1. Statistical thinking in antibiofilm research Cord Hamilton Al Parker Marty Hamilton MBL and SBML: 23 October 2008 1

2. 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

3. Log Reduction (LR) for a Quantitative Assay V c = viable cell density of biofilm grown in the absence of antimicrobial treatment V d = viable cell density of biofilm grown in the presence of the disinfectant Log Reduction = log 10 (V c ) - log 10 (V d ) 3

4. Numerical Example V c = 10 7 & V d = 10 Log Reduction = log 10 (10 7 ) - log 10 (10) LR = 7 - 1 LR = 6 Interpretation: disinfectant killed 99.9999% of the bacteria 4

5. Calculating LR when there are multiple coupons = mean of control log 10 densities = mean of disinfected log 10 densities Log Reduction = C D C - D 5

6. Example: Mean of logs for 3 disinfected coupons Coupon Density log 10 Density (i) cfu / cm 2 (D i ) 1 9.6·10 4 4.982 2 1.7·10 4 4.230 3 9.7·10 3 3.987 Mean = 4.400 = mean density = 4.09∙10 4 log of mean density = 4.61 D 6

7. Example: Control coupons Coupon log 10 Density (i) (C i ) 1 7.499 2 7.013 3 7.863 C = 7.458 7

8. Calculating LR when there are multiple coupons = 7.458 & = 4.400 Log Reduction = C D C - D = 7.458 - 4.400 LR = 3.058 8

9. Within-experiment standard error (SE) of the LR S c = variance of control log 10 densities S d = variance of disinfected log 10 densities n c = number of control coupons n d = number of disinfected coupons SE of LR = (within-experiment) S ncnc S ndnd c 2 d 2  2 2 9

10. Example: Calculating SE for single reactor experiment S c = 0.181865 and n c = 3 S d = 0.269272 and n d = 3 SE = 0.181865 3 0.269272 3  2 2 = 0.3878 10

11. Uncertainty in LR Estimate LR ± SE= 3.058 ± 0.388 or 3.06 ± 0.39 or 3.1 ± 0.4 11

12. 3 2 1 0 Log Reduction ± SE Experiment 12 RDR biofilm : 5 ppm chlorine for 10 minutes 12 345

13. Experiment repeated 3 times, each using three control and 3 disinfected coupons 13

14. Statistical summary for data from 3 experiments, with 3 control and 3 disinfected coupons per experiment log densitymean log densitySD log densityStandard error of Expcontroldisinfectedcontroldisinfected log reductioncontroldisinfectedlog reduction 16.738493.08115 16.820563.293266.832403.135463.696950.100360.138860.09892 16.938163.03196 26.662762.92334 26.739573.034886.714403.056563.657840.044730.145280.08776 26.740863.21146 36.915642.73748 36.745572.660186.852932.708054.144880.093410.041830.05909 36.897582.72651 Pooled within-experiment SD of the control log density:0.08326 Pooled within-experiment SD of the disinfected log density:0.11851 Between-experiment SD of the log reduction:0.25736 14

15. S n c m c 2 + Formula for the SE of the mean LR, averaged over experiments S c = within-experiment variance of control coupon LD S d = within-experiment variance of disinfected coupon LD S E = between-experiments variance of LR n c = number of control coupons n d = number of disinfected coupons m = number of experiments 2 2 2 S n d m d 2 + S m E 2 SE of mean LR = 15

16. Formula for the SE of the mean LR, using estimated standard deviations 0.0833 n c m + 0.1185 n d m 2 + 0.2574 m 2 SE of mean LR = 2 Pooled within-experiment SD of the control log density:0.0833 Pooled within-experiment SD of the disinfected log density:0.1185 Between-experiment SD of the log reduction:0.2574 16

17. 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 (n c ):23612 no. disinfected coupons (n d ):23612 no. experiments (m) 10.2770.2710.2640.261 20.1960.1910.1870.184 30.1600.1560.1520.151 40.1380.1350.1320.130 60.1130.1100.1080.106 100.0880.0860.0840.082 1000.0280.0270.026 17

18. Dilution series and drop plate technique Source: BiofilmsOnline Counted dilution 32 colonies 10 18

19. Find the fraction of initial beaker volume in each of the dilution tubes Source: BiofilmsOnline Beaker: contained all cells from coupon 0.10.010.0010.0001 fraction of beaker volume in tube 10 19

20. Estimated number of cells in beaker = cfu count divided by the volume fraction plated Beaker: contains all cells from coupon 10 -4 fraction in tube Plated 50 μl from tube; plate contains a fraction 50/10000 = 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 10 7 10 20

21. Dilution series and filter technique: pooling data from two tubes 9 ml filtered 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 -6 421 cfu 39 cfu 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 10 7 10 21