Summary Introduction/Synopsis Application 1: Great tit nesting behaviour (crossed r in addition to om effects) Application 1: Great tit nesting behaviour (crossed r in addition to om effects) Data background

Summary Introduction/Synopsis Application 1: Great tit nesting behaviour (crossed r in addition to om effects) Application 1: Great tit nesting behaviour (crossed r in addition to om effects) Data background www.phwiki.com

Summary Introduction/Synopsis Application 1: Great tit nesting behaviour (crossed r in addition to om effects) Application 1: Great tit nesting behaviour (crossed r in addition to om effects) Data background

Kennedy, Ginny, Host has reference to this Academic Journal, PHwiki organized this Journal Simple methods to improve MCMC efficiency in r in addition to om effect models William Browne, Mousa Golalizadeh, Martin Green in addition to Fiona Steele Universities of Bristol in addition to Nottingham Thanks to ESRC as long as supporting this work Summary Introduction. Application 1 – Clutch size in great tits. Method 1: Hierarchical centering. Method 2: Parameter expansion. Application 2 – Mastitis incidence in dairy cattle. Method 3: Orthogonal predictors. Application 3 – Contraceptive discontinuation in Indonesia. Conclusions. Introduction/Synopsis MCMC methods allow easy fitting of complex r in addition to om effects models The simplest (default) MCMC algorithms can produce poorly mixing chains. By reparameterising the model one can greatly improve mixing. These reparameterisations are easy to implement in WinBUGS (or MLwiN) The choice of reparameterisation depends in part on the model/dataset.

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Application 1: Great tit nesting behaviour (crossed r in addition to om effects) Original work was collaborative research with Richard Petti as long as (Institute of Zoology, London), in addition to Robin McCleery in addition to Ben Sheldon (University of Ox as long as d). Application 1: Great tit nesting behaviour (crossed r in addition to om effects) A longitudinal study of great tits nesting in Wytham Woods, Ox as long as dshire. 6 responses : 3 continuous & 3 binary. Clutch size, lay date in addition to mean nestling mass. Nest success, male in addition to female survival. Data: 4165 nesting attempts over a period of 34 years. There are 4 higher-level classifications of the data: female parent, male parent, nestbox in addition to year. We only consider Clutch size here Data background Note there is very little in as long as mation on each individual male in addition to female bird but we can get some estimates of variability via a r in addition to om effects model. The data structure can be summarised as follows:

MCMC efficiency as long as clutch size response The MCMC algorithm used in the univariate analysis of clutch size was a simple 10-step Gibbs sampling algorithm. To compare methods as long as each parameter we can look at the effective sample sizes (ESS) which give an estimate of how many ‘independent samples we have’ as long as each parameter as opposed to 50,000 dependent samples. ESS = of iterations/, Clutch Size Here we see that the average clutch size is just below 9 eggs with large variability between female birds in addition to some variability between years. Male birds in addition to nest boxes have less impact. Effective Sample sizes We will now consider methods that will improve the ESS values as long as particular parameters. We will firstly consider the fixed effect parameter.

Trace in addition to autocorrelation plots as long as fixed effect using st in addition to ard Gibbs sampling algorithm Hierarchical Centering This method was devised by Gelf in addition to et al. (1995) as long as use in nested models. Basically (where feasible) parameters are moved up the hierarchy in a model re as long as mulation. For example: is equivalent to The motivation here is we remove the strong negative correlation between the fixed in addition to r in addition to om effects by re as long as mulation. Hierarchical Centering In our cross-classified model we have 4 possible hierarchies up which we can move parameters. We have chosen to move the fixed effect up the year hierarchy as it’s variance had biggest ESS although this choice is rather arbitrary. The ESS as long as the fixed effect increases 50-fold from 602 to 35,063 while as long as the year level variance we have a smaller improvement from 29,604 to 34,626. Note this as long as mulation also runs faster 1864s vs 2601s (in WinBUGS).

