Joint Modelling Approaches in diabetes research. Francisco Gude Clinical Epidemiology Unit, Hospital Clínico Universitario de Santiago

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1 Jont Modellng Approaches n dabetes research Clncal Epdemology Unt, Hosptal Clínco Unverstaro de Santago

2 Outlne 1 Dabetes 2 Our research 3 Some applcatons

3 Dabetes melltus Is a serous lfe-long health condton that occurs when the amount of glucose (sugar) n the blood s too hgh because the body can t use t properly a. a

4 Type II Dabetes Internatonal Dabetes Federaton. IDF Dabetes Atlas. 6th ed. Brussels, Belgum, 2013

5 Dabetes complcatons

6 Dagnoss Fastng plasma glucose (fpg) 126mg/dL Glycated hemoglobn (a1c) 6.5 %

7 Control over tme Glycated hemoglobn (a1c) Fructosamne (fru)

8 Glycated protens Glycaton occurs n vvo when blood glucose reacts wth body protens. The hgher the level of blood sugar, the more sugar attaches to protens and the hgher the levels of protens whch are glycated.

9 How dabetes s controlled?

10 Why Blood Sugar Testng Matters?

11 a1c & Blood glucose relatonshp Many people ask what t means f I have a hgh a1c & normal blood glucose. On the other hand, many others ask what t means If I have healthy a1c and elevated fastng glucose

12 A-Estrada Glycaton and Inflammaton Study (AEGIS) AEGIS s a cross-sectonal, populaton-based study carred out n the muncpalty of A Estrada (NW, Span), whose data collecton and recrutng phase was completed n March It was amed to nvestgate glycaton and nflammaton processes and ther assocaton wth lfestyles and common dseases, manly dabetes. The total number of ndvduals ncluded n ths study was A number of 622 AEGIS partcpants underwent a 6-day perod of contnuous glucose montorng (CGM) procedures. Mean age of 52 years (range 18 to 91), 55 % were females and 187 (12 %) had been prevously dagnosed wth dabetes a1c and fru levels were hghly correlated (r = 0.72), and both protens were also correlated wth fpg levels (r = 0.72 and r = 0.56).

13 Prmary Care Bostatstcs Internal Medcne Clncal Epdemology Laboratory Allergy Cardology Endocrno

14 Case 1 AIM: To nvestgate whch factors may nfluence on glycated proten levels, ther varablty and dscordances between them. Bvarate response: Contnuous reponses (Y 1, Y 2 ) = (a1c, fru) Covarates: Gender, Age (n years), Glucose (glu n formulae) Body mass ndex (bm), Mean corpuscular volume (mcv), Albumn (alb).

15 Data analyss Bomedcal Problem: We need a more flexble framework, where: Two or more responses (a1c, fru) can be smultaneously modeled. The response dstrbuton s not restrcted. Dfferent dependence structures between margns. Statstcal Tool: Flexble copula regresson splne models. Bayesan: Structured addtve condtonal copula regresson models (Klen and Kneb, 2016). Frequentst: Bvarate copula addtve models for locaton, scale and shape (Marra and Radce, 2017).

16 Model Selecton Three key ponts Dstrbutons for both responses Relevant covarates n the addtve predctors A sutable copula functon

17 Fttng normal dstrbuton for a1c Aganst Ftted Values Aganst ndex Quantle Resduals Quantle Resduals Ftted Values ndex Densty Estmate Normal Q Q Plot Densty Sample Quantles Quantle. Resduals Theoretcal Quantles

18 Fttng normal dstrbuton for fru Aganst Ftted Values Aganst ndex Quantle Resduals Quantle Resduals Ftted Values ndex Densty Estmate Normal Q Q Plot Densty Sample Quantles Quantle. Resduals Theoretcal Quantles

19 Fttng lognormal dstrbuton for a1c Aganst Ftted Values Aganst ndex Quantle Resduals Quantle Resduals Ftted Values ndex Densty Estmate Normal Q Q Plot Densty Sample Quantles Quantle. Resduals Theoretcal Quantles

20 Fttng lognormal dstrbuton for fru Aganst Ftted Values Aganst ndex Quantle Resduals Quantle Resduals Ftted Values ndex Densty Estmate Normal Q Q Plot Densty Sample Quantles Quantle. Resduals Theoretcal Quantles

21 Model Lognormal dstrbuton for both a1c and fru wth a Gaussan copula to provde the best and most parsmonous ft. Margnal (a1c): η µ1 = β µ1 0 + f µ1 (glu) + f µ1 (age) + gender β µ1 1 + f µ1 (bm) + f µ1 (mcv) ; η σ2 1 = β σ f σ2 1 (glu) + f σ2 1 (age) + gender β σ2 1 1 Margnal (fru): η µ2 = β µ2 0 + f µ2 (glu) + f µ2 (age) + gender β µ2 + f µ2 (bm) + f µ2 (alb), η σ2 2 = β σ f σ2 2 (glu) + f σ2 2 (age) + gender β σ2 2 1, (Assocaton between a1c and fru): η θ = β θ 0 + f θ (glu) + f θ (age) + gender β θ 1 + f θ (mcv),

22 Results: Mean Mean a1c s(glucose,5.04) s(age,1) s(bm,2.19) s(mcv,6.36) Glucose (mg/dl) Age (years) BMI (Kg/ m 2 ) MCV (fl) Mean fru s(glucose,7.97) s(age,2.98) s(bm,5.21) s(albumne,9.53) Glucose (mg/dl) Age (Years) BMI ( Kg/ m 2 ) Albumn (g/dl)

23 Results: Varance Varance a1c s(glucose,6.27) s(age,1) Glucose (mg/dl) Age (years) Varance fru s(glucose,3.53) s(age,3.86) Glucose (mg/dl) Age (years)

