Tumor Promoter Studies with Soluble Nickel Compounds
8) Can animal-based respiratory toxicity data (used as a surrogate for tumor promotion effects) predict excess cancer risks seen in workers exposed to soluble nickel
7.8 FURTHER STATISTICAL ANALYSIS OF THE SLI 2000B 2- 2-GENERATION REPRODUCTIVE TOXICITY STUDY
Modified from an abstract to the International workshop on statistical modelling
. Generalized linear model with overdispersion
– a case study of the toxicogical effect of nickel sulphate hexahydrate on postimplantation-perinatal mortality rate in rats
Helle M. Sommer, Ph.D., M.Science, statistician, The Danish Veterinary and Food Administration; Ulla Hass, Ph.D, M. Science, reproductive toxicologist, The Danish Veterinary and Food Administration
Poul Thyregod, Associate Professor in Statistics, Danish Technical University
Keywords toxicity study, generalized linear model, binomial distribution, intralitter correlation, overdispersion.
This case study is based on the results of an oral (gavage) two-generation reproduction toxicity study in Sprague-Dawley rats with nickel sulphate hexahydrate (SLI 2000b).
In obtaining reliable results from toxicity studies one of the cornerstones is the choice of the statistical test method. Depending on the choice of method, different results may be obtained, however, some methods are usually more appropriate than others. In this presentation, an approach based upon binomial proportions (a generalized linear model with overdispersion) is described and compared to the method suggested by the industry (a Mann-Whitney test on the mortality rate per litter). A primary characteristic of the binomial overdispersion model is that it accounts for the structure of the type of data and for the important issue of the litter effect, and hence is believed to have greater power.
Toxicity study
Toxicity studies are performed to provide information concerning the toxic effects of a test substance, in this case the toxic effect of the presence of the substance in food / drinking water. The purpose of the study was to evaluate the potential effect of nickel sulphate hexahydrate on the integrity and performance of the male and female reproductive systems and the effects on peri-postnatal mortality rate. In this presentation, focus is on the mortality of the offspring due to maternal exposure during pregnancy and lactation. In order to determine the dose of the substance (given to the parents) that has an effect on the perinatal mortality of the offspring, the study was designed with a control group (group 1) and four test groups with increasing doses (group 2-5). The estimation of / decision on the dose that causes no toxic change, e.g., the no-observed-effect level, lies outside this presentation.
Each group contained between 25 to 28 female rats, which were given the substance orally dissolved in water at dosage levels of 1.0, 2.5, 5.0 or 10.0 mg/kg/day (group 2-5 respectively). Dosing began at 10 week prior to mating and continued until the day prior to scheduled euthanasia. Control animals were given water under the same experimental conditions. After the euthanasia of the maternal rats after weaning of the offspring, the peri-postnatal lethality in each litter was calculated (total number of implantations minus total number of live pups on the 4th day after birth in each litter).
The response is given, as the number of dead versus the total number of implantations in the litter. Each litter varies in size from 4 to 19, however there were no significant correlation between the size of the litter and the mortality rate. An example of the structure of the data is given in Table 1.
Group 1
(control) Group 2
1 mg/kg/day Group 3
2.5 kg/kg/day Group 4
5 mg/kg/day Group 5 10 mg/kg/day
Litter 1 0 / 6 2 / 12 2 /16 3 / 14 3 / 16
Litter 2 1 / 17 2 / 13 0 / 14 1 / 13 1 / 12
and so on … litter 28 … … … … …
“Mean mortality rate”
(percentage) 0,074 0,084 0,090 0,101 0,169
R312_0807_hh_chapter0124567 Statistical analysis
The response data can be described by binomial distributions (number of dead versus total number of implants).
The data material can be analysed in pairs (e.g. a certain group versus the control) or as a regression due to the increasing dose. In both cases the analysis is carried out in the generalized linear model with a logistic link function (ln(p/(1-p) =-) due to the binomial distributed response data. The canonic parameter - could now be described by the linear function:
- = ȕ + gj , j = 1,2…5 (comparing of groups) or - = ȕ + Ȗ·dj , j = 1,2…5 (regression)
where ȕ is the intercept, g is the deterministic effect from the groups, Ȗ is the slop of the regression line and d is the dose level. A goodness of fit test for the level model and for the regression analysis revealed a bad fit (P = 0.0005) and it could be concluded that there was more variation in the system than could be explained by the model and thereby by the binominal distribution.
