Chapter 5-4. Linear Regression Adjusted Means, ANOVA, ANCOVA, Dummy Variables,
and Interaction
In Stata version 11, there were some changes made on how to use categorical predictor variables,
which made it easier. Both the current Stata 11 approach and the earlier approach, which will be
called Stata 10, are given in this chapter, for the benefit of those with an earlier version.
For this discussion, we will use an example published by Nawata et al (2004). The data were
taken from the authors’ Figure 1, a scatterplot, and so only approximate the actual values used by
the authors.
File rmr.dta Codebook
group urinary excretion of albumin group (U-Alb)
a = U-Alb < 30 mg/d
b = 30 mg/d ≤ U-Alb ≤ 300 mg/d
c = 300 mg/d < U-Alb
lbm lean body mass (kg)
rmr
resting metabolic rate (kJ/h/m2)
In Chapter 5, we saw that the t test is identically a simple linear regression (univariable
regression, without covariates). The linear regression predicted the mean outcome (unadjusted
mean) for the two groups.
A more convincing analysis, however, is to model the adjusted means, which are the group
means adjusted for potential confounding variables.
Look at the Nawata (2004) article. Notice the unadjusted means for RMR are reported in their
Table 1, although significance between the means is tested both in an unadjusted fashion and
adjusted fashion (last two columns of Table 1). Next, notice the adjusted means are given in the
legend to Figure 1.
In the authors statistical methods section, they stated they used analysis of variance (ANOVA)
and analysis of covariance (ANCOVA).
Although there are special routines in Stata to fit them more easily, analysis of variance
(ANOVA) is just a linear regression with categorical predictor variables, and analysis of
covariance (ANCOVA) is just a linear regression with both categorical variables and continuous
variables. In ANOVA terminology, categorical variables are called factors and continuous
variables are called covariates.
_________________
Source: Stoddard GJ. Biostatistics and Epidemiology Using Stata: A Course Manual [unpublished manuscript] University of Utah
School of Medicine, 2010.
Chapter 5-4 (revision 16 May 2010)
p. 1
Opening the rmr dataset in Stata,
File
Open
Find the directory where you copied the course CD
Change to the subdirectory datasets & do-files
Single click on rmr.dta
Open
use "C:\Documents and Settings\u0032770.SRVR\Desktop\
Biostats & Epi With Stata\datasets & do-files\rmr.dta", clear
*
which must be all on one line, or use:
cd "C:\Documents and Settings\u0032770.SRVR\Desktop\"
cd "Biostats & Epi With Stata\datasets & do-files"
use rmr, clear
Click on the data browser icon, and we discover the group variable is alphabetic (called a string
variable, for “string of characters”). Stata displays string variables with a red font.
Since arithmetic cannot be done on letters, we first need to convert the string variable group into
a numeric variable,
Data
Create or change variables
Other variable transformation commands
Encode value labels from string variable
Main tab: String variable: group
Create a numeric variable: groupx
OK
encode group, generate(groupx)
To see what this did,
Statistics
Summaries, tables & tests
Tables
Twoway tables with measures of association
Row variable: group
Group variables: groupx
OK
tabulate group groupx
Chapter 5-4 (revision 16 May 2010)
p. 2
|
groupx
group |
a
b
c |
Total
-----------+---------------------------------+---------a |
10
0
0 |
10
b |
0
10
0 |
10
c |
0
0
12 |
12
-----------+---------------------------------+---------Total |
10
10
12 |
32
It looks like nothing happened. Actually the variable groupx has values 1, 2, and 3. It is just that
value labels were assigned that matched the original variable.
Actual value
1
2
3
Label
a
b
c
To see this crosstabulation without the value labels, we use,
Statistics
Summaries, tables & tests
Tables
Twoway tables with measures of association
Row variable: group
Group variables: groupx
Check Suppress value labels
OK
tabulate group groupx, nolabel
|
groupx
group |
1
2
3 |
Total
-----------+---------------------------------+---------a |
10
0
0 |
10
b |
0
10
0 |
10
c |
0
0
12 |
12
-----------+---------------------------------+---------Total |
10
10
12 |
32
This is still a nominal scaled variable, but that is okay for the ANOVA type commands in Stata,
which create indicator variables “behind the scenes”.
To test the hypothesis that the three unadjusted means are equal for rmr, similar to what Nawata
did for his Table 1, we can compute a one-way ANOVA, where one-way implies one predictor
variable, which is groupx.
Chapter 5-4 (revision 16 May 2010)
p. 3
Statistics
Linear models and related
ANOVA/MANOVA
One-way ANOVA
Response variable: rmr
Factor variable: group
<- string variable OK for oneway
Output: produce summary table
OK
oneway rmr group, tabulate
|
Summary of rmr
group |
Mean
Std. Dev.
Freq.
------------+-----------------------------------a |
136.08333
16.940851
12
b |
132.7
19.032428
10
c |
166.5
19.019129
12
------------+-----------------------------------Total |
145.82353
23.604661
34
Analysis of Variance
Source
SS
df
MS
F
Prob > F
-----------------------------------------------------------------------Between groups
7990.92451
2
3995.46225
11.91
0.0001
Within groups
10396.0167
31
335.355376
-----------------------------------------------------------------------Total
18386.9412
33
557.180036
Bartlett's test for equal variances:
chi2(2) =
0.1787
Prob>chi2 = 0.915
The p value from the Analysis of Variance table, “Prob > F = 0.0001”, is what Nawata reported
in Table 1 on the RMR row, P Value < 0.0001. (Nawata’s data were slightly different.)
