Testing and Plotting Simple Slopes of Interaction Effects

Today’s outline

•1) What is an interaction?

•2) Testing

•3) Plotting (let’s not get bogged down here)

•4) Probing (aka testing simple slopes)

•5) Cautions and Considerations

What is an interaction?• For whom, or when• An association differs based on some other

variable(s)

• It’s all about context

Low

Sib

ling

Intim

acy

Hig

h Si

blin

g In

timac

y1

1.251.5

1.752

Girls Boys

Dep

ress

ion

Low

Sib

ling

Intim

acy

Hig

h Si

blin

g In

timac

y1

1.251.5

1.752

Girls Boys

Dep

ress

ion

Testing• Preparing your variables – it’s all about ZERO• Centering is critical for all continuous variables

• X1Centered = X1 - Mean of X1

X Z XZ

1 1 1

2 4 8

3 7 21

4 10 40

5 13 65

M = 3 M = 7

rxxz = .98

rzxz = .98

X Z XZ

-2 -6 12

-1 -3 3

0 0 0

1 3 3

2 6 12

M = 0 M = 0rxxz = 0rzxz = 0

Testing

Plotting• You’ve found a significant interaction, now what?• Plot it

• Conflict X Sib Gender (0 = sister; 1 = brother)

• Two lines, solve for four points (-1, +1 SD of the continuous variable)

• yRefXH = bo + b1XH + b2MRef + b3XHMRef

• yRefXL = bo + b1XL + b2MRef + b3XLMRef

• yOneXH = bo + b1XH + b2MOne + b3XHMOne

• yOneXL = bo + b1XL + b2MOne + b3XLMOne

• Technically all control variables should be in the equation too• But if you’ve centered all continuous variables then you don’t need to

Plotting

• yRefXH = .30 + (.07*.63) + (.01*0) + (.06*.63*0)

• yRefXL = .30 + (.07*-.63) + (.01*0) + (.06*-.63*0)

• yOneXH = .30 + (.07*.63) + (.01*1) + (.06*.63*1)

• yOneXL = .30 + (.07*-.63) + (.01*1) + (.06*-.63*1)

Plotting

Plotting

Plotting• Now the 3-way interaction• 4 lines, 8 points (-1, +1 SD for each continuous variable)• yXHMHRef = bo + b1XH + b2MH + b3DRef + b4XHMH + b5XHDRef + b6MHDRef + b7XHMHDRef

• yXLMHRef = bo + b1XL + b2MH + b3DRef + b4XLMH + b5XLDRef + b6MHDRef + b7XLMHDRef

• yXHMLRef = bo + b1XH + b2ML + b3DRef + b4XHML + b5XHDRef + b6MLDRef + b7XHMLDRef

• yXLMLRef = bo + b1XL + b2ML + b3DRef + b4XLML + b5XLDRef + b6MLDRef + b7XLMLDRef

• yXHMHOne = bo + b1XH + b2MH + b3DOne + b4XHMH + b5XHDOne + b6MHDOne + b7XHMHDOne

• yXLMHOne = bo + b1XL + b2MH + b3DOne + b4XLMH + b5XLDOne + b6MHDOne + b7XLMHDOne

• yXHMLOne = bo + b1XH + b2ML + b3DOne + b4XHML + b5XHDOne + b6MLDOne + b7XHMLDOne

• yXLMLOne = bo + b1XL + b2ML + b3DOne + b4XLML + b5XLDOne + b6MLDOne + b7XLMLDOne

Plotting

You’ve plotted, now what?

1.2

1.3

1.4

1.5

1.6

1.7

1.8

-1

0

1

2

3

-5

-3

-1

1

3

5

1

1.5

2

2.5

3

Probing• Different than zero?

• But it looks significant!

• It’s not enough to plot the interaction

• You MUST probe it/test the simple slopes

Probing

• Remember, ZERO is important

• The main effect is the effect when everything else is at zero

• So, .07 is the slope for those with a sister (it is different than zero)

• The slope for brothers will be .07 + .06 (but we don’t know it’s standard error)

Probing• If we recode Sib Gender so

that brother = 0

• We see that the slope is .13• Now we know the SE (.03)

Probing• For the 3-way interaction

• Remember, it’s all about ZERO

• We need to recode our two moderators to adjust what zero means

• For Sib Gender

• 0 = 1

• 1 = 0

Probing• For Intimacy (continuous)

• Create two new variables to reflect high intimacy and low intimacy

• -1 & +1 SD• This is most common

• High Intimacy = mean centered intimacy – 1 SD of intimacy

• Low Intimacy = mean centered intimacy + 1 SD of intimacy

Not a typo

M = 2.97 +1 SD = .68

-1 SD = -.68

ZERO

M = 2.97 +1 SD = .68

-1 SD = -.68

M = 2.97 +1 SD = .68

-1 SD = -.68

Probing• Re-run your models with your combinations of re-coded

variables

• Must be done in the step where the interaction is the highest you are testing

• For a 3-way interaction you’ll end up testing 4 models

• One for each slope

Probing• Sibgen (0 = sister; 1 =

brother)

• SibgenR (0 = brother; 1 = sister)

• SibIntH (Intimacy @ +1 SD)

• SibIntL (Intimacy @ -1 SD)

• The re-coded variables must replace the old variable every time it is used in that model (the main effect and each interaction)

• Now we know which slopes are different from zero

• But there’s a whole lot more info here

• Conflict X Sib Gender

• The blue and grey slopes are different

• Intimacy X Conflict

• The green and grey slopes are different• Doesn’t map as cleanly as

conflict X sib gender• Difference from high to low is

greater than 1 (1.38)

• Mean differences

• At average levels of conflict, differences in depression based on high or low intimacy with a brother is -.09

• At average levels of conflict, differences in depression based on high or low intimacy with a sister is -.08

• The difference in depression based on having a brother or a sister is .02 for both high and low intimacy

• Mean differences

• At low levels of conflict, differences in depression based on high or low intimacy with a brother is -.13

• At low levels of conflict, differences in depression based on high or low intimacy with a sister is -.04

• The difference in depression based on having a brother or a sister at low intimacy is .04

• The difference in depression based on having a brother or a sister at high intimacy is .08

Cautions and Considerations• An interaction is like splitting your sample

• N = 100• 2-way interaction: N = 50

• 3-way interaction: N = 25

• 4-way interaction: N = 12.5

• Even with a larger sample• Some groups may be small when using categorical variables

• N = 157 (3 way should have ~39)

• One group had 20

Cautions and Considerations• How should you scale your figures?

• You want to accurately convey your findings

• Possible range• May make it hard to interpret, but is the absolute most honest

• Observed range• More realistic than the possible range, may be influenced by outliers

• 2 SDs• Often a good option giving the picture of what most of your data look

like

• 3 SDs• Often a good option giving a better picture of what most of your data

look like

Possible Range Observed Range

2 SDs 3 SDs

Cautions and Considerations• Non-significant

interactions• Should they stay or should

they go?• If part of a higher order

interaction they must stay

• Reason to take out• May inflate standard errors • Especially for probing slopes

• Reason to leave• More clear presentation of

analysis

• Better for reviewers/readers

• My preferred option

Cautions and Considerations• Checking work• It’s easy to make an error in

plotting or probing

• From your plotting:• Calculate the rise for each slope

• From your probing• Multiply 2 SDs by the

unstandardized coefficient for each association

• Results from plotting and probing should match

• Plotting with templates

• It’s really awesome

• Be careful

• Verify

• Check against probing