1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2022/2023
4. computer science and econometrics - First-cycle studies leading to Bachelor's degree
5. Experimental Design

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Experimental Design - course description

Aim of the course

To introduce the students to the theoretical and practical foundations of experimental design.

Prerequisites

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Passing the lectures on probability theory and elements of mathematical statistics.

Scope

Lecture

1. Univariate and multivariate normal distribution and related distributions. Random variable, random variable with normal distribution (repetition). Chi-square distribution of the quadratic form and theorems on the independence of linear and quadratic forms, Student's t distributions, F-Snedecor distributions (2 hours)

2. Linear model, definition and assumptions about the model (2 hours)

3. Estimators obtained using the least squares (LS) method and their relationship with estimability (2 hours)

4. Theorem on the characterization of estimable functions (2 hours)

5. Normal equations and properties of LS estimators (2 hours)

6. Probability distributions of LS estimators and their functions (2 hours)

7. Residuals in the linear model. Independence of the sum of squared residuals of LS estimators (2 hours)

8. Unbiased estimator for variance and its probability distribution (2 hours)

9. Theory of testing statistical hypotheses for linear functions of model parameters with the use of Student's t distribution (2 hours)

10. Analysis of variance table for testing complex hypotheses, F-Snedecor test (2 hours)

11. Confidence intervals for parametric functions, their interpretation (2 hours)

12. Prediction and confidence intervals of parametric functions and for prediction (2 hours)

13. Examples of optimal plans with a singular design matrix, linear restrictions on parameters (6 hours)

Laboratory

1. Repetition and development of knowledge about probability theory. Normal distribution and its properties. Multivariate normal distribution of random variables and its basic numerical characteristics. Functions of random variables and their distributions (2 hours)

2. Independence of variables. Determining and showing the independence of the mean and variance from a normal sample based on the theorem on the independence of linear and quadratic forms (2 hours)

3. Writing a linear model for one- and multivariate regression functions, using LS method to determine explicit formulas for estimating model parameters. Examples (4 hours)

4. Determination of the model residuals and the sum of squares of the residuals, as well as the variance estimator and confidence intervals for parameters and predictions (4 hours)

5. Analysis of variaance table for the above-mentioned model with an example (2 hours) Colloquium (2 hours)

6. Repeat exercise for points 3-5. for the one-way and multi-way analysis of variance model (10 hours)

7. Repeat the exercise from 3-5. for factorial designs 2^k (2 hours) Colloquium (2 hours)

Teaching methods

Traditional lecture (chalk and blackboard for the most important phrases only, computer examples), in laboratories, solving previously announced exercises (computation exercises, for given practical examples using using selected statistical packages).

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

1. Student's preparation for laboratories is verified by checking the knowledge (concept, properties, theorems) necessary to solve the next exercise on the list (lack of preparation for the laboratory is included in the final grade).

2. The final project, with varying degrees of difficulty, to assess whether the student has achieved the learning outcomes to a minimum degree.

3. A written project referring to concepts and theorems that check the understanding of the acquired knowledge based on this project

The subject grade consists of the laboratory grade (40%, including the project grade) and the project grade (60%).

The condition for taking the project is a positive grade from the laboratory. The condition for passing the lecture is a positive project grade.

Recommended reading

1. C. R. Rao, Linear Statistical Inference and its Applications, Wiley, Canada 2002.
2. H. Scheffe, The Analysis of Variance, Wiley, New York, 1959.
3.  D. C. Montgomery, Design and Analysis of Experiments, John Wiley & Sons, 1991

Further reading

1. E. L. Lehmann, Testing statistical hypothesis, Second edition. Wiley, New York 1986.

Notes

Modified by dr Ewa Synówka (last modification: 16-02-2024 17:40)