Aims: To discuss different LMM-based approaches applied in GWAS and software packages implementation and Classify different computational tools that applies LMM approaches according to their applicability and performance. To identify possible SNPs associated to a particular disease using different computational tools based on LMM approaches. To estimate genetic and residual variance parameters that account phenotypic variation of the disease. Study Design: Case control study Place and Duration of Study: The research was carried out in Tanzania at African Institute of Mathematical Science for six months. Methodology: Linear Mixed Models (LMMs) are widely applied in genomic wide associations studies (GWAS) owing to their effectiveness of correcting hidden relationship, population structure and family structure. This essay is aimed at exploring different mathematical approaches of LMMs in GWAS. These approaches are linear mixed model with inclusion of all markers (LMMi) and linear mixed model with exclusion of all markers (LMMe) when calculating genetic relationship matrix. LMMi is more efficient as compared to LMMe when applied in studies of randomly ascertained quantitative traits. The LMM approaches are classified based on their applicability and performance. Two computational GWAS tools namely, PLINK and EMMAX were used which were based on LMM approaches to analyze unpublished real data from West Africa (Gambia and Ghana). Genetic and residual variance parameters were estimated that accounted for the phenotypic variation of the disease to be 0.0594 and 0.0723. A total of 338408 variants and 959 people (484 males, 405 females and 70 missing phenotypes) pass filters and quality control using PLINK was used in the study. Among the remaining phenotypes, 864 are cases and 95 are controls. The performance of different mathematical approaches of LMMs and their software implementation, including EMMAX and Plink via the application to a GWAS of tuberculosis (TB) in 959 individuals in West Africa (Ghana and Gambia) was compared. Of these 864 cases of TB and 95 healthy individuals retained after quality control (QC) using Plink, and 329601 autosome single nucleotide polymorphisms (from chromosome 1 to chromosome 22) included in the analysis after 288 duplicands ID individuals removed after QC. The LMM approaches are classified based on their applicability and performance. Two computational GWAS tools, namely Plink and EMMAX were used in the analysis of data. Genetic and residual variance parameters were estimated that accounted for the phenotypic variation of the disease to be 0.0594 and 0.0723. Results: Result showed that SNPs associated with tuberculosis were  on chromosome 17 and SNP  on chromosome 13 with both having  false discovery rate with step up significance value. Plink failed to correct hidden relatedness. Although EMMAX reduced the false positive rate, it still exhibited very low presence of stratification. Conclusion: This study aimed at understanding and exploring different approaches of mixed models as applied in genetic studies. Overview of genetic variation, advantages, successes and application of mixed models and current challenges of mixed models in GWAS were discussed. Moreover, the study showed that SNPs was associated with a particular disease using computational tools that applies LMM approaches. The summary statistics from PLINK and EMMAX found two causal SNPs associated with the TB. These SNPs were rs7225581 on chromosome 17 and SNP rs4941412 on chromosome 13 with both having 0.69% FDR H. However, PLINK failed to correct hidden relatedness. This phenotypic variation showed that all common single nucleotide polymorphisms (SNPs) expressed approximately 18.52% of phenotypic variation of the disease.  

The Central limit theorem is a very powerful tool in statistical inference and Mathematics in general, since it has numerous applications such as in topology and many other areas. For the case of probability theory, it states that, “given certain conditions, the sample mean of a sufficiently large number or iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed”. In the research paper, three different statements of our theorem (CLT) are given. This research paper has data regarding the shoe size and the gender of the of the university students. The paper is aimed at finding if the shoe sizes converges to a normal distribution as well as find the modal shoe size of university students and to apply the results of the central limit theorem to test the hypothesis if most university students put on shoe size seven. The Shoe sizes are typically treated as discretely distributed random variables, allowing the calculation of mean value and the standard deviation of the shoe sizes. The sample data which is used in this research paper belonged to different areas of Kibabii University which was divided into five strata. From two strata, a sample size of 74 respondents was drawn and from the remaining three strata, a sample of 73 students per stratum was drawn at random which totaled to a sample size of 367 respondents. By analyzing the data, using SPSS and Microsoft Excel, it was vivid that the shoe sizes are normally distributed with a well-defined mean and standard deviation. We also proved that most university students put on shoe size seven by testing our hypothesis using the p-value. The modal shoe size for university students was found to be seven since it had the highest frequency (97/367). This research was aimed at enlightening shoe investors, whose main market is the university students, on the shoe sizes that are on high demand among university students.

