Download pdf - Applying multiple imputation on waterbird census data

Applying multiple imputation onwaterbird census dataComparing two imputation methods

ISEC, Montpellier, 1 july 2014Thierry Onkelinx, Koen Devos & Paul Quataert

Overzicht

1 Introduction

2 Testing the Underhill index

3 An alternative way of imputing

4 Testing multiple imputation using INLA

5 Conclusions

2 / 23

1 Introduction




5 Conclusions

3 / 23

Waterbird census in Flanders (Belgium)

• Aim: monitor winteringbirds

• Total number• Average over months

per winter

• Data collected byvolunteers

• 1200 sites• 23 winters• 6 months per winter

• Missing data on average26% (9% - 42%)

• Impution required

35°

40°

45°

50°

55°

60°

−10° 0° 10° 20° 30°

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Underhill-index

Described by Underhill and Prys-Jones (1994)Algorithm

1 Replace all missings with starting value

2 Fit model to imputed dataset

3 Predict missing data

4 Replace imputed value with rounded prediction when prediction islarger

5 Re-iterate from 2. until imputations are stable

Negative binomial GLM with global effects for winter, month and site

Potential problems• Imputed values can never decrease: risk for bias

• Imputing with model predictions: risk for reduced standard errors

5 / 23

Underhill-index

Described by Underhill and Prys-Jones (1994)Algorithm

1 Replace all missings with starting value

2 Fit model to imputed dataset

3 Predict missing data

4 Replace imputed value with rounded prediction when prediction islarger

5 Re-iterate from 2. until imputations are stable

Negative binomial GLM with global effects for winter, month and site

Potential problems• Imputed values can never decrease: risk for bias

• Imputing with model predictions: risk for reduced standard errors

5 / 23

1 Introduction




5 Conclusions

6 / 23

Test setup

• Generate dataset (40 sites, 24 winters, 6 months)• Remove 25% data completely at random• Impute missing data• Calculate total per winter and month over all sites• Model totals

• Poisson regression with overdisperion

• Estimate per winter

• Relative bias: exp(βUnderhill )exp(βcomplete)

• Relative SE:σβUnderhill

σβcomplete

1 Original Underhill-index, starting value = zero

2 Original Underhill-index, starting value = geometric mean

3 Altered Underhill-index, starting value = zero

4 Altered Underhill-index, starting value = geometric mean

Altered index: replace imputation with rounded predictions

7 / 23

Test setup






σβcomplete






7 / 23

Test setup






σβcomplete






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Data generating model

Counts follow negative binomial distribution

• Fixed size• Variable mean (defined on log-scale)

• Intercept

• Linear trend and random walk along winter

• Random intercept and random walk along winter per site

• Sine wave within a winter with variable phase among winters

• Gaussian noise at observation level

• Different dataset per simulation• All datasets based on the same hyperparameters

8 / 23

Example of simulated dataset

0

50

100

150

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time (winters)

True

mea

n pe

r si

te

Site 12 Site 13 Site 18 Site 27

Site 30 Site 31 Site 34 Site 8

10

20

30

40

468

10

510152025

20304050

10

20

30

40

4

8

12

16

1.01.52.02.53.0

2

3

4

5

2 4 6 2 4 6 2 4 6 2 4 6

Month

True

mea

n pe

r si

te

Year

1

2

3

4

5

Figure : Example of simulated dataset

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Evaluation of Underhill index

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Altered

Original

80% 100% 120% 140% 160%

Relative bias, 100% = complete dataset

Alg

orith

m

Startingvalue

Mean

Zero

Relative bias

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Altered

Original

70% 80% 90% 100%

Relative SE, 100% = complete dataset

Alg

orith

m

Startingvalue

Mean

Zero

Relative SE

Figure : Evaluation of the Underhill index

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1 Introduction




5 Conclusions

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Requirements

• Choose model that doesn’t require starting values

• We choose a negative binomial GLMM

• Winter and month as fixed effect (factors)• Site as random intercept• Fitted in R (R Core Team 2014) with INLA (Rue et al. 2009)

• Take the uncertainty of predictions into account

• Sample from negative binomial distribution

• Size

• Sample from distribution of hyperparameter

• Mean

• Sample from gaussian distribution• Mean and SE of prediction on the link scale

12 / 23

Requirements

• Choose model that doesn’t require starting values

• We choose a negative binomial GLMM

• Winter and month as fixed effect (factors)• Site as random intercept• Fitted in R (R Core Team 2014) with INLA (Rue et al. 2009)