24 Results: Assocaton between a1c and fru s(glucose,7.1) s(age,4.43) s(mcv,4.57) Glucose (mg/dl) Age (years) MCV (fl)

25 Another type of copula? Contour lnes of denstes for dfferent glucose levels Normo Glycemc ( ρ = 0.07 ) Predabetes ( ρ = 0.48 ) Dabetes ( ρ = 0.53 ) Fructosamne Fructosamne Fructosamne HbA1c HbA1c HbA1c

26 Case 2 AIM: To nvestgate whch factors nfluence on the dscordances found when dagnosng dabetes by two dfferent tests. A1c Crteron Normal Bellow 6,4% Dabetes 6,5% or hgher Fastng Plasma Glucose (fpg) Crteron Normal Bellow 125 mg/dl Dabetes 126 mg/dl or hgher

27 Model Bvarate Copula regresson splne models for bnary outcomes Bvarate Bnary Response: (Y 1, Y 2 ) = (a1c, fpg). Covarates: Gender, Age and BMI. Equaton 1 (a1c): η y1 = β y1 0 + gender β y1 1 + f y1 Proposed Model (age) + f y1 (bm) Equaton 2 (Fastng Plasma Glucose): η y2 = β y2 0 + gender β y2 1 + f y2 Equaton 3 (Assocaton): (age) + f y2 (bm) η θ = β θ 0 + gender β θ 1 + f θ (age) gender,

28 Results: Assocaton between both dagnostc tests The overall Kendall s ˆτ and ˆθ are postve and sgnfcant: ˆθ (0.793, 0.987) and ˆτ (0.625, 0.913). s(age,1) s(age,1) Age (Women) Age (Men)

29 Results: Dagnostc tests agreement, Kappa ndex ˆk = 2(P(Y 1 = Y 2 z) P(Y 1 Y 2 z)) P(Y 1 = 1 z)p(y 1 = 0 z) + P(Y 2 = 1 z)p(y 2 = 0 z) where, z s a covarate or a set of covarates, based on Lands & Koch (1977). Women Men Kappa ndex Kappa ndex Age, years Age, years

30 Case 3 AIM: Investgatng dscordances when dagnosng dabetes by usng three dfferent tests: Glycated haemoglobn test (a1c) Fastng plasma glucose test (fpg) Oral glucose tolerance test (ogtt) Multvarate copula regresson splne models for bnary outcomes Trvarate Bnary Response: (Y 1, Y 2, Y 3 ) = (a1c, fpg, ogtt). Covarates:???

31 Case 4 Patents may be clasfed accordng to three groups by usng two dfferent tests (a1c, fpg) AIM: Whch factors nfluences on the mssclasfcaton. FPG (mg/dl) A1c (%) Bvarate copula regresson splne models for multnomal outcomes Bvarate multnomal response: (Y 1, Y 2 ) =(a1c, fpg). Covarates:???

32 Case 5 A number of 622 AEGIS partcpants underwent a 6-day perod of contnuous glucose montorng procedures. Our clncal research focuses on the use of all the nformaton avalable from patent s glucose profles over the course of one week n a szeable populaton sample. From a statstcal pont of vew, these glucose profles wll be treated as functonal data. Glucose Profles Breakfast Glucose Indvdual #100 wth Dabetes Indvdual #108 wthout Dabetes Tme (Hours)

33 Case 5, current research As a startng pont, we have developed new methodologes whch enable glucose profles to be ncluded as a lnear functonal effect n dstrbutonal regresson models (Klen and Kneb, 2015). We assumed a semparametrc predctor, for any parameter θ of the response dstrbuton: η θ = β0 θ + v β θ + f1 θ (x 1 ) + + fq θ (x q ) + v r (s)βr θ (s)ds, where: η, transformed varable of the orgnal response. v β, parametrc effects β of the covarates, v. f 1 (x 1 ),..., f q (x q ), nonlnear, smooth effects of contnuous covarates. vr (s)β r (s)ds, represents functonal data wth dscrete observaton ponts, s S.

34 Case 5, future research To consder non lnear functonal effects. Glucose Tme

35 Case 5, future research Jont Modelng for bvarate responses ncludng Functonal Data. To extend the followng bvarate copula regresson models? Frequentst: Copula addtve models for locaton, scale and shape (CGAMLSS, Marra and Radce, 2017) Bayesan: Structured Addtve Copula Regresson Models (Klen and Kneb, 2016)

36 AEGIS II All partcpants ncluded n the AEGIS I, are subsequently observed as a cohort n the AEGIS II Ongong project, Predcton models for rsk of developng type 2 dabetes by usng contnuous glucose montorng. A follow up study Funded by grants from the Spansh Insttute of Health and European Regonal Development Fund All subjects are convened for evaluaton, whch comprses 1 Questonnares 2 Lfestyles 3 Blood samplng 4 Contnuous glucose montorng

37 Jont modelng for longtudnal data and survval events Contnuous Glucose Montorng ( ) Contnuous Glucose Montorng ( ) 1 year 2 years 3 years 4 years AEGIS 1 AEGIS 2 A1c FPG fru Renal functon Survval outcomes: - Dabetes - Cardovascular dsease - Retnopathy - Kdney dsease - Mortalty

38 Jont modelng for longtudnal data and survval events Interest on: Multvarate longtudnal outcomes Multvarate tme-to-event data Dynamc predctons Evaluaton of predctve accuracy AIM: To estmate the probablty of developng dfferent mcrovascular outcomes gven fastng glucose or glycated hemoglobn trajectores.

39 Jont Modelng of Functonal Data and Tme To event Follow-up study AEGIS 5 Years AEGIS II Contnuous Glucose Montorng Contnuous Glucose Montorng Functonal Data - Tme To Event

40

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