They extend standard linear regression models through the introduction of random effects and/or correlated residual errors. Residual analysis: Pearson, deviance, adjusted residuals, etc.Linear mixed models are a popular modelling approach for longitudinal or repeated measures data.Likelihood ratio test, and statistic, Δ G 2.Overall goodness-of-fit statistics of the model we will consider:.Confidence Intervals and Hypothesis tests for parameters.InferenceĪs before, the usual tools from the basic statistical inference are valid, and anything that holds for GLMs for example anything that we said for logistic regression. In general, there are no closed-form solutions, so the ML estimates are obtained by using iterative algorithms such as Newton-Raphson (NR), Iteratively reweighted least squares (IRWLS), etc. Like we saw in Logistic regression, the maximum likelihood estimators (MLEs) for (β 0, β 1 ... etc.) are obtained by finding the values that maximizes log-likelihood. Note that the interpretation of parameter estimates, α and β will stay the same you just need to multiple counts by t as well. This means that mean count is proportional to t. log(t) which is an observation and it will change the value of estimated counts: It is an adjustment term and a group of observations may have the same offset, or each individual may have a different value of t. The term –log( t) is referred to as an offset. Poisson loglinear regression model for the expected rate of the occurrence of event is: Random component: Response Y has a Poisson distribution, and t is index of the time or space more specifically the expected value of rate Y/ t, E( Y/ t)= 1/ t E( Y) = μ/ t If β Issue: can yield μ 0, then exp(β) > 1, and the expected count μ = E( y) is exp(β) times larger than when X = 0 This model is the same as that used in ordinary regression except that the random component is the Poisson distribution. Sometimes the identity link function is used in Poisson regression. For now let’s focus on a single variable X. Systematic component: Any set of X = ( X 1, X 2, … X k) explanatory variables. Random component: Response Y has a Poisson distribution more specifically the expected count Y, E( Y) = μ Convention is to call such model “Poisson Regression” Explanatory variables, X = ( X 1, X 2, … X k), can be continuous or a combination of continuous and categorical variables.Then the counts we are modeling are the counts in a contingency table, and the convention is to call such a model log-linear model (e.g., see Lesson 5.) Explanatory variables, X = ( X 1, X 2, … X k), can be ALL categorical.But we can also have Y/ t, the rate (or incidence) where t is an interval representing time, space or some other grouping. In Crab data we may ask (1) How does the number of satellites a female horseshoe crab has depend on the width of her back (2) What is the rate of satellites per unit width? In the Credit Card data, we may ask (1) What is the expected number of credit cards a person may have, given the his/her income, or (2) What is the sample rate of possession of credit cards? Variables We will first introduce a formal model and then look at the specific