Inflation; rates of interest [simple, compound (interest and discount), real, nominal, effective, dollar-weighted, time-weighted, spot, forward], term structure of interest rates; force of interest (constant and varying); equivalent measures of interest; yield rate; principal; equation of value; present value; future value; current value; net present value; accumulation function; discount function; annuity certain (immediate and due); perpetuity (immediate and due); stocks ( common and preferred); bonds (including zero-coupon bonds); other financial instruments such as mutual funds, and guaranteed investment contracts. Determining equivalent measures of interest; discounting; accumulating; determining yield rates; estimation the rate of return on a fund; and amortization.
Survival function, hazard functions, cumulative hazard function, censoring. Kaplan-Meier survival curve, parametric models. Comparison of two groups – log-ranked test.
Inclusion of covariates – Cox P.H. model, application of model checking. Competing risks – extensions of Cox’s model.
Organisation and Planning: Protocol, patient selection, response Justification of method for randomisation: Uncontrolled trials, blind trials, Placebo’s, ethical issues. The size of a clinical trial: Maintaining trials progress: Forms and data management, protocol deviations. Methods of data analysis: Binary responses, cross-over trials, survival data prognostic factors. Testing Hypothesis, Statistical Models: Inferential statistics-creating statistical hypothesis, the Z-test; designing a single variable experiment; errors in statistical decision making. Power of test used in clinical trials/maximizing tests power. Significance testing: t-test. Conducting two-way experiments and trials. Interpreting overall results of Clinical Trials. Nonparametric Procedures/Tests & Ranked Data: , Mann-Whitney U test, Kruskal-Willis, Friedman.
Stationary and non-stationary series: removal of trend and seasonality by differencing. Moments and auto-correlation. Models: simple AR and MA models (mainly AR(1), MA(1)): moments and auto-correlations; the conditions of stationarity: invertibility. Mixed (ARMA) models, and the AR representation of MA and ARMA models. Yule-Walker equations and partial auto-correlations (showing forms for simple AR, MA models). Examples showing simulated series from such processes, and sample auto-correlations and partial auto-correlations. (Other models, e.g., trend and seasonal). Model identification: Elementary ideas of identification of models based on simple acf and pacf showing difficulties with real series. Estimation of parameter: initial estimate based on sample acf and pacf only (least squares estimates by iterative method). Result for standard error of sample acf, pacf and estimators. Forecasting: use of the AR representation for forecasting. Minimum mean square error forecasts. Updating.
Formulating Linear Programming Models: Goal programming, Transportation problem, Case study. Mathematical Programming: Project planning and control, Dynamic programming,
Integer programming. Probabilistic Models: Application of queuing theory, Forecasting and simulation, Decision analysis (making hard decisions), Multi-criteria decision making.
Multivariate data summary and graphical displays. Multivariate normal distributions: Estimation of mean and covariance, one- and two-sample problems, analysis of variance.
Reduction of dimensionality: principal components and factor analyses. Discrimination and classification. Correlation; partial, multiple and canonical. Non-metric problems: clustering and scaling.
Criteria of choice, and optimality consideration, in respect of point estimation, hypothesis tests and confidence intervals. Likelihood methods with special consideration of maximum likelihood estimates (m.l.e.) and likelihood ratio tests including multiparameter problems (and linearisation methods). Specific techniques will include: Hypothesis Testing:
Pure significance tests, simulation tests, Neyman Pearson Lemma, UMP test. Point Estimations: Efficiency, consistency, minimum variance bound estimators. Determination of m.l.e’s including linearisation and asymptotic properties, maximum likelihood ratio tests and large-sample equivalents, asymptotic optimality. Score tests. Jackknifing, bootstrapping. Prior distributions: Representation of prior information via a prior distribution, substantial information, vague priors and ignorance, empirical Bayes ideas. Normal Models: Theory for unknown), prior-posterior-predictive, normal regression model. Comparisons: Comparisons of classical, Bayesian, decision-theory approaches and conclusions via specific examples.
The necessity and practical use of sample surveys: sample versus census, presentation and organisation of a survey. Methods of sampling. Simple random samples: techniques, estimation, choice of sample size. Ratio and regression estimators. Stratified random sampling: criteria for good stratification before or after sampling. Quota sampling. One-stage and two-stage cluster sampling. Systematic sampling. Comparison and choice of estimators. Estimation of treatment contrasts and their precision.
Exploratory Data Analysis: Data display, histograms, stem-and-leaf plots, box plots, data summary and description. Elementary Methods: Single-and two-sample problems, standard normal-theory tests and estimators, departures from assumptions, Poisson, Binomial and multinomial models, dispersion tests, goodness-of-fit, two-way contingency table. Regression Methods: Linear, multi-linear and polynomial regression, estimation of parameters, examination of residual, model checking. Analysis of variance: One- and two-way analyses of variance. Examination of residuals. Unbalanced case.