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Epi-on-the-Island Time Series Regression (TSR) 6-10 July 2015 Thursday 2: TSRs for infectious diseases Ben Armstrong, LSHTM, based strongly on: Chisato Imai* et al Env Res In press 2015 * Nagasaki University Standard TSR (sorry β terminology change!) The most common time series regression (TSR) is Poisson model described as; ππ‘ ~ ππππ π ππ(µπ‘ ) πππ ππ‘ = π0 + ππ₯π‘ + ππ π§π,π‘ + π π‘ π where f(t): smoothing function of time t to control seasonality and long term trends. xt : time varying variables of interest (e.g. temperature). π»π , π», πΌπ : regression coefficients. zp,t : other risk factors. Issues in using TSR for infectious disease 1. Strong autocorrelation βTrue contagionβ due to transmissions among individuals. 2. Immune and susceptible population The number of immune / susceptible population changes over time. 3. Unusual lag structures and association patterns E.g. intermediators (such as vectors) on the causal pathways 4. Control for seasonality and long term trends Time splines may impede finding longer lag effects of exposures. 5. Overdispersion Very large . Time series susceptible-infectiousrecovery(TSSIR) model* ππ‘ = πΌ π½π‘ ππ‘β1 ππ‘β1 ππ‘β1 π πΎ ππ‘ β ππ‘ = ππ‘ β ππ‘βπ π π π=0 where N: the total population size, S: the number of susceptible individuals, Ξ²: is pathogen transmissibility at time t, m: the max immune duration Ξ±, Ξ³: mixing parameters, and ΞΊ: the decay immune function Taking logs leaves something like a TSR log ππ‘ β log π½π‘ + πΌ log ππ‘β1 πΎ β ππ‘ π ππ‘βπ π π + log ππ‘ π=1 * Koelle K, Pascual M. Am Nat 2004;163(6):901-13. 1. Strong autocorrelation Usual TSR way to approach β Lagged model residuals Alternative suggested by TSSIR: ππ‘ = ππ‘β1 πΎ πΌ π½π‘ ππ‘β1 ππ‘ ππ‘β1 log ππ‘ β log π½π‘ + πΌ log ππ‘β1 β πΎ ππ‘ π ππ‘βπ π π + log ππ‘ π=1 οΆ i.e. use lagged log(Y) as x-variable 2. Strong autocorrelation A C B Tokyo influenza data residuals D 2. Immune & susceptible population A: The problem, an ad hoc solution, and idea Issue: disease -> immunity -> susceptibles βΌ > disease βΌ EG (Lopman 2009) Norovirus: ~ 1 year immune duration βImmune population factorβ, xipf = ο πππ π‘ π¦πππ ππ‘βπ Simplified form of Koelleβs TSSIR model. π π = 1 (last year) otherwise 0. π ππ‘βπ π π = π=0 ππ‘β1 πππ π‘ π¦πππ 2. Immune & susceptible population B: Options οΆ Rely on time smooth to take account of change in suscetibles οΆ Use some (weighted) sum of past cases as xvariable (eg, cases so far in season) οΆ Consider only timing of onset of peak of epidemic (eg does cold weather precipitate an early flu peak?) [~ survival analysis with time-dependent covariates] 3. Unusual lag structures and association patterns Example: Lyme disease ~ rain 2 years lag (Subak 2003) οΆ Be informed by prior knowledge. β Biological plausibility & preceding studies can be informative for the duration of delayed effects and the nature of association patterns (linear, U-, J-shaped) οΆ Be prepared to use flexible lag/shape functions. β DLNMs !~ Unusual lag and shape (cholera) 4. Control for seasonality and long term trends. οΆ Choose suitable unit of time data or respect restrictions (week or month rather than daily) β Time unit should make sense with biological plausibility (e.g. incubation period) of hypothesized association lag β Attentions to details : 52 weeks or 53 weeks p.a. οΆ Avoid depletion of the precision to estimate the longer lag effects of exposures by a time spline function o Separate functions for seasonal and long term patterns for long lag effects. (Use fourier terms and simple functions for long term trend)? 5. Overdispersion οΆ Consider different distribution models allowing for overdispersion ο± Quasi-Poisson and negative binomial ο± Zero-inflated Poisson and negative binomial For excessive number of zero counts ο± Gaussian linear model for log(Yt) If outcome counts are consistently large enough Choosing overdispersion model Ver Hoefβs (2007) method Approaches other than TSR β’ TSSIRs β’ ARIMA etc (including fractional ARIMA) β’ Wavelets Summary of models as fitted to Tokyo influenza data models TSR with quasi-Poisson standard TSR + autocorrelations (AC)d + AC + AC + AC + AC + immune term up to onsets up to peaks AC term immune term distribution temperature effect estimate % (95% CI)b dispersion parameter count - - QP -5.8 (-10.9, -0.5) 349.4 count πππ πππ’πππ‘β1 - QP -3.7 (-6.9, -0.5) 118.4 count count count count count count ππ‘β1 logβ‘ (ππ‘β1 + 1) logβ‘ (ππ‘β1 + 0.5) logβ‘ (ππ‘β1 + 1) logβ‘ (ππ‘β1 + 1) logβ‘ (ππ‘β1 + 1) Ξ£(cases*) - QP QP QP QP QP QP -4.8 (-8.6, -0.8) -5.5 (-7.5, -3.4) -5.5 (-7.5, -3.4) -6.7 (-8.7, -4.6) -5.7 (-12.0, 1.0) -4.5 (-8.9 to 0.1) 188.7 48.4 49.0 47.1 9.0 67.7 logβ‘ (ππ‘β1 + 1) logβ‘ (ππ‘β1 + 1) Ξ£(cases*) Ξ£(cases*) NB Gaussian -6.6 (-10.0, -3.0) -5.2 (-8.6, -1.6) na na onset (1,0) logβ‘ (ππ‘β1 + 1) - Bernoulli -19.5 (-49.9, 29.4) na onset (1,0) logβ‘ (ππ‘β1 + 1) - CB -16.0 (-46.7, 32.4) na peak (1,0) logβ‘ (ππ‘β1 + 1) - Bernoulli -50.7 (-75.3, -1.7) na peak (1,0) logβ‘ (ππ‘β1 + 1) - CB -51.6 (-78.3, 7.7) na outcome (Y) TSR with different distribution models negative binomial count linear regression log(count+1) non-TSR modelse Onset: logistic regression Onset: cox regression Peak: logistic regression Peak: cox regression predictors Conclusions and discussions οTSR models can be used for studies of infectious disease and weather, but may require modifying. οTSR is not dominant for ID TS Epi; TSSIRs and ARIMA as frequent. οFuture: TSSIR β TSR hybrid?