Model Based Statistics in Biology. Part V. The Generalized Linear Model. Chapter 16 Introduction

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1 Model Based Statistics in Biology. Part V. The Generalized Linear Model. Chapter 16 Introduction ReCap. Parts I IV. The General Linear Model Part V. The Generalized Linear Model 16 Introduction 16.1 Analysis of Count Data Binomial, Poisson, and Overdispersed Counts 16.2 Goodness of fit - Chisquare statistic Goodness of fit - G 2 Statistic Extrinsic hypothesis: Mendelian Ratios Intrinsic hypothesis: Two-way Contingency Test 16.3 Analysis of Deviance Improvement in Fit G 2 Analysis of Deviance Extrinsic hypotheses Intrinsic hypothesises: Two-way Contingency Test 16.4 GzLM for Normal Errors 16.5 GzLM Notation: Normal errors (GLM) Binomial, Poisson, and Overdispersed Counts Lognormal Data ReCap Part I (Chapters 1,2,3,4) Quantitative reasoning (Models and Measurement): Example of scallops, which combined models (what is the relation of scallop density to substrate?) with statistics (how certain can we be?) ReCap Part II (Chapters 5,6,7) Hypothesis testing uses the logic of the null hypothesis to make a decision about an unknown population parameter. Estimation is concerned with the specific value of an unknown population parameter. ReCap Part III (Ch 9, 10,11) The General Linear Model with a single explanatory variable. The General Linear Model. Advantage of learning unifying concepts rather than list of statistical tests. GLM a general procedure that is more useful and flexible than a collection of special cases. Today: Introduction to the Generalized Linear Models 1

2 Introduction to the Generalized Linear Model. In the previous sections we learned to write, execute, and interpret statistical models for linear model with a normal (fixed) errors. In this section we extend what we have learned to linear models with errors that are not homogeneous. The most important application will be count data, which is bounded at zero, and will show nonhomogeneous errors because the variance rises with the mean. This approach, the Generalized Linear Model (GzLM) underlies what was treated in 20th texts as separate topics, including logistic regression, contingency texts, multiway tables, and analysis of frequencies. Biological data often do not meet the assumptions for the General Linear Model. Residuals are rarely homogeneous for count data. Residuals are neither homogeneous nor normal for right skewed data, where extreme counts are concentrated in the right tail (above the mean). Transformations are traditionally recommended to remedy these problems. This remedy sometimes cures the statistical problems, but the side effects are unpleasant. They include uninterpretable models of the relation of the response to explanatory variable (David I. Warton and Francis K. C. Hui Ecology 92:3 10) and biased estimates of effect sizes ( Packard (2009. J. Theor. Biol. 257, ; Ballentyne 2013 Journal of Theoretical Biology 317: ) Problems with presentation of GzLM in 2003 rubrics Contrast of GzLM and GLM not clearly stated Notation of GzLM not clear Concepts of error model, structural model, and link not clearly distinguished. Links introduced (in quizzes) without explanation. No hands on examples in lab. For binomial response Proportion, odds, logodds, Odds ratio not clearly distinguished. Distinction of binomial and poisson response not clearly distinguished in lecture notes. Clearly distinguished in class in 2003 by defining the unit: Score each unit as present/absent > binomial Count in a defined unit > poisson. Too many concepts in the initial lecture 2

3 We already have seen one remedy, which is to do a randomization test. With this remedy we don t have to torture the data with a transformation. As we have seen this is a lot of work, especially if we want to report effect sizes with confidence limits. A side effect of this remedy is that our estimates of parameters (means, slopes, etc) may be biased because a normal error was used to make the estimates. For example, a few large counts, as from a process that generates right skewed data, will have a large influence on our estimate of the mean. The Generalized Linear model (GzLM) leaves the response variable on its original scale. In any science, analysis of data on its original scale, retaining units, is more appropriate than some arbitrary transformation chosen for statistical reasons. The Generalized Linear model was developed in 1972 (Nelder and Wedderburn) and presented in text form a decade later (McCullagh and Nelder 1983). The GzLM includes logistic regression (e.g. Menard 1993) and log linear models (Feinberg 1970, Bishop et al 1975, Agresti 1996) as special cases of the generalized linear model. Introductory texts in iology (e.g. Sokal and Rohlf 2013, Zar 2010) treat the analysis of count data under the heading of analysis of frequencies. The analysis of counts are sometimes presented as non-parametric tests, and hence free of assumptions. Hypothesis testing for count data employs the chi-squared distribution, which rests on the the same assumptions as its daughters F and t-distributions (Feller ref.). 3

