TESTING FOR SPATIAL AUTOCORRELATION IN MODEL FITTING. Mike Antolin. November 5, 2007

Transcription

1 Gettig Started TESTING FOR SPATIAL AUTOCORRELATION IN MODEL FITTING Mike Atoli November 5, 2007 The R package is supported olie by a series of servers that cotai the mai GUI (program), ad a series of additioal packages useful for specialized aalyses. We will first ope the R program the load packages oto each of your computers the ecessary. The most geerally useful package is i the basic R, the MASS library. # Load mai statistical library library(mass) Let s load the coral data (Bruo et al ) ad make some ice plots to see what they look like. First, chage the default directory to the place where you have the data file, by goig to the File pulldow ad the the Chage dir... butto. ### Import data coral<-read.table("gbr_ws_data.csv",sep=",",header=t) attach(coral) ames(coral) # create a graphics widow for each plot plot(lat_dd,lon_dd) Bruo, J.F., E.R. Selig, K.S. Casey, C.A. Page, B. L. Willis, C.D. Harvell, H. Sweatma A. M. Meledy Thermal Stress ad Coral Cover as Drivers of Coral Disease Outbreaks. PLoS Biology.

2 LON_DD LAT_DD Now let s make a couple of other eat lookig plots of the same data, just for fu. First, use the Packages pulldow to get the scatterplot3d ad geor packages. # Make some facy plots for exploratory data aalysis library(geor) library(scatterplot3d) # make data colums used by poits.geodata coral$coords = cbid(lat_dd,lon_dd) coral$data = CASES plot(lat_dd, LON_DD, type= "",mai="number of Cases", text(jitter(lat_dd), jitter(lon_dd),cases,cex=0.75) poits.geodata(coral,cex.var=coral_cover,cex.mi=.,cex.max=3, mai="coral Cover", poits.geodata(coral,cex.var=wssta,cex.mi=.,cex.max=3, mai="ocea Temperature", scatterplot3d(lat_dd, y=lon_dd, CASES, type="h", mai="number of Cases", Testig the data for fit to the Poisso ad egative biomial distributios: Parasite data are quite ofte NOT ormally distributed, so data should be plotted ad tested for fits to either Poisso (radom) or the egative biomial distributio (aggregated, or clumped). First fit to a Poisso distributio: # Make some defiitios ad descriptios of the data avecases<-mea(cases) # average cases defied varcases< var(cases) # variace of cases sd(cases) summary(cases) sample_size=legth(cases) 2

3 sample_size # Create data tables for aalysis frequecies<-table(cases) frequecies ucases<-as.vector(ames(frequecies)) # get levels from the table ito a vector ycases<-as.umeric(ucases) # chage them from a vector to a umber maxlev<-levels(as.factor(ycases)) # umber of levels i frequecy table maxlev # this will be used for plottig ad biig # plot the fit to Poisso distributio par(mfrow=c(,2)) barplot(frequecies,ylab="frequecy",xlab="obs. Cases") barplot(dpois(0:(maxlev-),avecases)*sample_size,ames=as.character(0:(maxlev-)), ylab="frequecy",xlab="exp. Cases") varmea=varcases/avecases varmea if (varmea>) text(2,25,"looks like this is NOT Poisso distributed!", cex=,col="red",fot=4) Now check the egative biomial distributio # plot the fit to to a egative biomial par(mfrow=c(,2)) barplot(frequecies,ylab="frequecy",xlab="obs. Cases") barplot(dbiom(0:(maxlev-),,mu=avecases)*sample_size,ames=as.character(0: (maxlev-)),ylab="frequecy",xlab="exp. Cases") k<-avecases^2/(varcases-avecases) if (k<) text(2,20,"looks like this could be egative biomial!", cex=,col="red",fot=4) Make a sweet plot! # Plot double a historgram with both observed ad expected o it exp<-dbiom(0:(maxlev-),,mu=avecases)*sample_size both<-umeric(2*maxlev) both[:(2*maxlev) %% 2!=0]<-frequecies both[:(2*maxlev) %% 2==0]<-exp lab<-character(2*maxlev) lab[:(2*maxlev) %% 2==0]<-as.character(0:(maxlev-)) barplot(both,col=rep(c("white","grey"),maxlev),ames=lab,ylab="frequecy",xlab="cases" ) leged(90,70,c("obs","exp"),fill=c("white","grey")) # make facy X-axis axis(,at=c(:(2*maxlev)), label=labels[:(2*maxlev) %% 2==0]<-as.character(0:((2*maxlev)-)), lwd=2,pos=-.5,cex.axis=0.5) How about a statistical test for goodess of fit? 3

4 ## Do a approximate test of goodess of fit to the egative biomial distributio # make categories cs<-factor(0:(maxlev-)) #levels(cs) # this will check how may "bis" you've made # make your expected ad observed categories, ad calculate chi-squared ef<-as.vector(tapply(exp,cs,sum)) of<-as.vector(tapply(frequecies,cs,sum)) chisq<-sum((of-ef)^2/ef) ef of Prob<--pchisq(chisq,(levels(cs)-3)) Prob # Write the result of your test o your plot mtext(paste("chi-squared = ",chisq," with probability P =",Prob), side=3,cex=,col="red",fot=4) Fittig a GLM: # Use GLM to test effects of temperature ad coral cover o whitig, betwee regios # origial model library(mass) # Fit a GLM usig quasi-poisso errors fit = glm(cases ~ WSSTA + CORAL_COVER + WSSTA*CORAL_COVER + factor(sector), family=quasipoisso) coef(fit) summary(fit) # Estimate the k parameter for the egative biomial distributio ybar=predict(fit,type="respose") theta.ml(cases, ybar) theta= # Fit a GLM with egative biomial errors fit2=glm(cases ~ WSSTA + CORAL_COVER + WSSTA*CORAL_COVER +factor(sector), family=egative.biomial(theta=theta)) coef(fit2) summary(fit2) plot(wssta,fit2$resid,xlab="wssta") Testig residuals of the model for additioal spatial autocorrelatio Mora s I, a measure of spatial autocorrelatio to idetify departures from spatial radomess over all samples i the system: I YZ ij i= j= = 2W w ( y y ) i= ( y y)( y i i 2 j y) 4

5 y i ad y j are values of your variable at ay poits i ad j is the umber of samplig poits, w ij is the spatial proximity betwee poits i ad j, W is the sum of all 2 values of w ij, the spatial weights matrix. A positive I would idicate that additioal local processes ot accouted for i the model could be actig. First load the spatial library: ##### Istall Spatial Stats library S_HOME="RSpatial/" source(paste(s_home,"spatial_start.r",sep="")) Spatial_start().First<-fuctio(){Spatial_start() Create the spatial weights matrix, i this case a rescale iverse of distace betwee poits. spt.wt<-spwtdist(lat_dd,lon_dd,rescale=t) Ru the spatial aalysis ad make the diagostic plots: morai(fit2$resid,spt.wt) #Plots to test for spatial autocorrelatio par(mfrow=c(2,2)) plot(wssta,fit2$resid,xlab="wssta") plot(fit2$resid, CASES -fit2$resid,xlab="residual",ylab="predicted value") title("predicted values vs residuals",cex=.7) plot(fit2$resid,spt.wt %*% fit2$resid, xlab="residual",ylab="weighted residuals") title("w * RESIDUALS VS. RESIDUALS",cex=.7) cor(fit2$resid,spt.wt %*% fit2$resid) 5