Bayesian inference for Cronbach's alpha
- Authors: Izally, Sharkay Ruwade
- Date: 2025-04-03
- Subjects: Bayesian inference , Bayesian statistical decision theory , Cronbach's alpha , Confidence distribution , Probability matching , Jeffreys prior , Random effects model
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/479919 , vital:78380 , DOI 10.21504/10962/479919
- Description: Cronbach’s alpha is used as a measure of reliability in fields like education, psychology and sociology. The reason for the popularity of Cronbach’s alpha is that it is computationally simple. Only the sample size and the variance components are needed and it can be computed for continuous as well as binary data. Cronbach’s alpha has been studied extensively using maximum likelihood estimation. Since Cronbach’s alpha is a function of the variance components, this often results in negative estimates of the variance components when the maximum likelihood method is considered as a method of estimation. In the field of Bayesian statistics, the parameters are random variables, and this can alleviate some of the problems of estimating negative variance estimates that often occur when the frequentist approach is used. The Bayesian approach also incorporates loss functions that considers the symmetry of the distribution of the parameters being estimated and adds some flexibility in obtaining better estimates of the unknown parameters. The Bayesian approach often results in better coverage probabilities than the frequentist approach especially for smaller sample sizes and it is therefore important to consider a Bayesian analysis in the estimation of Cronbach’s alpha. The reference and probability matching priors for Cronbach’s alpha will be derived using a one-way random effects model. The performance of these two priors will be compared to that of the well-known Jeffreys prior and a divergence prior. A simulation study will be considered to compare the performance of the priors, where the coverage rates, average interval lengths and standard deviations of the interval lengths will be computed. A second simulation study will be considered where the mean relative error will be compared for the various priors using the squared error, the absolute error and the linear in exponential (LINEX) loss functions. An illustrative example will also be considered. The combined Bayesian estimation of more than one Cronbach’s alpha will also be considered for m experiments with equal α but possibly different variance components. It will be shown that the reference and the probability-matching priors are the same. The Bayesian theory and results will be applied to two examples. The intervals for the combined model are however much shorter than those of the individual models. Also, the point estimates of the combined model are more accurate than those of the individual models. It is further concluded that the posterior distribution of α for the combined model becomes more important as the number of samples and models increase. The reference and probability matching priors for Cronbach’s alpha will be derived using a three-component hierarchical model. The performance of these two priors will be compared to that of the well-known Jeffreys prior and a divergence prior. A simulation study will be v vi considered to compare the performance of the priors, where the coverage rates, average interval lengths and standard deviations of the interval lengths will be computed. Two illustrative examples will also be considered. Statistical control limits will be obtained for Cronbach’s alpha in the case of a balanced one-way random effects model. This will be achieved by deriving the predictive distribution of a future Cronbach’s alpha. The unconditional posterior predictive distribution will be determined using Monte Carlo simulation and the Rao-Blackwell procedure. The predictive distribution will be used to obtain control limits and to determine the run-length and average run-length. Cronbach’s alpha will be estimated for a general covariance matrix using a Bayesian approach and comparing these results to the asymptotic frequentist interval valid under a general covariance matrix framework. Most of the results used in the literature require the compound symmetry assumption for analyses of Cronbach’s alpha. Fiducial and posterior distributions will be derived for Cronbach’s alpha in the case of the bivariate normal distribution. Various objective priors will be considered for the variance components and the correlation coefficient. One of the priors considered corresponds to the fiducial distribution. The performance of these priors will be compared to an asymptotic frequentist interval often used in the literature. A simulation study will be considered to compare the performance of the priors and the asymptotic interval, where the coverage rates and average interval lengths will be computed. , Thesis (PhD) -- Faculty of Science, Statistics, 2025
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- Date Issued: 2025-04-03
Extreme value theory with applications in finance
- Authors: Matshaya, Aphelele
- Date: 2024-10-11
- Subjects: Bitcoin , Bivariate analysis , Correlation (Statistics) , Extreme value theory , Generalized Pareto distribution , High frequency data , Tail risk
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/465047 , vital:76568