This was a test of the hypothesis that the three group means are the same:
Ho : 1  2  3
To get Nawata’s Adjusted P Value (for LBM), the last column of Table 1, we use an ANCOVA.
Stata 10:
Linear models and related
ANOVA/MANOVA
Analysis of variance and covariance
Dependent variable: rmr
Model: group lbm
Model variables: Categorical except the following
continuous variables: lbm
OK
anova rmr group lbm, continuous(lbm)
Chapter 5-4 (revision 16 May 2010)
// Stata version 10
p. 4
which gives an error message:
. anova rmr group
no observations
r(2000);
lbm, continuous(lbm) partial
Stata 11:
Linear models and related
ANOVA/MANOVA
Analysis of variance and covariance
Dependent variable: rmr
Model: group lbm
Model variables: Categorical except the following
continuous variables: lbm
OK
anova rmr group lbm
// Stata version 11
which gives an error message:
. anova rmr group lbm
group: may not use factor variable operators on string variables
r(109);
When you see the error message “no observations”, it usually means that you tried to use a string
variable where a numeric variable was required. (We used “group” instead of “groupx”, which
the oneway command allows but the anova command does not.)
Stata 10: Going back to change this,
Linear models and related
ANOVA/MANOVA
Analysis of variance and covariance
Dependent variable: rmr
Model: groupx lbm
Model variables: Categorical except the following
continuous variables: lbm
OK
anova rmr groupx lbm, continuous(lbm) // Stata version 10
Chapter 5-4 (revision 16 May 2010)
p. 5
Number of obs =
34
Root MSE
= 17.8777
R-squared
=
Adj R-squared =
0.4785
0.4264
Source | Partial SS
df
MS
F
Prob > F
-----------+---------------------------------------------------Model | 8798.55426
3 2932.85142
9.18
0.0002
|
groupx |
7734.4293
2 3867.21465
12.10
0.0001
lbm | 807.629752
1 807.629752
2.53
0.1224
|
Residual | 9588.38691
30 319.612897
-----------+---------------------------------------------------Total | 18386.9412
33 557.180036
Stata 11: Going back to change this, we use our numeric variable for group, groupx, and we put
a “c.” in front of our continuous variable, where the “c.” informs Stata it is a continuous variable.
Linear models and related
ANOVA/MANOVA
Analysis of variance and covariance
Dependent variable: rmr
Model: groupx c.lbm
OK
anova rmr groupx c.lbm // Stata version 11
Number of obs =
Root MSE
34
R-squared
= 17.8777
= 0.4785
Adj R-squared =
0.4264
Source | Partial SS
df
MS
F
Prob > F
-----------+---------------------------------------------------Model | 8798.55426
3 2932.85142
9.18
0.0002
|
groupx |
7734.4293
2 3867.21465
12.10
0.0001
lbm | 807.629752
1 807.629752
2.53
0.1224
|
Residual | 9588.38691
30 319.612897
-----------+---------------------------------------------------Total | 18386.9412
33 557.180036
We use the p value for the groupx row, which is “p=0.0001”, in approximate agreement with
Nawata’s p=0.0008 (differing only because the datasets differ slightly).
In this model, we tested the null hypothesis that the three group means are the same:
Ho : 1  2  3
versus
H A : 1  2  3
after adjustment for LBM (after controlling for LBM). In other words, we actually tested
Ho : adjusted 1  adjusted 2  adjusted 3
versus
H A : adjusted 1  adjusted 2  adjusted 3
Chapter 5-4 (revision 16 May 2010)
p. 6
Since p=.0001 < 0.05, we reject H0 and accept HA , concluding that at least two of the group’s
differ in their mean value.
Aside: Review of Hypothesis Testing
Recall, that the p value is the probability of obtaining the result that we did in our sample
if the null hypothesis H0 is true. If H0 is true, then we observed at least two means that
were different from each other just due to sampling variation. It can be attributed to
sampling variation, since under H0, no differences in means existed in the population we
sampled from, so the sample means differ simply because “they just happen to come out
that way” by the process of taking a sample (by chance). H0, then, can be viewed as the
mathematical expression consistent with sampling variation, which is what we want to
eliminate as an explanation for the observed effect.
If this probability is small, p < 0.05, we conclude that sampling variation was too unlikely
to be an explanation for the sample result. We then accept the opposite, the alternative
hypothesis HA, that there is a difference somewhere among the means in the population
we sampled from. We have to always keep in mind the dysjunctive syllogism of research
(Ch 5-2, p.1) when accepting HA, however, recognizing that the p value cannot rule out
bias or confounding by other confounders as alternative explanations for the observed
effect.
Chapter 5-4 (revision 16 May 2010)
p. 7
The same results could be derived using a linear regression model.
Stata 10: We can get the anova procedure to show us the regression model that matches the
ANOVA, using the regress option.
Linear models and related
ANOVA/MANOVA
Analysis of variance and covariance
Model tab: Dependent variable: rmr
Model: groupx lbm
Model variables: Categorical except the following
continuous variables: lbm
Reporting tab: Display anova and regression table
OK
anova rmr groupx lbm, continuous(lbm) regress anova // Stata 10
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-----------------------------------------------------------------------------_cons
142.1363
16.17229
8.79
0.000
109.1081
175.1645
groupx
1
-29.91667
7.305324
-4.10
0.000
-44.83613
-14.9972
2
-33.32727
7.660557
-4.35
0.000
-48.97222
-17.68233
3
(dropped)
lbm
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
-----------------------------------------------------------------------------Number of obs =
34
Root MSE
= 17.8777
R-squared
=
Adj R-squared =
0.4785
0.4264
Source | Partial SS
df
MS
F
Prob > F
-----------+---------------------------------------------------Model | 8798.55426
3 2932.85142
9.18
0.0002
|
groupx |
7734.4293
2 3867.21465
12.10
0.0001
lbm | 807.629752
1 807.629752
2.53
0.1224
|
Residual | 9588.38691
30 319.612897
-----------+---------------------------------------------------Total | 18386.9412
33 557.180036
Notice it reports that group 3 was dropped. What was actually dropped was the indicator
variable (or dummy variable) for group 3. Group 3 became the referent and was absorbed into
the intercept term (_cons). The anova command creates indicator variables for the categorical
variables, uses them in the calculation, but does not add them to the variables in the data browser.
