Journal of Clinical Epidemiology 65 (2012) 855e862Visualizing interaction effects: a proposal for presentationand interpretation

Claudia Lamina*, Gisela Sturm, Barbara Kollerits, Florian KronenbergDivision of Genetic Epidemiology, Department of Medical Genetics, Molecular and Clinical Pharmacology, Innsbruck Medical University,

Schopfstr. 41, A-6020 Innsbruck, Austria

Accepted 15 February 2012; Published online 30 May 2012AbstractObjective: Interaction terms are often included in regression models to test whether the impact of one variable on the outcome is modi-fied by another variable. However, the interpretation of these models is often not clear. We propose several graphical presentations andcorresponding statistical tests alleviating the interpretation of interaction effects.

Study Design and Setting: We implemented functions in the statistical program R that can be used on interaction terms in linear,logistic, and Cox Proportional Hazards models. Survival data were simulated to show the functionalities of our proposed graphical visu-alization methods.

Results: The mutual modifying effect of the interaction term is grasped by our presented figures and methods: the combined effect ofboth continuous variables is shown by a two-dimensional surface mimicking a 3D-Plot. Furthermore, significance regions were calculatedfor the two variables involved in the interaction term, answering the question for which values of one variable the effect of the other variablesignificantly differs from zero and vice versa.

Conclusion: We propose several graphical visualization methods to ease the interpretation of interaction effects making arbitrarycategorizations unnecessary. With these approaches, researchers and clinicians are equipped with the necessary information to assessthe clinical relevance and implications of interaction effects. 2012 Elsevier Inc. All rights reserved.

Keywords: Interaction; Visualization; Categorization; Cox models; Linear models; Logistic models1. Background

In epidemiology, interaction terms are often incorpo-rated into multivariable regression models to test whetherthe impact of one variable on the outcome is modified byanother variable. However, the interpretation of interactioneffects is often not clear. It depends on the scale of the vari-ables included in the interaction term (continuous and/orcategorical) and the scale of the model (linear regression,logistic regression, Cox Proportional Hazards model). Theinterpretation is rather straightforward, if interaction effectsbetween the two categorical variables or between one con-tinuous and one categorical variable are considered. Onesuch example would be an effect that can only be seen inmen, but not in women, in Europeans but not in Asiansetc. For such situations, there are also graphical solu-tions implemented in standard statistical programs [1]. If* Corresponding author. Tel.: 43-512-9003-70565; fax: 43-512-9003-73561.

E-mail address: claudia.lamina@i-med.ac.at (C. Lamina).

0895-4356/$ - see front matter 2012 Elsevier Inc. All rights reserved.doi: 10.1016/j.jclinepi.2012.02.013researchers are interested in interaction between two con-tinuous variables, the interpretation is less clear. It seemsthat interaction terms between continuous variables areavoided or continuous variables are categorized before-hand. If significant P-values of continuous interaction termsare reported, most authors switch to categorizations of theseformer continuous variables for interpretation purposes. Ingeneral, categorizations of continuous variables are oftendone rather arbitrarily by choosing percentiles of the re-spective interacting variables, for example, by dichotomiz-ing using the median [2]. Categorization of continuousvariables leads to significant loss of information and power,though [3]. Sometimes, clinically recognized and prede-fined cutpoints are chosen, such as a body mass index ofO30 kg/m2 for the definition of obesity. Even in this case,information is lost because the definition of these cutpointshas been designed for a totally different purpose. This ap-proach can lead to considerable bias. The problem evenworsens, if both continuous variables that constitute to aninteraction term are categorized [4].

In other cases, regression coefficients andP-values of inter-action terms are shown, but the reader is left alone with the

Delta:1_given nameDelta:1_given nameDelta:1_given nameDelta:1_surnameDelta:1_given nameDelta:1_given nameDelta:1_given nameDelta:1_surnameDelta:1_given nameDelta:1_given nameDelta:1_given nameDelta:1_surnamemailto:claudia.lamina@i-med.ac.athttp://dx.doi.org/10.1016/j.jclinepi.2012.02.013http://dx.doi.org/10.1016/j.jclinepi.2012.02.013

856 C. Lamina et al. / Journal of Clinical Epidemiology 65 (2012) 855e862What is new?

1. Epidemiological publications on interaction ef-fects in survival analyses mostly use categoriza-tions of variables for the ease of interpretation.This approach might lead to considerable bias,though.

