Statistical thinking in antibiofilm research

Cord HamiltonAl Parker

Marty Hamilton

MBL and SBML: 23 October 20081

Topics (presenter)

Calculating LR and the within-experiment standard error of LR (Cord)

Using data from repeated experiments to find more reliable LR values in the future (Al)

Analysis of dilution series counts (Marty)

2

Log Reduction (LR) fora Quantitative Assay

Vc = viable cell density of biofilm grown in the absence of antimicrobial treatment

Vd = viable cell density of biofilm grown in the presence of the disinfectant

Log Reduction = log10(Vc) - log10(Vd)

3

Numerical Example

Vc = 107 & Vd = 10

Log Reduction = log10(107) - log10(10) LR = 7 - 1 LR = 6

Interpretation: disinfectant killed 99.9999% of the bacteria

4

Calculating LR whenthere are multiple coupons

= mean of control log10 densities

= mean of disinfected log10 densities

Log Reduction =

C

D

C - D

5

Example: Mean of logsfor 3 disinfected coupons

Coupon Density log10Density (i) cfu / cm2 (Di) 1 9.6·104 4.982 2 1.7·104 4.230 3 9.7·103 3.987

Mean= 4.400 =

mean density = 4.09 10∙ 4

log of mean density = 4.61

D

6

Example: Control coupons

Coupon log10Density (i) (Ci)

1 7.499 2 7.013 3 7.863

C = 7.458

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Calculating LR whenthere are multiple coupons

= 7.458 & = 4.400

Log Reduction =

C D

C - D

= 7.458 - 4.400

LR = 3.058

8

Within-experiment standard error (SE) of the LR

Sc = variance of control log10 densities

Sd = variance of disinfected log10 densities

nc = number of control coupons

nd = number of disinfected coupons

SE of LR = (within-experiment)

Snc

Snd

c2

d2

2

2

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Example: Calculating SEfor single reactor experiment

Sc = 0.181865 and nc = 3

Sd = 0.269272 and nd = 3

SE = 0.181865

30.269272

3

2

2

= 0.3878

10

Uncertainty in LR Estimate

LR ± SE = 3.058 ± 0.388

or 3.06 ± 0.39

or 3.1 ± 0.4

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3

2

1

0Log

Redu

ction

± S

E

Experiment1 2

RDR biofilm: 5 ppm chlorine for 10 minutes

12

3 4 5

Experiment repeated 3 times, each using three control and 3 disinfected coupons

13

Statistical summary for data from 3 experiments, with 3 control and 3 disinfected coupons per experiment

log density mean log density SD log density Standard error ofExp control disinfected control disinfected   log reduction control disinfected log reduction

1 6.73849 3.081151 6.82056 3.29326 6.83240 3.13546 3.69695 0.10036 0.13886 0.098921 6.93816 3.03196

2 6.66276 2.923342 6.73957 3.03488 6.71440 3.05656 3.65784 0.04473 0.14528 0.087762 6.74086 3.21146

3 6.91564 2.737483 6.74557 2.66018 6.85293 2.70805 4.14488 0.09341 0.04183 0.059093 6.89758 2.72651

Pooled within-experiment SD of the control log density: 0.08326Pooled within-experiment SD of the disinfected log density: 0.11851

Between-experiment SD of the log reduction: 0.25736

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S

nc • m

c2

+

Formula for the SE of the mean LR, averaged over experiments

Sc = within-experiment variance of control coupon LD

Sd = within-experiment variance of disinfected coupon LD

SE = between-experiments variance of LR

nc = number of control coupons

nd = number of disinfected coupons

m = number of experiments

2

2

2

S

nd • m

d2

+S

m

E2

SE of mean LR =

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Formula for the SE of the mean LR, using estimated standard deviations

0.0833

nc • m+

0.1185

nd • m

2

+0.2574

m

2

SE of mean LR =

2

Pooled within-experiment SD of the control log density: 0.0833Pooled within-experiment SD of the disinfected log density: 0.1185

Between-experiment SD of the log reduction: 0.2574

16

Choosing the numbers of coupons and the number of experiments. Table cell is the the SE of the mean LR. Shaded SE values are designs requiring 24 coupons.

no. control coupons (nc): 2 3 6 12no. disinfected coupons (nd): 2 3 6 12

no. experiments (m)  1 0.277 0.271 0.264 0.2612 0.196 0.191 0.187 0.1843 0.160 0.156 0.152 0.1514 0.138 0.135 0.132 0.1306 0.113 0.110 0.108 0.106

10 0.088 0.086 0.084 0.082100 0.028 0.027 0.026 0.026

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Dilution series and drop plate technique

Source: BiofilmsOnline

Counted dilution32 colonies

10

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Find the fraction of initial beaker volume in each of the dilution tubes

Source: BiofilmsOnline

Beaker: containedall cells fromcoupon

0.1 0.01 0.001 0.0001

fraction of beaker volume in tube

10

19

Estimated number of cells in beaker = cfu count divided by the volume fraction plated

Beaker: containsall cells fromcoupon

10-4 fraction in tube

Plated 50 μl from tube;plate contains a fraction50/10000 = 5 x 10-3 of the volume in the tube.

f = (5 x 10-3) 10-4 = 5 x 10-7

Estimate:32/(5 x 10-7) = 6.4 x 107

10

20

Dilution series and filter technique: pooling data from two tubes

9 mlfiltered

10 ml filtered

Count 20 fields on each filter;corresponds to 0.02 of filter area

f = 0.001 x 0.9 x 0.02 = 1.8 x 10-5

f = 0.0001 x 1.0 x 0.02 = 2.0 x 10-6

421cfu

39cfu

The 460 cfu corresponds to this fraction of the beaker volume:f = 1.8x10-5 + 2.0x10-6

= 2.0 x 10-5

Estimate for beaker = 460/(2.0x10-5)= 2.3 x 107

10

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