How To Find Main Effects And Interactions

Factorial Designs Intro

Outline:

-- why nosotros do them

-- language

-- Principal Effects and Interactions

-- Definitions

-- Graphs

-- Math (ANOVA) approach

-- When the Math and Graph do not concur

Factorial Designs are those that involve more than one cistron (IV). In this course nosotros will only deal with 2 factors at a time -- what are chosen 2-mode designs.

-- why nosotros do them

-- t-examination allow us make comparisons betwixt ii groups -- two different levels of one 4

-- one-way ANOVA let united states of america compare multiple levels of one Four

The problem is that we are stuck with only using one IV so far. This is a problem because we seldom think that differences in a DV are just related to simply 1 matter. We usually think that things are a little more complex, that there are several things related to any man behavior.

In that location are three basic reasons for doing 2-way designs.

-- 2 IVs of interest

-- For command

-- Disquisitional Experiments

2 IVs (factors) of interest

This is the instance if we think that there are two or more variables related to the phenomenon and we want to look at both at the same fourth dimension. This way we tin come across how things piece of work together to crusade changes. Take for example my interest in how shared knowledge affects remembering. Shared knowledge consists of ane) knowledge of the to exist remembered material and 2) background knowledge (familiarity with partner). I've started on one result and suggested for our one-way design that we look at familiarity. How near if we want to study both IVs at the aforementioned fourth dimension?

Draw as a box

	IV_A: Shared Knowledge	of the Material
IV_B: Familiarity of Conversation Partners	Cognition of the Same Material	Not Shared Cognition
Stranger	Strangers remembering shared material	Strangers remembering different material
Roommates	Roommates remembering shared fabric	Roommates remembering different fabric

For control

Sometimes we are really interested in one Four but know that another IV (based on theories or previous research) is also related to the DV. Sometimes we include this other 4 for command purposes -- i) it volition more often than not decrease our MSE and go far easier to detect furnishings of the IV of involvement, and two) we can be certain that the IV of interest works the same way in all situations of interest.

i. For instance, based on Deborah Tannen's work and previous work I take washed with some students here I have reason to believe that men and women may talk about the by differently. Just doing the experiment without paying attending to gender may increase my within-group variability. If some men and some women talk to strangers and men and women differ, then I will have loftier variability in that grouping. If some men and some women talk to their roommates and men and women differ and then I will have high variability in that group. If I group by gender as well, notwithstanding, I will have lower within-grouping variability. Men talking to stranger will accept low variability inside groups, etc. This will make it easier to find an consequence of familiarity.

2. In addition, I can then be sure that men and women will acquit similarly in response to the variable of interest.

Describe as a box

	IV_A:	Gender
IV_B: Familiarity of Conversation Partners	Males	Females
Stranger	Male person Strangers remembering	Female person Strangers remembering
Roommates	Male person Roommates remembering	Female Roommates remembering

Critical Experiments

Sometimes nosotros are lucky and brilliant enough to be able to compare to theories in one experiment past using each theory to propose ane 4. Example is Chi's work on retention development. Equally kids go older they are able to remember more. The maturation theory says it has to do with the development of the encephalon and the ability to process information. This is direct tied to age. The expanding knowledge base theory suggests that as you know more you tin can learn more than. This is normally tied to historic period. But since it isn't straight tied to age information technology suggests that you can get some young kids who are experts in a given domain, and some adults who aren't experts in that domain. This is what Chi did in the domain of chess.

Draw as a box

	Four_A:	Age
IV_B: Chess Expertise	Young	College Students
Novices	Young Novices	Higher Historic period Novices
Experts	Young Experts	College Age Experts

For DVs, Chi measured the ability to remember the location of chess pieces on a chess board and the ability to do elementary memory tasks.

This nicely pits theories against one another and ane very likely will be rejected.

-- linguistic communication

IV (Independent Variable) = Factor = Handling (there tin can exist 2 or more than in factorial design)

Levels (each IV has two or more levels)

Cells (the specific confluence of the levels of all IVs)

The simplest instance is what is called a 2 x two design.

Draw as a box

	IV	A
Iv B	A = i	A = two
B = 1	A1, B1	A2, B1
B = ii	A1, B2	A2, B2

This is the simplest case of a ii fashion design,
each IVhas 2 levels.
IV A has 1 and ii.
IVB has 1 and 2.
There are 4 cells: A1B1, A1B2, A2B1, A2B2

This is a 2 ten 2 blueprint. 2x2 tells y'all a lot about the design:

there are two numbers and so in that location ii IVs

the outset number is a ii so the first Four has 2 levels

the second number is a 2 so the second IV has 2 levels

2 ten 2 = 4 and that is the number of cells

A 2x3 design

there are two numbers and then in that location 2 IVs

the first number is a 2 so the first IV has 2 levels

the second number is a 3 then the second 4 has three levels

ii x3 = 6 and that is the number of cells

A 2x2x3 design

there are three numbers and then there 3 IVs

the first number is a 2 so the first Iv has two levels

the second number is a two so the second IV has 2 levels

the third number is a iii so the third IV has 3 levels

2 x 2 x three = 12 and that is the number of cells

-- Main Effects and Interactions

When doing factorial design in that location are two classes of effects that nosotros are interested in: Primary Effects and Interactions

-- At that place is the possibility of a main outcome associated with each factor.

