# From first order interaction to second order interaction?

I am working on an experiment with two conditions

The experiment uses two pictures that are presented one after the other

The task is to identify the second picture

First condition is Time

the timing is very fast, e.g. first picture 40 milliseconds and second picture 40 milliseconds

and in the second option: the presentation time for the first picture is 60 milliseconds and for the second picture 20 milliseconds.

Second condition is position

Faces are shown looking to the right and looking to the left

The expectation is:

Firstly: when time becomes shorter, identification of the second picture will drop (Main effect)

Secondly, when the position of the first picture is the same with the position of the second picture, this will aid to the recognition of the second picture and when both positions are different the identification level will drop.

In the variance analysis, this will be shown as a main effect in the Time condition and as a first order interaction between Time X Position.

Now my question

when I introduce a third condition that is expected to be hierarchical related to the second condition

is my conclusion right that I will find in the Variance analysis a second order interaction between

Time X Position X The third condition?

The third condition look like this:

faces looking to the right wearing glasses

faces looking to the right wearing no glasses

faces looking to the left wearing glasses

faces looking to the left wearing no glasses

And how can we graphically show this second order interaction?

Thank you

Jan Sterenborg

A hierarchical relation between Position and another feature e.g. is that we first have tot establish the position of a face and after that we can establish another feature within that face e.g. the wearing of glasses.

When the position is not established, where do we have to look?

## Comments

I think there are complex approaches as well as relatively simple approaches. Personally, I would take a simple approach:

Questions 1, 2, 3, and 4:

(1) Is identification time different for short vs. long duration? (i.e., Is there a main effect of Factor A?)

(2) Is identification time different for matching than for non-matching spatial orientations? (i.e., Is there a main effect of Factor B?)

(3) Does the effect of duration on identification time depend on whether the spatial orientations do or don't match? (i.e., Is there a two-way, A-by-B interaction B?)

(4) Does the A-by-B interaction differ according to whether the faces are wearing glasses? (i.e., Is there a three-way, A-by-B-by-C interaction B?)

The pair of plots, below, show illustrate what a three-way ANOVA interaction might look like (and show less-explicitly what the main effects and two-way interactions might look like).

Note that the ANOVA will answer other questions for you tooo, such as: Is there two-way, A-by-C interaction?

R

Hello Anderson,

I expect something like above?

thanks for your reply

1 yes the identification time for the second picture is in one condition 40 and in the second condition 20 so I expect a main effect.

2 the orientation on the screen is the same for both pictures

3 idem as 2

4 impossible to answer because of 2?

Left to right is within the picture, not on the screen.

Actually, I don't think I sufficiently comprehend what your research design is. (Maybe some diagrams would help, instead of trying to describe it all in words?)

Relatedly, I think the graph you included, above, would be appropriate for a design in which there are only two factors: two levels of "duration" (short; long) plotted on the horizontal axis, and four levels of "condition" (++, +1, -+, and --) show in the legend. Thus, regarding ANOVA main effects and interactions, there would be the possibility of only a main effect of duration, a main effect of condition, and an interaction between duration and condition.

Alternatively, if you have three factors, I think that's most clearly represented as set of graphs with Factor A shown on the horizontal axis, Factor B shown in the legend, and Factor C shown as a row or column of graphs. If you have four factors then, if Factor C is shown as a row of graphs then Factor D should be shown as a column of graphs (and vice versa).

R

Thank you for your reaction Anderson

I will try it in words

I present two pictures, one after the other, no time in between. Pictures of males and females (en face).

Two presentation times:

first picture 40 ms and second picture 40 ms

and first picture 60 ms and second picture 20 ms

The instruction for the observer is to identify the second picture.

The identities of the first picture and the second picture are different. The set for the second picture contains identities all known to the observer. The set of the first pictures contains random individuals.

Expected effect are

An elaboration will be that the man and women are young or old, what could lead to a second order interaction: Presentation Time X Gender X Age? When there is a hierarchical relation between Gender and Age. This means first the Gender identification and in relation to that the Age identification.

Hi. It's still ambiguous whether you have Gender factor (male; female) or a Gender Correspondence factor (different; same). I'll assume the latter. Likewise, it's not clear whether your talking about an Age Category factor (young; old) or an Age Category Correspondence factor (same age-category; different age-category). I'll assume the former.