The intralitter correlation among responses from offspring in the same litter was suspected to be significant meaning that the biological variability between litters should not be ignored. Since the biological variability between litters turned out to be significant, the linear functions given above are insufficient in describing the system. This lack of fit is the so-called ‘litter effect’, and the tendency for fetuses from the same litter to respond more alike than fetuses from different litters has long been recognized. Consequently, the conventional binomial distribution provides poor fit to this kind of data, known as the “extra binomial variation” phenomenon, or overdispersion, and hence, in the framework of generalized linear models, this extra binomial variation could be modelled by introducing a dispersion parameter in the model. The dispersion parameter, ij, describes how many times larger the actual variance is compared to the variance of the binomial distribution.
Results
The main result shows a significant raise in the peri-postnatal mortality rate in the offspring due to maternal exposure of nickel sulphate hexahydratebefore and during pregnancy. This result was found both
1) when analysing for total homogeneity comparing all 5 groups and 2) when comparing group 1 against 5 and
3) when comparing 1, 2, 3, and 4 against group 5 and 4) when using the regression analysis for all 5 dose levels.
The reason for pooling group 1 – 4 is due to the industry’s concern of the control group, which in their opinion showed an unrealistically low mortality rate compared to historical control values.
The dispersion parameter was found to be 1.3, meaning that the actual variance is 30% larger than what could be explained by the binomial distribution due to biological variability between the litters.
The significance level (P-values in Wald tests) for the mortality rate for analysis 1), 2), 3) and 4) for modelling with and without the overdispersion was found to be:
Analyses P-value without overdispersion P-value with overdispersion
1) 0.0004 0.0115
2) 0.0002 0.0078
3) < 0.0001 0.0004
4) < 0.0001 0.0004
Programming in SAS
All analyses under the generalized linear model were performed using SAS Insight (fit option) with the number of dead offspring set as response variable (Y), and the group indicator as explanatory variable (X). The Response Distribution was chosen to binomial and the logit (or canonical) Link Function was chosen. The dispersion parameter (Scale) was set to Deviance and Quasi-likelihood was chosen as estimation method. The Binomial number of trials was set to the size of the litters.
Discussion
In the framework of experimental design, the litters act as blocks, and hence a possible litter effect might be modelled as a random effect. In studies of maternal exposure during pregnancy, the litter effect could be included by a hierarchical distribution structure. To account for the litter effect, the literature has suggested that the beta-binomial distribution be used as a means to describe the extra binomial variation (e.g. Williams 1975, Haseman and Kupper 1979).
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Lee and Nelder (2000) note that for generalized linear models, the variance functions under an adjustable dispersion parameter differs from that under a random effect model, and furthermore introduces a combined approach in a so-called hierarchical generalized linear model.
However, in the present study it was found that the small span of dosage levels did not warrant such an approach. Moreover an investigation of the deviance residuals did not suggest that a more detailed modelling was relevant.
Comparing with the Mann-Whitney method.
In the sector of industry, the Mann-Whitney U-test is the test analysis that is most often employed for these kinds of results from toxicity studies. Under this test, no significance was demonstrated for the present data set when comparing group 1 to group 5. However, the Mann-Whitney test simply does not use all information that lie within the data set and this fact contributes to a blurred picture compared to the generalized linear model and therefore the Mann-Whitney tests has less power to detect potential differences.
We find that the use of the Mann-Whitney test is not very appropriate for the present toxicity study. As it is a nonparametric test, it does not use information concerning the distribution of the data nor of the known sizes of the litters. The assumption on identical distribution of data in each group is clearly violated because of the varying litter sizes. Moreover a Mann-Whitney test assumes that there are no tied ranks (Altman 1991) which is not achieved for the present set of data. If there are many identical data values, complicated correlations should be applied to the large sample formula.
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