-- There is the possibility of an interaction associated with each human relationship among factors. (With a two-mode blueprint at that place is only one relationship, A ten B)

In a 2-manner pattern

-- Chief Effect of Gene A (1st IV): Overall departure amid the levels of A that is consistent across the levels of B. (Departure here mostly refers to direction, not to the size of the departure).

-- Main Consequence of Factor B (second IV): Overall difference among the levels of B that is consistent across the levels of A. (Difference here by and large refers to direction).

-- Interaction of AxB: Differences among the levels of ane Factor depend on the levels on the other Factor. (Difference here refers to direction and size of the upshot). This means, for case, that some departure between the levels of factor A may hold truthful at ane level of B only non at another level of B; or that the departure between two levels of A may be much stronger at 1 level of B than at another level of B, even though it is in the same direction.

An easy style to expect for Main Effects and Interactions is by graphing the Jail cell Means.

In each jail cell I have given yous the cell Mean = M_A,B

	IV	A
IV B	A = 1	A = 2
B = 1	Grand_1,one = 10	M_2,one = 15
B = 2	K_1,2 = 15	1000_two,2 = twenty

-- Definitions

-- Chief Effect of Cistron A (1st IV): Overall divergence among the levels of A that is consistent across the levels of B. (Divergence here more often than not refers to direction).

-- Main Effect of Factor B (2nd Iv): Overall difference amongst the levels of B that is consistent across the levels of A. (Divergence here by and large refers to management).

-- Interaction of AxB: Differences among the levels of one Factor depend on the levels on the other Gene. (Difference here refers to management and size of the event).

-- Graphs Look at the examples done in form

ME of A: Difference between A1 and A2 is in the aforementioned direction for both levels of B.

ME of B: Difference between B1 and B2 is in the aforementioned direction for both levels of A.

Interaction: The slopes of the lines are not parallel.

-- Math (ANOVA) approach Definitions

-- Main Effect of Cistron A (1st Four): Overall difference amongst the levels of A

-- Chief Consequence of Factor B (2nd IV): Overall difference among the levels of B

-- Interaction of AxB: Differences amidst the levels of one Factor depend on the levels on the other Cistron.

** Note what is missing: There is no business organization that the MEs are Consequent.

** Note what doesn't change: The definition of the interaction is constant.

New Terms

To make the judgements required to ascertain MEs and interactions by the math, I accept to introduce some more language.

The get-go information is Marginal Means: Marginal Means are the means for one level of an contained variable averaged beyond all level of the other IV. Thus yous have a Marginal Mean with A=1, which is the mean for everyone who experienced A at level 1, regardless of whether they experienced B at 1 or 2. In addition to the Marginal Means for each level of both IVs, you too accept a Full Mean, which is the average across the entire experiment.

	IV	A	.
Iv B	A = 1	A = 2	marginal Means
B = i	M_1,one = 10	M_2,ane = 15	One thousand_B=1= 12.5
B = 2	K_1,2 = xv	Grand_ii,2 = 20	One thousand_B=2= 17.v
marginal Means	G_A=1= 12.v	Chiliad_A=two = 17.5	M_T = 15.0

Now considering the three things we look for:

ME of A: Is in that location a large departure (compared to w/i group variability) among the A marginal means?

ME of B: Is at that place a large divergence (compared to westward/i group variability) amid the B marginal means?

Interaction: Deciding on the interaction you take to know if the 2 factors are additive. If additive, no interaction; if not-additive, interaction.

Additive ways that you can predict the prison cell means based on the marginal means. Here's an easy way to do that: Look at 3 of the prison cell means. Cover the fourth cell mean. in the example above, comprehend G_two,two = 20. Now endeavour to predict that based on the other 3 cell means. At Level B=1, going from A=i to A=2 adds five to the prison cell mean. Thus we should add 5 to the cell mean for B=ii, A=ane to get the prison cell hateful for B=2, A=2. Turns out to work this fourth dimension. Y'all tin predict and thus things are additive and at that place is no interaction.

-- When the Math and Graph practice non hold

Generally, believe whichever says no .

If the graph makes it look similar something is happening, but the math (ANOVA) says no, and then believe the math. The ANOVA is testing not merely to see if there is a difference, but that the difference is large compared to w/i group variability.

If the math says there is a main result, but looking at the graph indicates that there is not a consistent principal effect, then your principal upshot is an antiquity of the interaction. (Note, in gild for this to happen, in that location must and volition be an interaction.) Artifact: something created. In this instance, created by the interaction. That means information technology is created because the effects of one Factor go in different directions at different levels of the other Gene -- but that one of these is larger than the other and pulls the average (marginal ways) autonomously in ane management. In this case, when yous look at the marginal means, there is an overall difference, only if you wait at the cells it is non consistent. The true definition of a principal effect is a consistent overall difference, just the ANOVA only looks at the overall function. You, the researcher, accept to be concerned that the chief effect is consistent. Yous only go artifacts when you lot have an interaction.