Quantification of variability and uncertainty in systems medicine models
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![Page 1: STATSPUNE 1 Statistical ecology No variability - no statistics No uncertainty - no statistics Variability- heart of all natural phenomena Uncertainty –](https://reader035.fdocuments.us/reader035/viewer/2022062519/5697bfca1a28abf838ca9555/html5/thumbnails/1.jpg)
STATSPUNE
1
Statistical ecology
• No variability - no statistics
• No uncertainty - no statistics
• Variability- heart of all natural phenomena
• Uncertainty – rule of nature
• Laws of uncertainty – statistical models
• Stage 1 – deterministic laws
• Stage 2- probabilistic laws
We begin with illustration of stage1
A.P.Gore S.A.Paranjpe
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STATSPUNE
2
Population Dynamics
• Single population •exponential growth•Logistic growth
•Survivorship curves
•Age/ stage structured models•Leslie matrix
•Two populations –Lotka Volterra models•Competition•Predation•Symbiosis
A.P.Gore S.A.Paranjpe
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STATSPUNE
3
Understanding and Using Microbial Growth
Part I: Modeling growth –basic study
( logistic growth model and multiple regression)
Part II: Preventing growth- use of preservative(logistic regression)
Part III: Using growth – biodegradation of pesticide
(Factorial experiment)
A.P.Gore S.A.Paranjpe
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4
STATSPUNE
STATISTICAL ANALYSIS
&
MODELLING
IN
FOOD PRESERVATION
Part I
A.P.Gore S.A.Paranjpe
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5
STATSPUNE
•TARGET ORGANISM: Staphylococcus aureus (on CORIANDER LEAVES)
•AIM: STUDY EFFECT ON GROWTH •pH (5LEVELS) •WATER ACTIVITY (Aw) (10 levels)•(5 x 10 = 50 COMBINATIONS)
•DATA: OPTICAL DENSITY (OD)•HOURLY RECORD•150 HOURS•INCUBATED AT 370 C•Two strains: standard and wild
A.P.Gore S.A.Paranjpe
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6
STATSPUNE
MODELLING STEP 1.
•ONE pH x Aw COMBINATION
•FIT LOGISTIC GROWTH CURVE
•Nt = K / (1+q e-rt)
•q = (K-N0)/ N0
•Estimate K(SATURATION LEVEL)
•Estimate r ( GROWTH RATE)
A.P.Gore S.A.Paranjpe
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7
STATSPUNE
LOGISTIC GROWTH CURVE
TIME (HOURS)
OPT
ICA
L D
ENSI
TY
A.P.Gore S.A.Paranjpe
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8
STATSPUNE
MODELLING STEP 2.
METAMODEL
REGRESS K ON pH AND Aw
REGRESS r ON pH AND Aw
K= B0 + B1*pH + B2*Aw + B3*Aw2
A.P.Gore S.A.Paranjpe
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9
STATSPUNE
VALIDATION
PREDICT K AND r FOR
INTERMEDIATE UNUSED VALUES OF pH & Aw
CONDUCT EXPERIMENTS USING pH x Aw SPECIFIED
COMPARE OBSERVED K AND r WITH PREDICTION
A.P.Gore S.A.Paranjpe
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10
STATSPUNE
RESULTS STEP 1.
LOGISTIC MODEL FITTED TO OD DATApH=4.5
TIME(h)
OD-O
BSER
VED
AND
FITT
ED
AW 0.955
A.P.Gore S.A.Paranjpe
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11
STATSPUNE
LOGISTIC MODEL FITTED TO OD DATApH=4.5
0.088
0.09
0.092
0.094
0.096
0.098
0.1
0 20 40 60 80 100 120 140 160
TIME(h)
OD-O
BSER
VED
AND
FITT
ED
Aw=0.8
A.P.Gore S.A.Paranjpe
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STATSPUNE
12
RESULT STEP 2.
Kst = 55.6 + 0.0333*pH – 130*Aw + 76* Aw2
(R2 = 95%)
Kis = 52.1 + 0.0368* pH – 122*Aw + 71* Aw2
(R2 = 87%)
REGRESSION FOR r – INEFFECTIVE.
A.P.Gore S.A.Paranjpe
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STATSPUNE
13
MODEL VALIDATION(STANDARD)
-0.4
0
0.4
0.8
1.2
1.6
2
0 4 8 12SET NO.
SA
TU
RA
TIO
N O
D
Obs Pre LL UL
UL
LL
A.P.Gore S.A.Paranjpe
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STATSPUNE
14
MODEL VALIDATION(ISOLATE)
-0.3
0.3
0.9
1.5
2.1
0 4 8 12SET NO.
