1. Epi-on-the-Island Time Series Regression (TSR) 6-10 July 2015 Wed 1: DLNMs Ben Armstrong London School of Hygiene and Tropical Medicine

2. The trailer DLNMs ~200 papers 2011- At least one in vet epi: Morignat 2015 Env Res Easy R package!

3. The algebra terminology:- Recall general model form [In this example, temperature is the variable of interest.]

4. Distributed lag non-linear models (DLNMs) The shape story (Nonlinear Ms, eg splines) The lag story (DLMs) The lag-shape story (DLNMs)

5. The shape story: non-linear options: Cold effect Heat effect Thresholds

6. Other non-linear options

7. Net RR increment (%) over 28 days per degree above 20 (95% CI): 5.0(3.7,6.3) 4.9(3.6,6.2) 4.9(3.7,6.2) Recall: distributed lag models (DLM) The lag story :

8. The lag-shape story

9. The lag-shape problem: Seek combined virtues of: (a)Smooth temp-mort curves (b)Smooth effects of lag. Solution: assume DLMs for each coefficient of a basis variable ….

10. e.g.: RR by temperature (NCS) and lag(NCS)

11. Heat effects: Different slope for each lag DLNM: simple(r) case Use a linear threshold model Fit a distributed lag model (DLM) to the heat slope

12. Simple(r) to general DLNM: Shape flexibility + Lag flexibility = Total flexibilty Armstrong 2006 Basis variables

13. Extend this to any set of basis variables (eg splines) Armstrong 2006 Gasparrini 2010

15. DLNM R code example ( London data ) # MAKE DLNM BASIS VARIABLE basis.temp <- crossbasis(data$max_temp, argvar=list(fun="ns",df=3), 3 df spline as before arglag=list(fun=“ns”,knots=c(1,3,7)), constrained lag: 3 knots lag=20) max lag 20 (3-weeks) # FIT MODEL mdlnm2 <- glm(deaths~colag1+spl+basis.temp,… ) # GET PREDICTED RRS FOR A RANGE OF TEMPERATURES pred.temp <- crosspred(basis.temp, mdlnm2) # PLOT FITTED ASSOCIATION plot(pred.temp)

16. RR by temperature (NCS) and lag(NCS)

17. DLNM R code example ( London data ) … # PLOT FITTED ASSOCIATION IN VARIOUS WAYS plot(pred.temp,ptype=“3D”) plot(pred.temp,ptype="overall") plot(pred.temp,ptype="slices", lag=c(0,3,7,15)) plot(pred.temp,ptype="slices", var=c(0,35))

18. Ways of looking at DLNMs RR by lag for given temperature RR by temperature for given lag

19. - Adding graph of total (28 day) effect

20. dlnm shape options for x- variable and lag slopes "ns" and "bs" : natural cubic B-splines or B-splines of various degree. "poly" : polynomials functions. "strata" : indicator variables defining strata. (step function) "thr" : high, low or double linear threshold functions. "integer" : indicator variables for each integer value. "lin" : linear functions.

21. The y scale and “centering” in dlnms Y scale (usually RRs) is “centered”, eg to take the value RR=1 at some x value You can decide the centering value (“..,cen=25..“) or dlnm will use mean x. This affects CIs but not shape of x-y curves,

22. Overall association from DLNM: linear-threshold (Morignat 2015)

23. Two-stage multi-TS DLNMs (eg splines) Stage 1: Estimate DLNM for site “Reduce” to overall cumulative x-y curve Save coefficients of this spline curve and variance matrix Stage 2: Meta-analyse the spline coefficients using multivariate meta- analysis (R mvmeta) Plot or otherwise summarise pooled mean curve and report/explore heterogeneity Explore in practical exercise

24. Pooled mean overall DLNM curves for temperature vs mortality (Guo 2014)

25. Summaries of country pooled mean overall DLNM curves (Guo 2014)

26. Site-specific summaries (Guo 2014)

27. Another kind of summary (Gasparrini 2015)

28. DLNMs Summary 1.Fits TSR associations allowing flexible lag structures and/or x-y shapes 2.Extension of distributed lag model (DLM) 3.Smooth well-documented R package dlnm, including plotting options 4.Can be meta-analysed using mvmeta 5.Extends to designs beyond TSR (eg cohorts)