Andy Lin
IDRE Statistical Consulting


Regression models effects of IVs on DVs.

E.g. does amount of time exercising predict weight loss?

Can also model effect of IV modified by another IV
 moderating variable (MV)
 e.g is effect of exercise time on weight loss modified by the type of exercise?

Effect modification = interaction


Interactions are products of IVs

Typically entered with the IVs into regression

All we get out of regression is a coefficient

Not enough to understand interaction

What are the conditional effects?
 Simple effects and slopes
 Conditional interactions


Demonstrate methods to estimate, test and graph effects within an interaction

Specifically we will use PROC PLM to:

Calculate and estimate simple effects

Compare simple effects

Graph simple effects

weight = β
0
+ β s
SEX + β h
HEIGHT

Main effects models

IV effects constrained to be the same across levels of all other IVs in the model

Main effect of height is constrained to be the same across sexes
 Average of male and female height effect

weight = β
0
+ β s
SEX + β h
HEIGHT

Interaction models

Allow effect of an IV to vary with levels of another IV

Formed as product of 2 IVs

Now the effect of height may vary between sexes

And effect of sex may vary at different heights


From this equation

We can derive sex-specific regression equations

Males (sex=0)

Females (sex=1)



Each sex has its own height effect

Males (sex=0)

Females (sex=1)

These are the simple slopes of height within each group

Interaction coefficient is difference in simple slopes


We use proc plm for most of our analyses

Proc plm performs post-estimation analyses and graphing

Uses and “item store” as input

Contains model information (coefficients and covariance matrices)

Item store created in other procs
 Inlcuding glm, genmod, logistic, phreg, mixed, glmmix, and more


Important proc plm statement used in this seminar

Estimate statement

Forms linear combinations of coefficients and tests them against 0

Very flexible – linear combinations can be means, effects, contrasts, etc.

We use it to estimate and compare simple slopes

Syntax is a bit more difficult


Important proc plm statement used in this seminar

Slice statement

Specifically analyzes simple effects

Very simple syntax

Lsmestimate statement

Compare estimated marginal means, i.e. calculate simple effects

More versatile than slice


Lsmeans statement

Estimates marginal means and can calculate differences between them

Effectplot

Plots predicted values of the outcome across range of values on 1 or more predictors

Can visualize interactions

Many types of plots


Many of these statements found in regression procs

Why use PROC PLM?

Do not have to rerun model as we run code for interaction analysis

These statements sometimes have more functionality in
PROC PLM




Study of average weekly weight loss achieved by subjects in 3 exercise programs
900 subjects
Important variables:

Loss – continuous, normal outcome – average weekly weight loss

Hours – continuous predictor – average weekly hours of exercise

Effort – continuous predictor – average weekly rating of exertion when exercising, ranging from 0 to 50


Important variables cont:

Prog – 3-category predictor - which exercise program the subject followed, 1=jogging, 2=swimming,
3=reading (control)

Female – binary predictor - gender, 0=male,
1=female

Satisfied - binary outcome - subject’s overall satisfaction with weight loss due to participation in exercise program, 0=unsatisfied, 1=satisified


We first model the interaction of 2 continuous IVs

The effect of a continuous IV on the outcome is called a slope
 Expresses change in outcome pre unit increase in IV

With the interaction of 2 continuous variables, the slope of each IV is allowed to vary with the other IV
 Simple slopes


Let us look at model where Y is predicted by continuous X, continuous Z, and their interaction:

Be careful when interpreting β x
 and β z
They are simple effects (when interacting variable=0), not main effects


The coefficient β xz is interpreted as the change in the simple slope of X per unit-increase in Z

Equation for simple slope of X:

Continuous-by-continuous: example model

We regress loss on hours, effort, and their interaction

Is the effect of hours modified by the effort that the subject exerts?
 And the converse – is effect of effort modified by hours?