OBS PRED LL(PRED) UL(PRED)
A.P.Gore S.A.Paranjpe
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STATSPUNE
15
Conclusions
•Logistic model fits bacterial growth data well
•Maximum concentration reached can be explained in terms of ambient conditions
•Growth rate appears to be insensitive to ambient conditions
A.P.Gore S.A.Paranjpe
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STATSPUNE
16
Part IIPreventing bacterial growth- use of
preservative
Aim: Develop a ready reckoner for combination of
Aw , pH and preservative level that is safe
against a cock tail of 5 bacterial species
3 preservatives, 5 levels of each, 5 Aw levels and 6 pH levels
A.P.Gore S.A.Paranjpe
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STATSPUNE
17
Effect of P.paraben on bacterial growth
Level pH1 0.98 0.96 0.93 0.9 1 0.98 0.96 0.93 0.9
0 4 3 0 1 1 0 0 0 1 0 05 0 1 1 1 1 3 3 3 2 26 2 3 2 2 0 3 3 3 2 17 3 3 3 2 2 3 3 3 2 28 0 3 3 2 2 2 3 3 0 19 2 3 3 2 0 0 3 3 0 0
0.025 4 0 0 0 0 0 0 0 0 0 05 0 1 0 0 0 0 0 0 0 06 0 1 0 0 0 0 0 0 0 07 1 1 2 2 0 0 0 0 0 08 1 2 2 1 0 0 3 0 0 09 2 3 3 0 0 3 3 0 0 2
Water ActivityB. pumilus B.subtilis
Code : 0 -no growth cidal , 1- no growth cidal/static2- no growth static, 3 - growth
A.P.Gore S.A.Paranjpe
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STATSPUNE
18
Fitted 0 1 % CorrectObserved 0 96 4 96.00% 1 8 12 60.00% Overall 90.00%
Results for P. fluorescens with preservative P. paraben
Variable B S.E. Wald df Sig LEVEL -62.0601 17.5421 12.5159 1 .0004PH .7465 .2369 9.9282 1 .0016 AW 46.5984 13.4808 11.9484 1 .0005 Constant -50.1339 13.6899 13.4110 1 .0003
Fitting Logistic regression
A.P.Gore S.A.Paranjpe
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STATSPUNE
19
P. Paraben P. fluorescensPres level pH Aw gcode grecode Prob Pred
2 0 4 1 0 0 0.36596 02 0 5 1 0 0 0.54907 12 0 6 1 3 1 0.71979 12 0 7 1 3 1 0.84421 12 0 8 1 3 1 0.91956 12 0 9 1 3 1 0.96018 12 0 4 0.98 0 0 0.1852 02 0 5 0.98 0 0 0.32409 02 0 6 0.98 3 1 0.50286 12 0 7 0.98 3 1 0.6809 12 0 8 0.98 2 0 0.81823 12 0 9 0.98 3 1 0.90473 12 0 4 0.96 0 0 0.08215 02 0 5 0.96 3 1 0.15882 02 0 6 0.96 3 1 0.28485 02 0 7 0.96 3 1 0.4566 02 0 8 0.96 3 1 0.63933 1
Model performance
A.P.Gore S.A.Paranjpe
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STATSPUNE
20
Using the model to build ‘safe zone’
•Response 0 : if none of the 5 species grows1 : if at least one species grows
•Preservative: P. paraben
•Use interpolation to generate predictions for unobserved conditions
A.P.Gore S.A.Paranjpe
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STATSPUNE
21
Ready reckoner (Aw X Preservative level)
Aw 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.30.9 0 0 0 0 0 0 0 0 0 0 0 0 0
0.91 0 0 0 0 0 0 0 0 0 0 0 0 00.92 0 0 0 0 0 0 0 0 0 0 0 0 00.93 0 0 0 0 0 0 0 0 0 0 0 0 00.94 0 0 0 0 0 0 0 0 0 0 0 0 00.95 0 0 0 0 0 0 0 0 0 0 0 0 00.96 0 0 0 0 0 0 0 0 0 0 0 0 00.97 1 0 0 0 0 0 0 0 0 0 0 0 00.98 1 1 0 0 0 0 0 0 0 0 0 0 00.99 1 1 1 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 0 0 0 0
Aw 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.30.9 0 0 0 0 0 0 0 0 0 0 0 0 0
0.91 0 0 0 0 0 0 0 0 0 0 0 0 00.92 0 0 0 0 0 0 0 0 0 0 0 0 00.93 0 0 0 0 0 0 0 0 0 0 0 0 00.94 0 0 0 0 0 0 0 0 0 0 0 0 00.95 0 0 0 0 0 0 0 0 0 0 0 0 00.96 1 0 0 0 0 0 0 0 0 0 0 0 00.97 1 1 0 0 0 0 0 0 0 0 0 0 00.98 1 1 1 0 0 0 0 0 0 0 0 0 00.99 1 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 0 0 0 0
Aw 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.30.9 0 0 0 0 0 0 0 0 0 0 0 0 0
