Trajectories of Antibiotic Use in Early Life as a Marker of Susceptibility to Asthma

1. Danielle Belgrave - PhD Student - Work in Progress Trajectories of Antibiotic Use in Early Life as a Marker of Susceptibility to Asthma A Comparison of the Bayesian and Frequentist Approach

2. Background • Antibiotic exposure associated with development of asthma • By interfering with bacterial gut flora • By modifying the course of bacterial infections. • Further research required to determine whether observed associations are causal, the result of confounding by indication or reverse causation • Impact of antibiotic use on asthma as causal analysis

3. Hypothesis • Antibiotic use rather than being causally related to asthma may be a prognostic indicator of susceptibility to infections • Antibiotic use picks up a signal of something that occurs very early in life, and which is completed by 24 months of age or is a stable feature • Children can be characterized in terms of susceptibility based on patterns of antibiotic use • This phenotype is predictive of contemporaneous and future asthma and wheeze symptoms

4. Objectives • To establish homogeneous susceptibility classes based on differences in monthly patterns of Antibiotic use in the first 2 years of life, while considering the impact of associated factors using longitudinal latent class analysis • To validate latent phenotypic classes using external covariates • Compare Frequentist and Bayesian Machine Learning approaches to this analysis

5. Data Description • Manchester Asthma and Allergy Study • Population-based birth cohort study • 1185 subjects were recruited prenatally and followed prospectively • Antibiotic and Paracetamol prescriptions and symptoms of asthma/wheezing from the primary care medical records of 916 children • Oral Steroid • Hospital Admission • Exacerbation • Asthma/ Wheeze events

6. Statistical methods Frequentist Approach

7. Latent Class Analysis (1) • Children’s susceptibility characterized using latent class model • Assume that each child belongs to one of a set of N latent classes, with the number of classes and their size not known a priori • Other than random temporal fluctuation, each child’s pattern of antibiotic use is to be explained by their belonging to a particular class of susceptibility • Children in same class are similar with respect to the observed variables in the sense that their observed scores are assumed to come from the same probability distributions

8. Latent Class Analysis (2) • Latent Classes are homogeneous • Longitudinal Logistic Regression Model • Predicted antibiotic use at each month was given by a logistic model with predictors : • Age • Child attended nursery before receiving their first antibiotic • Number of older siblings • The susceptibility latent class

9. Latent Class Analysis (3) • Trajectory model: allows us to hypothesize that there may be subgroups of children who have changing levels of susceptibility over time • Random-intercept model: assume relative susceptibility remained constant over time Logit{Pr(yij= 1| x, ci = k)} = β0+ β1[t] + β2[tstart of day-care] + +β3[has older siblings] + ξk+ ξk[t] yij= Event of antibiotic use for child i at time j Pr(ci= k) is multinomial over k classes and independent across children

10. Latent Class Analysis (4) • Bayesian Information Criteria (BIC) goodness-of-fit statistics for ascertaining the “best” model with respect to number of classes and time complexity • BIC: a measure that combines the log-likelihood value with the number of parameters, penalizing models with additional parameters. • Validation Model: Cox regression model to asthma or wheeze occurrence using the latent class phenotypic definition of susceptibility as a predictor

11. Cox Regression Model • Question: Does susceptibility represent a higher risk of asthma severity? • A framework for investigating event occurrence • Examines whether variation in the risk of event occurrence varies systematically with predictors • Features of the data:- • We know the precise instant when events occur • There exist an infinite number of these events because any division of continuous time can always be made finer • The probability of observing any particular event time is infinitesimally small and approaches 0 as time’s divisions get finer

12. Describing Event Occurrence • Survival Function: S(tij) = Pr(‘surviving’ longer than time t) = Pr(Ti > tj) T is a continuous random variable Tiis individual i’s event time tjclocks the infinite number of instants when events can occur • The survival probability for individual i at time tj is the probability that his/her event time will exceed tj

13. Describing Event Occurrence • Hazard Function: Assesses the conditional risk – at a particular moment – that an individual who has not yet done so will experience the event • Cumulative Hazard Function: Assesses at each point in time, the total amount of accumulated risk that individual i has faced from the beginning of time until the present

14. Cox Regression Model • Proportional hazards assumption: the hazard for any individual is a fixed proportion of the hazard for any other individual log h(tij) = logh0(tij) + βClassi h(tij) = h0(tj)eβClassi • Assumptions • For each value of the predictors there is a postulated log hazard function • Each of these log hazard functions has an identical shape • The distance between each of these log hazard functions is identical at every instant

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24. Results

25. Comparison of Models

26. Characteristics of Classes Identified

27. Characteristics of Classes Identified

28. Latent Classes and Asthma/Wheeze in First 3 Years of Life (Time to 1st Event) • Risk of Asthma/Wheeze increases • with increasing susceptibility class • Hazard of asthma/wheeze for • children with ‘high susceptibility’ 3.6 times • than for ‘resilient’ class (p<0.001) • Hazard of asthma/wheeze for children • with ‘high susceptibility’ 1.7 times than for • ‘normal immune response’ class (p<0.001)

29. Latent Classes and Asthma/Wheeze After Age 3 (Time to 1st Event) • Risk of Asthma/Wheeze increases • with increasing susceptibility class • Hazard of asthma/wheeze for children • with ‘high susceptibility’ 2.7 times than for • ‘normal’ class (p<0.005) • Hazard of asthma/wheeze for children • with ‘high susceptibility’ 1.7 times than • for ‘normal immune response’ class • (p<0.005)

