APPLYING GENERALIZED LINEAR MODELS
CONTENTS GENERALIZED LINEAR MODELLING STATISTICAL MODELLING A MOTIVATING EXAMPLE HISTORY DATA GENERATING MECHANISMS AND MODELS DISTRIBUTIONS REGRESSION MODELS EXPONENTIAL DISPERSION MODELS EXPONENTIAL FAMILY EXPONENTIAL DISPERSION FAMILY MEAN AND VARIANCE LINEAR STRUCTURE POSSIBLE MODELS NOTATION FOR MODEL FORMULAE ALIASING THREE COMPONENTS OF A GLM RESPONSE DISTRIBUTION OR “ERROR STRUCTURE” LINEAR PREDICTOR LINK FUNCTION STANDARD MODELS EXTENSIONS INFERENCE EXERCISES DISCRETE DATA LOG LINEAR MODELS SIMPLE MODELS POISSON REPRESENTATION MODELS OF CHANGE MOVER–STAYER MODEL SYMMETRY DIAGONAL SYMMETRY LONG-TERM DEPENDENCE EXPLANATORY VARIABLES OVERDISPERSION HETEROGENEITY FACTOR RANDOM E?ECTS RASCH MODEL FITTING AND COMPARING PROBABILITY DISTRIBUTIONS FITTING DISTRIBUTIONS POISSON REGRESSION MODELS SETTING UP THE MODEL LIKELIHOOD FUNCTION FOR GROUPED DATA COMPARING MODELS SPECIAL CASES TRUNCATED DISTRIBUTIONS MIXTURE DISTRIBUTIONS MULTIVARIATE DISTRIBUTIONS GROWTH CURVES EXPONENTIAL GROWTH CURVES CONTINUOUS RESPONSE COUNT DATA LOGISTIC GROWTH CURVE GOMPERZ GROWTH CURVE MORE COMPLEX MODELS TIME SERIES POISSON PROCESSES POINT PROCESSES HOMOGENEOUS PROCESSES NONHOMOGENEOUS PROCESSES BIRTH PROCESSES MARKOV PROCESSES AUTOREGRESSION OTHER DISTRIBUTIONS MARKOV CHAINS REPEATED MEASUREMENTS SURVIVAL DATA GENERAL CONCEPTS SKEWED DISTRIBUTIONS CENSORING PROBABILITY FUNCTIONS “NONPARAMETRIC” ESTIMATION PARAMETRIC MODELS PROPORTIONAL HAZARDS MODELS EXPONENTIAL DISTRIBUTION WEIBULL DISTRIBUTION “SEMIPARAMETRIC” MODELS PIECEWISE EXPONENTIAL DISTRIBUTION COX MODEL EVENT HISTORIES EVENT HISTORIES AND SURVIVAL DISTRIBUTIONS COUNTING PROCESSES MODELLING EVENT HISTORIES TIME DEPENDENCE GENERALIZATIONS GEOMETRIC PROCESS GAMMA PROCESS SPATIAL DATA SPATIAL INTERACTION DIRECTIONAL DEPENDENCE CLUSTERING ONE CLUSTER CENTRE ASSOCIATION SPATIAL PATTERNS RESPONSE CONTOURS DISTRIBUTION ABOUT A POINT NORMAL MODELS LINEAR REGRESSION ANALYSIS OF VARIANCE NONLINEAR REGRESSION EMPIRICAL MODELS THEORETICAL MODELS 10 DYNAMIC MODELS 10 1 DYNAMIC GENERALIZED LINEAR MODELS 10 1 1 COMPONENTS OF THE MODEL 10 1 2 SPECIAL CASES 10 1 3 FILTERING AND PREDICTION 10 2 NORMAL MODELS 10 2 1 LINEAR MODELS 10 2 2 NONLINEAR CURVES 10 3 COUNT DATA 10 4 POSITIVE RESPONSE DATA 10 5 CONTINUOUS TIME NONLINEAR MODELS REFERENCES A MOTIVATING EXAMPLE ALTMAN (1991, P 199) PROVIDES COUNTS OF T4 CELLS/MM3 IN BLOOD SAMPLES FROM 20 PATIENTS IN REMISSION FROM HODGKIN’S DISEASE AND 20 OTHER PATIENTS IN REMISSION FROM DISSEMINATED MALIGNANCIES, AS SHOWN IN TABLE 1 1 WE 1 GENERALIZED LINEAR MODELLING TABLE 1 1 T4 CELLS/MM3 IN BLOOD SAMPLES FROM 20 PATIENTS IN REMISSION FROM HODGKIN’S DISEASE AND 20 PATIENTS IN REMISSION FROM DISSEMINATED MALIGNANCIES (ALTMAN, 1991, P 199) HODGKIN’S NON-HODGKIN’S DISEASE DISEASE WISH TO DETERMINE IF THERE IS A DI?ERENCE IN CELL COUNTS BETWEEN THE TWO DISEASES TO DO THIS, WE SHOULD ?RST DE?NE EXACTLY WHAT WE MEAN BY A DI?ERENCE FOR EXAMPLE, ARE WE SIMPLY LOOKING FOR A DI?ERENCE IN MEAN COUNTS, OR A DI?ERENCE IN THEIR VARIABILITY, OR EVEN A DI?ERENCE IN THE OVERALL FORM OF THE DISTRIBUTIONS OF COUNTS? A SIMPLE NAIVE APPROACH TO MODELLING THE DI?ERENCE WOULD BE TO LOOK AT THE DI?ERENCE IN ESTIMATED MEANS AND TO MAKE INFERENCES USING THE ES- TIMATED STANDARD DEVIATION SUCH A PROCEDURE IMPLICITLY ASSUMES A NORMAL DISTRIBUTION IT IMPLIES THAT WE ARE ONLY INTERESTED IN DI?ERENCES OF MEANS AND THAT WE ASSUME THAT THE VARIABILITY AND NORMAL DISTRIBUTIONAL FORM ARE IDENTICAL IN THE TWO GROUPS THE RESULTING STUDENT T VALUE FOR NO DI?ERENCE IN MEANS IS 2 11 BECAUSE THESE ARE COUNTS, A MORE SOPHISTICATED METHOD MIGHT BE TO AS- SUME A POISSON DISTRIBUTION OF THE COUNTS WITHIN EACH GROUP (SEE CHAPTER 2) HERE, AS WE SHALL SEE LATER, IT IS MORE NATURAL TO USE DI?ERENCES IN LOG- ARITHMS OF THE MEANS, SO THAT WE ARE LOOKING AT THE DI?ERENCE BETWEEN THE MEANS, THEMSELVES, THROUGH A RATIO INSTEAD OF BY SUBTRACTION HOWEVER, THIS 1 1 STATISTICAL MODELLING