Continuous-by-continuous: example model proc glm data=exercise; model loss = hours|effort / solution; store contcont; run ;

The “|” requests main effects and interactions
 solution requests table of regression coefficients
 store contcont creates an item store of the model for proc plm

Continuous-by-continuous: example model


Interaction is significant
Remember that hours and effort terms are simple slopes

Continuous-by-continuous: calculating simple slopes

Estimate statement used to form linear combinations of regression coefficients

Including simple slopes (and effects)

Very flexible

Understanding the regression equation very helpful in coding estimate statements


Estimate ‘label’ coefficient values / e
 e.g. to estimate expected loss when hours=2 and effort = 30 proc plm restore=contcont; estimate 'pred loss, hours=2, effort=30' intercept 1 hours 2 effort 30 hours*effort 60 / e; run ;

The regression coefficients are multiplied by their values and summed to form the estimate, which is tested against 0


We see that the values are correct

And a test against 0 (not interesting here)

Continuous-by-continuous: calculating simple slopes

Let’s revisit the formula for the simple slope of X moderated by Z

In the estimate statement, we will put a 1 after β x and the value of z after β zx

In our model, X = hours and Z=effort

Continuous-by-continuous: calculating simple slopes

What values of effort to choose to evaluate simple slopes of hours

Two common choices:

Substantively important values (education=12yrs, BMI=18, temperature = 98.6, etc.)

Data-driven values (mean, mean+sd, mean-sd)

There are no a priori important values of effort, so we choose (mean, mean+sd, mean-sd) = (26.66,
34.8, 24.52)

Continuous-by-continuous: calculating simple slopes proc plm restore=contcont; estimate 'hours, effort=mean-sd' hours 1 hours*effort 24.52
,
‘ hours, effort=mean' hours 1 hours*effort 29.66
,
'hours, effort=mean+sd' hours 1 hours*effort 34.8
/ e; run ;

Continuous-by-continuous: calculating simple slopes

We might be interested in whether those simple slopes are different, but we don’t need to test it

Why?

If the moderator is continuous and interaction is significant then simple slopes will always be different

We demonstrate a difference to show this

Continuous-by-continuous: calculating simple slopes

To get the difference between simple slopes, take the difference between values across coefficients in the estimate statement hours 1 hours*effort 29.66
- hours 1 hours*effort 24.52
hours 0 hours*effort 5.14

Continuous-by-continuous: calculating simple slopes

Coefficients with 0 values can be omitted: proc plm restore=contcont; estimate 'diff slopes, mean+sd - mean' hours*effort 5.14
; run ;

Same t-value and p-value as interaction coefficient

Continuous-by-continuous: graphing simple slopes

We use effectplot statement in proc plm

Plot predicted outcome across range of values of predictors

We will plot across range of 2 predictors to depict an interaction

proc plm source=contcont; effectplot contour (x=hours y=effort); run ;


Contour plots uncommon

Nice that both continuous variables are represented continuously

Simple slopes of hours are horizontal lines across graph

The more the color changes, the steeper the slope

proc plm source=contcont; effectplot fit (x=hours) / at(effort= 24.52 29.66 34.8
); run ;

Effort will not be represented continuously, so we must specify values what we want

A separate graph will be plotted for each effort


More easily understood

But why not all 3 on one graph?

Creating a custom graph through scoring

We can make the graph ourselves by getting predicted loss values across a range of hours at the
3 selected effort values (24.52, 29.66, 34.8) by:

Creating a dataset of hours and effort values at which to predict the outcome loss

Use the score statement in proc plm to predict the outcome and its 95% confidence interval

Use the scored dataset in proc sgplot to create a plot

Creating a custom graph through scoring data scoredata; do effort = 24.52
, 29.66
, 34.8
; do hours = 0 to 4 by 0.1
; output; end; end; run ; proc plm source=contcont; score data=scoredata out=plotdata predicted=pred lclm=lower uclm=upper; run ; proc sgplot data=plotdata; band x=hours upper=upper lower=lower / group=effort transparency= 0.5
; series x=hours y=pred / group=effort; yaxis label="predicted loss"; run ;

Creating a custom graph through scoring
Purty!


Special case of a continuous-by-continuous interaction

Interaction of IV with itself

Allows the (linear) effect of the IV to vary depending on the level of the IV itself

Models a curvilinear relationship between DV and IV


The regression equation with linear and quadratic effects of continuous predictor X:


β x is still interpreted as slope of X when X=0
β xx interpretation slightly different
 Represents ½ the change in the slope of X when X increase by 1 unit


To get formula for simple slope of X, we must use partial derivative:

Here we see that the slope of X changes by 2 β xx per unit-increase in X


We regress loss on the linear and quadratic effect of hours proc glm data=exercise order=internal; model loss = hours|hours / solution; store quad; run ;


Quadratic effect is significant

Negative sign indicates that slope becomes more negative as hours increases (inverted U-shaped curve)

