Modeling Growth  From Dinosaurs to Children
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Gates Notes
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Growthrate studies of dinosaurs reveal that the same statistical methods can be applied to better understanding a major issue in global health, child malnutrition.

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Slide 1  7 October 2014
 Modeling
 MODELING GROWTH: FROM DINOSAURS TO CHILDREN

Slide 3  To better understand their biology, metabolism, and behavior
 Why Model Dinosaur Growth?

Slide 4  4
 Commonalities with modeling child growth
 The importance of best practices, especially when data is imperfect
 Choosing appropriate variables
 Choosing the mathematical model that objectively works best
 Fitting curves appropriately
 Understanding and respecting the limitations of data and models

Slide 6  6
 Lines of arrested growth (LAGS)
 Velociraptor
 Tenontosaurus

Slide 8  8
 Estimating age and size from fossilized bone
 Lines of arrested growth (LAGs)
 Estimate age from LAG count
 Estimate mass from circumference of major bones:
 M = a (cfemur + chumerus)b
 for quadrupeds,
 a = 0.078, b = 2.73
 for bipeds,
 a = 0.16, b =2.73

Slide 10  Analyzed 31 Sets of Dinosaur Growth Data, Spanning 14 Taxa

Slide 12  When complete data is not available on individuals, aggregating data can enable reconstruction of plausible growth trajectories
 Some researchers do the fitting by eye
 Leastsquares optimization is replicable and more accurate
 Need to use noninteger scaling factors (not every individual has the same birthday!)
 12
 Constructing longitudinal seriesfrom sparse data

Slide 14  Testing Fits to Multiple Models
 14
 21 increasing functions
 14 asymptotic attenuating
 49 asymptotic sigmoidal
 84 functions × 31 data sets = 2,604 fits
 parameters
 functions
 1
 4
 2
 21
 3
 31
 4
 28
 Using more free parameters may improve the fit, but not always enough to justify the additional degrees of freedom

Slide 16  16
 Growth IS USUALLY (BUT NOT ALWAYS) heteroscedastic
 Wandering albatross
 Heteroscedastic growth: the distribution of body sizes widens substantially with age as individual growth trajectories diverge
 At age 50d, σ = 1,115; at age 250d, σ = 1,505

Slide 18  Increases the likelihood that an algorithm will find the best fit, and decreases the probability of algorithm failure
 Tested using synthetic data set and six commonly used global fitting algorithms from R and Mathematica
 A New Preprocessing transformation for Fitting Models

Slide 19  19
 From bone circumference and age estimates, we calculated the likely peak growth rate
 Example: Tyrannosaurus
 Maximum bone growth rate occurs at age 2
 Bone circumference (cm)
 Age (y)
 Peak mass growth rate: 365 kg/yr
 PER YEAR

Slide 21  Almost all previous analyses in the literature were affected by one or more methodological issues
 21
 Irreproducible fits
 Use of subjective, rather than objective, criteria to select among models
 Inappropriate choice of independent variable
 Incorrect or handtweaked data and curvefitting
 Attempted replications of prior published results
 Age
 Mass

Slide 23  23
 Choose the right independent variable
 Linear regression methods assume it has no error
 Monte Carlo analysis illustrates the consequence of choosing wrong: a different mean and a much greater standard deviation
 Lessons Applicable to Modeling ChilD Growth
 Age as the independent variable
 Time as the independent variable

Slide 24  24
 Choose the right independent variable
 Linear regression methods assume it has no error
 Monte Carlo analysis illustrates the consequence of choosing wrong: a different mean and a much greater standard deviation
 Lessons Applicable to Modeling ChilD Growth
 Age as the independent variable
 Time as the independent variable

Slide 26  26
 Try multiple alternative models
 Don’t overfit by using too many parameters
 Respect model constraints
 Use an objective criterion, such as AICc, to select the best model
 All goodnessoffit metrics are local to the data points—there is no legitimate prediction beyond the limits of the data
 Lessons Applicable to Modeling Child Growth

Slide 28  28
 If the shoe fits—it doesn’t necessarily mean anything
 Every growth model has a linear phase!
 Lessons Applicable to Modeling Child Growth

Slide 30  30
 Avoid unjustified extrapolation by estimating error and plotting confidence intervals
 Outside the bounds of the data, CIs grow wide quickly
 Lessons Applicable to Modeling Child Growth

Slide 32  Crosssectional data sets cannot accurately simulate longitudinal series
 Longitudinal vs. Crosssectional data
 Standard curvefitting algorithms assume uncorrelated errors
 This assumption is violated by longitudinal data sets that follow individuals over time

Slide 34  Crosssectional data sets cannot accurately simulate longitudinal series
 Longitudinal vs. Crosssectional data
 Source data: longitudinal growth measurements on 100 flamingos
 100 longitudinal fits: fit the same function to each individual series
 100 crosssectional fits: fit the same function as above to data sets of randomly selected points representing the
 entire life span
 Values of coefficient a b c

Slide 36  Following individuals for many years enables insights into the effects of path history
 Need to use models and measures designed for longitudinal series
 The Power of Longitudinal Data

Slide 38  38
 Weighted Least Squares Analysis

Slide 39  Improper measurement of the independent variable attenuates relationships.
 Jerry Hausman, M.I.T., in Journal of Economic Perspectives 15:4 (2001)
 39
 The Iron Law of econometrics
 Nigeria: +89%
 Oshodi BA. Energy Economics and Project Financing Options in Nigeria. Rochester, NY: Social Science Research Network, 2014
 Zambia: +24%
 Jerven M. Poor Numbers: How We Are Misled by African Development Statistics and What to Do about It. Ithaca, NY: Cornell University Press, 2013.
 Ghana: +63%
 Jerven M, Duncan ME. Revising GDP estimates in subSaharan Africa: lessons from Ghana. Afr Stat J 2012; 15: 13–24.
 GDP measurements are noisy and often inaccurate even in developed economies
 Changes to national accounting systems can cause large artificial changes

Slide 41  Consensus view: Stunting Falls as GDP Rises
 Absolute decline in prevalence
 Relative decline (% of previous prevalence)

Slide 43  Underweight Also goes down when GDP goes up
 Absolute decline in prevalence
 Relative decline (% of previous prevalence)

Slide 44
 Countries where GDP per capita was growing were 67% more likely to reduce the prevalence of child stunting

Slide 46
 Causal or Simply Correlated?
 Stunting and sanitation

Slide 48
 Causal or Simply Correlated?
 Stunting and maternal education