# Statistical Interaction

A Statistical Interaction is a Statistical Measure that describes the situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable.

**AKA:**Interaction.**Context:**- It can range from being a Quantitative Interaction to being a Qualitative Interaction.
- It can range from being an Additive Interaction to being a Multiplicative Interaction.

**Example(s):**- Clinical Interaction such as:
- interaction between stroke severity and the efficacy of a drug on patient survival;
- interaction between smoking and inhaling asbestos fibres;
- joint interaction effect of low-dose aspirin and warfarin for the prevention of ischemic heart disease;

- Biologic Interaction such as:
- interaction of species and air temperature and their effect on body temperature.

- Physical Interaction such as:
- interaction of temperature and time in cookie baking;

- Gene-Environment Interaction such as:
- interaction between genetic risk factors for type 2 diabetes and diet;

- Negative Mediated Interaction,
- ...
- …

- Clinical Interaction such as:
**Counter-Example(s):****See:**Synergism, Antagonism, Mediation Analysis, Statistics, Additive Map, Regression Analysis, Factorial Experiments, Exposure-Outcome Relationship, Cause-Effect Relationship, Exposure-Mediator Interaction.

## References

### 2022a

- (Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/Interaction_(statistics) Retrieved:2022-3-20.
- In statistics, an
**interaction**may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable (that is, when effects of the two causes are not additive).^{[1]}Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments.The presence of interactions can have important implications for the interpretation of statistical models. If two variables of interest interact, the relationship between each of the interacting variables and a third "dependent variable" depends on the value of the other interacting variable. In practice, this makes it more difficult to predict the consequences of changing the value of a variable, particularly if the variables it interacts with are hard to measure or difficult to control.

The notion of "interaction" is closely related to that of moderation that is common in social and health science research: the interaction between an explanatory variable and an environmental variable suggests that the effect of the explanatory variable has been moderated or modified by the environmental variable.

- In statistics, an

- ↑ Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. Oxford University Press. ISBN 978-0-19-920613-1.

### 2022b

- (Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/Glossary_of_clinical_research#I Retrieved:2022-3-20.
- QUOTE:
**Interaction**(Qualitative & Quantitative)- The situation in which a treatment contrast (e.g. difference between investigational product and control) is dependent on another factor (e.g. centre). A quantitative interaction refers to the case where the magnitude of the contrast differs at the different levels of the factor, whereas for a qualitative interaction the direction of the contrast differs for at least one level of the factor. (ICH E9)

- QUOTE:

### 2022c

- (Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/Interaction Retrieved:2022-3-20.
**Interaction**is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect. Closely related terms are interactivity and interconnectivity, of which the latter deals with the interactions of interactions within systems: combinations of many simple interactions can lead to surprising emergent phenomena.*Interaction*has different tailored meanings in various sciences.

### 2021

- (Bours, 2021) ⇒ Martijn J.L.Bours (2021). "Tutorial: A nontechnical explanation of the counterfactual definition of effect modification and interaction". In: Journal of clinical epidemiology, 134, 113-124.
- QUOTE: In other words, to study for whom and in which situation(s) causal effects occur. Studying the combined influence of two (or more) exposures on some outcome constitutes a more elaborate and refined type of causal question that refers to the concepts of
*effect modification*and*interaction*(...). What's more, effect modification and interaction may sometimes be confused with the concept of confounding. Confounding is concerned with separating causal from noncausal effects that cloud inferences about exposure-outcome relations (...). Unlike confounding though, effect modification and interaction are not unwanted biases to be eliminated, but are part of causal reality to be elucidated (...). Effect modification and interaction have been defined as follows within the field of epidemiology (...):

*Effect modification*: Variation in the selected effect measure for the factor under study across levels of another factor.*Interaction*: The interdependent, reciprocal, or mutual operation, action, or effect of two or more factors to produce, prevent, control, mediate, or otherwise influence the occurrence of an event.