2. To ease the interpretation of interaction effects be-tween two continuous variables, we proposea graphical visualization approach which we im-plemented in functions in the statistical programR. They can be used on Generalized LinearModels and Cox Proportional Hazards models.

3. The mutual modifying effect of both variablesconstituting the interaction effect can be shownby a two-dimensional twisted surface and interac-tion plots showing the effect estimator of one vari-able for varying values of the other variable.interpretation. The effects and P-values of the main effectsare also not interpretable without taking into account theconcurrent levels of the other constituting variable [5].

By including multiplicative interaction terms, condi-tional hypotheses are tested: An increase in one variableis associated with an increase in the outcome variable,when the second interacting variable is equal to a specificvalue. Inevitably, all constitutive terms of an interactionterm should be included in the regression model also indi-vidually. With the inclusion of an interaction term, theseformer average effects are conditional effects then andchange just by definition. Therefore, changes in effectsand/or P-values after including interaction terms cannotbe interpreted usefully. The conditional nature also appliesfor the P-values of the conditional effects, which are givenin the output tables of statistical programs: they test,whether the slope of one variable is significantly differentfrom zero, if the other constituting variable is zero. In mostcases, this test is not very useful, depending on the scale ofthe variable.

The typical reader will not make the necessary linearcombinations to calculate the effect of the first variablefor specific values of the second variable. It would be pos-sible, though, given the typical output table of a regressionmodel, including effect estimates, standard errors, andP-values. However, it is not possible to conclude on the sta-tistical significance of such a linear combination.

Therefore, it is hardly possible to interpret interactioneffects simply by looking at the output table from regres-sion models. The results of interaction terms have to be pre-sented in a different, more reader-friendly way.

Such examples can only rarely be found. We conducteda literature search on survival analyses publicationsincluding interaction terms between two continuous vari-ables (see Appendix on the journals Web site at www.jclinepi.com). It revealed that only one study group usedthe complete range of both continuous variables to interpretthe interaction effects: The authors presented interactionplots, showing the modifying effect of waist circumferenceon the relationship between several parameters (cholesterol,leptin, and adiponectin) and mortality [6,7].

The lack of such forms of presentations is certainly be-cause of unfamiliarity with the methodological backgroundand lack of easy to use methods. Therefore, such methodsare required in the clinical epidemiological setting to avoidcommon misinterpretations in interaction models.

To ease the interpretation of interaction effects betweentwo continuous variables, we propose several possibilitiesfor graphical presentation, which we implemented in func-tions in the statistical program R: These functions havebeen implemented for linear regression, logistic regression,and Cox Proportional Hazards models. To demonstrate thefunctionalities, one data set including survival data hasbeen simulated and analyzed in Cox models including lin-ear interaction effects between two continuous predictors.2. Methods

2.1. Formulating and interpreting multiplicativeinteraction terms in linear, logistic, and Cox models

For explanatory purposes, a linear regression model isassumed first. Interaction terms between two continuousvariables x1 and x2 on the continuous outcome variable Ycan be modeled in the following way:EYjx1;x25Xb5b0 b1x1 b2x2 b3x1 x21with b0 being the intercept, b1 and b2 the conditional ef-fects of x1 and x2 and b3 being the interaction effect onthe outcome Y. The linear combination on the right sideof the equation is the linear predictor Xb, to which furthercovariates can be added. The regression coefficients for x1and x2 are conditional effects because they depend on thevalues of the other constituting variable:

A one unit change of x1 corresponds to a change ofb1 b3x2 on Y.

A one unit change of x2 corresponds to a change ofb2 b3x1 on Y.

