Mathematics and Statistics II

- Having summarized and analyzed the information collected from a sample (topics covered in Mathematics and Statistics I), the goal now is, using Statistical Inference, to contrast whether a sample situation derives from a certain probability model and infer the available knowledge of that model to the population. In particular, from the data obtained using a random sample, we intend to apply the appropriate statistical procedures to infer unknown characteristics of the population and calculate the error of the estimates.
- To apply, through the use of a statistical package, the concepts of regression, hypothesis test, and confidence intervals to physical, chemical, biological and data from medical-pharmaceutical databases, and interpret the results.
- To provide an elementary capability for designing experiments according to statistical criteria.

UNIT 1: INTRODUCTION TO STATISTICAL INFERENCE. ESTIMATION
1.1 Population and sample.
1.2 Parameter. Statistics.
1.3 Distributions of several statistics. Central limit theorem.
1.4 Point estimators. Properties of the estimators.
1.5 Estimation by confidence intervals: basic concepts. Confidence level.
1.6 Confidence intervals for mean, variance and proportion.
1.7 Determination of sample size.

UNIT 2: HYPOTHESIS TESTS
2.1 Statistical hypothesis. Formulation and method.
2.2 Types of error. Decision criteria. Critical level or p-value. Power of a test.
2.3 Interpretation of a hypothesis test. Relationship between confidence intervals and hypothesis tests.
2.4 Contrasts with a sample: for a mean, for a proportion and for a variance.
2.5 Contrasts with two samples: comparison of two variances; comparison of two means (independent samples, paired samples); comparison of two proportions.
2.6 Confidence intervals for the difference of means, difference of proportions and quotient of variances.

UNIT 3: THE CHI-SQUARE TEST
3.1 Contrasts for categorical data: contingency tables. Chi-square test. 2 × 2 tables. Study design. Homogeneity tests. Independence tests.
3.2 Goodness of fit tests: Pearson's chi-square test; the Kolmogorov-Smirnov test; normality test.

UNIT 4: REGRESSION AND CORRELATION
4.1 Introduction. General concepts.
4.2 Regression: least squares method, regression lines.
4.3 Total variance. Residual variance and explained variance.
4.4 Correlation: linear correlation coefficient.
4.5 Other regression models: the exponential model and the potential model.
4.6 Hypothesis testing for regression parameters: ANOVA.

Basic
– Milton, J.S.,“Estadística para Biología y Ciencias de la Salud” Tercera edición ampliada. McGraw-Hill Interamericana, Madrid, 2007.

- Notes on the subject, available in the virtual course.

Complementary
– Cao Abad, R., Francisco Fernández M., y otros, Introducción a la estadística y sus aplicaciones, Ed. Pirámide (Grupo Anaya, S.A.), Madrid, 2001.

– Colton, T., Estadística en Medicina, Ed. Masson-Litle, Brown, S.A., Barcelona, 1995.

– Martín Andrés, A.; Luna del Castillo, J. de D., Bioestadística para las Ciencias de la Salud, Ed. Norma S.L. (4ª edición), Madrid, 1994.

– Peña Sánchez de Rivera, D., Estadística Modelos y métodos. I. Fundamentos, Alianza editorial, S.A., Madrid, 2000.

- Samuels, Myra L; Witmer, J., Fundamentos de Estadística para las Ciencias de la Vida, Ed. Pearson Educación, 2012.

– Sánchez M.; Frutos G.; Cuesta, P.L., Estadística y Matemáticas Aplicadas. Edición dirigida a los estudios de Farmacia, Editorial Síntesis S.A., Madrid, 1996.

Knowledge:
Con 17. How to design experiments based on statistical criteria.
Con 18. How to evaluate scientific data related to medicine and healthcare products. Use statistical analysis applied to pharmaceutical sciences.

Skills or abilities:
H/D 10. Apply physical and mathematical knowledge to pharmaceutical sciences.
H/D 11. Apply computational and data processing techniques to physical, chemical and biological data.

Competence:
Comp 01. Ability to analyze and synthesize.
Comp 05. Basic computer handling skills
Comp 07. Problem solving.

Since the subject is fundamentally practical, contents will be developed with simplicity, without sacrificing precision.

- Lectures for large groups: Some time will be devoted to the introduction, presentation or illustration of a theoretical question; the remaining time will be devoted to solving related problems or exercises.

- Interactive lectures in small groups: Students will be given exercise and problem sheets corresponding to each of the topics. Each student will try to solve them individually, or if necessary, in the classroom, with their active participation.

- Interactive lectures for small groups with computer: Data entry and coding (with EXCEL). Attendance to these lectures is compulsory. There will be an exam after the lectures.

- Tutorials in very small groups will be devoted, individually or in groups, to solving doubts and monitoring of students.

- The grading of each student will be carried out through continuous assessment and final tests set in the Faculty's exam schedule. It will be compulsory to have completed and passed computer practices.

-- In cases of fraudulent completion of exercises or tests, the “Regulations for evaluating the academic performance of students and reviewing grades” will apply.

-- The student's grade will be the sum of 80% of the final exam grade and 20% of the grade corresponding to the continuous assessment.

-- Continuous assessment will be carried out through written or computer tests, resolution of problem sheets, student participation during lectures and tutorials. Students will know their continuous assessment grade before the final exam.

-- The assessments will be set during school hours. They will have a maximum duration of 1 hour. The day, time and subject for each assessment will be announced in advance.

-- The final test will consist of solving problems similar to those explained during lectures.

-- The evaluation of skills will be carried out in the following ways:
- Exam: Comp 01, comp 07, Con 17, Con 18, H/D 10.
- Computer practices: Comp 05, H/D 11, Con 18.
- Interactive lectures: Comp 01, Comp 07, H/D 10, Con 18.

-- The same evaluation conditions and the grade of the continuous assessment of the first opportunity will be maintained for the second opportunity.

-- The computer science practices already completed and passed will remain as passed in successive academic years.

-- Students repeating this subject may request their continuous aqssessment grade from the previous course to be taken into account.

Theoretical teaching: 23 hours, 100% in-person.
Interactive teaching: 10 hours, 100% in-person.
Interactive laboratory/computer classroom teaching: 8 hours, 100% in person.
Student tutoring: 2 hours, 100% in-person.
Exams and review: 2 hours, 100% in-person.
Student personal work: 67.5 hours, 0% in-person.

Much time is spent in solving exercises. Therefore it is recommended:
- To try to solve the problems of the problem sheets.
- To use the bibliography to consolidate the knowledge and techniques acquired during problem solving.
- To attend tutorials to solve doubts that arise throughout the course.
- To use the virtual classroom of the USC to access the didactic material.