Conducting post-hoc tests of
compound coefficients using
simple slopes for a categorical by
categorical interaction
Jane E. Miller, PhD
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Overview
•
•
•
•
Simple slopes defined
Application of simple slopes to interactions
Calculation of standard errors for simple slopes
Charts to show conclusions of simple slope tests for
interactions
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Post-hoc probing
• Post-hoc (“after the fact”) probing is a way to
conduct formal statistical tests of differences that
were not specified for a priori inferential testing, e.g.,
– Comparisons other than against the reference category
• See podcast on testing statistical significance of differences across
coefficients (s)
– Those involving more than one , e.g., interactions
• Can be conducted from output that is easily obtained
from standard software packages
• Does not require respecifying the model
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Simple slopes calculation for
compound coefficients
• The simple slopes technique is a way to calculate the
point estimate of effect size for a combination of two
coefficients from the same model
• The point estimate of effect size for each of those
combinations is a compound coefficient
– A linear combination of two is
• Models involving main effects and interaction terms
require summing more than one  to obtain the
overall effect of the variables involved in the
interaction
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Review: Calculation of overall effect of
an interaction
• The predicted value of the dependent variable from a model
with a two-way interaction between predictors X1 and X2 can
be written as a function of the estimated coefficients βi:
Y = β0 + β1 X1 + β2X2 + β3 X1_X2
• Rearranging terms to group those that involve the focal
predictor (X1 ) yields:
(β0 + β2 X2) + (β1 +β3X2) X1 ,
simple intercept ώ0
simple slope ώ1
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Review: Focal and modifier variables
in an interaction
• In a statistical interaction between two independent
variables X1 and X2
– X1 is sometimes referred to as the focal predictor
– X2 is called the moderator or modifier variable because it
alters the association between X1 and Y
• Identifying which of the IVs in an interaction is the
modifier depends on the research question
– E.g., race/ethnicity is specified as the modifier because it is
hypothesized to change the association between education
and birth weight
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Standard errors for simple slopes
• Both the simple intercept and the simple slope are
compound coefficients
– Each is a linear combination of two estimated is
• Simple slope: ώ0 = β0 + β2 X2
• Simple intercept : ώ1 = (β1 +β3X2) X1
• The inferential statistical tests for a compound
coefficient require calculating the standard error of
the simple slope using the estimated variances and
covariances for those is
s.e. (ώ1 | X2) = √ [var(β1) + (2 × X2× cov(β1,β3)) + (X2)2× var(β3)]
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Calculating the standard error for a
compound coefficient
• s.e. (ώ1 | X2) is the standard error of the simple slope ώ1
s.e. (ώ1 | X2) = √ [var(β1) + (2 × X2× cov(β1β3)) + (X2)2× var(β3)]
• Where
 var(j) and var(k) are the variances of j and k, respectively
 cov(j, k) is the covariance between j and k
• The variance-covariance matrix can be requested as
part of a regression command
• Note that s.e. (ώ1 | X2) depends on the value of the
moderator variable in the interaction, X2
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Formulas to calculate overall interaction pattern
BW = f (NHB, MA, <HS, =HS,
NHB_<HS, NHB_=HS, MA_<HS, MA_=HS)
Subgroup
βs involved in overall
effect for that group*
Non-Hispanic white, < HS
= β<HS
Non-Hispanic white, = HS
= β=HS
Non-Hispanic white, > HS
NA (reference category)
Non-Hispanic black, < HS
= βNHB + β<HS + βNHB_<HS
Non-Hispanic black, = HS
= βNHB + β=HS + βNHB_=HS
Non-Hispanic black, > HS
= βNHB
Mexican American, < HS
= βMA + β<HS + βMA_<HS
Mexican American, = HS
= βMA + β=HS + βMA_=HS
Mexican American, > HS
= βMA
• More than one β
is involved in the
calculations of
overall effect for
any subgroup that
is not in the
reference
category for
either IV in the
interaction
* Difference in birth weight compared to reference category
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Estimated coefficients from an OLS model of birth
weight in grams
β
t-statistic
Main effects terms
Race (ref. = non-Hispanic white)
Non-Hispanic Black (NHB)
Mexican American (MA)
Mother’s ed. (ref. = > HS)
Less than high school (<HS)
High school graduate (=HS)
Interactions: race & education
NHB_<HS
MA_<HS
NHB_=HS
MA_=HS
F-statistic