Diminishing returns on increasing hours

Quadratic effect: calculating simple slopes

We construct estimate statements for simple slopes in the same way as before

BUT, we must be careful to multiply the value after the quadratic effect by 2


We will put a 1 after β x
β xx and the value of 2*x after
No a priori important values of hours, so we choose mean=2, mean+sd=2.5, and mean-sd=1.5

Quadratic effect: calculating simple slopes proc plm restore=quad; estimate 'hours, hours=mean-sd(1.5)' hours 1 hours*hours 3 ,
'hours, hours=mean(2)' hours 1 hours*hours 4 ,
'hours, hours=mean+sd(2.5)' hours 1 hours*hours 5 / e; run ;

Slopes decrease as hours increase, eventually non-significant

Quadratic effect: comparing simple slopes

Do not need to compare

Significance always same as interaction coefficient

Quadratic effect: graphing the quadratic effect

The “fit” type of effectplot is made for plotting the outcome vs a single continuous predictor proc plm restore=quad; effectplot fit (x=hours); run ;

Quadratic effect: graphing the quadratic effect
• Diminishing returns apparent
• Too many hours of exercise may lead to weight gain

Continuous-by-categorical: the model

We can also estimate the simple slopes in a continuous-by-categorical interaction

We will estimate the slope of the continuous variable within each category of the categorical variable

We could also look at the simple effects of the categorical variable across levels of the continuous

First, how do categorical variables enter regression models?

Categorical predictors and dummy variables



A categorical predictor with k categories can be represented by k dummy variables
Each dummy codes for membership to a category, where 0=non-membership and 1=membership
However, typically only k-1 dummies are entered into the regression model?

Each dummy is a linear combination of all other dummies -- collinearity

Regression model cannot estimate coefficient for a collinear predictor

Categorical predictors and dummy variables

Omitted category known as the reference category

All effects of a categorical variable in the regression table are comparisons with reference group

SAS by default will use the last category as the reference

Categorical predictors and dummy variables

Interaction of dummy variables and continuous variable

To interact the dummy variables with a continuous predictor, multiply each one by the continuous variable

Any interaction involving an omitted dummy will be omitted as well

Continuous-by-categorical: the model

Here is the regression equation for a continuous variable, X, interacted with a 3-category categorical predictor, M



β x is simple slope of X for M=3
β m1
β xm1 and β m2 and β xm2 are simple effects of M when X=0 r epresent differences in slopes of X when
M=1 and M=2, and differences in simple effects of M per unit change in X

Continuous-by-categorical: the model

Formulas for simple slopes

Continuous-by-categorical: example model

We regress loss on hours, prog (3-category) and their interaction proc glm data=exercise order=internal; class prog; model loss = hours|prog / solution; store catcont; run ;


Put prog on class statement to declare it categorical
Use order=internal to order prog by numeric value rather than formats

Continuous-by-categorical: example model
Interaction is significant overall
Notice the 0 coefficients for reference groups

Continuous-by-categorical: calculating simple slopes

Here are the formulas for our simple slopes again:


SAS will accept the first two formulas for estimates of the simple slopes in estimate statements
But the estimate statement for the slope of X when (M=3)
REQUIRES the inclusion of the coefficient for the interaction X and (M=3), even though it is constrained to 0

We don’t normally need to calculate the slope in the reference group, nor compare to other slopes, so not usually a huge problem

Continuous-by-categorical: calculating simple slopes proc plm restore = catcont; estimate 'hours slope, prog=1 jogging' hours 1 hours*prog 1 0 0 ,
'hours slope, prog=2 swimming' hours 1 hours*prog 0 1 0 ,
'hours slope, prog=3 reading' hours 1 hours*prog 0 0 1 / e; run ;

Notice the inclusion of the zero coefficient in the estimate of the slope when M=3

Continuous-by-categorical: calculating simple slopes

Increasing hours increases weight loss in jogging and swimming, lessens loss in reading program

Notice that last estimate appears in regression table as hours coefficient


If calculating a simple slope or effect, do not omit interaction coefficients

Otherwise, SAS will average over those coefficients

Let’s pretend we forgot to include the 0 interaction coefficient in the estimation of the hours slope when
M=3 proc plm restore = catcont; estimate 'hours slope, prog=3 reading (wrong)' hours 1 / e; run ;


