the regression coefficients and lower the power of (hierarchical) linear models. A modeling procedure is discussed to reduce the attenuation. The procedure consists of (1) an item response theory (IRT) model to map the discrete item responses to a continuous latent scale and (2) a generalizability theory (GT) model to separate the variance in the latent measurement into variance components of interest and nuisance variance components. It will be shown how measurements obtained from this mixture of IRT and GT models can be embedded in (hierarchical) linear models, both as predictor or criterion variables, such that error variance due to nuisance effects are partialled out. Using examples from the field of educational measurement, it is shown how general-purpose software can be used to implement the modeling procedure.

Development of Playfulness in Children with Low Executive Functions: The Role of Parental Playfulness and Parental Playtime with Their Child.

Children with low executive functions (EFs) are described as having lower levels of playfulness, the quality of children's play, compared to children with EFs within the normal range. However, how playfulness in children with low EFs develops over time remains unclear. Additionally, little is known about how parental playfulness and parental playtime with their child affect these developmental trajectories in children with low EFs. To address these research gaps, we measured playfulness in 62 children with low EFs and 62 children with EFs within the normal range aged 3 to 6 years at three time points over 2 years. We used the Children's Playfulness Scale, which captures multi-informant perspectives from parents and teachers. Moreover, the parents of children with low EFs reported their own playfulness and their playtime with their children at T1. Repeated-measures hierarchical linear models indicated significantly lower levels of playfulness in the children with low EFs than in the controls, with no significant changes observed over 2 years in either group. In the children with low EFs, we found a significant positive relationship between parental playfulness at T1 and children's playfulness 2 years later but a significant negative relationship between parental playtime at T1 and children's playfulness 2 years later. These results prompt a broad discussion on potential implications for the enhancement of playfulness in children with low EFs within the family environment.

Age, period and cohort effects in depression prevalence among Canadians 65+, 1994 to 2018: A multi-level analysis.

BACKGROUND: The nature of the Canadian population 65+ has changed considerably over the past several decades. They comprise a larger proportion of the population, are better educated, and are wealthier than previous generations. We estimate the contributions of chronological aging, temporal periods, and birth cohort effects on the trends in the major depressive episode (MDE) prevalence among Canadian seniors from 1994/1995 to 2017/2018. METHODS: Using data from two sets of national health surveys, the National Population Health Survey (NPHS) and the Canadian Community Health Survey (CCHS). Pooled data on 150,246 survey respondents aged 65+ from 16 repeated cross-sectional surveys are included. Hierarchical regression age-period-cohort models were used to visualize the linear and non-linear effects of age, period, and cohort trends in late-life depression. RESULTS: We found that: the prevalence of MDE in later life fluctuated non-significantly during the study time period; the probability of developing MDE declined with increasing age from 65 to 80+ (ß = -.32, p = .027). The significant quadratic birth cohort predictor showed a non-linear increasing association with the prevalence of MDE from the earlier to later-born cohorts (ß = .01, p = .049). We also found that females 65+ were consistently more likely to be depressed than males 65+ (ß = .47, p = .007). The significantly negative "age × female" interaction shows that age exerts a greater effect on females' probability of developing MDE than males (ß = -.09, p = .011). There were no consistent significant period effects but there were peaks in prevalence around 2001, 2008, and 2012 which corresponded to some recent historical events. Our moderation analysis documents that lower levels of education significantly contributed to the higher rates of depression among cohorts born earlier in the 20th century. CONCLUSIONS: Our findings show the presence of strong chronological age and cohort effects and weaker period effects on the prevalence of late-life depression in Canadian seniors.

Improvements in pain coping predict treatment success among patients with chronic primary pain.

OBJECTIVE: Given the increasing incidence and prevalence of chronic pain, effective treatments for chronic pain are needed. This study aimed to investigate the role of cognitive and behavioral pain coping regarding the prediction of treatment outcomes among inpatients with chronic primary pain participating in an interdisciplinary multimodal treatment program. METHODS: At intake and discharge, 500 patients with chronic primary pain completed questionnaires on pain intensity, pain interference, psychological distress, and pain processing. RESULTS: Patients' symptoms, cognitive and behavioral pain coping improved significantly after treatment. Similarly, separate cognitive and behaviroal coping skills improved significantly after treatment. Hierarchical linear models revealed no significant associations of pain coping with reductions in pain intensity. Whereas the overall level and improvements in cognitive pain coping predicted reductions in pain interference and psychological distress, the overall level and improvements in behavioral pain coping were associated with reductions in pain interference alone. DISCUSSION: Since pain coping seems to influence both pain interference and psychological distress, improving cognitive and behavioral pain coping during an interdisciplinary multimodal pain treatment seems to be a key component in the successful treatment of inpatients with chronic primary pain, enabling them to function better physically and mentally despite their chronic pain. Clinically, it might be worth fostering and exercising cognitive restructuring as well as action planning in treatment to reduce both pain interference and psychological distress levels post-treatment. In addition, practicing relaxation techniques might help reduce pain interference post-treatment, whereas making experiences of personal competence might help reduce psychological distress post-treatment.

Migration and health: Exploring the effect of destinations on the health outcomes among rural-urban migrants in India.

Conservative estimates of the census of India pegged the number of rural-urban migrants at 78 million, out of the total internal migrant population of 456 million, in 2011. Despite their sizable number, larger than several European nations, very little is known about whether the type of urban destination matters for the physical health of these largely poor populations. Using data from two waves of the India Human Development Survey (2004-05 and 2011-12), we conduct multi-level analyses to explore the impact of metropolitan versus non-metropolitan destinations on the odds of short-term and long-term illnesses among rural-urban migrants across residential durations. Findings show that rural-urban migrants to metropolitan cities experience higher likelihood of suffering from overall and pollution-related major illnesses relative to their counterparts in non-metropolitan urban areas. On the other hand, migrants to metropolitan cities experience lower odds of suffering from minor illnesses than non-metropolitan rural-urban migrants. However, these significant effects of urban destination disappear when we compare health outcomes between shorter versus longer-duration migrants. We subject these multi-level analytic findings to robustness checks that corroborate our foregoing mixed results. Our findings generate initial evidence on health disparities among rural-urban migrants by destination and duration. These findings underscore the importance of health needs among migrant populations, that require attention particularly in the short-term of their relocation to cities.

Solutions for Determining the Significance Region Using the Johnson-Neyman Type Procedure in Generalized Linear (Mixed) Models.

Researchers often compare the relationship between an outcome and covariate for two or more groups by evaluating whether the fitted regression curves differ significantly. When they do, researchers need to determine the "significance region," or the values of the covariate where the curves significantly differ. In analysis of covariance (ANCOVA), the Johnson-Neyman procedure can be used to determine the significance region; for the hierarchical linear model (HLM), the Miyazaki and Maier (M-M) procedure has been suggested. However, neither procedure can assume nonnormally distributed data. Furthermore, the M-M procedure produces biased (downward) results because it uses the Wald test, does not control the inflated Type I error rate due to multiple testing, and requires implementing multiple software packages to determine the significance region. In this article, we address these limitations by proposing solutions for determining the significance region suitable for generalized linear (mixed) model (GLM or GLMM). These proposed solutions incorporate test statistics that resolve the biased results, control the Type I error rate using Scheffé's method, and uses a single statistical software package to determine the significance region.