Chapter 5-4 (revision 16 May 2010)
p. 8
Stata 11: We can follow the anova procedure with a regress command to show us the
regression model that matches the ANOVA,
anova rmr groupx c.lbm // Stata version 11
regress
. anova rmr groupx c.lbm // Stata version 11
Number of obs =
34
Root MSE
= 17.8777
R-squared
=
Adj R-squared =
0.4785
0.4264
Source | Partial SS
df
MS
F
Prob > F
-----------+---------------------------------------------------Model | 8798.55426
3 2932.85142
9.18
0.0002
|
groupx |
7734.4293
2 3867.21465
12.10
0.0001
lbm | 807.629752
1 807.629752
2.53
0.1224
|
Residual | 9588.38691
30 319.612897
-----------+---------------------------------------------------Total | 18386.9412
33 557.180036
. regress
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------groupx |
2 | -3.410606
7.654802
-0.45
0.659
-19.0438
12.22258
3 |
29.91667
7.305324
4.10
0.000
14.9972
44.83613
|
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
112.2196
15.87451
7.07
0.000
79.79954
144.6397
------------------------------------------------------------------------------
Notice it reports that group 1 was dropped. What was actually dropped was the indicator
variable (or dummy variable) for group 1. Group 1 became the referent and was absorbed into
the intercept term (_cons). The anova command creates indicator variables for the categorical
variables, uses them in the calculation, but does not add them to the variables in the data browser.
Notice that the anova command provides a single p value to test the equality of the three groups,
whereas the regression model uses two p values (one for each of two group indicators). We will
see how to combine these into a single p value below.
Chapter 5-4 (revision 16 May 2010)
p. 9
When modeling a categorical predictor variable, one less indicator than the number of categories
is used in the model (see box).
______________________________________________________________________________
Why One Indicator Variable Must Be Left Out
Regression models use a data matrix which includes a column of 1’s for the constant term.
_cons group1 group2 group3 group
1
1
0
0
1
1
0
1
0
2
1
0
0
1
3
The combination of group1, group2, and group3 indicator variables predict the constant variable
exactly. This is, we have perfect colinearity, where
Constant = group1 + group2 + group3
With all three indicator variables in the model, it is impossible for the constant term to have an
“independent” contribution to the outcome when holding constant all of the indicator terms—the
constant term is completely “dependent” upon the indicator terms.
Leaving one indicator variable out resolves this problem. That indicator becomes part of the
constant (the constant being the combined referent group for all predictor variables in the model).
______________________________________________________________________________
Let’s create the indicator variables for group and verify this is what the anova command did.
Data
Create or change variables
Other variable creation commands
Create indicator variables
Variable to tabulate: groupx
New variables stub: group_
OK
tabulate groupx, generate(group_)
Notice in the Variables window that three new variables were added: group_1, group_2, and
group_3.
We can verify these new variables are indicator variables by looking at them in the data browser,
or by listing them:
Chapter 5-4 (revision 16 May 2010)
p. 10
Data
Describe data
List data
Main tab: Variables: groupx group_1-group_3
Options tab: Display numeric codes rather than label values
OK
list groupx group_1-group_3, nolabel
+--------------------------------------+
| groupx
group_1
group_2
group_3 |
|--------------------------------------|
1. |
1
1
0
0 |
2. |
1
1
0
0 |
3. |
1
1
0
0 |
...
10. |
1
1
0
0 |
|--------------------------------------|
11. |
2
0
1
0 |
12. |
2
0
1
0 |
13. |
2
0
1
0 |
...
20. |
2
0
1
0 |
|--------------------------------------|
21. |
3
0
0
1 |
22. |
3
0
0
1 |
23. |
3
0
0
1 |
...
32. |
3
0
0
1 |
+--------------------------------------+
You can use the space bar (display next page) or enter key (display next line) to scroll through
the output.
Chapter 5-4 (revision 16 May 2010)
p. 11
Stata 10: Now, to duplicate the above model, with the indicator for group 3 left out:
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-----------------------------------------------------------------------------_cons
142.1363
16.17229
8.79
0.000
109.1081
175.1645
groupx
1
-29.91667
7.305324
-4.10
0.000
-44.83613
-14.9972
2
-33.32727
7.660557
-4.35
0.000
-48.97222
-17.68233
3
(dropped)
lbm
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
-----------------------------------------------------------------------------
we use,
Statistics
Linear models and related
Linear regression
Dependent variable: rmr
Independent variables: group_1 group_2 lbm
OK
regress rmr group_1 group_2 lbm
// Stata version 10
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------group_1 | -29.91667
7.305324
-4.10
0.000
-44.83613
-14.9972
group_2 | -33.32727
7.660557
-4.35
0.000
-48.97222
-17.68233
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
142.1363
16.17229
8.79
0.000
109.1081
175.1645
------------------------------------------------------------------------------
Chapter 5-4 (revision 16 May 2010)
p. 12
Stata 11: Now, to duplicate the above model, with the indicator for group 3 left out:
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------groupx |
2 | -3.410606
7.654802
-0.45
0.659
-19.0438
12.22258
3 |
29.91667
7.305324
4.10
0.000
14.9972
44.83613
|
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
112.2196
15.87451
7.07
0.000
79.79954
144.6397
------------------------------------------------------------------------------
we use,
Statistics
Linear models and related
Linear regression
Dependent variable: rmr
Independent variables: group_2 group_3 lbm
OK
regress rmr group_2 group_3 lbm
// Stata version 11
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------group_2 | -3.410606
7.654802
-0.45
0.659
-19.0438
12.22258
group_3 |
29.91667
7.305324
4.10
0.000
14.9972
44.83613
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
112.2196
15.87451
7.07
0.000
79.79954
144.6397
------------------------------------------------------------------------------
If we used this regression approach, what p value would we use to report in Table 1 (similar to
Nawata’s single p value)?