The e option gives us the estimate coefficients

SAS applied values of .333 to all 3 interaction coefficients, averaging their effects

Continuous-by-categorical: calculating simple slopes

We again take differences in values across coefficients to test differences in simple slopes: hours 1 hours*prog 1 0 0
-hours 1 hours*prog 0 1 0 hours 0 hours*prog 1 -1 0

Continuous-by-categorical: calculating simple slopes proc plm restore = catcont; estimate 'diff slopes, prog=1 vs prog=2' hours*prog 1 1 0 ,
'diff slopes, prog=1 vs prog=3' hours*prog 1 0 1 ,
'diff slopes, prog=2 vs prog=3' hours*prog 0 1 1 / e; run ;


Slopes in prog=1 and prog=2 do not differ
Other 2 comparisons are regression coefficients

Continuous-by-categorical: graphing slopes

The slicefit type of effectplot plots the outcome against a continuous predictor on the X-axis, with separate lines by a categorical predictor (typically, but can be continuous) proc plm source=catcon; effectplot slicefit (x=hours sliceby=prog) / clm; run ;

The option clm adds confidence limits

Continuous-by-categorical: graphing slopes
Easy to see direction of effects, and that slopes in jogging and reading do not differ

Categorical-by-categorical: the model

The interaction of a categorical variables X with 2 categories and M with 3 produces 6 interaction dummies

Any interaction dummy formed by a omitted dummy will be omitted as well

4 of the 6 will be omitted because of collinearity

Categorical-by-categorical: the model

Categorical-by-categorical: the model

Categorical-by-categorical: the model

Regression equation modeling the interaction of X and
M



β x is simple effect of X (X=0 vs X=1) for M=3
β m1 and β m2 are simple effects of M when X=1
β x0m1 and β x0m2 r epresent differences in effects of X when M=1 and M=2, or differences in effects of M when X=0

Think of simple effects as differences in expected means

Simple effects represent differences between the mean outcome of 2 groups that belong to different categories on one predictor

For instance, the simple effect of X when M=1 is the difference between the mean outcome when
X=0,M=1 and the mean outcome when X=1,M=1

Simple effects expressed as differences in means

Categorical-by-categorical: example model proc glm data=exercise order=internal; class female prog; model loss = female|prog / solution; store catcat; run ;

Categorical-by-categorical: example model
Interaction is overall significant
Lots of omitted coefficients

Categorical-by-categorical: estimating simple effects with the slice statement

Slice statement designed for simple effect estimation

Syntax:
 slice interaction_effect / sliceby= diff
 interaction_effect is interaction to be decomposed

Sliceby= specifies variable at whose distinct levels the simple effects of the other variable will be estimated

Diff produces numerical estimates of the simple effect, instead of just a test of significance (default)

Categorical-by-categorical: estimating simple effects with the slice statement proc plm restore = catcat; slice female*prog / sliceby=prog diff adj=bon plots=none nof e means; slice female*prog / sliceby=female diff adj=bon plots=none nof e means; run ;





Estimates both sets of simple effects
Bonferroni adjustment due to multiple comparisons (adj=bon)
No plotting (hard to interpret and slow)
We suppress the somewhat redundant F-test “nof”
The means of each cell will be output with “means”

Categorical-by-categorical: estimating simple effects with the slice statement
All simple effects are significant except males vs females in reading program
So genders differ in other 2 programs
And programs differ within each gender

Estimating simple effects with the lsmestimate statement

The lsmestimate statement combines lsmeans and estimate statements

Used to estimate linear combinations of estimated
(marginal) means

From a balanced population

Simple effects can be estimated through linear combinations of marginal means

Estimating simple effects with the lsmestimate statement

Syntax:
 lsmestimate effect [value, level_x level_m]...

Effect is effect made up of only categorical predictors

Value is value to apply to mean in linear combination
 level_x and level_m are the ORDINAL levels of the categorical predictors defining target mean

