Regression to the mean (RTM) is a statistical phenomenon that happens when subjects having relatively high or low observations upon remeasurement are found closer to the population mean. RTM can erroneously influence the conclusion in a pre-post study design. Expressions are available for quantifying RTM when the distribution of pre-post variables is bivariate normal; however, these methods assume the pre and post observations are identically distributed and strictly positively correlated. This study generalizes previous results to include non-stationary, normally distributed random variables that are potentially negatively correlated, and also provides the decomposition of the conditional mean difference into RTM and unbiased intervention effects. In addition, the maximum likelihood estimators are derived and the unbiasedness, consistency and normality of these estimators are established. A simulation study is conducted to asses the accuracy of estimating the RTM and intervention effects using existing and the proposed methods. Data on the blood lead level in both intervention and placebo groups are used for decomposing the total change in blood lead level on pre-post occasions into RTM and intervention effects.

Toxoplasmosis is a zoonotic parasitic disease caused by T. gondii, an obligate intracellular apcomplexan zoonotic parasite that is geographically worldwide in distribution. The parasite infects humans and all warm-blooded animals and is highly prevalent in various geographical regions of the world, including Pakistan. The current study addressee prevalence of Toxoplasma infection in women in various geographical regions, mapping of endemic division and t district of Khyber Pakhtunkhwa province through geographical information system (GIS) in order to locate endemic regions, monitor seasonal and annual increase in prevalence of infection in women patients. Setting: Tertiary hospitals and basic health care centers located in 7 divisions and 24 districts of Khyber Pakhtunkhwa (KP) province of Pakistan. During the current study, 3586 women patients from 7 divisions and 24 districts were clinically examined and screened for prevalence of T. gondii infection. Participants were screened for Toxoplasma infection using ICT and latex agglutination test (LAT) as initial screening assay, while iELISA (IgM, IgG) was used as confirmatory assay. Mapping of the studied region was developed by using ArcGIS 10.5. Spatial analyst tools were applied by using Kriging/Co-kriging techniques, followed by IDW (Inverse Distance Weight) techniques. Overall prevalence of T. gondii infection was found in 881 (24.56%) patients. A significant (<0.05) variation was found in prevalence of infection in different divisions and districts of the province. Prevalence of infection was significantly (<0.05) high 129 (30.07%) in Kohat Division, followed by 177 (29.06%), 80 (27.87%), 287 (26.72%), 81 (21.21%), 47 (21.07%), and 80 (13.71%) cases in Hazara Division, D.I Khan Division, Malakand Division, Mardan Division, Bannu Division, and Peshawar Division. Among various districts, a significant variation (<0.05) was found in prevalence of infection. Prevalence of infection was significantly (0.05) seasonal and annual variation was found in prevalence of Toxoplasma infection. LAT, ICT and ELISA assays were evaluated for prevalence of infection, which significantly (<0.05) detected T. gondii antibodies. LAT, ICT and ELISA assays significantly (0.05) difference was found between positivity of LAT and ICT assays. A significant difference (<0.05) was found in positivity of Toxoplasma-specific (IgM), (IgG) and (IgM, IgG) immunoglobulin by ICT and ELISA assay. The current study provides comprehensive information about geographical distribution, seasonal and annual variation of Toxoplasmosis infection in various regions of Khyber Pakhtunkhwa province of Pakistan. Infection of T. gondii in women shows an alarming situation of disease transmission from infected animals in the studied region, which is not only a serious and potential threat for adverse pregnancy outcomes, but also cause socioeconomic burden and challenges for various public and animal health organizations in Pakistan and across the country.

Regression to the mean occurs when an unusual observation is followed by a more typical outcome closer to the population mean. In pre- and post-intervention studies, treatment is administered to subjects with initial measurements located in the tail of a distribution, and a paired sample t -test can be utilized to assess the effectiveness of the intervention. The observed change in the pre-post means is the sum of regression to the mean and treatment effects, and ignoring regression to the mean could lead to erroneous conclusions about the effectiveness of the treatment effect. In this study, formulae for regression to the mean are derived, and maximum likelihood estimation is employed to numerically estimate the regression to the mean effect when the test statistic follows the bivariate t -distribution based on a baseline criterion or a cut-off point. The pre-post degrees of freedom could be equal but also unequal such as when there is missing data. Additionally, we illustrate how regression to the mean is influenced by cut-off points, mixing angles which are related to correlation, and degrees of freedom. A simulation study is conducted to assess the statistical properties of unbiasedness, consistency, and asymptotic normality of the regression to the mean estimator. Moreover, the proposed methods are compared with an existing one assuming bivariate normality. The p -values are compared when regression to the mean is either ignored or accounted for to gauge the statistical significance of the paired t -test. The proposed method is applied to real data concerning schizophrenia patients, and the observed conditional mean difference called the total effect is decomposed into the regression to the mean and treatment effects.

Regression to the mean is said to have occurred when subjects having relatively high or low measurements are remeasured closer to the population mean. This phenomenon can influence the conclusion about the effectiveness of a treatment in a pre-post study design. The mean difference of the pre- and post-variables, conditioned on the initial variable being above or below a cut-point, is the sum of regression to the mean and treatment effects. Expressions for regression to the mean are available for the bivariate normal distribution under restrictive assumptions, and for the bivariate Poisson and binomial distributions, more generally. This article derives expressions for regression to the mean for any bivariate distribution while making fewer restrictive assumptions than previous methods. Maximum likelihood estimators are derived, and the unbiasedness, consistency, and asymptotic normality of these estimators are shown for exponential families, where possible. Data on the cholesterol levels in men aged 35-39 are used for decomposing the conditional mean difference in cholesterol level on pre-post occasions into regression to the mean and treatment effects. In another example, data on diastolic blood pressure for 341 patients are used to demonstrate the fraction of change due to regression to the mean and the treatment effects, respectively.