- Description: The development and implementation of extreme value theory models has been very significant as they demonstrate an application of statistics that is very much needed in the analysis of extreme events in a wide range of industries, and more recently the cryptocurrency industry. The crypto industry is booming as the phenomenon of cryptocurrencies is spreading worldwide and constantly drawing the attention of investors, the media, as well as financial institutions. Cryptocurrencies are highly volatile assets whose price fluctuations continually lead to the loss of millions in a variety of currencies in the market. In this thesis, the extreme behaviour in the tail of the distribution of returns of Bitcoin will be examined. High-frequency Bitcoin data spanning periods before as well as after the COVID-19 pandemic will be utilised. The Peaks-over-Threshold method will be used to build models based on the generalised Pareto distribution, and both positive returns and negative returns will be modelled. Several techniques to select appropriate thresholds for the models are explored and the goodness-offit of the models assessed to determine the extent to which extreme value theory can model Bitcoin returns sufficiently. The analysis is extended and performed on Bitcoin data from a different crypto exchange to ensure model robustness is achieved. Using Bivariate extreme value theory, a Gumbel copula is fitted by the method of maximum likelihood with censored data to model the dynamic relationship between Bitcoin returns and trading volumes at the extreme tails. The extreme dependence and correlation structures will be analysed using tail dependence coefficients and the related extreme correlation coefficients. All computations are executed in R and the results are recorded in tabular and graphical formats. Tail-related measures of risk, namely Value-at-Risk and Expected Shortfall, are estimated from the extreme value models. Backtesting procedures are performed on the results from the risk models. A comparison between the negative returns of Bitcoin and those of Gold is carried out to determine which is the less risky asset to invest in during extreme market conditions. Extreme risk is calculated using the same extreme value approach and the results show that Bitcoin is riskier than Gold. , Thesis (MSc) -- Faculty of Science, Statistics, 2024
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- Date Issued: 2024-10-11
Bayesian hierarchical modelling with application in spatial epidemiology
- Authors: Southey, Richard Robert
- Date: 2018
- Subjects: Bayesian statistical decision theory , Spatial analysis (Statistics) , Medical mapping , Pericarditis , Mortality Statistics
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/59489 , vital:27617
- Description: Disease mapping and spatial statistics have become an important part of modern day statistics and have increased in popularity as the methods and techniques have evolved. The application of disease mapping is not only confined to the analysis of diseases as other applications of disease mapping can be found in Econometric and financial disciplines. This thesis will consider two data sets. These are the Georgia oral cancer 2004 data set and the South African acute pericarditis 2014 data set. The Georgia data set will be used to assess the hyperprior sensitivity of the precision for the uncorrelated heterogeneity and correlated heterogeneity components in a convolution model. The correlated heterogeneity will be modelled by a conditional autoregressive prior distribution and the uncorrelated heterogeneity will be modelled with a zero mean Gaussian prior distribution. The sensitivity analysis will be performed using three models with conjugate, Jeffreys' and a fixed parameter prior for the hyperprior distribution of the precision for the uncorrelated heterogeneity component. A simulation study will be done to compare four prior distributions which will be the conjugate, Jeffreys', probability matching and divergence priors. The three models will be fitted in WinBUGS® using a Bayesian approach. The results of the three models will be in the form of disease maps, figures and tables. The results show that the hyperprior of the precision for the uncorrelated heterogeneity and correlated heterogeneity components are sensitive to changes and will result in different results depending on the specification of the hyperprior distribution of the precision for the two components in the model. The South African data set will be used to examine whether there is a difference between the proper conditional autoregressive prior and intrinsic conditional autoregressive prior for the correlated heterogeneity component in a convolution model. Two models will be fitted in WinBUGS® for this comparison. Both the hyperpriors of the precision for the uncorrelated heterogeneity and correlated heterogeneity components will be modelled using a Jeffreys' prior distribution. The results show that there is no significant difference between the results of the model with a proper conditional autoregressive prior and intrinsic conditional autoregressive prior for the South African data, although there are a few disadvantages of using a proper conditional autoregressive prior for the correlated heterogeneity which will be stated in the conclusion.
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- Date Issued: 2018