Statistical thinking in
antibiofilm research
Cord Hamilton
Al Parker
Marty Hamilton
1
MBL and SBML: 23 October 2008
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) for
a 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 when
there are multiple coupons
C = mean of control log10 densities
D = mean of disinfected log10 densities
Log Reduction = C
- D
5
Example: Mean of logs
for 3 disinfected coupons
Coupon
(i)
1
2
3
Density log10Density
cfu / cm2
(Di)
9.6·104
4.982
1.7·104
4.230
9.7·103
3.987
Mean = 4.400 = D
mean density = 4.09∙104
log of mean density = 4.61
6
Example: Control coupons
Coupon
(i)
log10Density
(Ci)
1
2
3
7.499
7.013
7.863
C = 7.458
7
Calculating LR when
there are multiple coupons
C = 7.458 & D = 4.400
Log Reduction =
C - D
= 7.458 - 4.400
LR = 3.058
8
Within-experiment standard
error (SE) of the LR
2
Sc = variance of control log10 densities
2
Sd = variance of disinfected log10 densities
nc = number of control coupons
nd = number of disinfected coupons
SE of LR =
(within-experiment)
2
c
2
d
S
S
+
nc nd
9
Example: Calculating SE
for single reactor experiment
2
Sc = 0.181865 and nc = 3
2
Sd = 0.269272 and nd = 3
SE =
0.181865
0.269272
+
3
3
= 0.3878
10
Uncertainty in LR Estimate
LR ± SE
= 3.058 ± 0.388
or 3.06 ± 0.39
or 3.1 ± 0.4
11
RDR biofilm: 5 ppm chlorine for 10 minutes
Log Reduction ± SE
3
2
1
0
1
2
3
4
5
Experiment
12
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
Exp
1
1
1
log density
control disinfected
6.73849
3.08115
6.82056
3.29326
6.93816
3.03196
mean log density
control disinfected log reduction
SD log density
Standard error of
control disinfected
log reduction
6.83240
3.13546
3.69695
0.10036
0.13886
0.09892
2
2
2
6.66276
6.73957
6.74086
2.92334
3.03488
3.21146
6.71440
3.05656
3.65784
0.04473
0.14528
0.08776
3
3
3
6.91564
6.74557
6.89758
2.73748
2.66018
2.72651
6.85293
2.70805
4.14488
0.09341
0.04183
0.05909
Pooled within-experiment SD of the control log density:
Pooled within-experiment SD of the disinfected log density:
Between-experiment SD of the log reduction:
0.08326
0.11851
0.25736
14
Formula for the SE of the mean LR,
averaged over experiments
2
Sc = within-experiment variance of control coupon LD
Sd2 = within-experiment variance of disinfected coupon LD
SE2 = between-experiments variance of LR
nc = number of control coupons
nd = number of disinfected coupons
m = number of experiments
SE of mean LR =
2
Sc
nc • m
+
2
Sd
nd • m
+
2
SE
m
15
Formula for the SE of the mean LR,
using estimated standard deviations
Pooled within-experiment SD of the control log density:
Pooled within-experiment SD of the disinfected log density:
Between-experiment SD of the log reduction:
2
SE of mean LR =
0.0833
nc • m
+
0.1185
nd • m
2
+
0.2574
0.0833
0.1185
0.2574
2
m
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):
no. disinfected coupons (nd):
2
2
3
3
6
6
12
12
no. experiments (m)
1
2
3
4
6
10
100
0.277
0.196
0.160
0.138
0.113
0.088
0.028
0.271
0.191
0.156
0.135
0.110
0.086
0.027
0.264
0.187
0.152
0.132
0.108
0.084
0.026
0.261
0.184
0.151
0.130
0.106
0.082
0.026
17
Dilution series and drop plate technique
10
Counted dilution
32 colonies
18
Source: BiofilmsOnline
Find the fraction of initial beaker volume
in each of the dilution tubes
10
Beaker:
contained
all cells
from
coupon
0.1
0.01
0.001
0.0001
fraction of beaker volume in tube
19
Source: BiofilmsOnline
Estimated number of cells in beaker = cfu
count divided by the volume fraction plated
10
Beaker:
contains
all cells
from
coupon
10-4 fraction in tube
Estimate:
32/(5 x 10-7)
= 6.4 x 107
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
20
Dilution series and filter technique:
pooling data from two tubes
The 460 cfu corresponds to this
fraction of the beaker volume:
f = 1.8x10-5 + 2.0x10-6
= 2.0 x 10-5
10
9 ml
filtered
421
cfu
10 ml
filtered
39
cfu
Estimate for
beaker
= 460/(2.0x10-5)
= 2.3 x 107
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
21