This paper is based on electricity consumption pattern in rental houses around Kibabii University (KU) situated in Western region of Kenya. Because of unexpected blackout faced by nonresident students at the time they need electricity most for their studies, this work intends to find out the directive measure to curb this crisis. Since the usage of electricity showed high relationship to the number of households sharing a common meter, Regression analysis prove to be the most effective method to model a solution to this problem. SPSS was used to analyze the data obtained. The results showed the consistency in linear trend of usage of electrical power on a monthly basis among students, it is observed also that the rate of consumption of power among nonresident students of KU is affected by the number of households sharing the meter. The consequence of this study is that with the correct data in place one is able to know the amount of power in kilowatt-hours needed for consumption throughout the semester and plan effectively so that power loss is not experienced. The results will be so useful to the KPLC (Kenya Power and Lighting Company) and KU fraternity for planning purposes.

The analysis of climate change, climate variability and their extremes has become more important as they clearly affect the human society and ecology. The impact of climate change is reflected by the change of frequency, duration and intensity of climate extreme events in the environment and on the economic activities. Climate extreme events, such as extreme rainfall threaten to environment, agricultural production and loss of people’s lives. Dodoma daily rainfall data exported from R-Instat software were used after being provided by Tanzania Meteorological Agency. The data were recorded from 1935 to 2011. In this essay, we used climate indices of rainfall to analyse changes in extreme rainfall. We only used 6 rainfall indices related to extremes to describe the change in rainfall extremes. Extreme rainfall indices did not show statistical evidence of a linear trend in Dodoma rainfall extremes for 77 years. Apart from the extreme rainfall indices, this essay utilized two techniques in extreme value theory namely the block maxima approach and peak over threshold approach. The two extreme value approaches were used for univariate sequences of independent identically distributed (iid) random variables. Using Dodomadaily rainfall data, this essay illustrated the power of the extreme value distributions in modelling of extreme rainfall. Annual maxima of Dodoma daily rainfall from 1935 to 2011 were fitted to the Generalized Extreme Value (GEV) model. Gumbel was found to be the best fit of the data after likelihood ratio test of GEV and Gumbel models. The Gumbel model parameters were considered to be stationary and non-stationary in two different models. The stationary Gumbel model was found to be good fit of Dodoma maximum rainfall. Later, the levels at which maximum Dodoma rainfall is expected to exceed once, on average, in a given period of time T = 2, 5, 10, 20, 30, 50 and 100 years, were obtained using stationary Gumbel model. Lastly, the data of exceedances were fitted to     the Generalized Pareto (GP) model under stationary climate assumption.

The purpose of this paper is to determine the time that a patient can spend waiting for service in Kibabii University Healthcare Clinic in Bungoma County (Kenya). The main objective was to provide necessary information to service facility managers, stakeholders, hospital staffs and other related institutions with the knowledge to improve the queuing system or to curb long waiting time of patients seeking services which can cause deterioration of the disease and sudden demise. This project also aims at providing suggestions to various factors identified to be the causes of long waiting time in the outpatient department at Kibabii University healthcare to help the smooth running of the clinic. The ODK tool was used in data collection procedure to capture the opinions of the respondents. The results from the ODK Tool was exported to XLS which is an export feature of data to excel, later data was exported to SPSS for analysis from excel.