Chapter 5-4 (revision 16 May 2010)
p. 13
The anova command provided such a single p value (p=0.0001).
Number of obs =
34
Root MSE
= 17.8777
R-squared
=
Adj R-squared =
0.4785
0.4264
Source | Partial SS
df
MS
F
Prob > F
-----------+---------------------------------------------------Model | 8798.55426
3 2932.85142
9.18
0.0002
|
groupx |
7734.4293
2 3867.21465
12.10
0.0001
lbm | 807.629752
1 807.629752
2.53
0.1224
|
Residual | 9588.38691
30 319.612897
-----------+---------------------------------------------------Total | 18386.9412
33 557.180036
We can get this from a post-estimation approach, after running the linear regression command as
well.
To test the hypothesis,
Ho : adjusted 1  adjusted 2  adjusted 3
we need to constuct a command that looks like:
Stata 10:
test (_cons+igroup1)=(_cons+igroup2)=_cons
Statistics
Postestimation
Tests
Test linear hypotheses
Specification 1: tests these coefficients: _cons+group_1 = _cons+group_2 = _cons
OK
test _cons+group_1 = _cons+group_2 = _cons // Stata Version 10
( 1)
( 2)
group_1 - group_2 = 0
group_1 = 0
F(
2,
30) =
Prob > F =
Chapter 5-4 (revision 16 May 2010)
12.10
0.0001
p. 14
Stata 11:
test (_cons+igroup2)=(_cons+igroup3)=_cons
Statistics
Postestimation
Tests
Test linear hypotheses
Specification 1: tests these coefficients: _cons+group_2 = _cons+group_3 = _cons
OK
test _cons+group_2 = _cons+group_3 = _cons
// Stata Version 11
. test (_cons+group_2 = _cons+group_3 = _cons)
( 1)
( 2)
group_2 - group_3 = 0
group_2 = 0
F(
2,
30) =
Prob > F =
12.10
0.0001
We see that the F statistic and p value match those in the above ANOVA model output.
Chapter 5-4 (revision 16 May 2010)
p. 15
Reporting Adjusted Group Means with Confidence Intervals
Reporting adjusted group means is rather clever, because you are able to show how groups differ
when controlling for potential confounders.
Exercise: Look at the article by Kalmijn et al (2002), which is another article that reports
adjusted group means.
1) Notice in their Statistical Analysis Section, they described the use of dummy variable
coding in their linear regression model as:
“…In addition, subjects who smoked >0-20 pack-years and >20 pack-years were
compared to never smokers by including two dummy variables in the regression
model. Confounders that were taken into account were age (continuous), sex,
education (four dummy categories), body mass index, total cholesterol level, and
systolic blood pressure…”
2) Notice in their Table 2 that they are reporting adjusted mean scores.
To get the adjusted group means following a regression, we might try:
Stata 10:
Statistics
Postestimation
Adjusted means and proportions
Main tab: Compute and display predictions for each level of variables: groupx
Options tab: Prediction: linear prediction
confidence or prediction intervals
OK
adjust, by(groupx) xb ci
Stata 11: The adjust command is no longer listed on the menu, but can be run as a command,
adjust, by(groupx) xb ci
Chapter 5-4 (revision 16 May 2010)
p. 16
------------------------------------------------Dependent variable: rmr
Command: regress
Variables left as is: lbm, group_1, group_2
---------------------------------------------------------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
136.083
[125.543
146.623]
b |
132.7
[121.154
144.246]
c |
166.5
[155.96
177.04]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
Notice, however, that these means are identical to the means from the above one-way ANOVA,
which did not adjust for lbm, since lbm was not in that one-way ANOVA model.
|
Summary of rmr
group |
Mean
Std. Dev.
Freq.
------------+-----------------------------------a |
136.08333
16.940851
12
b |
132.7
19.032428
10
c |
166.5
19.019129
12
------------+-----------------------------------Total |
145.82353
23.604661
34
Analysis of Variance
Source
SS
df
MS
F
Prob > F
-----------------------------------------------------------------------Between groups
7990.92451
2
3995.46225
11.91
0.0001
Within groups
10396.0167
31
335.355376
-----------------------------------------------------------------------Total
18386.9412
33
557.180036
Bartlett's test for equal variances:
Chapter 5-4 (revision 16 May 2010)
chi2(2) =
0.1787
Prob>chi2 = 0.915
p. 17
To get the adjusted mean, we need to hold the covariates constant at some value. Usually,
researchers choose the covariate’s overall mean value.