30. Latent Classes and Asthma/Wheeze Exacerbation in First 3 Years of Life • Risk of exacerbation increases with • increasing susceptibility class • Hazard of exacerbation for children • with ‘susceptibility to infection’ 5 times • than for ‘resilient’ class (p<0.001) • Hazard of exacerbation for children with ‘susceptibility to infection’ 1.93 times than • for ‘medium susceptibility’ class (p<0.01)

31. Latent Classes and Asthma/Wheeze Exacerbation After Age 3 • Risk of exacerbation not significantly • different for low susceptibility groups • Hazard of exacerbation for children • with ‘susceptibility’ 2.7 times than for • ‘resilient’ class (p<0.005) • Hazard of exacerbation for children with ‘high susceptibility’ 1.9 times than for • ‘normal’ class (p=0.02)

32. Validation of Latent Classes

33. Characteristics of Classes Identified

34. Confounding By Indication? Pearson χ²= 4.8772 Pr = 0.560

35. Statistical Analysis Alternative Approach: Bayesian Machine Learning using Infer.NET

36. Classification-Bayesian Machine Learning • Quantify uncertainty in model parameters by defining a prior belief of that parameters: uninformative conjugate priors • Assume that each child belongs to one of a set of N latent classes, with the number of classes and their size not known a priori • Children in same class are similar with respect to the observed variables in the sense that their observed scores are assumed to come from the same probability distributions

37. Mixture of Continuous-Time Logistic Regressions Logit{Pr(yij= 1| x, ci = k)} = βk+ β1kt +β2k[tstart of day-care] +β3k[has older siblings] Dir. p N Disc. c M C x V. Gauss. Dot Prod. b Ranges: N: Child C: Susceptibility M: Month Gauss. Variables: c: Latent class of child x: Feature vector (observed) (incl. bias, month in time window) b: Parameter vector y: Antibiotic use (observed) Logist. y

38. Hierarchical Mixture of Continuous-Time Logistic Regressions Dir. V. Gauss. p m N Disc. C c V. Gauss. M b x Dot Prod. Ranges: N: Child C: Susceptibility M: Month Gauss. Variables: c: Latent class of child x: Feature vector (observed) (incl. bias, month in time window) b: Parameter vector m:Shared mean of parameter vector y: Antibiotic use (observed) Logist. y

39. Results

40. Mixture of Discrete-Time Logistic Regressions Dir. Logit{Pr(yij= 1| x, ci = k)} = βkj+ β1kjt +β2kj[tstart of day-care] +β3kj[has older siblings] p N Disc. c M x C Dot Prod. V. Gauss. Ranges: N: Child C: Susceptibility M: Month b Gauss. Variables: c: Latent lass of child x: Feature vector (observed) (incl. bias) b: Parameter vector y: Antibiotic use (observed) Logist. y

41. Hierarchical Mixture of Discrete-Time Logistic Regressions Dir. p V. Gauss. N m Disc. c M x C Dot Prod. V. Gauss. Ranges: N: Child C: Susceptibility M: Month b Gauss. Variables: c: Latent lass of child x: Feature vector (observed) (incl. bias) b: Parameter vector m: Shared mean of parameter vector y: Antibiotic use (observed) Logist. y

42. Results • Resilient Class: Probability of receiving antibiotics increases over 1st 9 months and then decreases till age 24 • Susceptible Class: Probability of receiving antibiotics increases over 1st8months and then decreases till age 24

43. Logit{Pr(yij= 1| x, ci = k)} = βkw+ β1kw +β2kw[tstart of day-care] +β3kw[has older siblings] Mixture of Piecewise-Linear Logistic Regressions Dir. p N Disc. W c M x C Dot Prod. V. Gauss. b Ranges: N: Child C: Susceptibility W: Time Window M: Month in Time Window Gauss. Variables: c: Latent class of child x: Feature vector (observed) (incl. bias, month in time window) b: Parameter vector y: Antibiotic use (observed) Logist. y

44. Hierarchical Mixture of Piecewise-Linear Logistic Regressions Dir. V. Gauss. p m N Disc. C c W V. Gauss. x M b Dot Prod. Ranges: N: Child C: Susceptibility W: Time Window M: Month in Time Window Gauss. Variables: c: Latent class of child x: Feature vector (observed) (incl. bias, month in time window) b: Parameter vector m: Shared mean of parameter vector y: Antibiotic use (observed) Logist. y

45. Results

46. Characteristics of Classes (BML)

47. Characteristics of Classes (BML)

48. Latent Classes and Asthma/Wheeze in First 3 Years of Life (Time to 1st Event) • Hazard of asthma/wheeze for susceptible class 2.2 times than for ‘normal’ class • (p<0.001) • Hazard of asthma/wheeze for • children with ‘high susceptibility’ 3.6 times • than for ‘resilient’ class (p<0.001) • Hazard of asthma/wheeze for children • with ‘high susceptibility’ 1.7 times than for • ‘normal immune response’ class (p<0.001)

49. Latent Classes and Asthma/Wheeze After Age 3 (Time to 1st Event) • Hazard of asthma/wheeze for susceptible class 1.8 times than for ‘normal’ class • (p<0.001) • Hazard of asthma/wheeze for children • with ‘high susceptibility’ 2.7 times than for • ‘normal’ class (p<0.005) • Hazard of asthma/wheeze for children • with ‘high susceptibility’ 1.7 times than • for ‘normal immune response’ class • (p<0.005)

50. Latent Classes and Asthma/Wheeze Exacerbation in First 3 Years of Life • Hazard of asthma/wheeze for susceptible class 2 times than for ‘normal’ class • (p<0.001) • Hazard of exacerbation for children • with ‘susceptibility to infection’ 5 times • than for ‘resilient’ class (p<0.001) • Hazard of exacerbation for children with ‘susceptibility to infection’ 1.93 times than • for ‘normal susceptibility’ class (p<0.01)