Transcript statistics.ats.ucla.edu

ANALYZING AND
VISUALIZING INTERACTIONS
IN SAS 9.4
Andy Lin
IDRE Statistical Consulting
Background

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
Background

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
Purpose of seminar


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=β0+βsSEX+βhHEIGHT
Main effects vs interaction models

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=β0+βsSEX+βhHEIGHT
Main effects vs interaction models

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
Simple effects and slopes

From this equation

We can derive sex-specific regression equations
Males (sex=0)

Females (sex=1)


Simple effects and slopes

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
PROC PLM



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
more
glm, genmod, logistic, phreg, mixed, glmmix, and
PROC PLM


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
PROC PLM


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
PROC PLM

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
WHY PROC PLM


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
Dataset used in seminar



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
Dataset used in the seminar

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
Continuous-by-continuous: the model

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
Continuous-by-continuous: the model


Let us look at model where Y is predicted by
continuous X, continuous Z, and their interaction:
Be careful when interpreting βx and βz
 They
are simple effects (when interacting variable=0),
not main effects
Continuous-by-continuous: the model


The coefficient βxz 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 statement syntax


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 βx
and the value of z after βzx
 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

Simple slopes as contour plots
proc plm source=contcont;
effectplot contour (x=hours y=effort);
run;
Simple slopes as contour plots



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
Simple slopes as a fit plot
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
Simple slopes as a fit plot


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
series x=hours y=pred / group=effort;
yaxis label="predicted loss";
run;
transparency=0.5;
Creating a custom graph through
scoring
Purty!
Quadratic effect: the model


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
Quadratic effect: the model



The regression equation with linear and quadratic
effects of continuous predictor X:
βx is still interpreted as slope of X when X=0
βxx interpretation slightly different
 Represents
½ the change in the slope of X when X
increase by 1 unit
Quadratic effect: the model


To get formula for simple slope of X, we must use
partial derivative:
Here we see that the slope of X changes by 2βxx
per unit-increase in X
Quadratic effect: example model

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: example model

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 βx and the value of 2*x after
βxx
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
βx is simple slope of X for M=3
βm1 and βm2 are simple effects of M when X=0
βxm1 and βxm2 represent 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
Potential pitfall

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;
Potential pitfall


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
βx is simple effect of X (X=0 vs X=1) for M=3
βm1 and βm2 are simple effects of M when X=1
βx0m1 and βx0m2 represent 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'
adj=bon;
run;
[1, 1 1] [-1, 1 2] [-1, 2 1] [1, 2 2]/ e
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;
Graph of simple gender effects
No effect of
gender in the
reading
program
Graph of simple program effects
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 3rd 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'
run;
hours 1 hours*female 0 1 hours*prog 0 0 1 hours*female*prog 0 0 0 0 0 1 / e adj=bon;
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,
'diff hours slope, male-female prog=3'
hours*female 1 -1 hours*female*prog 0 0 1 0 0 -1 / e
adj=bon;
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


We need to partition our graphs by a thirdvariable now.
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


Imagine instead we are more interested in gender
differences across programs and at different hours
of weekly exercise?
Similar questions can be posed
What are the simple effects of X
across M and Z?



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
The “at” option allows us to specify hours
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'
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'
run;
[1, 1 3] [-1, 2 3] / e adj=bon at hours=1.51;
[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;
Estimating the simple effects of X
across M and Z using lsmestimate
Are the conditional interactions
significant?


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
The “joint” option on lsmestimate performs a joint Ftest
Are the conditional interactions
significant?

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

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
Logistic Regression

Binary (0/1) outcome
 Often


defined as success and failure
Models how predictors affect probability of the
outcome
Probability, p, is transformed to logit in logistic
regression
Logit transformation

Logit transforms probability to log-odds metric
 Can
take on any value (instead of restricted to 0
through 1)
Logistic regression

Logit of p (not p itself) is modeled as having a
linear relationship with predictors
Non-linear relationship between p and
predictors



Imagine simple logit model where estimate the log
odds of p when X=0 and X=1:
The difference between log odds estimate is:
Remembering our logarithmic identity and the
definition of odds:
Non-linear relationship between p and
predictors

We substitute and get:

Which we then exponentiate:
Odds ratios

Exponentiated logistic regression coefficients are
interpreted as odds ratios (ORs)
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
Differences in probabilities are not
 But ORs can be misleading without knowing the underlying
probabilities

Logistic regression, categorical-bycontinuous interaction: example model


We model how the odds (probability) of
satisfaction is predicted by hours of exercise,
program and their interaction
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;



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


The simple slope of hours in each program yields an
odds ratio when exponentiated
We use the oddsratio statement within proc logistic
to estimate these simple odds ratios
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;

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

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;

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
Predicted probabilities

Odds ratios summarize the effects of predictors in 1
number, but can be misleading because we don’t
know the underlying probabilities
 E.g.
OR for p=.001 and p=.003 is the same OR for
p=.25 and p=.5

Good idea to get sense of probabilities of outcome
across groups
The lsmeans statement for predicted
probabilities

The lsmeans statement is used to estimate marginal
means
 The
ilink option allows transformation back the original
response metric (here probabilities)
 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”
Graphs of predicted probabilities


The effectplot statement by default plots the
outcome in its original metric
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;
Graphs of predicted probabilities
Graphs of predicted probabilities
Concluding guidelines

Guidelines for using estimate statement to estimate simple slopes




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:










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)
Concluding guidelines

Guidelines for using lsmestimate statement to
estimate simple effects
 Think
of simple effects as differences between means
 Assign one mean the value 1 and the other -1
 Remember to use ordinal values for categorical
predictors, not the actual numeric values
 To compare simple effects, put two effects on the same
row and reverse the values for one of them
 Use joint for joint F-tests
 Use adj= for multiple comparisons
It’s over!

Thank you for attending!