Stata 10:
Statistics
Postestimation
Adjusted means and proportions
Main tab: Compute and display predictions for each level of variables: groupx
Variables to be set to there overall mean value: lbm
Options tab: Prediction: linear prediction
confidence or prediction intervals
OK
adjust lbm, by(groupx) xb ci // Stata version 10
Stata 11: Just running the command (not available on the menu),
adjust lbm, by(groupx) xb ci // Stata version 10
-------------------------------------------------------------------------------Dependent variable: rmr
Command: regress
Variables left as is: group_1, group_2
Covariate set to mean: lbm = 44.088235
----------------------------------------------------------------------------------------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
136.268
[125.725
146.81]
b |
132.857
[121.31
144.405]
c |
166.184
[155.637
176.732]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
These adjusted means are slightly different from those computed above:
---------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
136.083
[125.543
146.623]
b |
132.7
[121.154
144.246]
c |
166.5
[155.96
177.04]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
Chapter 5-4 (revision 16 May 2010)
p. 18
The adjusted mean is simply the predicted value from the regression equation. Copying the
regression equation from above,
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------group_1 | -29.91667
7.305324
-4.10
0.000
-44.83613
-14.9972
group_2 | -33.32727
7.660557
-4.35
0.000
-48.97222
-17.68233
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
142.1363
16.17229
8.79
0.000
109.1081
175.1645
------------------------------------------------------------------------------
and computing the adjusted group mean for group2,
display 142.1363 -29.91667*0 -33.32727*1 + 0.5454562*44.088235
132.85723
which agrees with the adjusted group mean computed above.
-------------------------------------------------------------------------------Dependent variable: rmr
Command: regress
Variables left as is: group_1, group_2
Covariate set to mean: lbm = 44.088235
----------------------------------------------------------------------------------------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
136.268
[125.725
146.81]
b |
132.857
[121.31
144.405]
c |
166.184
[155.637
176.732]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
What did this model look like?
To see how well the model fitted the data, we can overlay the fitted lines onto the original data.
First we need a variable containing the predicted values:
Statistics
Postestimation
Predictions, residuals, etc.
New variable name: rmr_hat
Produce: Fitted values (xb)
OK
predict rmr_hat, xb
Chapter 5-4 (revision 16 May 2010)
p. 19
We now want to overlay three scatterplots (one for each group) and three prediction line graphs
(one for each group).
We could use the graphics menu, which would require setting the the overlaying of six plots.
Alternatively, we can copy the following into the do-file editor, and run it (also in Ch 5-4.do)
100
120
140
RMR
160
180
200
sort lbm // always sort on x variable for line graphs
#delimit ;
twoway (scatter rmr lbm if groupx==1
, msymbol(square) mfcolor(green) mlcolor(green) msize(large))
(scatter rmr lbm if groupx==2
, msymbol(circle) mfcolor(blue) mlcolor(blue) msize(large))
(scatter rmr lbm if groupx==3
, msymbol(triangle) mfcolor(red) mlcolor(red) msize(large))
(line rmr_hat lbm if groupx==1
, clpattern(solid) clwidth(thick) clcolor(green))
(line rmr_hat lbm if groupx==2
, clpattern(solid) clwidth(thick) clcolor(blue))
(line rmr_hat lbm if groupx==3
, clpattern(solid) clwidth(thick) clcolor(red))
, legend(off)
ytitle(RMR)
xtitle(LBM)
;
#delimit cr
30
40
50
LBM
60
70
Notice, the regression procedure assumes all lines are parallel, just shifted vertically for each
group, unless interaction terms are added to the model.
Chapter 5-4 (revision 16 May 2010)
p. 20
Look at Figure 1 in the Nawata article. Our model did not fit the data well at all, compared to
what Nawata noticed about these data. (The red line should be slanting downward.)
These data are said to interact. That is, the best fitting lines for each group are not parallel. It
would be better to let each group have its own slope, as well as its own intercept.
We need to first create some interaction variables.
Data
Create or change variables
Create new variable
(Version 10): New variable name : lbmXgroup1
(Version 11): Variable name : lbmXgroup1
(Version 10): Contents of new variable: lbm*group_1
(Version 11): Contents of variable: lbm*group_1
OK
generate lbmXgroup1 = lbm*group_1
and modifying the command for group 2,
generate lbmXgroup2 = lbm*group_2
Fitting a new model with these interaction terms added,
Statistics
Linear models and related
Linear regression
Dependent variable: rmr
Independent variables: igroup1 igroup2 lbm lbmXgroup1 lbmXgroup2
OK
regress rmr group_1 group_2 lbm
lbmXgroup1 lbmXgroup2
Source |
SS
df
MS
-------------+-----------------------------Model | 10820.5192
5 2164.10384
Residual | 7566.42198
28 270.229356
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 5,
28)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
8.01
0.0001
0.5885
0.5150
16.439
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------group_1 | -122.4433
34.5581
-3.54
0.001
-193.2324
-51.65425
group_2 | -100.1493
39.63594
-2.53
0.017
-181.3399
-18.95877
lbm | -.8470919
.6166423
-1.37
0.180
-2.110226
.4160425
lbmXgroup1 |
2.085718
.7644016
2.73
0.011
.5199118
3.651523
lbmXgroup2 |
1.498063
.8819738
1.70
0.100
-.3085782
3.304704
_cons |
204.3368
27.94916
7.31
0.000
147.0855
261.588
------------------------------------------------------------------------------
Chapter 5-4 (revision 16 May 2010)
p. 21
The interaction term for group 1 is significant, and so should be kept in the model. For group 2,
it is approaching significant, so it could be kept in or not. Keeping this second term in would not
improve the fit by much.
Keeping both interaction terms in, let’s see what happens graphically.
First saving the new predicted values,
Statistics
Postestimation
Predictions, residuals, etc.
