A Practical Approach, Second Edition=Ronald D. Ho.pdf

706 DEVELOPMENTAL REPRODUCTIVE TOXICOLOGY: A PRACTICAL APPROACH, SECOND EDITIONA. Likelihood and Quasi-likelihood MethodsLet n ij be an observed count from the jth animal in the ith group 1 < i < g and 1 < j < m i . The countsn ij are often modeled by a Poisson distribution,nij −µ i epn ( ij) =n !ijµ i, n = 012 , , ,…,ijthe mean and variance of n ij are E( nij) = Var( nij)= µ i . A common complication in the analysis ofcount data is that the observed variation often exceeds or falls behind the variation that is predictedfrom a Poisson model. A generalized variance for n ij is of the form Var( n ij ) = µ i ( 1 + φ i µ i ).If φ i > 0,then n ij has an extra-Poisson variation; if φ i < 0, then n ij has a sub-Poisson variation; and if φ i = 0,then n ij becomes a Poisson. In the Poisson model, the mean function is often modeled by a loglinearfunction. The dose-response model for trend test is µ i = exp( β0 + β1di). This analysis isreferred to as the Poisson regression.The classical extra-Poisson modeling assumes that the mean of the Poisson has a gammadistribution, which leads to a negative binomial (gamma-Poisson) distribution for the observed data,−1ijΓ( nij+ φi) ⎛ φµ ⎞ ⎛i ipn ( ij)=−1Γ( n + 1) Γ( φ ) ⎝⎜ 1 + φµ ⎠⎟ 1 ⎞⎝⎜ 1 + φµ ⎠⎟ijiiinii1φiwhere φ i > 0. The maximum likelihood estimation of the negative binomial model was describedin detail by Lawless. 22 As in the beta-binomial model, the restriction on φ≥0 can limit its application.For example, the number of implantations or the number of corpora lutea may exhibit asub-Poisson variation. The quasi-likelihood approach 23,24 provides a method to model both extraorsub-Poisson variation data. The quasi-likelihood approach assumes that the mean and varianceof count data are of a negative binomial form, E( n ) = µ and Var( n ) = µ ( 1 + φ µ ).B. Exampleij i ij i i iThe example is the hydroxyurea data considered in Section III. In this example, the effect ofhydroxyurea exposure on the number of implants was analyzed. Because the exposure occurredafter implantation, the numbers of implants would not be expected to differ among the four groups.The means and variances for the four groups areµ ˆ = 10. 29 ( 3. 72) , µ ˆ = 10. 76 ( 5. 19) , µ ˆ = 12. 75 ( 3. 07) , µ ˆ = 10. 50 ( 3.21)1 2 34The means are much larger than the variance in all four groups. The data exhibited sub-Poissonvariation, so the negative binomial model is not appropriate for the analysis.Table 17.6 contains the summary of the analyses using the litter-based analysis, Poisson model(Poisson regression), and quasi-likelihood approach. In the litter-based analysis, the numbers ofimplants were transformed by a square root transformation for data normalization. The litter-basedand quasi-likelihood approaches gave similar results in all three tests at the 5% significance level.The Poisson model shows larger p-values in all tests because it does not adjust for sub-Poissonvariation. The SAS PROC GLM and PROC GENMOD procedures were used in the analysis.© 2006 by Taylor & Francis Group, LLC