the common scenario in time - to - event analysis is represented by a cohort of n individuals , free from a specific disease of interest d at time t = 0 , who are observed during a follow - up period to evaluate the disease - free survival . survival percentiles can be defined as the time points by which specific proportions of the study population have experienced the event d. for example , the time by which the first 50% of the individuals have experienced the event is defined as 50th survival percentile or median survival . the survival curve depicts a complete summary of the entire range of observed survival percentiles , presenting the proportion of events during the follow - up time . a common approach to evaluate the survival function is to fix a specific time t  usually the end of follow - up  and to estimate survival probabilities or rates of the event d in the time interval [ 0 , t ] , possibly according to levels of specific exposures or risk factors . in a percentile approach , on the other hand , the incidence proportion p is fixed to a specific level and the outcome to be evaluated is the corresponding survival percentile , the time t by which the study population reaches the specific fraction of events p. let g and e be two binary exposures , which can take values 0 or 1 , and are both risk factors for the event d. figure 1 presents a possible survival experience for the four combinations of the two exposures ( i.e. , g = 0 , e = 0 ; g = 1 , e = 0 ; g = 0 , e = 1 ; g = 1 , e = 1 ) . given a fixed proportion of events p , the pth survival percentiles for each of the four groups ( t00 , t10 , t01 , t11 ) are displayed in the figure . the difference ( t11 
  t00 ) represents the difference in the pth survival percentile between participants with both exposures and participants with neither . the quantities ( t10 
  t00 ) and ( t01 
  t00 ) are the differences between participants with only exposure g or e , respectively , and participants with neither exposure . following the conventional notation introduced in terms of risk , the following difference represents an intuitive measure of interaction at the pth percentile ( ip ) : survival percentiles . given a fixed fraction of cases , survival percentiles are defined as the time points by which different subpopulations reach the same proportion of events . the time points t00 , t10 , t01 , and t11 are the pth percentiles in each of the four possible combinations of two binary exposures g and e. this difference , calculated as an additive measure , can be rewritten as t11 
  t10 
  t01 + t00 and represents a measure of interaction on the metric of survival time . it expresses to what extent the difference in survival due to the presence of both exposures exceeds the sum of survival differences due to each specific exposure . comparing this measure with 0 , we can define the interaction as superadditive , if greater than 0 , or subadditive otherwise . unadjusted differences in survival percentiles can be obtained from the kaplan  meier method , deriving the confidence intervals of the differences via bootstrap . when adjustment for other covariates is needed , other methods to evaluate survival percentiles are required . laplace regression was introduced as a method for estimating the conditional percentiles of a potentially censored outcome , and can be used in time - to - event analysis to evaluate adjusted survival percentiles . differently from other survival analysis techniques , laplace regression directly models the percentiles of the time variable as a linear combination of the predictors . coefficients estimates can be interpreted as differences in time ( i.e. , years , months , days ) by which different subpopulations reach the same fraction of events . as any regression method , laplace can model the outcome ( i.e. , survival time ) as a function of multiple covariates , possibly including continuous exposures with flexible transformations . situations of heteroskedasticity can be accommodated by letting the scale parameter depend on one or more covariates . the simultaneous estimation and plotting of different percentiles might require some smoothing , depending on the variability across percentiles , and algorithms such as the lowess can be applied . nevertheless , this parametric assumption , shared by other methods in quantile regression , has been shown not to influence the performances of the model under different data distributions . simulations studies from previous articles have documented good performances of the model in terms of computational speed , precision , robustness of standard errors , and coverage of confidence intervals that was close to the nominal value . performance of the model was further improved after the introduction of a gradient search maximization algorithm , currently implemented for the estimation of the model . a flexible user - friendly program for the estimation of laplace regression , which makes use of this algorithm , is available in stata . recent developments have increased the potential of laplace regression in the epidemiologic context , presenting how to use the method to derive and draw adjusted survival curves , and to estimate percentile of attained age at the event of interest . this latter study presented the meaning and estimation of survival percentiles with different time scales , exploring the statistical properties of laplace in estimating percentiles of attained age , and discussing the advantages that this application may accrue in time - to - event analysis . to evaluate the impact of the two binary exposures g and e and their interaction on the pth survival percentile of the time variable t , we can fit the following model the time variable t is defined