New variable name: rmr_hat2
Produce: Fitted values (xb)
OK
predict rmr_hat2, xb
and repeating graph, using the new predicted values
sort lbm // always sort on x variable for line graphs
#delimit ;
twoway (scatter rmr lbm if groupx==1
, msymbol(square) mfcolor(green) mlcolor(green) msize(large))
(scatter rmr lbm if groupx==2
, msymbol(circle) mfcolor(blue) mlcolor(blue) msize(large))
(scatter rmr lbm if groupx==3
, msymbol(triangle) mfcolor(red) mlcolor(red) msize(large))
(line rmr_hat2 lbm if groupx==1
, clpattern(solid) clwidth(thick) clcolor(green))
(line rmr_hat2 lbm if groupx==2
, clpattern(solid) clwidth(thick) clcolor(blue))
(line rmr_hat2 lbm if groupx==3
, clpattern(solid) clwidth(thick) clcolor(red))
, legend(off)
ytitle(RMR)
xtitle(LBM)
;
#delimit cr
Chapter 5-4 (revision 16 May 2010)
p. 22
100
120
140
RMR
160
180
200
This looks much more like Nawata’s model, except for the one outlying blue point added to the
dataset for illustration.
30
40
50
LBM
60
70
Clearly, now, the adjusted mean difference depends very much on what value we hold lbm
constant at, since the groups differ more at low values of LBM than at high values.
Stata 10:
Statistics
Postestimation
Adjusted means and proportions
Compute and display predictions for each level of variables: groupx
Variables to be set to there overall mean value: lbm
OK
adjust lbm, by(groupx) xb ci
Stata 11:
adjust lbm, by(groupx) xb ci
Chapter 5-4 (revision 16 May 2010)
p. 23
-------------------------------------------------------------------------------Dependent variable: rmr
Command: regress
Variables left as is: group_1, group_2, lbmXgroup1, lbmXgroup2
Covariate set to mean: lbm = 44.088235
----------------------------------------------------------------------------------------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
135.797
[126.067
145.527]
b |
132.456
[121.801
143.11]
c |
166.99
[157.242
176.738]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
Compared to the earlier model, without the interaction terms,
-------------------------------------------------------------------------------Dependent variable: rmr
Command: regress
Variables left as is: group_1, group_2
Covariate set to mean: lbm = 44.088235
----------------------------------------------------------------------------------------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
136.268
[125.725
146.81]
b |
132.857
[121.31
144.405]
c |
166.184
[155.637
176.732]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
it did not change much.
This is because regression lines always passes through the group means (X-variable mean and Yvariable mean) and so the slope pivots around it. At the mean, then, the slope is of no
consequence.
If you wanted to know the adjusted mean values for a point far away from the mean however, it
would have a big effect.
Stata 10:
Statistics
Postestimation
Adjusted means and proportions
Compute and display predictions for each level of variables: groupx
Variables to be set to there overall mean value: <leave blank this time>
Variables to be set to a specified value: Variable: lbm Value: 57
OK
adjust lbm=57, by(groupx) xb ci
Chapter 5-4 (revision 16 May 2010)
p. 24
Stata 11:
adjust lbm=57, by(groupx) xb ci
-------------------------------------------------------------------------------Dependent variable: rmr
Command: regress
Variables left as is: group_1, group_2, lbmXgroup1, lbmXgroup2
Covariate set to value: lbm = 57
----------------------------------------------------------------------------------------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
124.859
[105.505
144.214]
b |
121.518
[101.735
141.302]
c |
156.053
[137.69
174.415]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
Compared to the earlier model, without the interaction terms,
-------------------------------------------------------------------------------Dependent variable: rmr
Command: regress
Variables left as is: group_1, group_2
Covariate set to mean: lbm = 44.088235
----------------------------------------------------------------------------------------------------------------------------groupx |
xb
lb
ub
----------+----------------------------------a |
136.268
[125.725
146.81]
b |
132.857
[121.31
144.405]
c |
166.184
[155.637
176.732]
---------------------------------------------Key: xb
= Linear Prediction
[lb , ub] = [95% Confidence Interval]
is very different.
Stata’s New Categorical Variable Facility
With Version 11, Stata has a much easier way to work with categorical variables. You simply
put an “i.” in front of each categorical variable, and Stata will create indicator variables behind
the scenes to use in the regression model.
The command,
regress rmr group_2 group_3 lbm
can now be specified as,
regress rmr i.groupx lbm
Chapter 5-4 (revision 16 May 2010)
// Stata version 11
p. 25
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------groupx |
2 | -3.410606
7.654802
-0.45
0.659
-19.0438
12.22258
3 |
29.91667
7.305324
4.10
0.000
14.9972
44.83613
|
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
112.2196
15.87451
7.07
0.000
79.79954
144.6397
------------------------------------------------------------------------------
By default, it uses the first category of groupx as the referent group.
To specify the second category as the referent group, we can use, “ib2.”, which the “b” stands for
“base level”, which is another name for referent category,
regress rmr ib2.groupx lbm
// Stata version 11
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------groupx |
1 |
3.410606
7.654802
0.45
0.659
-12.22258
19.0438
3 |
33.32727
7.660557
4.35
0.000
17.68233
48.97222
|
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
108.809
16.05747
6.78
0.000
76.01528
141.6028
------------------------------------------------------------------------------ o
We see that category 2 of groupx was left out this time, which turns it into the referent category.