For X=0 and X=1, specify 1 for X=0 and 2 for X=1

Estimating simple effects with the lsmestimate statement proc plm restore=catcat; lsmestimate female*prog 'male-female, prog = jogging(1)' [ 1 , 1 1 ] [1 , 2 1 ],
'male-female, prog = swimming(2)' [ 1 , 1 2 ] [1 , 2 2 ],
'male-female, prog = reading(3)' [ 1 , 1 3 ] [1 , 2 3 ],
'jogging-reading, female = male(0)' [ 1 , 1 1 ] [1 , 1 3 ],
'jogging-reading, female = female(1)' [ 1 , 2 1 ] [1 , 2 3 ],
'swimming-reading, female = male(0)' [ 1 , 1 2 ] [1 , 1 3 ],
'swimming-reading, female = female(1)' [ 1 , 2 2 ] [1 , 2 3 ],
'jogging-swimming, female = male(0)' [ 1 , 1 1 ] [1 , 1 2 ],
'jogging-swimming, female = female(1)' [ 1 , 2 1 ] [1 , 2 2 ] / e adj=bon; run ;

Estimating simple effects with the lsmestimate statement
Same estimates as slice statement

Comparing simple effects with the lsmestimate statement

Only the lsmestimate and not the slice statement can compare simple effects

To compare, place 2 simple effects on same row and reverse values for 1
[1, 1 1] [-1, 2 1]
-[1, 1 2] [-1, 2 2]
[1, 1 1] [-1, 2 1] [-1, 1 2] [1, 2 2]

Comparing simple effects with the lsmestimate statement proc plm restore=catcat; lsmestimate prog*female 'diff m-f, jog-swim’ [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 2 ] [ 1 , 2 2 ],
'diff m-f, jog-read' [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 3 ] [ 1 , 2 3 ],
'diff m-f, swim-read' [ 1 , 1 2 ] [1 , 2 2 ] [1 , 1 3 ] [ 1 , 2 3 ],
'diff jog-read, m - f' [ 1 , 1 1 ] [1 , 1 3 ] [1 , 2 1 ] [ 1 , 2 3 ],
'diff swim-read, m - f' [ 1 , 1 2 ] [1 , 1 3 ] [1 , 2 2 ] [ 1 , 2 3 ],
'diff jog-swim, m - f' [ 1 , 1 1 ] [1 , 1 2 ] [1 , 2 1 ] [ 1 , 2 2 ]/ e adj=bon; run ;

Comparing simple effects with the lsmestimate statement
All differences are significant – although only one we already didn’t know

Categorical-by-categorical: graphing simple effects

The interaction type effectplot is used to plot the outcome vs two categorical predictors.

The connect option is used to connect the points proc plm restore=catcat; effectplot interaction (x=female sliceby=prog) / clm connect; effectplot interaction (x=prog sliceby=female) / clm connect; run ;

No effect of gender in the reading program

Effect of program seems stronger from females

3-way interactions: categorical-bycategorical-by-continuous

Interaction of 3 predictors can be decomposed in many more ways than the interaction of 2.

Imagine we interact 2-category X with 3-category
M and continuous Z

How can we decompose this interaction?

3-way interactions, categorical-bycategorical-by-continuous: the model

We can estimate the conditional interaction of X and Z across levels of M

Do X and Z interact at each level of M?

Are the X and Z interactions different across levels of
M?
 We can further decompose the conditional interactions of X and Z

What are the simple slopes of Z across X and simple effects of X across Z?

3-way interactions, categorical-bycategorical-by-continuous: the model

We could then look at interaction of X and M across levels of Z

Do X and M interact at various values of Z?

Are these interactions different?
 Within each conditional interaction of X and M, what are the simple effects of X and M?

We can also look at the interaction of M and Z across X

3-way interactions, categorical-bycategorical-by-continuous: the model

Regression equation can be intimidating


Single variable coefficients are still simple effects and slopes (but now for 2 reference levels each)
2-way interaction coefficients are conditional interactions (at reference level of 3 rd variable)

3-way interactions: example model

We regress loss on female (2-category), prog (3category) and hours (continuous) proc glm data = exercise order=internal; class female prog; model loss = female|prog|hours / solution; store catcatcon; run ;

3-way interactions: example model
3-way interaction is significant

3-way interactions: example model
Not very easy to interpret!

3-way interactions: simple slope focused analysis

Imagine our focus is estimating which groups benefit the most from increasing the weekly number of hours of exercise.

This analysis is focused on the simple slopes of hours

We approach this section by addressing questions the researcher might ask, starting with the lowest level and building up

What are the simple slopes of Z across levels of X and M?