This means that b1 itself is the effect of x1 on Y, if x2 isequal to zero and vice versa. Thus, the interpretation of re-gression coefficients cannot be done without taking thelevels of the interacting variable into account. Accordingto Rothman [8], the statistical term interaction is usedto refer to departure from the underlying form of a statisticalmodel. For a linear model, a significant interaction termthus implies deviation from additivity, meaning the additiveeffect of the individual effects b1x1 b2x2 on the outcome.

http://www.jclinepi.comhttp://www.jclinepi.com

857C. Lamina et al. / Journal of Clinical Epidemiology 65 (2012) 855e862The interpretation based on the direction of effect estimatesis only unambiguous, if both the conditional effect, for ex-ample of x1, and the interaction effect have the same effectdirection. In this case, the absolute value of the overall x1effect increases for increasing values of x2. If all effect es-timates are positive, this implies that the interaction leads tomore than an additive effect. If the interaction effect andconditional effect have opposing algebraic signs, the condi-tional effect is attenuated by increasing values of the otherinteracting variable and can even be switched into the otherdirection, depending on the size of the corresponding esti-mates. Obviously, interpretation of interaction effects basedon the table of regression coefficients alone is hard to ac-complish and requires at least some linear combinationsto be made.

The corresponding P-values are interpreted accordingly:They also reflect the x1 effect on Y for x25 0 and vice ver-sa. This may or may not be useful depending on the scaleand ranges of the variables. One may want to test, however,whether the effect of x1 significantly differs from zero givena specific value of x2. The equation for this simple slopestest requires the regression coefficients to be known andthe corresponding variance-covariance matrix [5], whichcan be given by the estimated regression model.

If you want to test the effect of x1 for a specific x2 value,the slope to be tested would be given by:slopex15b1 b3x2 2

and the corresponding standard error byseslopex15ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis11 2 x2s13 x2s33

p3with s11 being the variance of b1, s33 the variance of b3, ands13 the covariance of b1 and b3.

Given the slope and its standard error, a t-test can thenbe constructed:t5slopex1seslopex1

|tn k 1 4with n being the number of cases and k the number of vari-ables not including the intercept. The simple slopes test ofx2 effects for specific x1 values can be derived likewise.

The interpretations and formulas on effect estimatesgiven in this paragraph can be generalized to logistic re-gression or Cox models by substituting the expectation ofthe outcome variable by a general function. The linearpredictor Xb5b0 b1x1 b2x2 b3x1 x2 is then mod-eled by a monotone link function as the logit functionlnp=1 p for logistic regression models, with p beingthe probability of the outcome. In this case, the intuitiveeffect measure is the odds ratio (OR), which is estimatedby ORi5 exp(bi) for variable xi. Thus, the regression func-tion can be rewritten to p=1 p5expb0 expb1x1expb2x2 expb3x1 x2. On this scale, a significantinteraction effect would imply a deviation from multiplica-tivity of individual effects [9], in contrast to interactioneffects in a linear model, which shows departure fromadditivity.

The Cox Proportional Hazards model [10] models thetime to a specific event such as death accounting for cen-sored observation. The hazard function h(t) is the instanta-neous probability of an event at time t given that theindividual has survived until time t and is defined asfollows:ht5h0texpXb 5with h0(t) being the underlying baseline hazard, which can-not be estimated. For each of the covariates in the linear pre-dictor, the factor exp(bi) gives the so-called hazard ratio(HR), which is a measure of relative risk. It characterizesthe shift in the hazard function as a result of a shift of oneunit in the corresponding variable xi. The interpretation oninteraction effects on the linear predictor Xb in a Cox modelcan be done analogical to the linear regression model. ForCox models, it might be more meaningful to look at thechange in HR, though. As for logistic regression models,on the scale of the relative risk estimate, the inclusion of in-teraction terms tests the deviation from multiplicativity.2.2. Description of the simulated survival data setincluding interaction effects