To specify the last category as the referent group, we can use, “ib(last).”,
regress rmr ib(last).groupx lbm
Chapter 5-4 (revision 16 May 2010)
// Stata version 11
p. 26
Source |
SS
df
MS
-------------+-----------------------------Model | 8798.55426
3 2932.85142
Residual | 9588.38691
30 319.612897
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 3,
30)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
9.18
0.0002
0.4785
0.4264
17.878
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------groupx |
1 | -29.91667
7.305324
-4.10
0.000
-44.83613
-14.9972
2 | -33.32727
7.660557
-4.35
0.000
-48.97222
-17.68233
|
lbm |
.5454562
.3431357
1.59
0.122
-.1553203
1.246233
_cons |
142.1363
16.17229
8.79
0.000
109.1081
175.1645
------------------------------------------------------------------------------
This left group 3, the last category, out of the model to serve as the referent category.
To get the interaction terms, we use the “#” operator. To get the model specified by this
command,
regress rmr group_1 group_2 lbm
lbmXgroup1 lbmXgroup2
we use,
regress rmr ib(last).groupx lbm i.groupx#c.lbm // Stata version 11
Source |
SS
df
MS
-------------+-----------------------------Model | 10820.5192
5 2164.10384
Residual | 7566.42198
28 270.229356
-------------+-----------------------------Total | 18386.9412
33 557.180036
Number of obs
F( 5,
28)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
34
8.01
0.0001
0.5885
0.5150
16.439
-----------------------------------------------------------------------------rmr |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------groupx |
1 | -122.4433
34.5581
-3.54
0.001
-193.2324
-51.65425
2 | -100.1493
39.63594
-2.53
0.017
-181.3399
-18.95877
|
lbm | -.8470919
.6166423
-1.37
0.180
-2.110226
.4160425
|
groupx#c.lbm |
1 |
2.085718
.7644016
2.73
0.011
.5199118
3.651523
2 |
1.498063
.8819738
1.70
0.100
-.3085782
3.304704
|
_cons |
204.3368
27.94916
7.31
0.000
147.0855
261.588
------------------------------------------------------------------------------
Chapter 5-4 (revision 16 May 2010)
p. 27
How to Interpret the Interaction Term
The file Ch5-4.dta contains the following practice data. This is just for illustration, so we will
ignore that the sample size is too small for three predictor variables.
1.
2.
3.
4.
5.
6.
7.
8.
+---------------------------+
| group
x
y
xgroup |
|---------------------------|
|
0
10
200
0 |
|
0
12
201
0 |
|
0
14
202
0 |
|
0
17
203
0 |
|---------------------------|
|
1
9
320
9 |
|
1
15
350
15 |
|
1
20
400
20 |
|
1
24
475
24 |
+---------------------------+
where group = group variable (0 or 1)
x = a continuous predictor variable
y = a continuous outcome variable
xgroup = x by group interaction, created by multiplying x by group.
We will see why this strange interaction variable produces something meaningful, and learn how
to interpret this interaction term.
Chapter 5-4 (revision 16 May 2010)
p. 28
The regression model without the interaction term looks like:
0
50
100 150 200 250 300 350 400 450 500
Y
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
8.356792
2.016724
4.14
0.009
3.172639
13.54095
group |
153.412
19.17877
8.00
0.000
104.1114
202.7126
_cons |
90.7725
29.48489
3.08
0.028
14.97919
166.5658
------------------------------------------------------------------------------
10
15
20
25
X
With just these two “main effect” terms (main effects as opposed to interaction terms), the model
is constrained to fit the data as best it can with the limitation that the slopes are equal (lines
parallel) and all the group variable can do is shift the line up or down (change the intercept).
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
8.356792
2.016724
4.14
0.009
3.172639
13.54095
group |
153.412
19.17877
8.00
0.000
104.1114
202.7126
_cons |
90.7725
29.48489
3.08
0.028
14.97919
166.5658
------------------------------------------------------------------------------
Y = a + b1(X) + b2(group) = 90.77 + 8.36 (X) + 153.41(group) from the regression table
Y = a + b1(X) + b2(group) = 90.77 + 8.36 (X) + 153.41 (0)
= 90.77 + 8.36 (X)
Y = a + b1(X) + b2(group) = 90.77 + 8.36 (X) + 153.41 (1)
= 90.77 + 8.36 (X) + 153.41
for group 0
for group 1
(shifted up by 153.41)
while for either group the slope for X remains 8.36.
Chapter 5-4 (revision 16 May 2010)
p. 29
With the interaction term, the model looks like:
0
50
100 150 200 250 300 350 400 450 500
Y
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
.4299065
3.16022
0.14
0.898
-8.34427
9.204083
group |
19.77166
49.9967
0.40
0.713
-119.0414
158.5847
xgroup |
9.609776
3.479546
2.76
0.051
-.0509926
19.27054
_cons |
195.8037
42.66296
4.59
0.010
77.35236
314.2551
------------------------------------------------------------------------------
10
15
20
25
X
With both the two “main effect” terms and the interaction term, the model is permitted to have
different slopes. Now the distance between the two lines, or the group difference, depends on the
covariate X.
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
.4299065
3.16022
0.14
0.898
-8.34427
9.204083
group |
19.77166
49.9967
0.40
0.713
-119.0414
158.5847
xgroup |
9.609776
3.479546
2.76
0.051
-.0509926
19.27054
_cons |
195.8037
42.66296
4.59
0.010
77.35236
314.2551
------------------------------------------------------------------------------
Y = a + b1(X) + b2(group) + b3(X  group)
= 195.8 + 0.4(X) +19.8(group) + 9.6 (X  group)
= 195.8 + 0.4 (X) +19.8 (0) + 9.6 (0)
= 195.8 + 0.4 (X)
for group 0
Y = a + b1(X) + b2(group) + b3(X  group)
= 195.8 + 0.4 (X) +19.8 (group) + 9.6(X  group)
= 195.8 + 0.4 (X) +19.8 (1) + 9.6 (X  1)
= 195.8 + 0.4 (X) +19.8+ 9.6 (X)
= 195.8 + (0.4+9.6)(X) +19.8
for group 1
Chapter 5-4 (revision 16 May 2010)
p. 30
For the reference group, group 1, a simple regression line is fitted.
