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Radiation is applied and we take a sample of cells. Due to radiation, some of these cells are dicentric. I am modelling the ratio of dicentrics per cell against dose. The experiment is repeated in 2 labs. Each laboratory (Ed or Pittsburgh) has 2 type of cultures: immediate analysis of cells after radiation, or waiting 48 hours after radiation. Here are my data:

Dose Cells Dicentrics Laboratory Culture
0.1 328 2 Edinburgh Immediate
0.2 723 1 Edinburgh Immediate
0.5 513 4 Edinburgh Immediate
1.0 369 4 Edinburgh Immediate
1.5 280 7 Edinburgh Immediate
2.0 361 19 Edinburgh Immediate
3.0 238 26 Edinburgh Immediate
0.1 847 1 Edinburgh Stored
0.2 1404 3 Edinburgh Stored
0.5 492 3 Edinburgh Stored
1.0 489 10 Edinburgh Stored
1.5 533 26 Edinburgh Stored
2.0 1112 98 Edinburgh Stored
3.0 422 74 Edinburgh Stored
0.1 172 1 Pittsburgh Immediate
0.2 345 1 Pittsburgh Immediate
0.5 277 3 Pittsburgh Immediate
1.0 364 8 Pittsburgh Immediate
1.5 235 5 Pittsburgh Immediate
2.0 307 15 Pittsburgh Immediate
3.0 238 20 Pittsburgh Immediate
0.1 585 1 Pittsburgh Stored
0.2 1002 3 Pittsburgh Stored
0.5 472 5 Pittsburgh Stored
1.0 493 14 Pittsburgh Stored
1.5 408 30 Pittsburgh Stored
2.0 690 75 Pittsburgh Stored
3.0 291 46 Pittsburgh Stored

What I want to do:

  1. Compare 2 models
  2. Test dependence on location and analysis type

I plotted prediction lines for the data which looks like:

enter image description here

The choice of quadratic model is: $E(Y)=C(\beta_0+\beta_1 D + \beta_2 D^2)$ where $Y$ is the number of dicentrics, $D$ is dose and $C$ is cells.

Laboratory Dependence: I fit 4 subset models independently, using subsets 1:7,8:14,15:21,22:28. The total deviance of 4 independent models is $D=9.067335$ with $16$ df.

I plotted another 2 more models which ignore dependence on the lab, and only include culture. These correspond to culture=immediate and culture=stored respectively. I found that the deviance is $D=15.3982$ with $22$ df.

Hence the null hypothesis of no dependence on lab is tested by comparing $15.3982-9.067335=6.33$ with $\chi ^2_{22-16}=12.59$. Hence we have insufficient evidence to reject the null hypothesis, and labs could be independent.

Why I don't think this is right: By looking at the plot, it seems that the lab should be highly significant. When I do an analysis of deviance on paper, it seems I am comparing the number of parameters and not degrees of freedom. So in this case I should compare it with $\chi^2_1$ since only lab parameter is different in those models?

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From your description, I have no idea what you did. It doesn't sound right, but it's hard to say. Here is how I'd analyze your data:

d = read.table(text="  Dose Cells Dicentrics Laboratory Culture
                        0.1 328 2 Edinburgh Immediate
                        ...
                        3.0 291 46 Pittsburgh Stored", header=T)

The first thing to notice is that these are binomial data. That is, they are counts of 'successes' out of a known total number of 'trials'. You need to use a logistic regression model here. I start with a full model with all interactions (in part because I have no idea what this is), and I include a squared effect of Dose (because you seem to want that).

m = glm(cbind(Dicentrics, Cells-Dicentrics)~poly(Dose,2)*Culture*Laboratory, 
        d, family=binomial)
summary(m)
# output omitted