There are a total of 6 groups made up by X and M, and we can estimate the slope of hours in each
We use estimate statements again

Place a 1 after the coefficient for the slope variable by itself (e.g. hours)

Place a 1 after each 2-way interaction coefficient involving the slope variable and either of the 2 factor groups (e.g. hours*(female=0) and hours*(prog=1))

Place a 1 after the 3-way interaction coefficient involving the slope variable and both of the factor groups (e.g. hours*(female=0,prog=1))

3-way interaction: estimating simple slopes using estimate statement proc plm restore=catcatcon; estimate 'hours slope, male prog=jogging' hours 1 hours*female 1 0 hours*prog 1 0 0 hours*female*prog 1 0 0 0 0 0 ,
'hours slope, male prog=swimming' hours 1 hours*female 1 0 hours*prog 0 1 0 hours*female*prog 0 1 0 0 0 0 ,
'hours slope, male prog=reading' hours 1 hours*female 1 0 hours*prog 0 0 1 hours*female*prog 0 0 1 0 0 0 ,
'hours slope, female prog=jogging' hours 1 hours*female 0 1 hours*prog 1 0 0 hours*female*prog 0 0 0 1 0 0 ,
'hours slope, female prog=swimming' hours 1 hours*female 0 1 hours*prog 0 1 0 hours*female*prog 0 0 0 0 1 0 ,
'hours slope, female prog=reading' hours 1 hours*female 0 1 hours*prog 0 0 1 hours*female*prog 0 0 0 0 0 1 / e adj=bon; run ;

3-way interaction: estimating simple slopes using estimate statement
Increasing number of weekly hours of exercise significantly increases weight loss in all groups except those in the reading program, where it decreases weight loss (not significantly for females in the reading program after Bonferroni adjustment)

Are the (X*Z) conditional interactions significant?

We can now compare the simple slopes hours of between genders within each program

This is a test of whether hours and gender interact within each program

As always, we test differences in effects by subtracting values across coefficients

Are the (X*Z) conditional interactions significant?
proc plm restore=catcatcon; estimate 'diff hours slope, male-female prog=1' hours*female 1 1 hours*female*prog 1 0 0 1 0 0 ,
'diff hours slope, male-female prog=2' hours*female 1 1 hours*female*prog 0 1 0 0 1 0 , adj=bon;
'diff hours slope, male-female prog=3' hours*female 1 1 hours*female*prog 0 0 1 0 0 1 / e run ;
Males and female benefit differently from increasing the number of hours in jogging and reading programs.
One of these interactions appears in the regression table? Which one?

Are the (X*Z) conditional interactions different?

We can test if the conditional interactions are different from one another

Do the way males and females benefit differently by increasing hours of exercise VARY between programs?

Take differences between conditional interactions

Notice only the 3-way interaction coefficient is left

Are the (X*Z) conditional interactions different?
proc plm restore=catcatcon; estimate 'diff diff hours slope, male-female prog=1-prog=2' hours*female*prog 1 1 0 1 1 0 ,
'diff diff hours slope, male-female prog=1-prog=3' hours*female*prog 1 0 1 1 0 1 ,
'diff diff hours slope, male-female prog=2-prog=3' hours*female*prog 0 1 1 0 1 1 / e; run ;
All of the comparisons are significant. The differential benefit from increasing exercise hours between genders differs between all 3 programs.

3-way interaction: graphing simple slopes
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We need to partition our graphs by a thirdvariable now.
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We can use the plotby= option, to plot separate graphs across levels of a variable proc plm restore=catcatcon; effectplot slicefit (x=hours sliceby=female plotby=prog) / clm; run ;

3-way interaction: graphing simple slopes
Easy to see slopes, differences between slopes, and interactions

3-way interaction, simple effects focused analysis
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Imagine instead we are more interested in gender differences across programs and at different hours of weekly exercise?
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Similar questions can be posed

What are the simple effects of X across M and Z?
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We use lsmestimate statements to estimate simple effects of female at each level of prog at the mean, mean-sd and mean+sd of hours
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The “at” option allows us to specify hours
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For this question we could use slice or lsmestimate