We simulated survival data to explain the interpretationof interaction effects and show the utilities of our graphicalpresentations in a Cox model. One thousand right-censoredobservations were generated with three continuous and nor-mally distributed variables with mean 0 and variance 1. Inaddition, one N(0,1)-distributed nuisance parameter hasbeen included, that is ignored in the following Cox model.Therefore, the linear predictor was constructed as follows:Linear predictor Xb5 2 x1 0 x2 1 x1 x2 1 x3 N0;1The effect estimates were chosen to represent two possi-ble interaction scenarios: a strong main effect that attenu-ates for increasing values of the other interacting variableis reflected by b1, whereas b25 0 reflects an effect, thatwould not be observed in a model not including an interac-tion term because the x2 effect even changes direction forvarying values of x1.

The simulation was performed using function Sim-surv of the library prodlim in R [11]. For each observa-tion, a Weibull distributed survival time was generated withbaseline risk set to 0.2. The censoring time was simulatedto follow an exponential distribution with a baseline censor-ing risk set to 3. The baseline risks for censoring and sur-vival were chosen to yield a high censoring rate. Theminimum of the two simulated times was taken as observa-tion time. The event indicator was set to 1, if the survivaltime was shorter than the censoring time, otherwise the in-dicator was set to 0.

858 C. Lamina et al. / Journal of Clinical Epidemiology 65 (2012) 855e8622.3. Implementation of the graphical utilities andcalculations in R

Several functions have been implemented in R to allevi-ate the interpretation of multiplicative interaction effects.They can be downloaded from http://www3.i-med.ac.at/genepi/ under Analysis tools. First, a generalized linearmodel (GLM) or Cox model has to be fitted using thefunction glm or cph (from the Design library). Then, thefunction PlotInteraction3D (or PlotInteraction3DCox) cal-culates predicted values of the linear predictor for eitherthe GLM or Cox model for given ranges of both variablesconstituting the interaction term. Additional covariates andalso factor variables can be included in the models. By de-fault, mean values are chosen for continuous covariates. Forthe factor variables, one factor level has to be chosen, forwhich predicted values should be calculated. The resultingmatrix of predicted values is spanned as a two-dimensionalsurface into a three-dimensional grid. Alternatively, a col-ored contour plot can be given out. Consecutively, two-dimensional slices can be cut out of this plot for specificvalues of one of the constituting variables (function Plo-tInteraction2D). Each slice is the linear predictor of theregression model for varying values of one variable keepingthe other variable fixed. Assuming a linear model, for ex-ample, this corresponds to the predicted outcome. Settingx2 to the fixed value c, it can be derived from formula (1)by y5b0 b1x1 b2c b3cx1. The function PlotInterac-tion2D uses the R-function predict, which gives outthese fitted values and the corresponding standard errors(se.fit). In addition, point-wise 95% confidence intervalsare calculated for each value of, for example, x1 byy6z0:975 se:fit, which are plotted for the complete rangeof x1 values, with z0:975 being the 97.5% quantile of the nor-mal distribution (|1.96). It is recommended to look at thoseplots for a set of values spanning the range of the variable,for example, for the 5th, 25th, 50th, 75th, and 95th quantile.By means of the functions getRegion or getRegion-Cox, significance regions can be calculated. For a givenrange of values for x1, for example, the slope of the x2 vari-able with the corresponding P-value is printed. An interac-tion effect plot [12] is also given out, which shows how theeffect of one variable changes with the changing values ofthe other variable. This corresponds to plotting theeffect estimate of one variable (slopex1) for a given valueof the other variable as in formula (1). The respectivepoint-wise 95% confidence intervals for the x1 effectsfor given values of x2 are derived from (1) and (2) byslopex16z0:975 seslopex1. Both, effect estimates for x1

Table 1. True and estimated effect estimates of the conditional and interac

Variable True effect, b Estimated effect, b se x1 2 1.8834 0.10x2 0 0.0227 0.09x1 * x2 1 1.0378 0.08x3 1 0.8826 0.07

