Applied Bayesian Modelling - Free

Table 2.5 Couples with female partner as smokers: predicted cycledistribution under mixture modelMean St. devn. 2.50% 97.50%P 0.29 0.04 0.22 0.37a 3.36 3.75 1.16 11.61b 9.23 13.33 2.25 36.77CycleENSEMBLE ESTIMATES: POOLING OVER SIMILAR UNITS 511 28.9 3.4 22.5 35.62 18.2 0.9 16.4 19.83 12.2 0.8 10.6 13.64 8.6 0.8 7.1 10.15 6.3 0.7 5.0 7.76 4.7 0.6 3.7 5.97 3.6 0.5 2.8 4.68 2.8 0.4 2.1 3.69 2.2 0.3 1.7 2.910 1.8 0.2 1.4 2.311 1.4 0.2 1.1 1.912 1.2 0.2 0.9 1.5Over 12 8.2 2.2 4.2 12.9In practice, couple characteristics (e.g. the woman's age) might be expected to affectthe chance of conception, and one might then link the p ik 's in Equation (2.22) topredictors via a logit regression. Remaining variability between couples might then bemodelled via a Normal error in the logit link. In fact, this approach might be adoptedfor all 586 couples using the female smoking status S ik (ˆ1 for smokers, 0 otherwise) asa predictor, so thatwith u ik Normal.logit(p ik ) ˆ g 0 g 1 S ik u ik2.3.2 Smoothing methods for continuous dataFor metric data, assumed initially at least to be approximately Normal, a typicalproblem involves two way data with i ˆ 1, : : , n j replicated observations y ij withingroups j ˆ 1, : : , K. Sometimes the observed data may be provided only as groupaverages y j aggregating over individual observations, though with details on the variabilitywithin groups (or on 95% intervals for the mean). Assuming the observed meansare derived from similar observation settings or similar types of unit, they may beregarded as draws from an underlying common density for the unknown true meansm j . This assumption leads to a hierarchical model, with the first stage specifying thedensity of the observations, and the second (and maybe higher) stages specifyinghyperparameters which underlie the observations. Typically, the underlying cell meansm j are taken to differ by group but the variance s 2 (and so also the precision t ˆ s 2 ) isassumed constant over groups. In analysis of variance situations, the goal may be toassess additionally whether the underlying group means are equal. For data in a onegroup y i , i ˆ 1, : : , n, the higher stage density involves a single mean and variance.