Validation of Predictive Models: Acceptable Prediction Zone Method

Thomas P. Oscar, Ph.D.USDA, Agricultural Research ServiceMicrobial Food Safety Research UnitUniversity of Maryland Eastern Shore

Princess Anne, MD

Background Information

Terminology

• Performance evaluation

– Process of comparing observed and predicted

values.

• Validation

– A potential outcome of performance evaluation.

– Requires establishment of criteria.

Criteria

• Test Data– Interpolation

– Extrapolation

• Performance– Bias

– Accuracy

– Systematic Bias

Secondary Models

Predictive Modeling

PrimaryModel

PrimaryModel

Nmax

Model

max

Model

Model

No

Model

Observed No Predicted No

Observed Predicted

Observed max Predicted max

Observed Nmax Predicted Nmax

PredictedN(t)

ObservedN(t)

TertiaryModel

PredictedN(t)

Stage 3

Stage 2

Stage 1

Performance Evaluation

Goodness-of-fitPrimary/Secondary Models

VerificationTertiary Models

InterpolationAll Models

ExtrapolationAll Models

Test Data CriteriaInterpolation

• Independent data.

• Within the response surface.

– Uniform coverage.

• Collected with same methods.Incomplete and biased evaluation

Model data (10 to 40C) versus

Test data (25 to 40C)

Test Data CriteriaExtrapolation

• Independent data.

• Outside the response surface.

– Only one variable differs.

• Collected with same methods.Confounded comparison

Strain A in broth versus

Strain B in food

Acceptable Prediction Zone MethodDescription

Relative Error (RE)

RE for = (predicted - observed)/predicted

RE for N(t), No, max and Nmax = (observed - predicted)/predicted

RE < 0 are “fail-safe”

RE > 0 are “fail-dangerous”

"Acceptable"

"Overly Fail-safe"

"Overly Fail-dangerous"

4 5 6 7 8 9 10 11-1.2

-0.8

-0.4

-0.0

0.4

0.8

1.2

1.6

Predicted N(t) (log CFU/g)

Rel

ativ

e er

ror

Performance Factor %RE = REIN/RETOTAL

Performance Criteria

• Acceptable Predictions

-0.30 < RE < 0.15 for max

-0.60 < RE < 0.30 for

-0.80 < RE < 0.40 for N(t), No, Nmax

• Acceptable Performance

%RE => 70

Acceptable Prediction Zone MethodDemonstration

Model Development Design

• Salmonella Typhimurium

– No = 4.8 log CFU/g

• Sterile cooked chicken

– 10, 12, 14, 16, 20, 24, 28, 32,

36, 38, 40C

• Viable counts

– BHI agar

– 12 per growth curve

Performance Evaluation DesignSecondary Models (Interpolation)

• Salmonella Typhimurium

– No = 4.8 log CFU/g

• Sterile cooked chicken

– 11, 13, 15, 18, 22, 26, 30, 34,

37, 39C

• Viable counts

– BHI agar

– 12 per growth curve

Primary ModelLogistic with Delay

N = No if t

N = Nmax/(1+[(Nmax/No)-1]exp[-max (t-)]) if t >

0 10 20 30 404

5

6

7

8

9

10

11

Dependent (goodness-of-fit)

32C

Time (h)

N (

log

CF

U/g

)

Primary Model PerformanceGoodness-of-fit

4 5 6 7 8 9 10 11-1.2

-0.8

-0.4

-0.0

0.4

0.8

1.2

1.6%RE = 93.8

Predicted N(t) (log CFU/g)

Rel

ativ

e er

ror

Secondary Model for No

No = mean No

5 10 15 20 25 30 35 40 454

5

6

7

8

9

10

11

Independent (interpolation)Dependent (goodness-of-fit)

No

(log

CF

U/g

)

Temperature (C)

No Model Performance

Type of EvaluationDependent (goodness-of-fit)Independent (interpolation)

%RE100100

4.70 4.75 4.80 4.85 4.90-1.0-0.8-0.6-0.4-0.2-0.00.20.40.60.81.0

Predicted No (log CFU/g)

Rel

ativ

e er

ror

Secondary Model for Hyperbola with Shape Factor

= [41.47/(T - 7.325)]1.44

5 10 15 20 25 30 35 40 451

10

100

Independent (interpolation)Dependent (goodness-of-fit)

