# Interaction (statistics)

In statistics, an interaction12 may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive. Most commonly, interactions are considered in the context of regression analyses.

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.1

## Introduction

An "interaction variable" is a variable constructed from an original set of variables to try to represent either all of the interaction present or some part of it. In exploratory statistical analyses it is common to use products of original variables as the basis of testing whether interaction is present with the possibility of substituting other more realistic interaction variables at a later stage. When there are more than two explanatory variables, several interaction variables are constructed, with pairwise-products representing pairwise-interactions and higher order products representing higher order interactions.

The binary factor A and the quantitative variable X interact (are non-additive) when analyzed with respect to the outcome variable Y.

Thus, for a response Y and two variables x1 and x2 an additive model would be:

$Y = c + ax_1 + bx_2 + \text{error}\,$

In contrast to this,

$Y = c + ax_1 + bx_2 + d(x_1\times x_2) + \text{error}, \,$

is an example of a model with an interaction between variables x1 and x2 ("error" refers to the random variable whose value is that by which Y differs from the expected value of Y; see errors and residuals in statistics).

## Interaction variables in modeling

### Interactions in ANOVA

A simple setting in which interactions can arise is a two-factor experiment analyzed using Analysis of Variance (ANOVA). Suppose we have two binary factors A and B. For example, these factors might indicate whether either of two treatments were administered to a patient, with the treatments applied either singly, or in combination. We can then consider the average treatment response (e.g. the symptom levels following treatment) for each patient, as a function of the treatment combination that was administered. The following table shows one possible situation:

B = 0 B = 1
A = 0 6 7
A = 1 4 5

In this example, there is no interaction between the two treatments — their effects are additive. The reason for this is that the difference in mean response between those subjects receiving treatment A and those not receiving treatment A is −2 regardless of whether treatment B is administered (−2 = 4 − 6) or not (−2 = 5 − 7). Note that it automatically follows that the difference in mean response between those subjects receiving treatment B and those not receiving treatment B is the same regardless of whether treatment A is administered (7 − 6 = 5 − 4).

In contrast, if the following average responses are observed

B = 0 B = 1
A = 0 1 4
A = 1 7 6

then there is an interaction between the treatments — their effects are not additive. Supposing that greater numbers correspond to a better response, in this situation treatment B is helpful on average if the subject is not also receiving treatment A, but is more helpful on average if given in combination with treatment A. Treatment A is helpful on average regardless of whether treatment B is also administered, but it is more helpful in both absolute and relative terms if given alone, rather than in combination with treatment B.

### Qualitative and quantitative interactions

In many applications it is useful to distinguish between qualitative and quantitative interactions.3 A quantitative interaction between A and B refers to a situation where the magnitude of the effect of B depends on the value of A, but the direction of the effect of B is constant for all A. A qualitative interaction between A and B refers to a situation where both the magnitude and direction of each variable's effect can depend on the value of the other variable.

The table of means on the left, below, shows a quantitative interaction — treatment A is beneficial both when B is given, and when B is not given, but the benefit is greater when B is not given (i.e. when A is given alone). The table of means on the right shows a qualitative interaction. A is harmful when B is given, but it is beneficial when B is not given. Note that the same interpretation would hold if we consider the benefit of B based on whether A is given.

B = 0 B = 1 B = 0 B = 1
A = 0 2 1 A = 0 2 6
A = 1 5 3 A = 1 5 3

The distinction between qualitative and quantitative interactions depends on the order in which the variables are considered (in contrast, the property of additivity is invariant to the order of the variables). In the following table, if we focus on the effect of treatment A, there is a quantitative interaction — giving treatment A will improve the outcome on average regardless of whether treatment B is or is not already being given (although the benefit is greater if treatment A is given alone). However if we focus on the effect of treatment B, there is a qualitative interaction — giving treatment B to a subject who is already receiving treatment A will (on average) make things worse, whereas giving treatment B to a subject who is not receiving treatment A will improve the outcome on average.

B = 0 B = 1
A = 0 1 4
A = 1 7 6

### Unit treatment additivity

In its simplest form, the assumption of treatment unit additivity states that the observed response yij from experimental unit i when receiving treatment j can be written as the sum yij = yi + tj.456 The assumption of unit treatment additivity implies that every treatment has exactly the same additive effect on each experimental unit. Since any given experimental unit can only undergo one of the treatments, the assumption of unit treatment additivity is a hypothesis that is not directly falsifiable, according to Coxcitation needed and Kempthorne.citation needed

However, many consequences of treatment-unit additivity can be falsified.citation needed For a randomized experiment, the assumption of treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit treatment additivity is that the variance is constant.citation needed

The property of unit treatment additivity is not invariant under a change of scale,citation needed so statisticians often use transformations to achieve unit treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance.7 In many cases, a statistician may specify that logarithmic transforms be applied to the responses, which are believed to follow a multiplicative model.58

The assumption of unit treatment additivity was enunciated in experimental design by Kempthornecitation needed and Coxcitation needed. Kempthorne's use of unit treatment additivity and randomization is similar to the design-based analysis of finite population survey sampling.

