Generalised linear models. Response variable can take a number of different formats

Transcription

1 Generalised linear models Response variable can take a number of different formats

2 Structure Limitations of linear models and GLM theory GLM for count data GLM for presence \ absence data GLM for proportion data

3 Format of response variable Response = Explanatory variable(s) + Error In linear models explanatory variables categorical or continuous Response variables always continuous ( real ) numbers of any value; 2.31, 7.93, etc. But what if your response variable differs? Yes\No, Presence\Absence, True\False, Dead\Alive Whole number counts Count data expressed as proportions 0-1, Ordinal (ranked) data, first, second, third.

4 Why do we need to generalise? What goes wrong if we stick to linear model? Make assumptions about the form of the data no longer met (see previous lectures) Normally distributed residuals, response should be real numbers (i.e. with decimal places possible). Problems obvious in residual plots Example with count data: count of hedgehogs from 40 samples, two different sites

5 Counts of hedgehogs have to be whole numbers; cannot have 1.75 hedgehogs!

6 QQ plot also indicates problems of using a linear model with these data. Points no longer on roughly straight line

7 Data transformation? Number of eggs laid by loggerhead turtles in response to changes in food supply Only whole numbers of eggs can be laid; zero minimum Log data better? problem: unequal variances

8 Limitations of linear model Raw data in form of counts indicates linear model is unsuitable; log data still a problem. Only have whole number counts (no fractions) but normal error model assumes continuous An attempt to do a linear model shows error with, for example, strong curve on QQ plot We need something more flexible than a linear model for a range of error structures: normal, count, proportion, presence\absence or ordinal data

9 Generalised linear models theory Three changes, based on what you have learnt already 1. Error Structure 2. Linear Predictor 3. Link Function This gives generalised linear models much greater flexibility over linear models

10 1. Error structure Linear models only have normal errors Now a wide choice, to best match your data. Error distributions (or error family ) you are most likely to use are: normal errors (already seen; continuous data) poisson errors (count data where variance mean) binomial errors (binary 1\0 data; dead\alive) binomial errors (proportion data, percentages)

11 2. Linear predictor Think of a linear model with 3 continuous predictors (multiple regression) predictors, x 1, x 2, x 3. Coefficients β 0, β 1, β 2, β 3 y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 Now generalise this equation, so to cope with any number of p predictors, where Σ is sum of all calculations each predictor: y = p j=1 β j x j

12 2. Linear predictor (cont.) BUT can usually no longer predict y directly as response may not be normally distributed, but based on another distribution such as poisson, or binomial. These error distributions more complicated to relate to observed response variable. Calculate η (eta) as an intermediate variable: η = p j=1 β j x j

13 3. Link function Calculate predicted values of y from η to give complete flexibility If response data are normally distributed real numbers, normal error model is used, and the values of y and η are identical. R automatically selects the best link function ( canonical function) to convert from η to y based on error structure so you usually do not need to worry about specifying it explicitly.

14 Canonical link functions The default link functions R selects for the most common errors structures are: Normal errors : identity link (no change) Poisson errors : log link Binomial errors : logit link Generally you won t have to worry about these as the best ones are the default, but it is worth knowing how a GLM is structured

15 GLM for count responses Change to syntax for earlier hedgehog example: hhog.glm <- glm(count~site, family=poisson, data=hhogdat) Now use glm rather than lm function, and add a family term to indicate what sort of error model is needed. Poisson errors best for count data. QQ plot now looks less distorted

16 Normal errors Poisson errors

17 GLM output from R > summary(hhog.glm) Call: glm(formula = count ~ site, family = "poisson", data = hhogdat) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) siteb *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for poisson family taken to be 1) Null deviance: on 79 degrees of freedom Residual deviance: on 78 degrees of freedom AIC:

18 Interpretation Highly significant site predictor indicated Coefficients are the logs of the counts Site A, mean hedgehog count is exp(0.2429) = 1.28 Site B, mean hedgehog count is exp( ) = 2.28 Note addition in calc for Site B; summary output from R shows difference from baseline site (Site A). AIC is Akaike Information Criterion; lower values better

19 GLM for binary responses May arise in many ecological studies: dead or alive present or absent occupied or empty Response data often simply a column of 0s and 1s. e.g. presence or absence of a bird spp depending on area of island logistic regression

20 > birdmod.glm = glm(incidence ~ area, binomial, birddat) > summary(birdmod.glm) Call: glm(formula = incidence ~ area, family=binomial, birddat) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) ** area *** --- Signif. codes: 0 *** ** 0.01 * 0.05 (Dispersion parameter for binomial family taken to be 1) Null deviance: on 49 degrees of freedom Residual deviance: on 48 degrees of freedom AIC:

21 Predicted incidence Can use results of birdmod.glm to predict probability of birds being present on island of given size Procedure Create a vector in R with range of possible x-values to predict against Use the predict command with glm output plus x- values to predict for, and store the response plot the predicted values as a curve against area plot observed presence-absence on same plot

22 xv = seq(0,9,0.01) # Create values 0, 0.01, 0.02, yv = predict(birdmod.glm, list(area=xv), type="response") plot(incidence ~ area) lines(yv ~ xv, col="red") Points: observed presence or absence of birds Red line: predicted incidence based on fitted model output Note: x-axis is area of island y-axis is probability or incidence

23 GLM for proportional responses Data might be proportion of nests occupied in a wood Proportion of flowers visited by pollinators Proportion of badger road deaths on A roads, B roads or unclassified roads Note that proportions are bounded: you cannot have less than 0 or greater than 100% in data.

24 Error model for proportions Again use binomial error model Difference is that response variable now consists of two columns; each containing the numbers found in the two classes Use R command cbind can help

25 GLM proportions example Sex ratio in insects; is the number of males greater than females depending on density?; head of dataset is:

26 Graphical study Proportion of males appear to increase with density

27 Procedure Create response variable y that contains numbers of males and females, and enter into glm y <- cbind(males, females) Two models, using raw and logged density: model <- glm(y ~ density, binomial) model2 <- glm(y ~log(density), binomial)

28 Proportion GLM output: raw density Residual deviance should be similar to residual d.f.

29 Proportion GLM: log density Residual deviance now similar to residual d.f. Second analysis, using log density, better Residual d.f. now similar to residual deviance Lower AIC value (38.2 vs 54.6) also better

30 Predicted values Similar procedure to presence\absence response data xv = seq(0,6,0.01) yv = predict(model2, list(density=exp(xv)), type="response") p = males / (males + females) plot(p ~ log(density), ylab= Proportion male ) lines(yv ~ xv, col="red") Note: predict line uses exp(xv) since this is the model with log density as explanatory variable y axis of observed data is proportion males (p)

31 Predicted and observed proportions

32 General comments on GLM You can have multiple explanatory variables It is possible to use interaction terms (interpretation is the same as for linear models). If you have a series of possible GLM models on the same response data, and you want to select best model, choose the one with lowest AIC value. stepaic function (MASS library) helps

33 Summary GLM when form of response data more complex. Count, presence\absence and proportion data Matthiopolous. How to be a quantitative ecologist (2011) Chapter 11, section 11.7 Crawley. Statistics: An introduction using R (2015) Chapters 12 to 15 Crawley. The R Book (2013) for more in-depth Chapter 13 GLM introduction Chapter 14 Count data Chapter 16 Proportion data Chapter 17 Binary data