Degrees of freedom (df)
** denotes p < 0.01
–168.1**
–104.2**
–5.66
–2.16
–54.2**
–62.0**
–2.35
–3.77
–38.5
99.4
18.4
93.7
65.59
13
–0.88
1.72
0.47
1.49
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Example calculation of a simple slope
ώ1 = (β1 +β3X2) × X1
• Where X1 = NHB and X2 = <HS
ώ1 = (β<HS +β<HS_NHB) × NHB
• From the model results on the previous slide
 β<HS = –54.2
 β<HS_NHB = –38.5
• For non-Hispanic black infants, NHB = 1
• Substituting those values into the equation:
ώ1 = (–54.2 + (–38.5)) × 1 = –92.7
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Example: Calculation of the standard error of the
simple slope for one race_education category
Variance-covariance matrix for the
estimated coefficients (βs)
βNHB
β<HS
β<HS_NHB
βNHB
882.10
β<HS
522.94
β <HS_NHB
–461.22 1,918.81
• In this example,
 <HS is X1, coded
 1 = non-Hispanic black
 0 = other racial/ethnic groups
 NHB is X2, coded
 1 = non-Hispanic black
 0 = other racial/ethnic groups
s.e. (ώ1 | NHB) = √ [var(β<HS)+(2×NHB × cov(β<HSβ<HS_NHB))+(NHB)2 × var(β<HS_NHB)]
= √ [522.94 + 2 × 1× (–461.22) + (1)2× 1,918.81] = 39.0
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Predicted birth weight with 95% CI,
by race and educational attainment
<HS
=HS
>HS
Difference in birth weight (grams)
50
0
-50
-100
Non-Hispanic white
-150
Mexican-American
-200
Non-Hispanic black
-250
-300
-350
-400
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Which contrasts are possible with
non-Hispanic white, > HS as the reference category
Predicted birth weight (grams) by
mother’s education and race/ethnicity,
United States, 1988–1994 NHANES III
Mother’s educational
attainment
Race/ethnicity
< HS
= HS
> HS
Non-Hispanic white 3,044 3,044 3,112
Mexican American
3,027 3,031 3,003
Non-Hispanic black
2,820 2,883 2,937
• The circled cell is the reference
category (Non-Hispanic white,
mother’s education > HS)
• Yellow-shaded cells can be
compared to the reference
category based on the standard
errors of the associated main
effects terms alone
• Green-shaded cells can be
compared to the reference
category using standard errors
calculated using the simple slope
for a compound coefficient
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Ballpark assessment of other contrasts
• If there is substantial overlap between confidence
intervals for two values, you can usually safely
conclude that they are not statistically significantly
different from one another
– It is very unlikely that taking the covariance between the
two βs into account when computing the standard error of
their simple slopes would alter that conclusion
• If the confidence intervals do not overlap or overlap
only slightly, formally test the statistical significance
of that difference by respecifying the model with a
different reference category
– See the podcast on that topic
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Visual assessment of other contrasts based
on confidence intervals from simple slope
<HS
Difference in birth weight (grams)
• The confidence intervals for
non-Hispanic white and
Mexican American infants
overlap considerably in the <
HS and = HS groups
• Even if these birth weight
differences were statistically
significant, they are so small
that they would be of trivial
substantive interest
=HS
>HS
50
0
-50
-100
-150
-200
-250
-300
-350
-400
Non-Hispanic white
Mexican-American
Non-Hispanic black
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Another contrast to test with a different
model specification
• Re-specify the model with
non-Hispanic black as the
reference category
<HS
Difference in birth weight (grams)
• The confidence intervals for
non-Hispanic black infants
don’t overlap very much in
the < HS and > HS groups
• The birth weight difference
between these groups is
>100 grams, so it is worth
testing whether that
difference is statistically
significant
=HS
>HS
50
0
-50
-100
-150
-200
-250
-300
-350
-400
Non-Hispanic white
Mexican-American
Non-Hispanic black
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Online calculator for simple slopes
• Preacher has created an online calculator to compute
simple intercepts, simple slopes, and their standard
errors from regression output
– Coefficients (βs)
– Variances and covariances of the βs
– Values of the moderator variable for which to calculate
standard errors of the simple slope
– Degrees of freedom for the model
• Can also graph the shape of the interaction with
confidence intervals, given values of the focal predictor
and moderator variables
• See http://www.quantpsy.org/interact/index.html
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Presenting results of post-hoc tests
• In methods section, mention use of simple slopes technique to
calculate standard errors for compound coefficients.