0.91 0 0 0 0 0 0 0 0 0 0 0 0 00.92 0 0 0 0 0 0 0 0 0 0 0 0 00.93 0 0 0 0 0 0 0 0 0 0 0 0 00.94 1 0 0 0 0 0 0 0 0 0 0 0 00.95 1 1 0 0 0 0 0 0 0 0 0 0 00.96 1 1 1 1 0 0 0 0 0 0 0 0 00.97 1 1 1 1 1 0 0 0 0 0 0 0 00.98 1 1 1 1 1 1 0 0 0 0 0 0 00.99 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 0 0 0 0
Aw 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.30.9 0 0 0 0 0 0 0 0 0 0 0 0 0
0.91 0 0 0 0 0 0 0 0 0 0 0 0 00.92 0 0 0 0 0 0 0 0 0 0 0 0 00.93 0 0 0 0 0 0 0 0 0 0 0 0 00.94 0 0 0 0 0 0 0 0 0 0 0 0 00.95 1 0 0 0 0 0 0 0 0 0 0 0 00.96 1 1 0 0 0 0 0 0 0 0 0 0 00.97 1 1 1 0 0 0 0 0 0 0 0 0 00.98 1 1 1 1 0 0 0 0 0 0 0 0 00.99 1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 0 0 0
pH 4
pH9pH 6
pH 7
Safe zone contracts as Aw/pH increaseA.P.Gore S.A.Paranjpe
Preservative level Preservative level
Preservative level Preservative level
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STATSPUNE
22
Optimizing
Biodegradation of Dimethoate
in
Industrial Effluents by
Brevundimonas sp.
A Factorial Experiment
Part III
A.P.Gore
STATSPUNE
S.A.Paranjpe
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STATSPUNE
23
Factors and Levels• Temperature(T) : 30 , 40 degrees Celsius
• pH (p) : 5 ,7
• Aeration(A) : Yes , No
• Inoculum(I) : 105, 109 cells/ml
• Substrate Conc. (mg/l) (S) : 2000 , 500
• Total # of factor combinations 25=32
• Response : % removal of Diamethoate
A.P.Gore S.A.Paranjpe
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STATSPUNE
24
•Initial trial 8 runs
•Quarter replicate
•Substrate effect found negligible
•Reduce number of factors to 4
•Combinations 16
•Second trial : Full experiment
A.P.Gore S.A.Paranjpe
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STATSPUNE
25
Pareto Diagram of Main Effects21
6.2 5.2 5.1
0
5
10
15
20
25
Inoculum Temperature pH Aeration
% Re
mov
al of
Di
met
hoat
e
Relative Importance of Factors
Informative graph not generated by soft-waresA.P.Gore S.A.Paranjpe
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STATSPUNE
26
Pareto Diagram for Interactions
4 3.8 3.83.4 3.2 3
2.3 2.2 2
0
1
2
3
4
5
p X I X A T X I p X T X I p X I p X A p X T X A T X A p X T I X A
Factor combination
% R
emov
al o
f D
imet
hoat
e
Relative Importance of Interactions
A.P.Gore S.A.Paranjpe
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STATSPUNE
27
pH X Temperature Interaction
35
40
45
50
55
pH5 pH7pH
% R
emov
al T30
T40
pH X Aeration Interaction
30
35
40
45
50
55
Air NoairAeration
% R
emov
al pH5
pH7
No Interaction
Mild Interaction
A.P.Gore S.A.Paranjpe
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STATSPUNE
28
Three Factor Interaction
The Way Two Factor Interaction Depends
on
Level of Third Factor
Deserves inclusion in soft wares
A.P.Gore S.A.Paranjpe
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STATSPUNE
29
Interaction pH X Temperature
15
19
23
27
pH5 pH7
% R
emo
val
T30
T40
Interaction pH X Temperature
60
64
68
72
76
pH5 pH7
% R
emov
al
T30
T40
Inoculum Low
Inoculum High
A.P.Gore S.A.Paranjpe
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STATSPUNE
30
Choosing Best Factor Combination
pH Temp Inoculum Aeration % Removal5 30 L No 13.675 30 L Yes 24.895 30 H No 60.935 30 H Yes 78.615 40 L No 15.295 40 L Yes 19.795 40 H No 46.865 40 H Yes 75.17 30 L No 23.567 30 L Yes 28.847 30 H No 62.667 30 H Yes 87.45
7 40 L No 16.27 40 L Yes 20.077 40 H No 50.837 40 H Yes 82.96
A.P.Gore S.A.Paranjpe
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STATSPUNE
31
Follow up:
Remedial potential of a single bacterial species was found
to be enhanced in a mixture
A.P.Gore S.A.Paranjpe