Estimating the simple effects of X across M and Z using lsmestimate proc plm restore=catcatcon; lsmestimate female*prog 'male-female, prog=jogging(1) hours=1.51' [ 1 , 1 1 ] [1 , 2 1 ],
'male-female, prog=swimming(2) hours=1.51' [ 1 , 1 2 ] [1 , 2 2 ],
'male-female, prog=reading(3) hours=1.51' [ 1 , 1 3 ] [1 , 2 3 ] / e adj=bon at hours= 1.51
; lsmestimate female*prog 'male-female, prog=jogging(1) hours=2' [ 1 , 1 1 ] [1 , 2 1 ],
'male-female, prog=swimming(2) hours=2' [ 1 , 1 2 ] [1 , 2 2 ],
'male-female, prog=reading(3) hours=2' [ 1 , 1 3 ] [1 , 2 3 ] / e adj=bon at hours= 2 ; lsmestimate female*prog 'male-female, prog=jogging(1) hours=2.5' [ 1 , 1 1 ] [1 , 2 1 ],
'male-female, prog=swimming(2) hours=2.5' [ 1 , 1 2 ] [1 , 2 2 ],
'male-female, prog=reading(3) hours=2.5' [ 1 , 1 3 ] [1 , 2 3 ] / e adj=bon at hours= 2.5
; run ;

Estimating the simple effects of X across M and Z using lsmestimate

Are the conditional interactions significant?
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The overall test of each conditional interaction of female and program (at a fixed number of hours) involves tests of 2 coefficients (which are differences in simple effects), so must be tested with a joint F-test
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The “joint” option on lsmestimate performs a joint Ftest

Are the conditional interactions significant?
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 proc plm restore=catcatcon; lsmestimate female*prog 'diff male-female, prog=1 - prog=2, hours=1.51' [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 2 ] [ 1 , 2 2 ],
'diff male-female, prog=1 - prog=3, hours=1.51' [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 3 ] [ 1 , 2 3 ],
'diff male-female, prog=2 - prog=3, hours=1.51' [ 1 , 1 2 ] [1 , 2 2 ] [1 , 1 3 ] [ 1 , 2 3 ] / e at hours= 1.51
joint; lsmestimate female*prog 'diff male-female, prog=1 - prog=2, hours=2' [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 2 ] [ 1 , 2 2 ],
'diff male-female, prog=1 - prog=3, hours=2' [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 3 ] [ 1 , 2 3 ],
'diff male-female, prog=2 - prog=3, hours=2' [ 1 , 1 2 ] [1 , 2 2 ] [1 , 1 3 ] [ 1 , 2 3 ] / e at hours= 2 joint; lsmestimate female*prog 'diff male-female, prog=1 - prog=2, hours=2.5' [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 2 ] [ 1 , 2 2 ],
'diff male-female, prog=1 - prog=3, hours=2.5' [ 1 , 1 1 ] [1 , 2 1 ] [1 , 1 3 ] [ 1 , 2 3 ], diff male-female, prog=2 - prog=3, hours=2.5' [ 1 , 1 2 ] [1 , 2 2 ] [1 , 1 3 ] [ 1 , 2 3 ] / e at hours= 2.5
joint; run ;

Are the conditional interactions significant?
Female and prog significantly interact at hours = 1.5, 2 and 2.5

3-way interaction: graphing simple effects
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We add the plotby= option to an interaction plot proc plm restore=catcatcon; effectplot interaction (x=female sliceby=prog) / at(hours = 1.51 2 2.5
) clm connect; run ;

3-way interaction: graphing simple effects
Interaction more pronounced at lower numbers of hours

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Binary (0/1) outcome
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Often defined as success and failure
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Models how predictors affect probability of the outcome
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Probability, p, is transformed to logit in logistic regression

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Logit transforms probability to log-odds metric
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Can take on any value (instead of restricted to 0 through 1)

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Logit of p (not p itself) is modeled as having a linear relationship with predictors

Non-linear relationship between p and predictors
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Imagine simple logit model where estimate the log odds of p when X=0 and X=1:
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The difference between log odds estimate is:
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Remembering our logarithmic identity and the definition of odds:

Non-linear relationship between p and predictors
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We substitute and get:
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Which we then exponentiate:

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Exponentiated logistic regression coefficients are interpreted as odds ratios (ORs)
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By what factor is the odds changed per unit increase in the predictor
Or, what is the percent change in the odds per unit increase in the predictor
Odds ratios are constant across the range of the predictor
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Differences in probabilities are not
But ORs can be misleading without knowing the underlying probabilities

Logistic regression, categorical-bycontinuous interaction: example model
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We model how the odds (probability) of satisfaction is predicted by hours of exercise, program and their interaction
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We can create an item store in proc logistic for proc plm