Trace in addition to autocorrelation plots as long as fixed effect using hierarchical centering as long as mulation Parameter Expansion We next consider the variances in addition to in particular the between-male bird variance. When the posterior distribution of a variance parameter has some mass near zero this can hamper the mixing of the chains as long as both the variance parameter in addition to the associated r in addition to om effects. The pictures over the page illustrate such poor mixing. One solution is parameter expansion (Liu et al. 1998). In this method we add an extra parameter to the model to improve mixing. Trace plots as long as between males variance in addition to a sample male effect using st in addition to ard Gibbs sampling algorithm

Autocorrelation plot as long as male variance in addition to a sample male effect using st in addition to ard Gibbs sampling algorithm Parameter Expansion In our example we use parameter expansion as long as all 4 hierarchies. Note the parameters have an impact on both the r in addition to om effects in addition to their variance. The original parameters can be found by: Note the models are not identical as we now have different prior distributions as long as the variances. Parameter Expansion For the between males variance we have a 20-fold increase in ESS from 33 to 600. The parameter exp in addition to ed model has different prior distributions as long as the variances although these priors are still ‘diffuse’. It should be noted that the point in addition to interval estimate of the level 2 variance has changed from 0.034 (0.002,0.126) to 0.064 (0.000,0.172). Parameter expansion is computationally slower 3662s vs 2601s as long as our example.

Trace plots as long as between males variance in addition to a sample male effect using parameter expansion. Combining the two methods Hierarchical centering in addition to parameter expansion can easily be combined in the same model. Here we per as long as m centering on the year classification in addition to parameter expansion on the other 3 hierarchies. Effective Sample sizes As we can see below the effective sample sizes as long as all parameters are improved as long as this as long as mulation while running time remains approximately the same.

Applications 2 & 3: Multilevel discrete-time survival analysis Used to model the durations until the occurrence of events e.g. length of time to death. Different from st in addition to ard regression due to right-censoring in addition to time varying covariates. In many applications events may be repeatable in addition to the outcome is duration of continuous exposure to the risk of an event. In our applications cows may suffer mastitis more than once in addition to women can initiate in addition to discontinue use of contraceptives several times. We begin with two levels of data episodes nested within individuals. With a discrete time approach we exp in addition to the data to create a lower level – time interval, which represents a fixed period of time in addition to is nested within episode. Data Expansion Suppose we have time intervals (e.g. days, weeks) indexed by t = 1, ,K where K is the maximum duration of an episode. Let tij denote the number of intervals as long as which individual i is observed in episode j. Be as long as e modelling we now need to exp in addition to the data as long as each episode ij to obtain tij records. We have ytij=0 as long as t=1, ,tij-1 in addition to the response as long as t= tij is 1 if an episode ends in an event in addition to 0 as long as censored episodes. Example: Individual observed as long as 7 months in addition to has event after 4 months: Response vector (0,0,0,1,0,0,0), Time intervals (1,2,3,4,1,2,3) Modelling Having restructured the data we can analyse the event occurrence indicator ytij using st in addition to ard methods as long as clustered binary response data. Here we model the probability of an event occurrence tij as a function of duration (zt) in addition to covariates xtij with uj being an individual specific r in addition to om effect representing unobserved time-invariant individual characteristics.

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MCMC algorithm as long as r in addition to om effect logistic regression models Consider the following simple model: A st in addition to ard MCMC algorithm (as used in MLwiN (Rasbash et al. (2000), Browne (2003)) is Step 1: Update using r in addition to om walk Metropolis sampling. Step 2: Update using r in addition to om walk Metropolis sampling. Step 3: Update from its inverse Gamma full conditional using Gibbs Sampling Hierarchical centered as long as mulation We can reparameterise by replacing the residuals uj with r in addition to om effects uj= 0+uj. Here the uj are (hierarchically) centred around 0. This as long as mulation allows Gibbs sampling to be used as long as the fixed effect 0 in addition to the r in addition to om effects variance. It is interesting to consider how an MCMC algorithm as long as this centered as long as mulation can be expressed in terms of the original parameterisation. MCMC Algorithm At iteration t+1 Step 1: Update 0 from it’s normal conditional distribution: Adjust to keep uj fixed. The other steps are unchanged by reparameterisation as the r in addition to om walk Metropolis sampling as long as uj is equivalent to the same step as long as uj in addition to the final step is unchanged.