- QUOTE: In other words, to study for whom and in which situation(s) causal effects occur. Studying the combined influence of two (or more) exposures on some outcome constitutes a more elaborate and refined type of causal question that refers to the concepts of

### 2019a

- (Brankovic et al., 2019) ⇒ Milos Brankovic, Isabella Kardys, Ewout W. Steyerberg, Stanley Lemeshow, Maja Markovic, Dimitris Rizopoulos, and Eric Boersma (2019). "Understanding of interaction (subgroup) analysis in clinical trials". In: European journal of clinical investigation, 49(8), e13145.
- QUOTE: A statistical interaction can be assessed in two ways: by
*stratification*—when treatment effects are assessed across subgroups defined by a baseline/demographic factor; or by*interaction modelling*—when the treatment and the baseline/demographic factor are included together with an interaction term into a statistical model (treatment + baseline factor + treatment × baseline factor).

- QUOTE: A statistical interaction can be assessed in two ways: by

### 2019b

- (Mutlu et al., 2019) ⇒ Unal Mutlu, Mohammad Arfan Ikram, and Mohammad Kamran Ikram (2019). "Clinical interpretation of negative mediated interaction". In: International Journal of Epidemiology, 48(4), 1286-1293.
- QUOTE: Negative interaction or negative mediation alone has already been discussed extensively with appropriate biological or mechanical interpretations.
^{[1]},^{[2]}The presence and direction of interaction depends on the scale that is used, i.e. additive or multiplicative. Briefly, negative interaction on the additive scale implies that the effect of the combined action of two exposures is smaller than the sum of their individual effects, whereas negative interaction on the multiplicative scale implies that the effect of the combined action of two exposures is smaller than the product of their individual effect. Interaction on the additive scale has been widely recognized by epidemiologists as the appropriate scale to assess biologic interaction.

- QUOTE: Negative interaction or negative mediation alone has already been discussed extensively with appropriate biological or mechanical interpretations.

- ↑ VanderWeele TJ, Knol MJ. A tutorial on interaction. Epidemiol Methods 2014;3:33–72
- ↑ MacKinnon DP, Fairchild AJ, Fritz MS. Mediation analysis. Annu Rev Psychol 2007;58:593–614.

### 2017

- (Corraini et al., 2017) ⇒ Priscila Corraini, Morten Olsen, Lars Pedersen, Olaf M. Dekkers, and Jan P. Vandenbroucke (2017). "Effect modification, interaction and mediation: an overview of theoretical insights for clinical investigators". In: Clinical epidemiology, 9, 331.
- QUOTE: The notions of effect modification, interaction and mediation represent conceptually different, although potentially interdependent notions. These subtle different notions address different research aims, which are related to different aspects of an exposure–outcome relationship (Box 1).

Type of assessment | Aim of the assessment |
---|---|

Effect modification | Separate exposure effects according to another variable (...) |

Interaction | Evaluate individual and joint effects of exposures (...) |

Mediation | Evaluate direct and indirect effects of exposures (...) |

- The clinical motivation behind the assessment of effect modification is to identify whether the effect of a treatment (or exposure) is different in groups of patients with different characteristics. If the effects are the same, the treatment (or exposure) effect is called homogeneous; if the effects are different, they are called heterogeneous(...).
Assessing effect modification may also help to identify a subset of patients who would not benefit from an intervention at all(...).

Interaction is of interest when researchers want to obtain the joint effect of two (or more) exposures on a disease or outcome.

^{[1]}To be considered a synergistic interaction, the joint effect has to be higher than the effect expected by the sum of their individual effects. Conversely, there is an antagonistic interaction between exposures, when the joint effect is less than the sum of their individual effects. This is in contrast to effect modification, where the effect of an exposure on an outcome is assessed in different strata of a third variable, but a joint effect is not assessed.From a clinical perspective, to assess interaction is particularly important when a disease can be treated by a combination of two or more treatments.

- The clinical motivation behind the assessment of effect modification is to identify whether the effect of a treatment (or exposure) is different in groups of patients with different characteristics. If the effects are the same, the treatment (or exposure) effect is called homogeneous; if the effects are different, they are called heterogeneous(...).

- ↑ Rothman KJ. Synergy and antagonism in cause-effect relationships. Am J Epidemiol. 1974;99(6):385–388.