In recent years, it has become commoncitation needed to use the terminology of Donald Rubin, which uses counterfactuals. Suppose we are comparing two groups of people with respect to some attribute y. For example, the first group might consist of people who are given a standard treatment for a medical condition, with the second group consisting of people who receive a new treatment with unknown effect. Taking a "counterfactual" perspective, we can consider an individual whose attribute has value y if that individual belongs to the first group, and whose attribute has value τ(y) if the individual belongs to the second group. The assumption of "unit treatment additivity" is that τ(y) = τ, that is, the "treatment effect" does not depend on y. Since we cannot observe both y and τ(y) for a given individual, this is not testable at the individual level. However, unit treatment additivity imples that the cumulative distribution functions F1 and F2 for the two groups satisfy F2(y)  = F1(y − τ), as long as the assignment of individuals to groups 1 and 2 is independent of all other factors influencing y (i.e. there are no confounders). Lack of unit treatment additivity can be viewed as a form of interaction between the treatment assignment (e.g. to groups 1 or 2), and the baseline, or untreated value of y.

### Categorical variables

Sometimes the interacting variables are categorical variables rather than real numbers and the study might then be dealt with as an analysis of variance problem. For example, members of a population may be classified by religion and by occupation. If one wishes to predict a person's height based only on the person's religion and occupation, a simple additive model, i.e., a model without interaction, would add to an overall average height an adjustment for a particular religion and another for a particular occupation. A model with interaction, unlike an additive model, could add a further adjustment for the "interaction" between that religion and that occupation. This example may cause one to suspect that the word interaction is something of a misnomer.

Statistically, the presence of an interaction between categorical variables is generally tested using a form of analysis of variance (ANOVA). If one or more of the variables is continuous in nature, however, it would typically be tested using moderated multiple regression.9 This is so-called because a moderator is a variable that affects the strength of a relationship between two other variables.

### Designed experiments

[Genichi Taguchi]] contendedcitation needed that interactions could be eliminated from a system by appropriate choice of response variable and transformation. However George Box and others have argued that this is not the case in general.10

### Model size

Given n predictors, the number of terms in a linear model that includes a constant, every predictor, and every possible interaction is $\tbinom{n}{0} + \tbinom{n}{1} + \tbinom{n}{2} + \cdots + \tbinom{n}{n} = 2^n$. Since this quantity grows exponentially, it readily becomes impractically large. One method to limit the size of the model is to limit the order of interactions. For example, if only two-way interactions are allowed, the number of terms becomes $\tbinom{n}{0} + \tbinom{n}{1} + \tbinom{n}{2} = 1 + \tfrac{1}{2}n + \tfrac{1}{2}n^2$. The below table shows the number of terms for each number of predictors and maximum order of interaction.

Number of terms
Predictors Including up to m-way interactions
2 3 4 5
1 2 2 2 2 2
2 4 4 4 4 4
3 7 8 8 8 8
4 11 15 16 16 16
5 16 26 31 32 32
6 22 42 57 63 64
7 29 64 99 120 128
8 37 93 163 219 256
9 46 130 256 382 512
10 56 176 386 638 1,024
11 67 232 562 1,024 2,048
12 79 299 794 1,586 4,096
13 92 378 1,093 2,380 8,192
14 106 470 1,471 3,473 16,384
15 121 576 1,941 4,944 32,768
20 211 1,351 6,196 21,700 1,048,576
25 326 2,626 15,276 68,406 33,554,432
50 1,276 20,876 251,176 2,369,936 1015
100 5,051 166,751 4,087,976 79,375,496 1030
1,000 500,501 166,667,501 1010 1012 10300

## Examples

Real-world examples of interaction include:

• Interaction between adding sugar to coffee and stirring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does.
• Interaction between adding carbon to steel and quenching. Neither of the two individually has much effect on strength but a combination of the two has a dramatic effect.
• Interaction between smoking and inhaling asbestos fibres: Both raise lung carcinoma risk, but exposure to asbestos multiplies the cancer risk in smokers and non-smokers. Here, the joint effect of inhaling asbestos and smoking is higher than the sum of both effects.11
• Interaction between genetic risk factors for type 2 diabetes and diet (specifically, a "western" dietary pattern). The western dietary pattern was shown to increase diabetes risk for subjects with a high "genetic risk score", but not for other subjects.12

## Notes

1. ^ a b Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. Oxford University Press. ISBN 0-19-920613-9.
2. ^ Cox, D.R. (1984). "Interaction". International Statistical Review (International Statistical Review / Revue Internationale de Statistique) 52 (1): 1–25. doi:10.2307/1403235. JSTOR 1403235.
3. ^ Peto, DP (1982). Statistical aspects of cancer trials (first ed.). Chapman and Hall, London.
4. ^ Kempthorne (1979)
5. ^ a b Cox (1958), Chapter 2
6. ^ Hinkelmann & Kempthorne (2008), Chapters 5-6
7. ^ Hinkelmann and Kempthorne (2008), Chapters 7-8
8. ^ Bailey on eelworms.
9. ^ Overton, R. C. (2001). "Moderated multiple regression for interactions involving categorical variables: a statistical control for heterogeneous variance across two groups". Psychol Methods 6 (3): 218–33. doi:10.1037/1082-989X.6.3.218. PMID 11570229
10. ^ George E. P. Box (1990). "Do interactions matter?". Quality Engineering 2: 365–369
11. ^ Lee, P. N. (2001). "Relation between exposure to asbestos and smoking jointly and the risk of lung cancer". Occupational and Environmental Medicine 58 (3): 145–53. doi:10.1136/oem.58.3.145. PMC 1740104. PMID 11171926
12. ^ Lu, Q. et al. (2009). "Genetic predisposition, Western dietary pattern, and the risk of type 2 diabetes in men". Am J Clin Nutr 89 (5): 1453–1458. doi:10.3945/ajcn.2008.27249.