Bayesian Model Averaging to Account for Model Uncertainty in Estimates of a Vaccine’s Effectiveness

Figures & data

Table 1 Study Definitions for Study on the Effectiveness of Lyme Vaccine

Variable	Definition
Definite Lyme disease	Physician diagnosed erythema migrans measuring >5 cm, or a positive laboratory test result for antibodies to Borrelia burgdorferi and objective evidence of either early disseminated or of late Lyme disease (eg, facial nerve palsy, meningitis, carditis, or arthritis).
Case	Residents of Connecticut, 15–70 years of age, diagnosed with first episode of “definite Lyme disease” between 2000 and 2013.
Control	Individuals without signs or symptoms of Lyme disease at focal time, matched to a case by age (±5 years) and geographic area (same telephone exchange).
Focal time	The common date between cases and controls. For controls, the focal time will be the date of telephone interview. For cases, the focal time will be the date of their diagnosis of Lyme disease.
Exclusion criteria	Significant immunosuppression (because of either an illness or a medication) or prior treatment for Lyme disease.
Fully immunized	Received >3 doses of Lyme vaccine at least 1 month prior to focal time.

Table 2 Characteristics of Participants Enrolled in Lyme Vaccine Study by Case Status, N=912

Variables	Cases, N (%)		Matched Controls, N (%)		Standardized Mean Difference*
Total	358	39%	554	61%	–
Demographics
Age (years), median (range)	46.0	(36.0–55.0)	47.0	(37.0–55.0)	0.025
Non-Hispanic White	348	(97.2%)	540	(97.5%)	0.017
Female	188	(52.5%)	365	(65.9%)	0.274
Some college education	249	(69.6%)	424	(76.5%)	0.158
Currently unemployed	41	(11.5%)	71	(12.8%)	0.042
Risk Factors
Has an occupational exposure to ticks	52	(14.5%)	50	(9.0%)	0.171
Lives near wooded area	340	(95.0%)	512	(92.4%)	0.105
Engages in frequent outdoor activities	307	(85.8%)	448	(80.9%)	0.131
Has pets at home	222	(62.0%)	370	(66.8%)	0.100
Protective Measures
Uses protective clothing while outdoors	157	(43.9%)	325	(58.7%)	0.299
Use of tick repellent on skin or clothing	98	(27.4%)	189	(34.1%)	0.146
Performs checks for ticks post-exposure	270	(75.4%)	419	(75.6%)	0.005
Sprays property with tick acaricides	13	(3.6%)	38	(6.9%)	0.145

Note: *The standardized mean difference (SMD) is the difference in means between cases and controls in units of the pooled standard deviation.

Figure 1 Distribution of the vaccine’s effectiveness across all possible models.

Figure 2 Vaccine effectiveness for model-averaged and the top subset of candidate models. *Excluded from BMA if PIP <0.01.

Abbreviations: AOR, adjusted odds ratio; VE, vaccine effectiveness; BMA, Bayesian model averaging; PIP, posterior inclusion probability.

Figure 3 Composition and posterior inclusion probabilities of confounders among the top subset of models.

Notes: In this heat map, each row corresponds to a potential confounder, and each column corresponds to a particular model. The best model is on the left, and the worst model is on the right. The shade of each square represents the posterior probability of each potential confounder. The darker shades represent higher posterior probabilities, and unshaded squares represent variables that were not included in a given model. Since the immunization status was included in all models a priori, it is checkered, and the PIP for this variable is set to 1.

Figure 4 Comparison of single-model selection and Bayesian model-averaging approaches.

Abbreviations: Vx, vaccine status; Pe, has pets at home; Cl, wears protective clothing while outdoors; Sx, male or female sex; Ed, at least some college education; Sp, sprays property with acaricide; Jb, has job exposure to ticks; Ou, engages in frequent outdoor activities; Rp, use of tick repellent on skin or clothing; Tk, use of tick repellent on skin or clothing; SE, standard error; aOR, adjusted odds ratio; BMA, Bayesian model averaging; AIC, Akaike information criterion.