Sumários

Class 13, Module II

19 Abril 2021, 14:30 • Elsa Maria Félix Gonçalves

Online class Friday, April 30, from 10h00 and 12h30

Linear Mixed Models. Some typical examples involving mixed models. The general linear mixed model: description and properties, estimation of covariance parameters (restricted maximum likelihood estimation).

Aula Extra (dúvidas de preparação para o teste de 19/4/21)

16 Abril 2021, 17:00 • Jorge Filipe Campinos Landerset Cadima

Aula extra de dúvidas, online, sexta-feira 16/4/21, das 17h00 às 20H10
The Linear Model test was taken on Monday April 19, during class hours. On the previous Friday (April 16), a 3-hr. extra session was held, during which last year's test was solved.

Class 12, Module II

13 Abril 2021, 14:30 • Jorge Filipe Campinos Landerset Cadima

Onli ne class Monday, April 26, from 14h30 and 17h00
[GLM slides 114-173] (The slides 174 - 202 are suplementary material, which is not considered course material). Exercises GLM 1 and 10: ANCOVA-type GLM models, in a probit regression context. Interpretation in a toxicological context and the likelihood ratio test. The Akaike Information Criterion (AIC) in a GLM context. Subselect selection heuristics using the AIC criterion: the R function 'step'. An example. The Gamma distribution: a parametrization suited for GLMs; the natural and dispersion parameters of a Gamma distribution as a member of the exponential family of distributions; the relation between the variance and the mean in a Gamma distribution; the canonical link for Gamma distributions and some specific contexts with a single numerical predictor. The problem of estimating unknown dispersion parameters, phi. Deviance and scaled deviances: definition; the case of Normal distribution and of Gamma distribution. Residuals in GLMs: the concept of the variance function; Pearson residuals and deviance residuals. How each of these concepts depends on the distribution of Y and on the link function used. The generalized Pearson statistic, X^2.  An estimator for the dispersion parameter phi, based on the generalized Pearson statistic. Standardized residuals. A very brief discussion of the use of residuals and other diagnostics for model checking in GLMs. Log-linear models and contingency tables: the case of two-way contingency tables with Poisson counts; an ANOVA-type log-linear model (without interaction effects) and independence; the bridge between this model and the chi-squared test for independence. A brief word on log-linear models for contingency tables with three or more factors.

Class 11, Module II

12 Abril 2021, 14:30 • Jorge Filipe Campinos Landerset Cadima

Online class on Tuesday, April 20, from 14h30 to 17h00.
[GLM Slides 58-113] A few words on numerical algorithms for parameter estimation in GLMs: the Newton-Raphson type algorithms. Further examples of GLMs for Bernoulli/Binomial response variables: the Probit model (definition, origins in toxicology, the use of a Normal cdf); the complementary log-log model (definition, the use of the Gumbel cdf). Applications with R to the Hosmer & Lemeshow data set. Asymptotic inference for Maximum Likelihood estimators: the general results and their application for GLMs. Asymptotic confidence intervals and hypothesis test for any linear combination of the model parameters. The profiling alternative for confidence intervals. Examples in R. GLMs for Poisson response variables: the canonical link and log-linear models. Exercise GLM 5. Measuring the goodness-of-fit: the deviance (definition, interpretation, the concept of a saturated model and expressions for Poisson, Bernoulli ansd 'Binomial/n' response variables). Comparing models and submodels: Wilk's Theorem and the Likelihood Ratio test. The goodness-of-fit test in Exercise 5.

Class 10, Module II

6 Abril 2021, 14:30 • Jorge Filipe Campinos Landerset Cadima

Online class on Tuesday, April 13, 14h30-17h00
Generalized Linear Models [Slides 1-57] Bibliography. Introduction and motivating example. The 3 components of a GLM. The definition of the (2-parameter) exponential family of distributions: the Normal, Poisson, Bernoulli and 'Binomial/n' as special cases of the exponential family of distributions. Link functions and the concept of a canonical link function. The Linear Model as GLM. The Logistic Regression as a GLM for Bernoulli or Binomial/n random component, with the canonical link. The R command glm and its arguments and auxiliary functions. The Hosmer & Lemeshow example. Properties of the logistic curve and of a Logistic Regression. Estimating parameters in a GLM by Maximum Likelihood: an overview of the problem; the nature of the log-likelihood as a function of the model parameters beta_j; the system of normal equations in a Logistic Regression and the need for numerical algorithms to find the maximum-likelihood estimators.