• Take the uncertainty of predictions into account

• Sample from negative binomial distribution

• Size

• Sample from distribution of hyperparameter

• Mean

• Sample from gaussian distribution• Mean and SE of prediction on the link scale

12 / 23

1 Introduction




5 Conclusions

13 / 23

Test setup

• Same test datasets as for testing Underhill-index• Fit INLA model to observed dataset• Generate M sets of imputed values• For each set m

• Calculate total per winter and month over all sites

• Model totals

• Save the regression parameters βi m and their SE σi m

• Aggregate over all M sets (Rubin 1987)

β̄i =1

M

M∑m=1

βi m

σ̄i =

√√√√ 1

M

M∑m=1

σ2i m +

M + 1

M

M∑m=1

(βi m − β̄i )2

M − 1

14 / 23

Test setup

• Same test datasets as for testing Underhill-index• Fit INLA model to observed dataset• Generate M sets of imputed values• For each set m

• Calculate total per winter and month over all sites

• Model totals

• Save the regression parameters βi m and their SE σi m

• Aggregate over all M sets (Rubin 1987)

β̄i =1

M

M∑m=1

βi m

σ̄i =

√√√√ 1

M

M∑m=1

σ2i m +

M + 1

M

M∑m=1

(βi m − β̄i )2

M − 1

14 / 23

Evaluation of multiple imputation

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1%

5%

25%

50%

75%

60% 80% 100% 120% 140%


Pro

port

ion

of m

issi

ng d

ata

Relative bias

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1%

5%

25%

50%

75%

90% 110% 130%


Pro

port

ion

of m

issi

ng d

ata

Relative SE

Figure : Evaluation of multiple imputation

15 / 23

Effect of design

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10

20

40

80

75% 100% 125% 150%


Num

ber

of s

ites Winters

5

10

20

Relative bias

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10

20

40

80

80% 100% 120% 140% 160%


Num

ber

of s

ites Winters

5

10

20

Relative SE

Figure : Effect of design

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Effect of model for imputation

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Constantsite

Randomwalk

per site

Truemean

70% 80% 90% 100% 110% 120% 130%


Mod

el fo

r im

puta

tion

Relative bias

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Constantsite

Randomwalk

per site

Truemean

90% 100% 110% 120% 130%


Mod

el fo

r im

puta

tion

Relative SE

Figure : Effect of imputation model

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Real life examples

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Anser brachyrhynchus: 10% missing, 18 sites, 4 months Numenius arquata: 24% missing, 208 sites, 6 months

Anas platyrhynchos: 40% missing, 852 sites, 6 months Haematopus ostralegus: 42% missing, 270 sites, 6 months

0

10,000

20,000

0

2,000

4,000

6,000

8,000

0

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50,000

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0

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20,000

1995 2000 2005 2010 2015 2000 2005 2010 2015

1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

Winter

Win

ter

aver

age

Figure : Reallife examples

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1 Introduction




5 Conclusions

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Underhill-index

• Must use zero as starting value

• Otherwise biased upward

Underestimates standard errors• Incorrect Type I errors!

• Too optimistic

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Multiple imputation

• Unbiased estimates

Increased standard errors• Imputation = more uncertainty

• Implies lower power• Sample size actual dataset < sample size complete dataset

• Increase of standard error depends on

1 Proportion of missing data

2 Size of dataset

3 Imputation model

• Marginal improvement with increased number of imputations

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Questions?

ReferencesR Core Team. 2014. “R: A Language and Environment for StatisticalComputing.” Vienna, Austria: R Foundation for Statistical Computing.http://www.r-project.org/.Rubin, D. B. 1987. Multiple Imputation for Nonresponse in Surveys. NewYork, NY: John Wiley & Sons, Ltd.Rue, H\aa vard, Sara Martino, Finn Lindgren, Daniel Simpson, and AndreaRiebler. 2009. INLA: Functions Which Allow to Perform Full BayesianAnalysis of Latent Gaussian Models Using Integrated Nested LaplaceApproximation.Underhill, L. G., and R. P. Prys-Jones. 1994. “Index Numbers for WaterbirdPopulations. I. Review and Methodology.” Journal of Applied Ecology 31(3): 463–480. doi:10.2307/2404443.

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http://www.r-project.org/

Bibliografie

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