On 11 February 2016, the RSS Medical Section and Primary Healthcare Special Interest Group held a joint meeting at the Royal Statistical Society to celebrate the award of the Bradford Hill Medal to Professor Tim Cole. The Bradford Hill Medal is awarded to statisticians who have made an outstanding contribution to Medical Statistics.
Tim’s contributions have been numerous, notably his LMS (lambda-mu-sigma) method for fitting centile curves to summarise the relationship between height and age, which has been widely adopted for the development of national and international growth charts in children. Thus this afternoon meeting was based around ‘Recent developments in growth assessments: the LMS method, GAMLSS and beyond’.
Tim kicked off the afternoon by introducing the LMS method with a whistle-stop tour of its development. He introduced growth centile charts and four important properties affecting their estimation (extrapolation, nonlinearity, heteroscedasticity and skewness), which the LMS method addressed. Tim gave credit to the seminal Box-Cox transformation paper (JRSS Series B, 1964) and van’t Hof’s paper on age references for skinfold data (Human Biology 1985), on which he based his initial ideas. Tim wrote this method up as an article for the RSS Series A journal in 1988, which was presented for the Royal Statistical Society as a ‘read paper’. Based on Peter Green’s suggestion (as a read paper discussant), the two published an extension of the method estimated by maximum penalised likelihood (Stat Med, 1992). Tim then went on to publish extensively about the LMS method in clinical journals so that it would be widely adopted and to develop the British 1990 growth reference centiles.
Tim further discussed how the L, M and S curves themselves can be informative, and how not accounting for kurtosis may only affect estimates at the extreme centiles. In response to a question, he suggested that a sample size of around 2,000 is adequate for fitting growth centiles for ages 0-20.
Next, Professor Mikis Stasinopoulos presented his extension of the LMS method to GAMLSS (Generalized Additive Models of Location, Scale and Shape), where one need not assume normality of the y-values as with the LMS method and where one may even include a parameter for kurtosis of the data. Generalized linear models and link functions are employed for some transformed variable Z, eg, a power exponentially distributed Z is assumed for the methods adopted by the World Health Organization. These GAMLSS models can be fitted using the gamlss() function in R, and to date up to 80 different distributions can be assumed (described in more detail on the GAMLSS website). Mikis demonstrated the use of gamlss() in data on lung function, assuming four different distributions, and assessing fit using fitted Q-statistics and worm plots.
Gichuru Phillip explained two scoring approaches for the binary responses of normally developing children to the Malawi Development Assessment Tool (or ‘MDAT’). The first item by item approach finds the age at which there is a given probability of passing an item using flexible Generalized Additive Model (GAM) methods to create reference charts. The objective is to give age estimates at which a 'normally developing child' is likely to pass an item given their age. The second scoring approach is a total score approach that uses all the binary responses across items to give a holistic score. While demonstrating the strong correlation between 'naive' total scores and age, Phillip explained how the GAMLSS framework can be used to: firstly obtain the expected score that a normally developing child of a given age should achieve, and secondly, eliminate the effect of age using Z-scores computed using smoothed mean and standard deviations. Phillip used three samples of ‘normal’, malnourished, and disabled children to demonstrate the elimination of the age effect so that comparisons between children of different ages could be made, for example, in a clinical trial. Receiver Operating Characteristic curves indicated good sensitivity of smoothed Z-scores using the GAMLSS model.
Finally, Professor Stef van Buuren described why he personally liked the LMS method, as well as its application in creating reference charts for several growth-related variables in Dutch children (anthropometric measures, Tanner stages, developmental scores, etc). Stef illustrated how worm plots and Q-Q plots were used to assess the fit of growth curves in these data (Stat Med, 2001). He then explored questions that interest health professionals and parents about child development, which growth charts can answer. For example, for 'given what I know of the child, how will he/she develop in future?', he demonstrated how predictions may be made by matching a child’s current growth trajectory to those of other similar, but older, children. Stef ended his talk by describing the ‘broken stick’ model, which fits linear B-splines to Z-values, with knots placed at particular ‘break’ ages, which can be fit using lmer(), but for which there is also an R package broken stick in development.
Following the talks, small-group discussions addressed several questions posed by the speakers. After such a series of engaging talks on the LMS method and its developments, it’s perhaps not surprising that discussants were clearer on the pros of LMS relative to quantile regression – although proponents of quantile regression would surely be able to advise on its advantages.
Tim reassured participants on one concern raised about LMS, explaining that the use of longitudinal data does not present independence issues since LMS methods estimate models for each age group, where each subject is only included once per group. It was considered that confidence intervals for the LMS models were not generally of interest, but that prediction intervals (similar to those presented by Stef) could be useful when monitoring the development of individuals, provided sufficient care was taken to identify appropriate matches and to explain their use and meaning.
The discussions also highlighted the need for a gamlss(), or similar, implementation in statistical software packages other than R.