Why should hierarchical centering be useful as long as discrete time survival models As we have seen discrete time survival models are a special case of r in addition to om effects logistic regression models. Expansion results in many higher level predictors that can be (hierarchically) centered around. Hierarchical centering should improve mixing but also speed up estimation. Will be particularly useful when we have many level 1 units per level 2 unit in addition to large level 2 variance, where the speed up will be greatest – see application 2. Not always useful but other potential solutions – see application 3. Application 2: Mastitis incidence in dairy cattle Mastitis – inflammation of mammary gl in addition to of dairy cows, usually caused by a bacterial infection. Infections arising in dry (non-lactating) period often result in clinical mastitis in early lactation. Green et al. (2007) use multilevel survival models to investigate how cow, farm in addition to management factors in the dry period correlate with mastitis incidence. Data – 2 years x 52 dairy farms – 103 farm years – 8,710 cow lactation periods following dry periods exp in addition to s to 256,382 records in total! Mastitis model The Model has many predictors. The duration terms zt consists of an intercept plus polynomials in log time to order 3. The 4 levels are farm, farm year, cow dry period in addition to weekly obs. although no r in addition to om effects occur at the dry period level. The model can easily be converted to a hierarchical centered as long as mulation by centering around the 10 farm-year level fixed effects

References Browne, W.J. (2003). MCMC Estimation in MLwiN. London: Institute of Education, University of London Browne, W.J. (2004). An illustration of the use of reparameterisation methods as long as improving MCMC efficiency in crossed r in addition to om effect models Multilevel Modelling Newsletter 16 (1): 13-25 Gamerman D. (1997) Sampling from the posterior distribution in generalized linear mixed models. Statistics in addition to Computing. 7, 57-68. Gelf in addition to , A.E., Sahu, S.K. in addition to Carlin, B.P. (1995) Efficient parameterisations as long as normal linear mixed models. Biometrika 83, 479-488 Gelman, A., Huang, Z., van Dyk, D., in addition to Boscardin, W.J. (2007). Using redundant parameterizations to fit hierarchical models. Journal of Computational in addition to Graphical Statistics (to appear). Green, M.J., Bradley, A.J., Medley, G.F. in addition to Browne, W.J. (2007) Cow, Farm in addition to Management Factors during the Dry Period that determine the rate of Clinical Mastitis after calving. Journal of Dairy Science 90: 3764-3776. Hills, S.E. in addition to Smith, A.F.M. (1992) Parameterization Issues in Bayesian Inference. In Bayesian Statistics 4, (J M Bernardo, J O Berger, A P Dawid, in addition to A F M Smith, eds), Ox as long as d University Press, UK, pp. 227-246. References cont. Liu, C., Rubin, D.B., in addition to Wu, Y.N. (1998) Parameter expansion to accelerate EM: The PX-EM algorithm. Biometrika 85 (4): 755-770. Liu, J.S., Wu, Y.N. (1999) Parameter Expansion as long as Data Augmentation. Journal Of The American Statistical Association 94: 1264-1274 Papaspiliopoulos, O, Roberts, G.O. in addition to Skold, M. (2003) Non-centred Parameterisations as long as Hierarchical Models in addition to Data Augmentation. In Bayesian Statistics 7, (J M Bernardo, M J Bayarri, J O Berger, A P Dawid, D Heckerman, A F M Smith in addition to M West, eds), Ox as long as d University Press, UK, pp. 307-32 Papaspiliopoulos, O, Roberts, G.O. in addition to Skold, M. (2007) A General Framework as long as the Parametrization of Hierarchical Models. Statistical Science 22, 59-73. Rasbash, J., Browne, W.J., Healy, M, Cameron, B in addition to Charlton, C. (2000). The MLwiN software package version 1.10. London: Institute of Education, University of London. Steele, F., Goldstein, H. in addition to Browne, W.J. (2004). A general multilevel multistate competing risks model as long as event history data, with an application to a study of contraceptive use dynamics. Statistical Modelling 4: 145-159 Van Dyk, D.A., in addition to Meng, X-L. (2001) The Art of Data Augmentation. Journal of Computational in addition to Graphical Statistics. 10, 1-50.

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