### 2014

- (VanderWeele & Knol, 2014) ⇒ Tyler J. VanderWeele, and Mirjam J. Knol (2014). "A Tutorial on Interaction". In: Epidemiologic methods, 3(1), 33-72. DOI:10.1515/em-2013-0005.
- QUOTE: There are a number of practical and theoretical considerations that motivate the study of interaction. One of the most prominent of these is that, in a number of settings, resources to implement interventions may be limited. It may not be possible to intervene on or treat an entire population. Resources may only be sufficient to treat a small fraction. If this is the case, then it may be important to identify the subgroups of individuals in which the intervention or treatment is likely to have the largest effect. As will be discussed below, methods for assessing additive interaction can help determine which subgroups would benefit most from treatment. Other more sophisticated methods can help identify groups of individuals, based on a large number of covariates, who would or would not benefit, or who would benefit to the greatest extent, from treatment. Even in settings in which resources are not limited and it is possible to intervene on everyone, it may be the case that a particular intervention is beneficial for some individuals and harmful for others. In such cases, it is very important to identify those groups for which treatment may be harmful and refrain from treating such persons. Techniques for assessing such so-called “qualitative” or "crossover" interactions are discussed in this tutorial are useful in this regard.
Another reason sometimes given for empirically assessing interaction is that it may provide insight into the mechanisms for the outcome.

- QUOTE: There are a number of practical and theoretical considerations that motivate the study of interaction. One of the most prominent of these is that, in a number of settings, resources to implement interventions may be limited. It may not be possible to intervene on or treat an entire population. Resources may only be sufficient to treat a small fraction. If this is the case, then it may be important to identify the subgroups of individuals in which the intervention or treatment is likely to have the largest effect. As will be discussed below, methods for assessing additive interaction can help determine which subgroups would benefit most from treatment. Other more sophisticated methods can help identify groups of individuals, based on a large number of covariates, who would or would not benefit, or who would benefit to the greatest extent, from treatment. Even in settings in which resources are not limited and it is possible to intervene on everyone, it may be the case that a particular intervention is beneficial for some individuals and harmful for others. In such cases, it is very important to identify those groups for which treatment may be harmful and refrain from treating such persons. Techniques for assessing such so-called “qualitative” or "crossover" interactions are discussed in this tutorial are useful in this regard.

### 2009a

- (De Mutsert et al., 2009) ⇒ Reee de Mutsert, Kitty J. Jager, Carmine Zoccali, and Friedo W. Dekker (2009). "The effect of joint exposures: examining the presence of interaction". In: Kidney international, 75(7), 677-681.
- QUOTE: Statistical interaction refers to the inclusion of a product term of the two risk factors under study in a statistical model, which is explained below. In many studies, the presence of interaction between two risk factors is assessed by testing whether the regression coefficient of such a product term is statistically significant, representing the excess risk due to interaction of the exposures. However, in this way, the presence of interaction is tested on the underlying scale of the model(...)
The regression equation of the linear regression model including a product term, or interaction term, is

- QUOTE: Statistical interaction refers to the inclusion of a product term of the two risk factors under study in a statistical model, which is explained below. In many studies, the presence of interaction between two risk factors is assessed by testing whether the regression coefficient of such a product term is statistically significant, representing the excess risk due to interaction of the exposures. However, in this way, the presence of interaction is tested on the underlying scale of the model(...)

$E\left(y\right)=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_1X_2$

- Where $E(y)$ is the estimated effect, $\beta_0$ is the intercept that can be interpreted as the background risk, $\beta_1$ and $\beta_2$ are the regression coefficients of the risk factors $X_1$ and $X_2$. By including the product term $\left(X_1 \times X_2\right)$ the interaction effect is estimated through estimation of the regression coefficient $\beta_3$ (...)

### 2009b

- (VanderWeele, 2009) ⇒ Tyler J. VanderWeele (2009). "On the Distinction Between Interaction and Effect Modification". In: Epidemiology, 20(6), 863-871.
- QUOTE: Interaction is defined in terms of the effects of 2 interventions whereas effect modification is defined in terms of the effect of one intervention varying across strata of a second variable. Effect modification can be present with no interaction; interaction can be present with no effect modification.