I don't see anything hierarchical about your design, since all factors are completely crossed with all other factors. The dependent variable (or criterion variable) is accuracy. The design is a . . .

2(Timing Configuration [40-40 or 60-20]) by2(Gender Correspondence[different gender or same gender]) by2(Age Category[young; old]) factorial. It is analyzable as a standard ANOVA, the flavor of which will depend whether some or all of the factors vary within-subjects. Below, I assume all that all factors vary within subjects . . .R

Hello Anderson

this is good, but I have to think this through

the difference between e.g. Gender factor and Gender correspondence factor is new to me.

A second thing is that this research points to the individual N=1 and not a group or more individuals.

The hierarchical part is crucial: how can I express that more clearly

When we are looking at one condition e.g Gender or Age apart from each other

the expectation is when this concept is used by the observer, there will be a first order interaction between Presentation time and Concept (Gender or Age)

When the observer does not use the concept or does not have the concept available, there will only be a presentation time main effect

Now the hierarchical part:

When the observer has both concepts of Gender and Age present

and Gender will be first established in the perception of a picture and in relation to that the Concept Gender Age will be established than we speak of a Hierarchical relation. And this will be confirmed in the analysis by a second order interaction.

When the observer does not use or have the concept Age available, then there will be no second order interaction.

It could also be that Age will be first established by the observer and in relation to that Gender

but from research of dr. Gé Calis we learn that Gender is more easily established than Age

I think your use of the term "hierarchical" has a completely different meaning than "hierarchical data" in statistics. An example of hierarchical data would be if age group could be young or old in the 40-40 condition but was restricted to young-only in the 60-20 condition. Your data don't have that characteristic so I think the term "hierarchy" is not at all relevant to your statistical question. Also not relevant is whether you have a "gender" or "gender correspondence" factor. Also not relevant is whether your randomly sampled data are from a single person (with the intend to statistically generalize to the person's non-sampled data), or from a group of persons (with the intent to statistically generalize to non-sampled members of the group). The fact is, you have the potential for a "main effect" of each of the three factors, the potential for "two-way" interactions between factors A & B, B & C, and A & C, as well as the potential for a "three-way" interaction between factors A, B, and C. That's all (with the exception of follow-up post-hoc tests or contrast you might perform).

As a reference, see https://onlinestatbook.com/2/analysis_of_variance/multiway.html

R

Hello Anderson.

thank you for your reply, I think there is a concept misunderstanding. I try to fix that, see my attachment.

Our methodology professor told us: before performing a statistical analysis, look first at the raw data and look whether the data point in the direction of your expectation. When that is not the case, the analysis is of no use. An analysis can be sensible in finding another cause or explanation.

My data tells me that there is a difference between the research defined Gender+ (man-man, woman-woman combinations) and the research defined Gender- (man-woman, woman-man combinations), when presentation time is 20 ms. When the presentation time is 40 ms there is no differentiation.

This is before the analysis.

After the ANOVA analysis, there is a significant main effect and a significant first order interaction. So my conclusion for this observer is: this observer uses a gender concept in the process of identifying a known person.

This seems to me very straight forward?

My next research step will be to investigate two research defined concepts e.g. male-female and young-old and try to show that there is a hierarchical relation between the two concepts. Hierarchical in the theoretical sense: first establishing gender and then age.

In terms of statistics I hope to find a main effect with presentation time, a first order interaction between presentation time and gender and a second order interaction between presentation time X gender X age. And I hope to find a way to present this graphically in an understandable way.

This is not an easy way because of the time relation involved. The aim is to show a dynamic process in action. Does the observer use certain concepts in action. What is going on before identification has taken place. When the identification process is completed, it's easy to describe the person at hand.

I hope that I could reach you in my struggle. To me, it's a good thing to reformulate the problem to get a more proper description of the problem and a better understanding...

Thanks for your effort!

Hi. "Two-way interaction" means the same thing as "1st order interaction" (though I believe the former term is more commonly used). "Three-way interaction" means the am thing as "second order interaction."

I think the additional discussion you provided does not change anything, statistically. Also, if desired each main effect and interaction can be graphed separately as shown in the following JASP file: Not_Hierarchical_Temp.jasp

R