Abbreviations: HR, hazard ration; CI, confidence interval.and corresponding confidence intervals are plotted for vary-ing values of x2 over their complete range. For logistic andCox models, OR or HR can be given alternatively by expo-nentiating the effect estimates and the corresponding confi-dence intervals. Thus, two figures, one for each of the twointeracting variables, are sufficient to summarize the inter-action effect over the complete range of both continuousvariables.3. Results

Eighty-two percent of the simulated observations arecensored. The results of the simulated Cox model are givenin Table 1. The risk experiencing an event is significantlydecreasing for increasing values of x1 (HR5 0.1521,P! 0.001), if x25 0. If x15 0, x2 does not have an effecton the hazard. Other implications that can directly be drawnfrom the numbers in the table, are that there is a significantinteraction effect, meaning, that the effects of x1 and x2 varyfor varying values of the other variable. Other slopes can becalculated by linear combinations of the effect estimates:The effect estimate of x2 for specific x1 values can be cal-culated by 0.0227 1.0378*(x1 value). For x150.6456(525% quantile), this would be 0.6473, for x15 1.772(595% quantile), it is 1.862. For the calculation ofP-values, however, specific covariances are needed fromthe output of the model. Thus, they cannot be deriveddirectly from Table 1, but can be given out by applyingthe test given in formula (4) implemented in the functiongetRegionCox: P5 4.4e 14 (for x150.6456) andP! 10e 20 (for x15 1.772).

A rough overview of the impact of the interaction can begiven by the twisted surface in Fig. 1: The highest risk canbe found at consecutive low values of both variables. Thevariable x1 does have a negative impact on the risk forlow values of x2. For increasing values of x2, the x1 effectis attenuated. For low values of x1, the variable x2 is posi-tively associated with the outcome variable, which is timeto event in this example (increasing slope), whereas x2 is in-versely associated for high values of x1 (decreasing slope).The surface is twisted, indicating that the direction of the x2effect even changes direction for varying values of x1. Al-ternatively, a colored contour plot (Fig. 2) can depict thesame information: The highest risk can be found in areas,which are displayed in pink (both x1 and x2 low), whereasthe risk decreases for darkening blue color.

Fig. 3 shows slices through the three-dimensional plotfor specific values of x1 or x2. The left panel shows the x1tion effects in the simulated Cox model

b HR 95% CI of HR P-value92 0.1521 [0.1228; 0.1884] !0.00176 1.023 [0.8449; 1.2386] 0.81683 2.823 [2.3744; 3.3564] !0.00177 2.417 [2.0757; 2.8148] !0.001

http://www3.i-med.ac.at/genepi/http://www3.i-med.ac.at/genepi/

Fig. 1. Interaction surface plot on the linear predictor of the hazardfunction. The z-axis shows the linear predictor of the hazard functionfor varying values of x1 and x2, illustrating their interacting effect onthe simulated event variable.

Fig. 2. Contour plot displaying the linear predictor of the hazard func-tion. The linear predictor of the hazard function for varying values ofx1 and x2 is displayed in different colors, ranging from blue (low risk)to pink (high risk).

859C. Lamina et al. / Journal of Clinical Epidemiology 65 (2012) 855e862effect, given x2 is equal to the 5th, 25th, 75th, and 95thquantile with its corresponding confidence intervals. Theright panel shows the x2 effects for the same given valuesof x1. These plots give the impression of flipping througha flip book, showing the twisting of effects in mutual de-pendence of the other constituting variable. Significance re-gions can be given out by the function getRegionCox.The x1 effect is significantly different from zero for x2values 1.55 (594.1% quantile). The x2 effect is signifi-cantly different from zero with a negative slope for x1values !0.19 (542.16% quantile) and significantly dif-ferent from zero with a positive slope for x1 valuesO0.17 (555.9% quantile). The interaction plots in Fig. 4show the HRs for x1 for varying x2 values with point-wise confidence intervals for each plotted value. It can beseen that the HR is significantly lower than 1 for the vastmajority of x2 values. The significance region can also beread from this plot. The x2 effect changes from an HR thatis smaller than 1 to higher than 1 almost exactly at themedian of x1 values. This plot can be printed for the linearpredictor, which is the default option, but also for the HR,which has been chosen here in this example.4. Discussion