We can use analysis of deviance tests (i.e., a likelihood ratio test of a nested model) to test the three-way interaction and the need for the squared term:

anova(update(m, .~poly(Dose,1)*Culture*Laboratory, d), m, test="LRT")
# Analysis of Deviance Table
# 
# Model 1: cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 1) + Culture + 
#     Laboratory + poly(Dose, 1):Culture + poly(Dose, 1):Laboratory + 
#     Culture:Laboratory + poly(Dose, 1):Culture:Laboratory
# Model 2: cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 2) * Culture * 
#     Laboratory
#   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
# 1        20     65.812                          
# 2        16      6.457  4   59.355 3.963e-12 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
anova(update(m, .~.-poly(Dose,2):Culture:Laboratory, d), m, test="LRT")
# Analysis of Deviance Table
# 
# Model 1: cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 2) + Culture + 
#     Laboratory + poly(Dose, 2):Culture + poly(Dose, 2):Laboratory + 
#     Culture:Laboratory
# Model 2: cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 2) * Culture * 
#     Laboratory
#   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
# 1        18     6.6571                     
# 2        16     6.4569  2  0.20016   0.9048

You seem to be right about the need for a polynomial fit, but there really doesn't seem to be any need for the three-way interaction.

At this point, modeling philosophies diverge: many will drop the interaction, but some advise against it. I am generally uncomfortable with dropping variables just because they are not significant, but this is sufficiently non-significant that I don't see much problem. In particular, dropping the interaction can make testing and interpreting the model simpler.

m2 = glm(cbind(Dicentrics, Cells-Dicentrics)~(poly(Dose,2)+Culture+Laboratory)^2, 
         d, family=binomial)
summary(m2)
# output omitted

This fits a model with three two-way interactions, but not the three-way. We can conduct subsequent tests of these two-way interactions.

anova(update(m2, .~.-poly(Dose,2):Culture, d), m2, test="LRT")
# Analysis of Deviance Table
# 
# Model 1: cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 2) + Culture + 
#     Laboratory + poly(Dose, 2):Laboratory + Culture:Laboratory
# Model 2: cbind(Dicentrics, Cells - Dicentrics) ~ (poly(Dose, 2) + Culture + 
#     Laboratory)^2
#   Resid. Df Resid. Dev Df Deviance Pr(>Chi)  
# 1        20    12.9341                       
# 2        18     6.6571  2    6.277  0.04335 *
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
anova(update(m2, .~.-poly(Dose,2):Laboratory, d), m2, test="LRT")
# Analysis of Deviance Table
# 
# Model 1: cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 2) + Culture + 
#     Laboratory + poly(Dose, 2):Culture + Culture:Laboratory
# Model 2: cbind(Dicentrics, Cells - Dicentrics) ~ (poly(Dose, 2) + Culture + 
#     Laboratory)^2
#   Resid. Df Resid. Dev Df Deviance Pr(>Chi)  
# 1        20    12.0733                       
# 2        18     6.6571  2   5.4163  0.06666 .
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
anova(update(m2, .~.-Culture:Laboratory, d), m2, test="LRT")
# Analysis of Deviance Table
# 
# Model 1: cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 2) + Culture + 
#     Laboratory + poly(Dose, 2):Culture + poly(Dose, 2):Laboratory
# Model 2: cbind(Dicentrics, Cells - Dicentrics) ~ (poly(Dose, 2) + Culture + 
#     Laboratory)^2
#   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
# 1        19     7.5582                     
# 2        18     6.6571  1   0.9011   0.3425

Here we see that there seems to be a Dose by Culture interaction (which I suspect makes scientific sense), no evidence of a Culture by Laboratory interaction (which also seems plausible), and a Dose by Laboratory interaction that doesn't quite meet the traditional criterion for 'statistical significance' (I'm not sure whether its existence makes sense). Although a p-value of $0.34$ might be kind of low, the observed value of the test statistic is $0.9$, which is less than the expected value of a $\chi^2$ random variable on $1$ df. I am slightly uncomfortable with dropping this, but willing to do so in the name of enhanced simplicity and interpretability. I would not drop the Dose by Laboratory interaction, however. (So the model remains somewhat harder to interpret.) The existence of the interaction that includes Laboratory implies that it is potentially relevant.

