Linear Models for Minimum Inhibitory Concentration (MIC) Data on Anti-microbial Resistance Huanhuan Wu & Carl James Schwarz Department of Statistics and Actuarial Science Simon Fraser University Nancy de With BC Ministry of Agriculture and Lands cschwarz@stat.sfu.ca Bacterial resistance to anti-microbial agents is often assessed using dilution series methods with in a highly mechanized fashion. The typical results are Minimum Inhibitory Concentration (MIC) values. For example, a series may contain the anti-microbial agent at concentrations of .12, .25, .50, 1, 2, 4 and 8 mg/L. A particular sample may show inhibition of growth at 2, 4 and 8 mg/L but growth at lower concentrations. The reported MIC value would then be 2 mg/L implying that a concentration of 2 mg/L showed inhibition, but a concentration of 1 mg/L did not show inhibition. In this presentation, we show how to fit linear models with fixed and random effects to this type of censored data using likelihood and Bayesian methods. AIC and DIC are used to discriminate among a set of models for a particular experiment. An example of comparing antimicrobial resistance between two different diets in poultry will be presented. Multiple barns of each type of diet were repeatedly visited. Fecal samples were collected on days 0, 10, 25, and 40 and the samples tested against an array of anti-microbial agents. 1 MIC data collection – I 2 MIC data collection - II 3 MIC data collection - III 4 MIC data examples Husbandry Practice No antibiotics No antibiotics … No antibiotics … No antibiotics … Antibiotics Farm Visit Sample 1 1 1 MIC (Ceftifour) .25 .50 1 1 2 .50 1.0 1 2 1 .00 .25 2 1 1 1.0 2.0 1 1 1 16.0 5 Important Assumptions • Tests are carried out without measurement error. • Tests are monotonic in the sense that if the inhibition is detected at concentration C then it is detected at any C’, where C’>C. • The MIC value of the anti-microbial agent when tested against a bacteria has a log-normal distribution. 6 Linear Model Let yi log(MICi ) f (yi ) Ui Li 2 y 1 i i exp f d dyi 2 2 2 2 where i X i Z i 7 Linear Model – Model Fitting How to fit? Censoring and random effects - if NO random effects -> Tobit model * SAS – Proc Lifereg; R – KMSurv - if random effects * Likelihood inference difficult (but see “data cloning” by Lele (2008)) * Bayesian methods WinBUGS and MCMC Standard priors used • Model selection using DIC - DIC = -2xloglikelihood + 2(effective parameters) - smaller DIC is “better” • Model diagnostics - trace plots - RGB statistics on mixing of chains - residual plots - sensitivity to priors - posterior predictive plots (Bayesian p-values) 8 Model Fitting Avian Project - Ceftiofur Effect of diet & visit 9 Model Fitting Avian Project - Ceftiofur Effect of farms (median at each visit plotted) 10 Model Fitting Avian Project – Ceftiofur • A total of 14 different models fit Model MF1: [Y]=Visit(C) Type(C) Visit(C)*Type(C) Farm(Type)(CR) MF3: [Y]=Visit(C) Type(C) Farm(Type)(CR) MF2: [Y]=Visit Type(C) Farm(Type)(CR) … M1: [Y]=Visit(C) Type(C) Visit(C)*Type(C) M2: [Y]=Visit Type(C) M3: [Y]=Visit(C) Type(C) …. Eff DIC Parm DIC 1596.0 18.0 0.0 1608.4 14.5 12.4 1608.6 11.1 12.5 1628.4 10.6 32.3 1638.1 4.0 42.1 1638.4 7.1 42.4 11 Model Fitting Avian Project - Ceftiofur Avg Diet effect Model M1 No farm effect Est (SD) -0.30 (0.59) Model MF1 With farm effect Est (SD) -0.18 (0.79) Diet effect - visit 1 Diet effect - visit 2 Diet effect - visit 3 Diet effect - visit 4 Diet effect - visit 5 -0.77 (0.61) -0.08 (0.55) -3.03 (0.66) 0.04 (0.50) -0.30 (0.58) -0.92 (0.82) -0.09 (0.75) -3.01 (0.88) 0.01 (0.73) -0.15 (0.79) 2.00 (0.11) na 1.88 (0.10) 0.69 (0.21) residual farm 12 Model Fitting Avian Project - Ceftiofur 13 Model Fitting Avian Project - Ceftiofur 14 Model Fitting Avian Project - Ceftiofur 15 Summary • MIC is interval censored - if NO random effects, fitting linear models is “easy” - WITH random effects, fitting is more difficult * need to integrate over random effects * “data cloning” method * Bayesian solution • Ignoring random effects usually leads to UNDERestimation of precision of answers • More information: * M.Sc. thesis of Wu http://www.stat.sfu.ca/people/alumni/ * Submission to Applied Statistics http://www.stat.sfu.ca/~cschwarz * Software Code also available at above web sites 16