– Provide citation to that method
• Create a table to present the is, standard errors, and
goodness-of-fit statistics from the multivariate model
• Conduct post-hoc tests for your substantive hypotheses behind
the scenes
– Calculate the simple slope
– Calculate the standard error of the simple slope
• Create a table or chart to report predicted values and
confidence intervals calculated from is and standard errors
– Use symbols or prose to convey which contrasts other that against the
reference category are statistically significantly different from one
another
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Summary
• Overall pattern of an interaction involves calculation of
compound coefficients
• Standard errors for compound coefficients can be
calculated using the simple slope technique based on
variance-covariance matrix from regression output
• Comparison of predicted values of the dependent
variable are against the reference category, with all
continuous independent variables set to 0
– To conduct contrasts for other values of IVs in the interaction
• Respecify the model with different reference categories
• Use the all-interaction-dummies approach
Separate podcasts
• Conduct calculations behind the scenes, present the
conclusions of those tests in the text
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Suggested resources
• Cohen, Jacob, Patricia Cohen, Stephen G. West, and Leona S. Aiken.
2003. Applied Multiple Regression/Correlation Analysis for the
Behavioral Sciences, 3rd Edition. Florence, KY: Routledge, chapters
7, 8, and 9.
• Figueiras, Adolfo, Jose Maria Domenech-Massons, and Carmen
Cadarso. 1998. Regression Models: Calculating the Confidence
Interval of Effects in the Presence of Interactions. Statistics in
Medicine 17: 2099–2105.
• Preacher, Kristopher J., Patrick J. Curran, and Daniel J. Bauer. 2006.
“Computational Tools for Probing Interaction Effects in Multiple
Linear Regression, Multilevel Modeling, and Latent Curve Analysis.”
Journal of Educational and Behavioral Statistics.31: 437–448.
• Miller, J. E. 2013. The Chicago Guide to Writing about Multivariate
Analysis, 2nd Edition. University of Chicago Press, chapter 16.
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Suggested online resources
• Preacher, Kristopher J. 2011. “Probing Interactions in
Multiple Linear Regression, Latent Curve Analysis, and
Hierarchical Linear Modeling: Interactive Calculation
Tools for Establishing Simple Intercepts, Simple Slopes,
and Regions of Significance.”
• Available online at
http://www.quantpsy.org/interact/index.html
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Suggested online resources, cont.
• Podcasts on
– Calculating the overall shape of an interaction from OLS
coefficients
– Approaches to testing statistical significance of interactions
– Using alternative reference categories to test statistical
significance of interactions
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Suggested practice exercises
• Study guide to The Chicago Guide to Writing about
Multivariate Analysis, 2nd Edition.
– Question #5 in the problem set for chapter 16
– Suggested course extensions for chapter 16
• “Reviewing” exercise #2
• “Applying statistics and writing” exercise #2
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.
Contact information
Jane E. Miller, PhD
jmiller@ifh.rutgers.edu
Online materials available at
http://press.uchicago.edu/books/miller/multivariate/index.html
The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.