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.
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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  
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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.
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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 
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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)
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Model Fitting
Avian Project - Ceftiofur
Effect of diet & visit
9
Model Fitting
Avian Project - Ceftiofur
Effect of farms (median at each visit plotted)
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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
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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
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Model Fitting
Avian Project - Ceftiofur
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Model Fitting
Avian Project - Ceftiofur
14
Model Fitting
Avian Project - Ceftiofur
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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
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