We have implemented functions in R providing severaldifferent graphical presentations and corresponding statisti-cal tests to ease the interpretation of interaction effects be-tween continuous variables. We have successfully appliedone of these approaches in a prospective study on inci-dent dialysis patients, which was very useful for theinterpretation of an interaction effect: we observed a signif-icant interaction between time-varying values of albuminand phosphorus on overall mortality in our cohort [13]:for high albumin values, increasing phosphorus concentra-tions were significantly associated with an increased mor-tality risk, whereas the beneficial effect of increasingalbumin values was only observed for low phosphoruslevels. We concluded that decisions about phosphorus-lowering therapy should also take albumin values into ac-count and epidemiological studies and guidelines shouldconsider this interplay. The proposed graphical presentationand the calculation of the significance regions helped in thediscussion with clinicians and interpretation of results.4.1. Categorizing continuous variables in interactionterms: pros and cons

Most of the interaction models between continuous vari-ables that are reported are broken down to subgroup analysisand/or categorizations of at least one of the two continuousvariables. A qualitative assessment of epidemiological pub-lications [2] found that 86% of papers evaluating continuousrisk factors on health outcomes categorize these risk factors.Most of them use quantiles, others equally spaced categories(e.g., 10-year-age categories) or external criteria, such asclinically respected cutpoints. The reasons for the catego-rization approaches were not mentioned in most of the pub-lications. There seems to be a gap between statisticalcapabilities and the ability to interpret interaction effectsin a meaningful way. Therefore, simplicity of interpretationmight be the most crucial point. However, this simplicity is

Fig. 3. Slices through the interaction surface plot for specific values. The y-axis shows the linear predictor of the hazard function for varying valuesof x1, holding x2 fixed at specific values (5%, 25%, 50%, 75%, and 95% quantiles; left panel) and the linear predictor of the hazard function forvarying values of x2, holding x1 fixed at specific values (5%, 25%, 50%, 75%, and 95% quantiles; right panel).

860 C. Lamina et al. / Journal of Clinical Epidemiology 65 (2012) 855e862gained at the cost of power loss. Dichotomization of data iseven equal to discarding one third of the data [3]. Using cut-point models in epidemiology is in best cases just unrealisticand conservative. Considerable variability in the categoriesis lost and artificial steps in risk are created. In regressionmodels where both originally continuous variables are

Fig. 4. Interaction plot. The y-axis gives the hazard ratio (HR) with point-wise 95% confidence intervals for x1 on mortality for varying values of x2(on the left) and for x2 for varying values of x1 (on the right).

861C. Lamina et al. / Journal of Clinical Epidemiology 65 (2012) 855e862dichotomized, however, there is not only the chance fora conservative error (i.e., power loss), but the probabilityof false positive findings might also be increased dramati-cally [4]. In situations, where both explanatory variablesare correlated with each other, but the partial correlation be-tween one variable and the outcome conditioning on theother continuous variable is close to zero, dichotomizationleads to bias: the effect of this variable, which would notbe associated in a regression model including both variablescontinuously, is then overestimated. Spurious significant in-teraction can also occur in such a situation. An additionalnonlinear effect in one of the explanatory variables createsthe illusion of interaction after dichotomization.

The extent of bias worsens the higher the continuousvariables are correlated. In epidemiological studies, thereis hardly a situation, where two explanatory variables inone model are not correlated with each other. These find-ings are also true for categorizations with more than twogroups [14] or if only one of the continuous variables is cat-egorized. This was shown for building categorized agegroups [15], an often-observed analysis in epidemiologicalstudies. The more groups are chosen for categorization,however, the less bias is likely to occur.