as time between entry into the study and experiencing the event d. an implicit assumption to allow interpretability of equation ( 2 ) is that parameters are constrained to keep the right - hand side of the equation positive . this should hold for all the observed , and possibly for all the potentially observable , combinations of covariate patterns . to improve interpretability , we prefer to code g and e so that their effects are in the same direction , that is , 1(p ) and 2(p ) have the same sign . from equation ( 2 ) , it is possible to estimate the pth survival percentiles for the four combinations of the two exposures , corresponding to the time points displayed in figure 1 . it simply follows that the measure of additive interaction ( 1 ) is estimated by the parameter 3(p ) . if 3(p ) > 0 , we are in the presence of superadditive interaction between g and e. if 3(p ) < 0 , the interaction is subadditive . the statistical test associated with the parameter 3(p ) in this situation , the interpretation of the product term coefficient as a measure of additive interaction is still valid when conditioning on the additional covariates . the fact that in a percentile approach the incidence proportion p is fixed at a specific value , as displayed in figure 1 , implies that the measure of additive interaction depends on p. this allows the study of interaction between two risk factors according to the fraction of events considered . we evaluated the interaction between smoking status ( 0 = current smoker ; 1 = never smoker ) and educational level ( 0 = primary education ; 1 = high school / university education ) in predicting overall mortality . we used data from the cohort of swedish men and the swedish mammography cohort , two large cohorts of ~80,000 men and women from central sweden , aged 4583 at baseline , established in 1997 and largely described elsewhere . after exclusions we considered in these analyses 71,238 participants who were followed up for 16 years between january 1 , 1998 , and december 31 , 2013 . during this period , an overall 23% of the study participants died ( n = 16,346 ) . because of the different fractions of cases across strata of the exposures , we focused our main analysis on the 10th survival percentile , the time by which the first 10% of subjects have died . first , we evaluated the following laplace regression model on the 10th percentile , with the two binary exposures and their interaction : predicted values of the 10th survival percentile for each of the four subpopulations , calculated by combining the obtained coefficients estimates , are presented in the table . the product term 3 , which estimates the additive interaction presented in equation ( 1 ) , has a simple and intuitive interpretation , as it represents the excess in survival when both predictors are equal to 1 . in our example , the presence of both predictors ( nonsmokers and highly educated ) was associated with 2.1 additional years of survival ( 3 = 2.1 years , 95% ci : 1.2 , 2.9 ) . this excess is larger than 0 , suggesting the presence of super additive interaction in predicting mortality between being nonsmoker and highly educated . tenth survival percentiles , in years , by levels of smoking and education we next adjusted for age at baseline as a 5-year categorical variable to evaluate the age - adjusted additive interaction between smoking and education in predicting overall mortality . this adjustment , which can be done simply by including the additional covariate in model ( 3 ) , strongly changed the estimate of the product - term coefficient to 0.8 years ( 95% ci : 1.4 , 0.1 ) . this suggests that the crude interaction was probably explained by the different distribution of age across strata of the two exposures . in particular , the median age at baseline ranged from 53 years for current smokers with high education to 66 years for never smokers with low education . the evaluation of additive interaction can be extended to other fractions of events rather than the first 10% . for example , we fitted age - adjusted laplace regression models to evaluate the interaction between smoking and education for percentiles 1 to 15 . this was done by evaluating model ( 3 ) , further adjusted for age at baseline , varying p from 1 to 15 . figure 2a presents the survival curves for the four combinations of smoking and education , further adjusted for age at baseline . the estimated coefficients of the interaction term from these models , with confidence interval , are reported in figure 2b . when looking at the first percentiles , which represent the early cases , the interaction is positive , later decreasing to a subadditivity . a , age - adjusted survival curves by levels of smoking ( current , never ) and education ( high , low ) . curves are calculated from age - adjusted laplace regression models , with fraction of events between 1 and 15 . b , the estimates of the interaction term , with confidence interval , smoothed by applying the lowess algorithm with a bandwidth of 0.6 . in this study , we introduced the topic of interaction in the context of survival percentiles . in a percentile approach to time - to - event outcomes , a measure of additive interaction can be estimated using laplace regression to model conditional survival percentiles , including a product term between two exposures of interest in the model . the regression coefficient of the product term represents the excess / decrease in survival due to the presence of both the exposures of interest . evaluating . a detailed tutorial , summarizing the wide discussion on the topic , has recently been published . . statistical interaction , which is the focus of this article , arises from a statistical model and should