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Mortality of Cancer Cells

Keywords: logistic regression, random effects, overdispersion

Description

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.

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Data File (tab-delimited text)

Source

Analysis

> 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 )
[1] 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)
[1] 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.