Keywords: logistic regression, random effects, overdispersion
The data comes from an experiment to measure the mortality of cancer cells under radiation under taken in the Department of Radiology, University of Cape Town. Four hundred cells were placed on a dish, and three dishes were irradiated at a time, or occasion. After the cells were irradiated, the surviving cells were counted. Since cells would also die naturally, dishes with cells were put into the radiation chamber without being irradiated, to establish the natural mortality. This data gives only these zero-dose data.
|Occasion||Irradiation occasion (1-27)|
|Survived||Number of cells surviving out of 400 placed on dish|
Data File (tab-delimited text)
|Schall, R. (1991). Estimation in generalized linear models with random effects. Biometrika 78, 719-727.|
|Data originally provided by Dr G. Blekkenhorst, Department of Radiology, University of Cape Town.|
> radiatio <- read.table("radatio.txt",header=T) > attach(radiatio) > glm.null <- glm(Survived/400~1,family=binomial,weights=rep(400,27)) > summary(glm.null,cor=F) Call: glm(formula = Survived/400 ~ 1, family = binomial, weights = rep(400,27)) Deviance Residuals: Min 1Q Median 3Q Max -0.3680785 -0.1112356 -0.03785896 0.140906 0.4413129 Coefficients: Value Std. Error t value (Intercept) -0.7186734 0.08926807 -8.050733 (Dispersion Parameter for Gaussian family taken to be 18.96031 ) Null Deviance: 495.6308 on 26 degrees of freedom Residual Deviance: 495.6308 on 26 degrees of freedom Number of Fisher Scoring Iterations: 0 # Pearson chi-squared statistic > sum( glm.null$weights*glm.null$residuals^2 )  492.9681
Note that this value is larger than the 470.34 quoted by Schall (1991). Perhaps there was a printing error in the published data.
> glm.fix <- glm(Survived/400~factor(Occasion),family=binomial,weights=rep(400,27)) > summary(glm.fix,cor=F) Call: glm(formula = Survived/400 ~ factor(Occasion), family = binomial, weig hts = rep(400, 27)) Deviance Residuals: Min 1Q Median 3Q Max -2.453395 -0.7923871 0 0.698892 2.432414 Coefficients: Value Std. Error t value (Intercept) -0.752886507 0.021229483 -35.4641936 factor(Occasion)1 -0.452228137 0.043209561 -10.4659275 factor(Occasion)2 -0.332045156 0.028699949 -11.5695384 factor(Occasion)3 0.013786385 0.018195352 0.7576872 factor(Occasion)4 0.052071660 0.013632398 3.8196992 factor(Occasion)5 -0.003070936 0.011402175 -0.2693290 factor(Occasion)6 0.090665012 0.009060228 10.0069239 factor(Occasion)7 -0.080510224 0.009319560 -8.6388438 factor(Occasion)8 -0.001549775 0.007333325 -0.2113332 (Dispersion Parameter for Binomial family taken to be 1 ) Null Deviance: 495.6308 on 26 degrees of freedom Residual Deviance: 32.79446 on 18 degrees of freedom Number of Fisher Scoring Iterations: 4 > anova(glm.fix,test="Chi") Analysis of Deviance Table Binomial model Response: Survived/400 Terms added sequentially (first to last) Df Deviance Resid. Df Resid. Dev Pr(Chi) NULL 26 495.6308 factor(Occasion) 8 462.8364 18 32.7945 0 > 1-pchisq(32.79,18)  0.01769375>
There is strong evidence for differences between the occasions (Chisquare = 462.8 on 8 df, P = 0), and some evidence for overdispersion even once differences between occasions have been accounted for (Chisquare = 32.8 on 18 df, P = 0.018). That is, variation between the 3 dishes on each occasion also seems greater than binomial variation. This is somewhat to be expected, as the survival of each cell could be expected to be positively associated with the survival of the surrounding cells.
Schall (1991) introduced random effects for occasion and at the dish level. See the reglm function example for further analysis.