Logistic regression, categorical-bycontinuous interaction: example model proc logistic data = exercise descending; class prog / param=glm order=internal; model satisfied = prog|hours / expb; store logit; run ;
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 descending tells SAS to model probability of 1 instead of 0, the default param=glm ensures we use dummy coding (rather than effect coding, the default) expb exponentiates the regression coeffients – although not all are interpreted as odds ratios

Logistic regression, categorical-bycontinuous interaction: example model
Interaction is significant

Logistic regression cat-by-cont, calculating and graphing simple ORs
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The simple slope of hours in each program yields an odds ratio when exponentiated
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We use the oddsratio statement within proc logistic to estimate these simple odds ratios
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A nice odds ratio plot is produced by default

Logistic regression cat-by-cont, calculating and graphing simple ORs proc logistic data = exercise descending; class prog / param=glm order=internal; model satisfied = prog|hours / expb; oddsratio hours / at(prog=all); store logit; run ;
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The at(prog=all) option requests that oddsratio for hours be calculated at each level of prog

Logistic regression cat-by-cont, calculating and graphing simple ORs
Increasing weekly hours of exercise increases odds of satisfaction in jogging and swimming groups

Simple odds ratios can be compared in estimate statements
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This code produces the simple odds ratios in an estimate statement proc plm restore=logit; estimate 'hours OR, prog=1' hours 1 hours*prog 1 0 0 ,
'hours OR, prog=2' hours 1 hours*prog 0 1 0 ,
'hours OR, prog=3' hours 1 hours*prog 0 0 1 / e exp cl; run;
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This code compares them proc plm restore=logit; estimate 'ratio hours OR, prog=1/prog=2' hours*prog 1 1 0 ,
'ratio hours OR, prog=1/prog=3' hours*prog 1 0 1 ,
'ratio hours OR, prog=2 /prog=3' hours*prog 0 1 1 / e exp cl; run ;

Simple odds ratios can be compared in estimate statements
The exponentiated differences between simple slopes (exponentiated interaction coefficient) yields a ratio of odds ratios
ORjog/ORswim = Ratio of ORs
4.109/5.079 = .809

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Odds ratios summarize the effects of predictors in 1 number, but can be misleading because we don’t know the underlying probabilities
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E.g. OR for p=.001 and p=.003 is the same OR for p=.25 and p=.5
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Good idea to get sense of probabilities of outcome across groups

The lsmeans statement for predicted probabilities
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The lsmeans statement is used to estimate marginal means
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The ilink option allows transformation back the original response metric (here probabilities)
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The at option allows specification of continuous covariates for estimation of means

The lsmeans statement for predicted probabilities proc plm source = logit; lsmeans prog / at hours= 1.51
ilink plots=none; lsmeans prog / at hours= 2 ilink plots=none; lsmeans prog / at hours= 2.5
ilink plots=none; run ;

The lsmeans statement for predicted probabilities
Predicted probabilities are the column “Mean”

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The effectplot statement by default plots the outcome in its original metric
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We can get an idea of the simple effects and simple slopes in the probability metric with 2 effectplot statements proc plm restore=logit; effectplot interaction (x=prog) / at(hours = 1.51 2 2.5
) clm; effectplot slicefit (x=hours sliceby=prog) / clm; run ;

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Guidelines for using estimate statement to estimate simple slopes
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Always put a 1 after the coefficient for slope variable
If interacted with continuous IV (not quadratic), put value of continuous IV after interaction coefficient
If interacted with categorical, put a 1 after relevant interaction dummy
If interacted with 3 way, make sure to include:
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 the coefficient alone both 2-way coefficients involving slope and either interactor
The 3-way coefficient involving all interactors
Follow the second rule above if interaction involves continuous (unless both are continuous, in which case apply the product of the 2 continuous interactors)
Follow the third rule if the interaction involves only dummy variables
To estimate differences, subtract values across coefficients
Use “e” to check values and coefficients
Use “joint” to perform a joint F-test
Use adj= to correct for multiple comparisons
Use exp to exponentiate estimates (for logistic and other non-linear models)

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Guidelines for using lsmestimate statement to estimate simple effects
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Think of simple effects as differences between means
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Assign one mean the value 1 and the other -1
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Remember to use ordinal values for categorical predictors, not the actual numeric values
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To compare simple effects, put two effects on the same row and reverse the values for one of them
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Use joint for joint F-tests
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Use adj= for multiple comparisons

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Thank you for attending!