Temperature (C)

(h

)

Model Performance

0 10 20 30 40 50 60-1.0-0.8-0.6-0.4-0.2-0.00.20.40.60.81.0

Type of EvaluationDependent (goodness-of-fit)Independent (interpolation)

%RE100100

Predicted (h)

Rel

ativ

e er

ror

Secondary Model for max

Modified Square Root

max = 0.01885 if T

11.43

max = 0.01885 + [0.004325(T – 11.43)]1.306 if T > 11.43

5 10 15 20 25 30 35 40 450.0

0.1

0.2

0.3

0.4

0.5Dependent (goodness-of-fit)Independent (interpolation)

Temperature (C)

max

(h-1

)

max Model Performance

Type of EvaluationDependent (goodness-of-fit)Independent (interpolation)

%RE100100

0.0 0.1 0.2 0.3 0.4-1.0-0.8-0.6-0.4-0.2-0.00.20.40.60.81.0

Predicted max (h-1)

Rel

ativ

e er

ror

Secondary Model for Nmax

Asymptote Model

Nmax = exp(2.348[((T – 9.64)(T – 40.74))/((T – 9.606)(T – 40.76))])

5 10 15 20 25 30 35 40 455

6

7

8

9

10

11

Independent (interpolation)Dependent (goodness-of-fit)

Temperature (C)

Nm

ax (

log

CF

U/g

)

Nmax Model Performance

Type of EvaluationDependent (goodness-of-fit)Independent (interpolation)

8 9 10 11-1.0-0.8-0.6-0.4-0.2-0.00.20.40.60.81.0 %RE

100100

Predicted Nmax (log CFU/g)

Rel

ativ

e er

ror

Secondary Models

Predictive Modeling

PrimaryModel

PrimaryModel

Nmax

Model

max

Model

Model

No

Model

Observed No Predicted No

Observed Predicted

Observed max Predicted max

Observed Nmax Predicted Nmax

PredictedN(t)

ObservedN(t)

TertiaryModel

PredictedN(t)

Tertiary Model PerformanceVerification

4 5 6 7 8 9 10 11-1.2

-0.8

-0.4

-0.0

0.4

0.8

1.2

1.6

Predicted N(t) (log CFU/g)

Rel

ativ

e er

ror

%RE = 90.7

Comparison of Models

Model REIN REOUT RETOTAL

Primary 121 8 129

Tertiary 117 12 129

Total 238 20 258

Fisher’s exact test; P = 0.48, not significant.

Performance Evaluation DesignTertiary Model (Interpolation)

• Salmonella Typhimurium

– No = 4.8 log CFU/g

• Sterile cooked chicken

– 11, 13, 15, 18, 22, 26, 30, 34, 37,

39C

• Viable counts

– BHI agar

– 4 per growth curve

Tertiary Model Performance Interpolation

0 5 10 15 20 254

5

6

7

8

9

10

11

Time (h)

N (

log

CF

U/g

)

Tertiary Model Performance Interpolation

4 5 6 7 8 9 10 11-1.0-0.8-0.6-0.4-0.2-0.00.20.40.60.81.0

Predicted N(t) (log CFU/g)

Rel

ativ

e E

rror

%RE = 97.5

Should the validated tertiary model be used to predict chicken safety?

• Evaluation for extrapolation to:

– other initial densities (No)

– other strains

– other chicken products

Performance Evaluation DesignTertiary Model (Extrapolation)

• Salmonella Typhimurium

– No = 0.8 log CFU/g

• Sterile cooked chicken

– 10, 12, 14, 16, 20, 24, 28, 32,

36, 40C

• Viable counts

– BHI agar

– 4 per growth curve

Tertiary Model Extrapolation to low No

0 10 20 30 400123456789

1011

Time (h)

N (

log

CF

U/g

)

4 5 6 7 8 9 10 11-10123456789

10 24 RE > 10

Predicted N (log CFU/g)

Rel

ativ

e E

rror

Tertiary Model PerformanceExtrapolation to low No

%RE = 2.5

Conclusions

• Criteria are important for evaluating performance of models.

• Consensus on validation would improve the quality and use of predictive models in the food industry.