4 References The Generalized Linear Model (Nelder and Wedderburn 1972) is one of the most important contribution to statistics in the last half of the 20th century. Texts range from highly mathematical to more accessible. Here is a lightly annotated list. Agresti, A. 1996, 2007 Introduction to Categorical Data Analysis. NY: John Wiley and Sons. Presents multiway tables and GzLM, examples from social science and biology. Bishop, Y.M.M., Feinberg, S.E., Holland Discrete Multvariate Analysis. MIT Press. Extensive treatment of multiway tables, examples from social, health, and biological sciences. Crawley, M.J GLIM for Ecologists. Blackwell. Stresses parameter estimation and model evaluation instead of hypothesis testing. The GLIM programming language has been replaced by R/S+ Dobson, A.J An Introduction to Generalized Linear Models. Chapman and Hall. Feinberg, S.E. 1977, The Analysis of Cross-Classified Categorical Data. MIT Press. Early treatment of log-linear models (Poisson error, log link) as multiway tables. Examples mostly social science. Hoffmann, J.P. Generalized Linear Models. An Applied Approach. Pearson Education Inc (Allyn and Bacon). Highly readable, includes SPSS, SAS and Stat code. Examples mostly from social science. Menard, S Applied Logistic Regression Analysis. London: Sage Publications. Lindsey, J.K. Applying Generalized Linear Models. NY: Springer Texts in Statistics. Extensive data sets and exercises from biology, social science, health science, engineering. Madsen, H. and Thyregood, P Introduction to General and Generalized Linear Models. CRC Press. Emphasis on math, circa numbered equations per chapter. Extensive R code. Question guided analysis of 5 data situations in Chapter 7. McCullagh, P. and J.A. Nelder (1st edition) 1989 (2nd edition). Generalized Linear Models. Chapman and Hall. The first text (1983) followed by a 2nd edition that remains the authoritative text. Myers, R.H. Montgomery, D.C. Vining, G.G Generalized Linear Models with Applications in Engineering and Sciences. Wiley. Extensive use of math, many data sets and exercises, mostly from engineering. Nelder, J.A., Wedderburn, R.W.M Generalized Linear Models. Journal of the Royal Statistical Society A 135: The publication that introduced the GzLM Smithson, M., Merkle, E.C Generalized LinearModels for Categorical and Continuous Limited Dependent Variables. CRC Press. Data files available online, mostly social science. Ca 7 exercises per chapter. Detailed description of complex analyses, in R and Stata. 4

5 Advantages of GzLM. Learning the GzLM has many advantages, compared to learning special cases such as logistic regression, log-linear models, etc. Carryover. Concepts already learned for the GLM can be applied to count data. This is more effective than learning new (and apparently different) sets of tests. Improved quality of statistical analyses. Likelihood ratios and p-values are more reliable if the appropriate error structure is used. Parameter estimates (means, ratios, odds) are more accurate if the appropriate error structure is used. Transformations become unnecessary. Transforming the response variable does not necessarily remedy failure to meet assumptions. It always changes the relation of the response to explanatory variable, With the generalized linear model the response variable remains on its original scale. Effect sizes can be interpreted in the same units as the measurement of the response variable. Assumptions become explicit. Unfortunately, standard goodness of fit tests do not distinguish whether the errors arise from counts from a known number of trials (Binomial) or from an unknown number of trials (Poisson). Greater flexibility in analysis. The generalized linear model permits us to apply the model based approach we have learned with the GLM (normal errors) to the analysis of count data. Model based statistics lead naturally to analysis of counts relative to the same full suite of explanatory variables, both categorical and ratio scale. In order to apply the generalized linear model, we need to make a few modifications of the generic recipe for applying the general linear model. 5

6 Biology 4605/7220. Generic Recipe for Data Analysis with the Generalized Linear Model. 1. Construct model. Begin with verbal and graphical model. Distinguish response from explanatory variables Assign symbols, state units and type of measurement scale for each. Write out structural model (explanatory variables) 2. Execute model Place data in model format, code model statement. Specify error structure and link function. Compute fitted values from parameter estimates. Compute residuals and plot against fitted values. 3. Evaluate the model, using residuals. If straight line inappropriate, revise the model (back to step 1). If errors not homogeneous, revise error structure and link (step 2) 4. State population and whether the sample is representative 5. Decide on mode of inference. Is hypothesis testing appropriate? If yes step 6, otherwise, skip to step State H o /H A pair (some analyses may require several pairs). State test statistic, its distribution (Chisquare), and tolerance of Type I error. 7. ANODEV: Partition df according to model. Table Source, df, G, p-value from Chisquare distribution. 8. Evaluate sensitivity to deviations from assumptions, and compute p-values and estimates by randomization if necessary. 9. Declare decision about model terms: If p < α then reject H o and accept H A If p > α then accept H o and reject H A Report conclusion with evidence: Either the ANODEV table or statistics ( G, df and p-value) for each model term of interest. 10. Report and interpret effect sizes of biological interest (means, slopes, odds ratios) along with one measure of uncertainty (standard error or confidence intervals). Use appropriate distribution to compute confidence limits. 6