As there are graphical methods available for interpreta-tion purposes (see Figs. 1e4) categorization of continuousdata is not necessary, anyway [16]. Another reason for cat-egorizations might be to detect nonlinear effects. To checkfor nonlinearity, however, many groups are needed. Never-theless, the real underlying structure can be missed. There-fore, flexible smoothing techniques should be preferred forthis purpose. Linear and logistic regression models can begeneralized to generalized additive models incorporatingsmooth functions in the R-package mgcv [17]. For inter-action effects, tensor products of smooth functions can beconstructed [18] leading to interaction surfaces reflectingthe nonlinear relationship. For survival models, this tech-nique is, to our knowledge, not implemented in commonstatistics programs. However, this can be done using Bayes-ian adaptive regression in the program BayesX [19], ap-proximating the surface by a tensor product of BayesianP-Splines [20]. These methods, however, are statisticallymore complex and this may limit their widespread use inclinical epidemiology. There is also the problem of overin-terpretation and uncertainties introduced by model instabil-ities. In conclusion, if the linearity assumption holds, thehighest statistical efficacy can be reached by modeling in-teraction terms continuously.4.2. Understanding and interpreting interaction effects

As already stated, regression models including multipli-cative interaction terms cannot be interpreted usefully justby looking at the typical output table as in Table 1. Meancentering of the variables has been proposed to make thisinterpretation easier. This does not alleviate the problemthat the effect estimate and P-value only refers to one spe-cific value of the other interacting variable, which is themean value in that case. Mean centering has also been saidto reduce the multicollinearity problem in interactionmodels. However, centering is just an algebraic transforma-tion and alters nothing important, not even multicollinearityissues [12,21]. It does result in different coefficients andstandard errors, but the only difference is the following:Not centering means: b1 is the effect of x1, when x2 is zero.Centering means: b1 is the effect of x1, when x2 is at itsmean. The latter does also not help with the interpretationof interaction effects. In our example, the conditional effectfor x2 was simulated to be 0 and estimated as |0.02. This isthe effect for x1 equal to 0. As this is also the mean of x1,this is equal to the conditional effect after mean centering.However, Fig. 4 and the output of our proposed functionsreveal that the effect of x2 is significantly different from0 for 98.1% of the observed data. In conclusion, presentingsuch a table as Table 1 at all is not very useful for interpre-tation purposes, no matter if centering was performed ornot. One needs methods spanning the complete range ofvalues for proper interpretation of effects on variables,which are involved in interaction effects.

For this purpose, we propose different graphical visual-ization methods, which can be used for GLMs or Cox

862 C. Lamina et al. / Journal of Clinical Epidemiology 65 (2012) 855e862models. The twisted plane in Fig. 1 shows how the effectsof both interacting variables change with changing valuesof the other respective variable. However, there is a dangerof overinterpretation at the tails of the distributions. Effectsthat can only be seen for very rare high or low values mightbe overrated. By cutting out slices out of this twisted planefor specific values (Fig. 3) together with its confidencebands, one can see, if the interaction is only triggered bya small percentage of the data at the tails of the distributionor if it affects most of the observations. Conditional effectswith their corresponding P-values or confidence intervalscan also be given out for a specified range of values (func-tion getRegion or getRegionCox, see Fig. 4). The in-teraction plots exemplified in Fig. 4 are sufficient to showthe effects for the complete range of both constituting vari-ables together with their point-wise confidence intervals.Reporting the significance region and the percentage ofthe observations falling into this region can help to betterjudge the possible implications: Are the ranges of variableswhere the interaction applies realistic and relevant? An in-teraction effect might not be interesting especially for clin-ical interpretation, if it is only driven by a small percentageof observations.5. Conclusion

As the interpretation of interaction effects between thetwo continuous variables is not straightforward, continuousvariables are categorized in most clinical epidemiologicalpublications leading to a loss of power and information.We give an overview over common misconceptions in theinterpretation of interaction effects and propose severalgraphical visualization methods to ease this interpretation.Together with the calculation of significance regions, theseplots can give researchers and clinicians the informationneeded to better assess the clinical relevance and implica-tions of interaction effects.Appendix

Supplementary material

Supplementary data related to this article can be foundonline at doi:10.1016/j.jclinepi.2012.02.013.References

[1] Barthel FM-S, Royston P. Graphical representation of interactions.