Here I fit the final model:

m3 = glm(cbind(Dicentrics, Cells-Dicentrics)~poly(Dose,2)*Laboratory+
                                             poly(Dose,2)*Culture, 
         d, family=binomial)
summary(m3)
# Call:
# glm(formula = cbind(Dicentrics, Cells - Dicentrics) ~ poly(Dose, 
#     2) * Laboratory + poly(Dose, 2) * Culture, family = binomial, 
#     data = d)
# 
# Deviance Residuals: 
#     Min       1Q   Median       3Q      Max  
# -1.1235  -0.3427   0.1043   0.3425   0.9214  
# 
# Coefficients:
#                                     Estimate Std. Error z value Pr(>|z|)    
# (Intercept)                          -4.2729     0.1682 -25.405   <2e-16 ***
# poly(Dose, 2)1                        6.7074     0.8008   8.376   <2e-16 ***
# poly(Dose, 2)2                       -0.7665     0.6117  -1.253   0.2101    
# LaboratoryPittsburgh                  0.2929     0.1786   1.641   0.1009    
# CultureStored                         0.2206     0.1849   1.193   0.2328    
# poly(Dose, 2)1:LaboratoryPittsburgh  -1.0325     0.8358  -1.235   0.2167    
# poly(Dose, 2)2:LaboratoryPittsburgh  -0.4245     0.5749  -0.738   0.4603    
# poly(Dose, 2)1:CultureStored          2.1885     0.8796   2.488   0.0128 *  
# poly(Dose, 2)2:CultureStored         -1.3988     0.6500  -2.152   0.0314 *  
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# (Dispersion parameter for binomial family taken to be 1)
# 
#     Null deviance: 759.2740  on 27  degrees of freedom
# Residual deviance:   7.5582  on 19  degrees of freedom
# AIC: 132.86
# 
# Number of Fisher Scoring iterations: 4

To evaluate and interpret this model, I make a plot of the data with the model's predicted values overlaid. To get a quick sense of the variability associated with each point (proportion dicentric), I compute the normal approximation of the binomial's standard error and plot that for each point as well.

xseq = seq(0,3, by=.1)
prop = with(d, Dicentrics/Cells)
pSE  = with(d, sqrt(prop*(1-prop)/Cells))

pchs = with(d, ifelse(Laboratory=="Edinburgh"&Culture=="Immediate",   1,
               ifelse(Laboratory=="Pittsburgh"&Culture=="Immediate", 16,
               ifelse(Laboratory=="Edinburgh"&Culture=="Stored",      2, 17))))
windows()
  with(d, plot(Dose, Dicentrics/Cells, pch=pchs))
  with(d, arrows(x0=Dose, y0=prop-pSE, y1=prop+pSE, code=3, length=.05, angle=90))
  lines(xseq, lty=1, y=predict(m3, data.frame(Dose       =xseq, 
                                              Laboratory ="Edinburgh",
                                              Culture    ="Immediate" ), type="r"))
  lines(xseq, lty=2, y=predict(m3, data.frame(Dose       =xseq, 
                                              Laboratory ="Pittsburgh",
                                              Culture    ="Immediate" ), type="r"))
  lines(xseq, lty=3, y=predict(m3, data.frame(Dose       =xseq, 
                                              Laboratory ="Edinburgh",
                                              Culture    ="Stored" ), type="r"))
  lines(xseq, lty=4, y=predict(m3, data.frame(Dose       =xseq, 
                                              Laboratory ="Pittsburgh",
                                              Culture    ="Stored" ), type="r"))
  legend("topleft", lty=1:4, pch=c(1,16,2,17),
         legend=c("Edinburgh Immediate", "Pittsburgh Immediate",
                  "Edinburgh Stored", "Pittsburgh Stored"))

enter image description here

From the plot, we can see that the interaction with Laboratory is necessitated by different effects at higher doses when the cells were stored.

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  • $\begingroup$ Thanks a lot, I'm going to play around with this tomorrow $\endgroup$
    – GRS
    Nov 11 '16 at 0:00

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