# Interaction variable

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In statistics, an **interaction variable** is one of variables often used in regression analysis. It is formed by the multiplication of two independent variables.

Usage of interaction variables, while offering some useful data, also raises the issues of multicollinearity.

## Example

There is a group of participants. Their postural control is being analysed and scored. Each participant performs 4 trials. These trials are based on two manipulated factors.

The two factors are stance (feet apart [S1], feet together [S2]) and vision (normal vision [V1], blindfold [V2]). So the four trials each participant completes are S1+V1, S1+V2, S2+V1, S2+V2.

Now with a repeated measures analysis of variance (ANOVA-RM), the data will be based on two within-subject factors (ie factors which the same subject can perform, as opposed to between subject factors such as gender, age group, intervention protocol, et cetera).

The ANOVA-RM will produce values indicating the probability that the difference is due to their being an actual difference (as opposed to the difference being the result of coincidence). Significance values will be obtained for each individual factor (ie main effects) as well as the significance of factor interrelationships (ie interaction effects). So an example of a main effect might be the ANOVA-RM returns a significance level of p=0.001 for the factor 'stance'. This means that there was a significant difference in performance, assuming p=0.001 (or a 0.1% probability that the result is due to chance and not an actual difference) is an acceptable level of significance. What this means is that 99.9% of the time, the difference between the feet apart and feet together conditions will be a result of there being an actual difference in performance associated with the stance (0.1% of the time it is due to coincidence).

With an interaction effect, a significant p-value is returned when multiple factors are considered. For example if stance*vision returns p=0.03. Whilst the significance of the interaction effect is less than that of the main effect, if the p=0.03 (97% of the time the result is due to the conditions, not coincidence) meets the pre-determined acceptable significance levels, interaction effects will take precedence over the main effects.

This means a main effect for stance or main effect for vision cannot be considered without acknowledging that an interaction effect was present.

What the interaction effect actually means in this example is that an interrelationship between the factors was found. So in this case, stance influences vision and/or vision influences stance (at least in some of the trials). Assuming the vision influencing stance is true, the postural control system will be more dependent upon vision when the stance condition is more difficult.

That is, in the feet together stance, there is a reduced base of support (so it is a more difficult stance to maintain). In this condition of a reduced base of support, the body might have to compensate by increasing the influence that visual feedback has on the balance control system.

Subsequently, even though the feet together trial might produce a reduced level of postural control compared to the feet apart trial (stance condition - main effect), the stance*vision interaction effect indicates that differences in performance associated with stance are influenced by visual feedback.

Where an interaction effect is present, it is necessary to run a post-hoc test (for example Scheffe, Tukey, Bonferroni) to determine where abouts the interaction effect occurs. It may be found that there is only an interaction effect present between one or a few of the variables. In the case of a 2x2 design, there will be six possible interactions: V1S1-V1S2, V1S1-V2S1, V1S1-V2S2, V1S2-V2S1, V1S2-V2S2, V2S1-V2S2. For the present example it may be found that the post-hoc test identifies the significant values producing an interaction effect are V1S2-V2S2, V1S1-V1S2 and V2S1-V2S2. It is from here that conclusions can be drawn as to what has caused the interaction effect. It may be easier to identify an interaction effect where they are graphically represented.

This is demonstrated in the plot to the right (where PL/s or path length per second represents the velocity at which the centre of pressure is deviating in mm/s). It can be seen that the amount of change in score between vision and blindfolded conditions is not consistent between feet apart and feet together conditions.

## References

## External links

- Credibility and the Statistical Interaction Variable: Speaking Up for Multiplication as a Source of Understanding
- Fundamentals of Statistical Interactions: What is the difference between "main effects" and "interaction effects"?

## Further reading

- James J. Jaccard, Robert Turrisi,
*Interaction Effects in Multiple Regression*, Sage Publications, 2003, ISBN 0761927425

## See also

{enWP|Interaction variable}}