Class Notes for 22^nd and 27^th February

Reminders:

(1) Homework 3 on Design due on Monday, 27^th;

(2) First Exam is Wednesday 1^st March.

Wednesday 22^nd Class

· Ex. 4.7 – old approach: Breslow-Day declares odds ratios (OR’s) are the same for the two faces, but CMH test declares this common value is not one; point est. is 18.65 and 95% CI is 7.46 to 46.6; note interpretation on p.17;

· Ex. 4.7 – new approach: HA log-linear model (4.5), we focus on the interactions; accepting that an interaction term is zero is equivalent to accepting that the OR is 1; this model fits the data – that the 3-way interaction is declared zero is equivalent to the above findings of the Breslow-Day test (same OR for 2 faces)

· Ex. 4.8 – a Poisson counterpart of the two-sample t-test

· Ex. 4.9 – a Poisson counterpart of the paired t-test; uses a conditional argument: conditional on n = y₁ + y₂, y₁ has a binomial dist., and since p = m₁/(m₁ + m₂), testing p = ½ is the null hypothesis here.

· Ex. 4.10 – a Poisson counterpart of ANOCOV – the “offset” – log(t) here – is the “covariate.”

Monday 27^th Class

· Nominal outcomes (hair color, one of four nucleotides) versus ordinal outcomes (poor, fair, good, excellent) – outcome = Y

· Three models are proposed and one in Appendix: BCL in 4.7, PO (extended to UPO in Appendix) in 4.8, AC (above PO), and CRA and CRB (on p.31); these models transform the p’s on the LHS in different ways; RHS is a linear model in the parms.

· Our focus here is on the PO model!

· The above models look similar but some fit a given data set better than others, and the interpretations differ in predicted values and odds-ratio interpretations

· Ex. 4.11 – 4 ordered outcomes (chronic respiratory disease) – PO model is fit – 3 categorical explanatory variables using 4 dummy variables – Output 4.11b tests all variables can be dropped using LR, Score and Wald tests – Output 4.11c “Class” analysis: hard to understand but proportionality is accepted (p = 0.1479) – Output 4.11d is used for odds-ratio interpretations

· Those with no job exposure to pollution have odds of being in the less serious (as opposed to more serious) respiratory direction 2.37 times the odds for those exposed to pollution on the job

· Equivalently, those with job exposure to pollution have odds of being in the more serious (as opposed to less serious) respiratory direction 2.37 times the odds for those not exposed to pollution on the job

· Ex. 4.12 – sometimes PO and BCL models don’t fit – GOF tests are rejected in both cases – CRB model fits these data best and fit is shown on p.33

· 4.5.1 on p.29: A “hyper-parameter” to help us pick the scale - q₆ in eqns. 4.9-4.10 – when q₆ = 1 use usual scale, when q₆ = 0 use log scale (any base is okay) – for budworms, output 4.13 indicates CI (-1.74,0.54) for F and (-0.70,1.15) for M – if they both included 1, we’d use dose scale – both CI’s include 0 so use log scale.