## Abstract

The synergistic effect of tobacco smoking and alcohol consumption on the risk of head and neck cancers has been mainly investigated as a cross-product of categorical exposure, thus leading to loss of information. We propose a bi-dimensional logistic spline model to investigate the interacting dose–response relationship of two continuous exposures (i.e., ethanol intake and tobacco smoking) on the risk of head and neck cancers, representing results through three-dimensional graphs. This model was applied to a pool of hospital-based case–control studies on head and neck cancers conducted in Italy and in the Vaud Swiss Canton between 1982 and 2000, including 1569 cases and 3147 controls. Among never drinkers and for all levels of ethanol intake, the risk of head and neck cancers steeply increased with increasing smoking intensity, starting from 1 cigarette/day. The risk associated to ethanol intake increased with incrementing exposure among smokers, and a threshold effect at approximately 50 g/day emerged among never smokers. Compared to abstainers from both tobacco and alcohol consumption, the combined exposure to ethanol and/or cigarettes led to a steep increase of cancer risk up to a 35-fold higher risk (95 % confidence interval 27.30–43.61) among people consuming 84 g/day of ethanol and 10 cigarettes/day. The highest risk was observed at the highest levels of alcohol and tobacco consumption. Our findings confirmed a combined effect of tobacco smoking and alcohol drinking on head and neck cancers risk, providing evidence that bi-dimensional spline models could be a feasible and flexible method to explore the pattern of risks associated to two interacting continuous-exposure variables.

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## Acknowledgments

This work was supported by grant from the Italian Association for Cancer Research and the Swiss League and Swiss Research against Cancer [Grants Number KFS-700 and OCS-1633]. Dr. Di Maso’s work was partially supported by a grant from Fondazione Umberto Veronesi. The authors wish to thank Drs. Renato Talamini, Silvia Franceschi, Luigi Barzan, Eva Negri, Paola Zambon, and Prof. Fabio Barbone for their help in study coordination, and Mrs Luigina Mei for editorial assistance.

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## Appendix: univariate logistic spline model

### Appendix: univariate logistic spline model

Let *y* = [*y*
_{1},…, *y*
_{
n
}]′ be a *n* × 1 vector of realizations of a random variable *Y* with binomial distribution so that *π* = *Pr*{*Y* = 1} and *x* = [*x*
_{1},…, *x*
_{
n
}]′ be a *n* × 1 vector of related realizations of a continuous predictor *X* of *π*. The additive logistic regression model:

where *f* is an arbitrary spline function, i.e. a piecewise polynomial truncated function defined as:

where *D* is the degree of polynomial, *t*
_{
k
} is the \(k\text{th}\) knot location, *K* is the number of knots, and

indicates a value of zero for negative values of the argument. The intercept *α* is absorbed into *f* and the continuity on knots is guaranteed by contrasts (·)_{+}. The log-likelihood function of the additive logistic regression model as:

where \(\beta = \left[ {\beta_{0} ,\beta_{1,1} , \ldots ,\beta_{1,D} ,\beta_{2,1} , \ldots ,\beta_{2,K} } \right]^{\prime}\) is the (*D* + *K* + 1) × 1 vector of parameters and \(\overset{\lower0.2em\hbox{$\smash{\scriptscriptstyle\smile}$}}{x}_{i} = \left[ {1, x_{i} ,x_{i}^{2} , \cdots ,x_{i}^{D} ,\left( {x_{i} - t_{1} } \right)_{ + }^{D} , \cdots ,\left( {x_{i} - t_{K} } \right)_{ + }^{D} } \right]^{{^{\prime} }}\) is the (*D* + *K* + 1) × 1 vector of covariates for the \(i\text{th}\) observation.

The maximum likelihood estimator of parameters \(\widehat{\beta }\) can be found solving:

The pointwise (1 − *α*) confidence band is \(\widehat{\beta }X \pm z_{{\left( {1 - {\raise0.7ex\hbox{$\alpha $} \!\mathord{\left/ {\vphantom {\alpha 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}} \left( {XVX^{\prime} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}\)where

is a (*D* + *K* + 1) × (*D* + *K* + 1) covariance matrix.

### Bi-dimensional logistic spline model

Let *x* = [*x*
_{1},…, *x*
_{
n
}]′ and *z* = [*z*
_{1},…, *z*
_{
n
}]′ be two *n* × 1 vector of related realizations of two continuous predictors *X* and *Z*. The corresponding spline function:

where, *D*
_{
x
} and *D*
_{
z
} are the degrees of the polynomials, \(t_{{k_{x} }}\) and \(t_{{k_{z} }}\) are the \(k_{x}\text{th}\) and \(k_{z}\text {th}\) knot location, *K*
_{
x
} and *K*
_{
z
} are the number of knots in the distribution, respectively, of *X* and *Z* continuous predictors,

and

indicate the zero value for negative values of the argument for *X* and Z respectively.

The parameters and the corresponding confidence intervals will be estimated through the methodology described above.

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Dal Maso, L., Torelli, N., Biancotto, E. *et al.* Combined effect of tobacco smoking and alcohol drinking in the risk of head and neck cancers: a re-analysis of case–control studies using bi-dimensional spline models.
*Eur J Epidemiol* **31**, 385–393 (2016). https://doi.org/10.1007/s10654-015-0028-3

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DOI: https://doi.org/10.1007/s10654-015-0028-3