Multivariate Normal and Bioassay Example

ManufacturerProductLineFlatInteriorNon-Flat*FlatExteriorNon-Flat*Primer/SealerManufacturerPhoneNumberManufacturerWeb SiteWhere to BuyRecommended VOC Limit (g/l) 100 150 100 150 200Sherwin-WilliamsHarmony ê ê - - êCheck paint can for VOC content. Certain other interior & exteriorproducts meet the VOC limit, depending on color & finish.800-336-1110www.sherwinwilliams.comSherwin-Williams stores.Too many to list. Use storelocator on manufacturer’swebsite.Notes:1. Non-flat paint includes eggshell, satin, semi-gloss, and gloss.2. The Clean Air Counts recommended VOC limit is based on the South Coast Air Quality Management District's VOC limit effective7/1/2002. The U.S. EPA's current regulations limit VOCs to 250 g/l and 380 g/l for flat and non-flat paint, respectively.3. This is not an exhaustive list of low -VOC paint manufacturers. The inclusion of a manufacturer or product in this list should not beconstrued as an endorsement by Clean Air Counts.4. Though every attempt was made to ensure the accuracy of the information, there may still be errors. Please check with themanufacturer for more detailed product information.Page 4 of 4

Analysis of bioassay experimentIn drug test, bioassay experiment is performed on animals.Let X be the dose level.Animals response: alive or dead (tumor or no tumor)θ = P(death) = F (x|η) : a cdf., η : unknown parameter.Commonly,orF (x|η) =F (x|η) =1, x ∈ R, µ ∈ R (1)1 + e−(x−η) 11 + e −(α+βX i ) , x i ∈ R, α ∈ R, β > 0 (2)or F (x|η) = Φ(x − η), x ∈ R, η ∈ Ror F (x|η) = Φ( x − µσ ) = Φ(−µ σ + 1 x), x ∈ R, µ ∈ R, σ > 0σ

Analysis of bioassay experimentA rough estimate of (α, β) :Since logit[E( y in i|α, β, x i )] = α + βx iMay fitbut y i = 0 or n i is not well defined.⎧⎪⎨ change y i = 1 2 if y i = 0logit( y in i) = α + βx i (3)⎪⎩ change y i = n i − 1 2 if y i = n iFit linear model (3) ⇒ (ˆα, ˆβ) = (.1, 2.9)Obtaining a contour plot of the joint posterior density (ignoringthe normalizing constant) to get posterior mode or other numericalmethods. See figure 3.4(a), posterior mean: (ˆα, ˆβ) = (1.37, 11.93)

Analysis of bioassay experimentLD50 (lethal dose level) or ED50 (effective dose level) = the doselevel at whoch the prob. of death is 50%.⇒ logit(θ i ) = log( .51−.5 ) = α + βx i ⇒ x i = − α β⇒ the posterior dist. of LD50 = the posterior dist. of (− α β )In practice, if we can simulate a large number of obs. of (α, β)then we can get the obs. of (− α β) from the posterior⇒ hist. of the post. of (− α β )Note that: if (X , Y ) have joint density p(x, y) we want to find thedensity of −X /Y , may use transformation, or we may simulate it.Let (X i , Y i ), i = 1,...,n, iid ∼ p(x, y)Then the histogram of −X i /Y i ≈ the density of −X /Y (because−X i /Y i ∼ density of −X /Y .)

Analysis of bioassay experimentAlternatively, we may use normal approximaton if ∑ n i is large.Here n i is not too large.We may derive the analytic form of the observed Fisherinformation for (α, β)Posterior mode = the MLE because of uniform prior.Normal approximation need find (ˆα, ˆβ), J(ˆα, ˆβ) =?Find J(α, β) : exerciseContour plot if normal approximation: figure 4.1(ˆα, ˆβ) = (.88, 7.93) mean of normal approximationActual posterior mean E(α, β|y) = (1.37, 11.93)Normal approximation is not very good because of the smallsample.