not be used to draw biologic conclusions . statistical interaction is usually assessed by including in the model a product - term between the two exposures of interest . this implies that the evaluation and interpretation of interaction depend on the scale of the model chosen , which can be either additive or multiplicative . in the context of time - to - event analysis , the multiplicative nature of the cox proportional - hazard model implies that interaction analysis is commonly limited to the multiplicative scale . various studies have underlined the important public health meaning of additive interaction , which can be used to assess which subgroups of individuals are to be treated . presenting both additive and multiplicative measures of interaction has been widely recommended , but this practice is still very uncommon . moreover , the absence of interaction on one scale implies the presence of interaction on the other scale . in survival analysis , measures of additive interaction such as the relative risk due to interaction , the synergy index , or the attributable proportion due to interaction can be calculated after fitting a cox model . among these , the relative risk due to interaction is the most frequently utilized approach , even if it can only be used to assess the direction of the additive interaction without any indication on the magnitude . as an alternative approach , the use of the additive hazard regression has been proposed . to the best of our knowledge , however , this method is seldom used in epidemiologic research . when dealing with time - to - event outcomes , the risk of the event of interest often varies across follow - up . all the available measures of interaction , whether additive or multiplicative , are calculated on the risk scale , and commonly assume a time - fixed effect . evaluating percentiles of survival represents a possible alternative for the analysis of time - to - event outcomes . common approaches fix the follow - up time to a specific value and evaluate the risk of the event within that time interval . by focusing on survival percentiles , on the other hand , a specific survival probability is fixed and the time by which that proportion is achieved is estimated . this change of perspective clarifies the connection between incidence proportion and time , and allows evaluating how the joint effect of two exposures is changing over time . in this article , we have chosen to use laplace regression to model survival percentiles because of its stability and computational speed . an option often used in epidemiologic studies is to derive the adjusted survival curves after estimating a cox model , and calculating the quantiles . this method , however , is computationally demanding and unclear with respect to standard error derivation . assumptions such as proportionality of hazards strongly influence the shape of the derived survival curve . other methods to evaluate quantiles of censored outcomes have been proposed and are available in r and sas . these methods make different assumptions and are valid semi - parametric alternatives to laplace regression . in this article , we showed how a measure of additive interaction expressed in the metric of time can be derived by evaluating survival percentiles . an advantage of the presented approach is that the additive interaction depends on the specified fraction of events . within a follow - up time , it is therefore possible to investigate how the interaction between two exposures changes according to the fraction of cases , or equivalently by time . currently available methods typically provide a single product - term coefficient , implicitly assuming that main effects and their interaction are constant over the entire follow - up time . , we introduced the concept of interaction analysis in the metric of time , which can be investigated by switching the focus from survival probabilities to survival percentiles . when laplace regression is used to model survival percentiles , a measure of additive interaction
evaluation of statistical interaction in time - to - event analysis is usually limited to the study of multiplicative interaction , via inclusion of a product term in a cox proportional - hazard model . 
 measures of additive interaction are available but seldom used . 
 all measures of interaction in survival analysis , whether additive or multiplicative , are in the metric of hazard , usually assuming that the interaction between two predictors of interest is constant during the follow - up period . 
 we introduce a measure to evaluate additive interaction in survival analysis in the metric of time . 
 this measure can be calculated by evaluating survival percentiles , defined as the time points by which different subpopulations reach the same incidence proportion . using this approach , 
 the probability of the outcome is fixed and the time variable is estimated . 
 we also show that by using a regression model for the evaluation of conditional survival percentiles , including a product term between the two exposures in the model , interaction is evaluated as a deviation from additivity of the effects . in the simple case of two binary exposures , 
 the product term is interpreted as excess / decrease in survival time ( i.e. , years , months , days ) due to the presence of both exposures . 
 this measure of interaction is dependent on the fraction of events being considered , thus allowing evaluation of how interaction changes during the observed follow - up . 
 evaluation of interaction in the context of survival percentiles allows deriving a measure of additive interaction without assuming a constant effect over time , overcoming two main limitations of commonly used approaches .