For group 1, the line is shifted up or down by the correct amount at the point X=0 (it extrapolated
in this case) and then adds an increment, or decrement, to the slope for the covariate X.
Thus the interaction term is how much needs to be added or substracted from the reference group
slope to modify it to be the correct slope for group 1.
Notice in the main effects model, the covariate X is significant.
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
8.356792
2.016724
4.14
0.009
3.172639
13.54095
group |
153.412
19.17877
8.00
0.000
104.1114
202.7126
_cons |
90.7725
29.48489
3.08
0.028
14.97919
166.5658
------------------------------------------------------------------------------
However, in the model with the interaction term, the covariate X is no longer significant.
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x |
.4299065
3.16022
0.14
0.898
-8.34427
9.204083
group |
19.77166
49.9967
0.40
0.713
-119.0414
158.5847
xgroup |
9.609776
3.479546
2.76
0.051
-.0509926
19.27054
_cons |
195.8037
42.66296
4.59
0.010
77.35236
314.2551
------------------------------------------------------------------------------
Should we drop the X variable from the model, then, because it is not significant?
Let’s see what happens if we do.
0
50
100 150 200 250 300 350 400 450 500
Y
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------group |
14.0754
24.49134
0.57
0.590
-48.88159
77.03239
xgroup |
10.03968
1.305393
7.69
0.001
6.684064
13.3953
_cons |
201.5
7.326498
27.50
0.000
182.6666
220.3334
------------------------------------------------------------------------------
10
15
20
25
X
Chapter 5-4 (revision 16 May 2010)
p. 31
-----------------------------------------------------------------------------y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------group |
14.0754
24.49134
0.57
0.590
-48.88159
77.03239
xgroup |
10.03968
1.305393
7.69
0.001
6.684064
13.3953
_cons |
201.5
7.326498
27.50
0.000
182.6666
220.3334
------------------------------------------------------------------------------
Y = a + b1(group) + b2(X  group)
= 201.5 +14.1(group) + 10.0(X  group)
= 201.5 +14.1(0) + 10.0(X  0)
= 201.5
for group 0
Y = a + b1(group) + b2(X  group)
= 201.5 +14.1(group) + 10.0(X  group)
= 201.5 +14.1(1) + 10.0(X  1)
= 201.5 +14.1 + 10.0(X)
for group 1
The model seems to do something reasonable for group 1, providing a correct single slope value.
However, it forces group 0 to just be a constant. That isn’t too bad for this example, since the
slope of X was so close to 0 for group 0 anyway. In a dataset where the slope of the referent
group is not close to 0, however, this will provide a terrible fit.
Chapter 5-4 (revision 16 May 2010)
p. 32
This example illustrates the following rule:
Rule About Interaction Terms
When an interaction term is included in the model, all of the main effects for the variables
that are multiplied to produce the interaction term must be included in the model as well,
whether significant or not.
If a greater than 2-way interaction term is included, all of the lower-order interaction
terms as well as the main effects must remain in the model. (If a three-way interaction is
included, then, the model must contain: A , B , C , A*B , A*C , B*C , A*B*C.
A formal description of why this rule is needed in found in Chapter 5-8, p.21.
How to Report Models With Interaction Terms
Interaction terms are difficult to discuss and would confuse nearly any reader if shown as a line
in a table that shows all of the coefficients for the multivariable model. What researchers do,
then, is state in the text that a significant interaction was found, along with a p value, but never
discuss the coefficient associated with the interaction term. Instead, the researcher will go on to
show a stratified analysis, stratifying by one of the variables composing the interaction term.
This approach is particularly useful, since an interaction means the effect is different for the
different levels of the covariate in the group x covariate interaction. Showing these individual
subgroups, or strata, is the most informative way to present an interaction.
Chapter 5-4 (revision 16 May 2010)
p. 33
Exercise. Look at the Kalmijn et al (2002) paper. Under the heading Alcohol Consumption in
the Results section (p.939), they state:
“There was a significant interaction between sex and alcohol consumption in relation to
speed (p = 0.008), indicating that, for women, the association between alcohol
consumption and pscyhomotor speed was positive and linear (p-trend < 0.001), whereas
for men it was absent (table 4).”
Then, in Table 4, they show the results stratified by gender.
They actually used alcohol as a ordinal scale variable in the model, rather than 5 indicator
variables, to obtain the p-trend. So they did not use interaction terms like we have done in this
chapter. The concept of reporting an interaction is the same, however.
Linear trend tests are covered in Chapter 5-26 “Trend Tests.”
References
Chatterjee S, Hadi AS, Price B. (2000). Regression Analysis by Example. 3rd ed. New York,
John Wiley and Sons.
Hamilton LC. (2003). Statistics With Stata. Updated for Version 7. Belmont CA, Wadsworth
Group/Thomson Learning.
Kalmijn S, van Boxtel MPJ, Verschuren MWM, et al. (2002). Cigarette smoking and alcohol
consumption in relation to cognitive performance in middle age. Am J Epidemiol
156(10):936-944.
Nawata K, Sohmiya M, Kawaguchi M, et al. (2004). Increased resting metabolic rate in patients
with type 2 diabetes mellitus accompanied by advanced diabetic nephropathy.
Metabolism 53(11) Nov: 1395-1398.
Chapter 5-4 (revision 16 May 2010)
p. 34