Obtaining appropriate interval estimates for age when multiple indicators are used: evaluation of an ad-hoc procedure

Abstract

When an estimate of age is needed, typically multiple indicators are present as found in skeletal or dental information. There exists a vast literature on approaches to estimate age from such multivariate data. Application of Bayes’ rule has been proposed to overcome drawbacks of classical regression models but becomes less trivial as soon as the number of indicators increases. Each of the age indicators can lead to a different point estimate (“the most plausible value for age”) and a prediction interval (“the range of possible values”). The major challenge in the combination of multiple indicators is not the calculation of a combined point estimate for age but the construction of an appropriate prediction interval. Ignoring the correlation between the age indicators results in intervals being too small. Boldsen et al. (2002) presented an ad-hoc procedure to construct an approximate confidence interval without the need to model the multivariate correlation structure between the indicators. The aim of the present paper is to bring under attention this pragmatic approach and to evaluate its performance in a practical setting. This is all the more needed since recent publications ignore the need for interval estimation. To illustrate and evaluate the method, Köhler et al. (1995) third molar scores are used to estimate the age in a dataset of 3200 male subjects in the juvenile age range.

Addendum

Summary of to estimate age for subject with ordinal scores Y _i  = Y_i1, Y_i2, Y_i3, Y_i4

• Step 1: Fit for each of the four ordinal scores (Y₁, Y₂, Y₃, Y₄) separately a continuation-ratio model in the training dataset.

• Step 2: Assuming a uniform prior distribution the age range 15 to 23 years, the posterior is proportional to the conditional distribution f(age|Y_i) ∝ P(Y_i|age). Assume conditional independence to calculate the multivariate distribution P(Y_i1, Y_i2, Y_i3, Y_i4|age)

  $$ P\left({\mathrm{Y}}_{\mathrm{i}1},{\mathrm{Y}}_{\mathrm{i}2},{\mathrm{Y}}_{\mathrm{i}3},{\mathrm{Y}}_{\mathrm{i}4}\Big|\mathrm{age}\right)=P\left({\mathrm{Y}}_{\mathrm{i}1}\Big|\mathrm{age}\right)\times P\left({\mathrm{Y}}_{\mathrm{i}2}\Big|\mathrm{age}\right)\times P\left({\mathrm{Y}}_{\mathrm{i}3}\Big|\mathrm{age}\right)\times P\left({\mathrm{Y}}_{\mathrm{i}4}\Big|\mathrm{age}\right) $$

  • Obtain the point estimate age_ML as the value of age maximizing this function.

  • Normalize this distribution to interpret a surface as a probability and to visualize the posteriors corresponding to different score patterns.

  • Obtain a (1-α)100 % PI as the (1-α)100 % values with the highest function value and/or obtain the API by applying expression (2).

  • The quantiles from the posterior (and thus also the PI and the API) are only valid if the conditional independence assumption holds.

• Step 3: In the training dataset, calculate the variance of the ${z}_i=\pm \sqrt{\left|2 \ln \left({\mathrm{LR}}_i\right)\right|}$, with LR _i  = f(age _i |Y _i )/f(age ^ML _i |Y _i ). Values of the variance higher than 1 refer to a violation of the conditional independence assumption.

• Step 4: Obtain the APIC for a specific pattern of four ordinal scores

  • In expression (2) replace f(age _i |Y _i )/f(age ^ML _i |Y _i ) by ${\left[f\left(\mathrm{age}\Big|{Y}_i\right)/f\left({\mathrm{age}}^{\mathrm{ML}}\Big|{Y}_i\right)\right]}^{1/var\left({z}_i\right)}$. The 95 % APIC is given by the range of all values of age for which the adapted expression (2) is smaller than or equal to 3.84.

  • Evaluate the cumulative distribution function of the standard normal at ${z}_i=\pm \sqrt{\left|2 \ln \left({\mathrm{LR}}_i\right)\right|}$ with $L{R}_i={\left[f\left(\mathrm{age}=b\Big|{Y}_i\right)/f\left({\mathrm{age}}^{\mathrm{ML}}\Big|{Y}_i\right)\right]}^{1/var\left({z}_i\right)}$ to obtain the probability that a subject with scores Y_i is younger than b years

• Step 5: Assure that the posterior distribution of age for a specific combination of scores Y_i can be interpreted as a probability density function in the age range considered, and calculate the APICn.

  • Calculate at each age ${z}_i=\pm \sqrt{\left|2 \ln \left({\mathrm{LR}}_i\right)\right|}$, with $L{R}_i={\left[f\left(\mathrm{age}\Big|{Y}_i\right)/f\left({\mathrm{age}}^{\mathrm{ML}}\Big|{Y}_i\right)\right]}^{1/var\left({z}_i\right)}$ instead of LR _i  = f(age _i |Y _i )/f(age ^ML _i |Y _i ) and its corresponding probability density.

  • Normalize the probability density values such that the total surface in the considered age range equals one.

  • Using the normalized values, calculate an alternative CI based on the (1-α)100 % values with the highest function value (APICn) and derive probabilities to be younger than a specific age