In addition to Simon Wood's reading of the methods I am adding a minor note on terminology:

What was reported was the ratio in risk per a difference equal to the interquartile range (of the predictor variable), and the 95% CI for such an estimate. They were not reporting a 95% CI for the interquartile range (of anything). Since it was a linear estimate, it would apply anywhere along the range of observed values of the predictor.

Followed by a cautionary note on methods;

Further I would question their approach to construction of that estimate. They apparently did a simple multiplication of the beta term by the IQR range and called that the "per cent change per IQR". They should have used predict.gam() at the locations of the IQR to get risk predictions since the beta coefficient is estimated on the log-risk scale.

I have seen similar errors in reporting results from Cox models and the justification given was "the SAS manual said I could do that". It is true that the effects of predictors can be linearly approximated by simple transformations of beta coefficients, but it is not true that they can be extended across a range as large as that offered by the IQR. Not everything that appears in the published literature reflects good statistical practice.

--
David.

On Feb 23, 2011, at 9:40 AM, clc wrote:


In one of the papers...

We developed core models with a generalized additive Poisson regression allowing for over-dispersion in the model (Wood, 2006). For each mortality
outcome, variations in seasonality, trends, mean temperature, and mean
humidity of current and previous days (lag 0–1) were fitted with penalized
cubic regression splines. Dummy variables were used to control the
variations for days of the week, holidays, and influenza epidemics. We added
a dummy variable for the 2003 severe acute respiratory syndrome (SARS)
epidemic. We chose 4 degrees of freedom (df) per year for smoothing function of the trends and 3 df for temperature and humidity. The choice of df for each smoothing function in the core models was made on the basis of observed residual autocorrelations using partial autocorrelation function (PACF). For
the core models fitted to the mortality data, time variant confounding
factors were considered as adequately controlled if absolute values of PACF
coefficients were <0.1 for the first two lag days and there were no
systematic patterns in the PACF plots.

Following the construction of an adequate core model for each mortality outcome, we entered visibility as a linear term into the regression model and examined the effects of visibility on mortality for single day lags 0–5 days, lag 0–1, and distributed lag 0–4 days ([Schwartz, 2000] and [Zanobetti et al., 2000]). The distributed lag effect take into account the possibility that visibility can affect deaths occurring on the same day and on several subsequent days. The net effect of visibility was the sum of the effect estimates for all six days. We expressed the effect of visibility as the percentage change in daily mortality with a decrease in the interquartile range (IQR) of visibility as 100%×IQR×β, where β is the estimated Poisson
regression coefficient, and referred to as the excess risk (ER%).



in one of the figures, they reported "Estimated excess risks (ER%) for daily mortality and associated 95% confidence intervals per interquartile range decrease in visibility (6.5 km) at single lags 0–5, mean lag 0–1 (0–1) and
distributed lag (DL) for lag 0–4 days"


What do they mean??! Thanks a lot!
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