Stata J 2006;6:348e63.

[2] Turner EL, Dobson JE, Pocock SJ. Categorisation of continuous risk

factors in epidemiological publications: a survey of current practice.

Epidemiol Perspect Innov 2010;7:9.

[3] Royston P, Altman DG, Sauerbrei W. Dichotomizing continuous pre-

dictors in multiple regression: a bad idea. Stat Med 2006;25:127e41.

[4] Maxwell SE, Delaney HD. Bivariate median splits and spurious sta-

tistical significance. Psychol Bull 1993;113:181e90.

[5] Aiken LS, West SG. Multiple regression: testing and interpreting in-

teractions. Thousand Oaks, CA: SAGE Publications; 1991.

[6] Postorino M, Marino C, Tripepi G, Zoccali C. Abdominal obesity

modifies the risk of hypertriglyceridemia for all-cause and cardiovas-

cular mortality in hemodialysis patients. Kidney Int 2011;79:765e72.

[7] Zoccali C, Postorino M, Marino C, Pizzini P, Cutrupi S, Tripepi G.

Waist circumference modifies the relationship between the adipose

tissue cytokines leptin and adiponectin and all-cause and cardiovas-

cular mortality in haemodialysis patients. J Intern Med 2011;

269:172e81.[8] Rothman KJ. Epidemiology; an introduction. New York, NY: Oxford

University Press; 2002.

[9] de Mutsert R, Jager KJ, Zoccali C, Dekker FW. The effect of joint

exposures: examining the presence of interaction. Kidney Int

2009;75:677e81.

[10] Therneau TM, Grambsch PM. Modeling survival datadextending

the Cox model. New York, NY: Springer; 2000.

[11] Bender R, Augustin T, Blettner M. Generating survival times to sim-

ulate Cox proportional hazards models. Stat Med 2005;24:1713e23.

[12] Brambor T, Clark WR, Golder M. Understanding interaction models:

improving empirical analyses. Polit Anal 2006;14:63e82.

[13] Zitt E, Lamina C, Sturm G, Knoll F, Lins F, Freistatter O, et al. In-

teraction of time-varying albumin and phosphorus on mortality in in-

cident dialysis patients. Clin J Am Soc Nephrol 2011;6:2650e6.

[14] Taylor JMG, Yu MG. Bias and efficiency loss due to categorizing an

explanatory variable. J Multivar Anal 2002;83:248e63.

[15] Chen H, Cohen P, Chen S. Biased odds ratios from dichotomization

of age. Stat Med 2007;26:3487e97.

[16] Altman DG. Categorizing continuous variables. In: Armitage P,

Colton T, editors. Encyclopedia of biostatistics. Chichester, UK: John

Wiley and Sons; 1998:563e7.

[17] Wood SN. Generalized additive models: an introduction with R. Boca

Raton, FL: Chapmann & Hall; 2006.

[18] Wood SN. Thin plate regression splines. J R Stat Soc B 2003;

65:95e114.

[19] Brezger A, Lang S. Generalized structured additive regression based

on Bayesian P-splines. Comput Stat Data Anal 2006;50:967e91.

[20] Fahrmeir L, Kneib T, Lang S. Regression: Modelle, Methoden und

Anwendungen. Berlin, Germany: Springer; 2007.

[21] Echembadi R, Hess JD. Mean-centering does not alleviate collinear-

ity problems in moderated multiple regression models. Market Sci

2007;26:438e45.

http://dx.doi.org/doi:10.1016/j.jclinepi.2012.02.013

Visualizing interaction effects: a proposal for presentation and interpretation1. Background2. Methods2.1. Formulating and interpreting multiplicative interaction terms in linear, logistic, and Cox models2.2. Description of the simulated survival data set including interaction effects2.3. Implementation of the graphical utilities and calculations in R

3. Results4. Discussion4.1. Categorizing continuous variables in interaction terms: pros and cons4.2. Understanding and interpreting interaction effects

5. ConclusionAppendix. Supplementary materialReferences