Title stata.com
contrast — Contrasts and linear hypothesis tests after estimation
Description Quick start Menu Syntax
Options Remarks and examples Stored results Methods and formulas
References Also see
Description
contrast tests linear hypotheses and forms contrasts involving factor variables and their interactions
from the most recently fit model. The tests include ANOVA-style tests of main effects, simple effects,
interactions, and nested effects. contrast can use named contrasts to decompose these effects into
comparisons against reference categories, comparisons of adjacent levels, comparisons against the
grand mean, orthogonal polynomials, and such. Custom contrasts may also be specified.
contrast can be used with svy estimation results; see [SVY] svy postestimation.
Contrasts can also be computed for margins of linear and nonlinear responses; see [R] margins,
contrast.
Quick start
Contrasts for one-way models
Test the main effect of categorical variable a after regress y i.a or anova y a
contrast a
Reference category contrasts of cell means of y with the smallest value of a as the base category
contrast r.a
Same as above, but specify a = 3 as the base category for comparisons
contrast rb3.a
Report tests instead of confidence intervals for each contrast
contrast r.a, pveffects
Report tests and confidence intervals for each contrast
contrast r.a, effects
Contrasts of the cell mean of y for each level of a with the grand mean of y
contrast g.a
Same as above, but compute grand mean as a weighted average of cell means with weights based on
the number of observations for each level of a
contrast gw.a
User-defined contrast comparing the cell mean of y for a = 1 with the average of the cell means for
a = 3 and a = 4
contrast {a -1 0 .5 .5}
1
2 contrast — Contrasts and linear hypothesis tests after estimation
Contrasts for two-way models
Test of the interaction effect after regress y a##b or anova y a##b
contrast a#b
Test of the main and interaction effects
contrast a b a#b
Same as above
contrast a##b
Individual reference category contrasts for the interaction of a and b
contrast r.a#r.b
Joint tests of the simple effects of a within each level of b
contrast a@b
Individual reference category contrasts for the simple effects of a within each level of b
contrast r.a@b
Orthogonal polynomial contrasts for a within each level of b
contrast p.a@b
Reference category contrasts of the marginal means of y for levels of a
contrast r.a
Same as above, but with marginal means for a computed as a weighted average of cell means, using
the marginal frequencies of b rather than equal weights for each level
contrast r.a, asobserved
Contrasts of the marginal mean of y for each level of a with the previous level—reverse-adjacent
contrasts
contrast ar.a
Contrasts for models with continuous covariates
Test of the interaction effect after regress y a##c.x or anova y a##c.x
contrast a#c.x
Reference category effects of a on the slope of x
contrast r.a#c.x
Reference category effects of a on the intercept
contrast r.a
Contrasts for nonlinear models
Orthogonal polynomial contrasts of log odds across levels of a after logit y i.a
contrast p.a
Test the main and interaction effects after logit y a##b
contrast a##b
contrast — Contrasts and linear hypothesis tests after estimation 3
Simple reference category effects for a within each level of b
contrast r.a@b
Contrasts for multiple-equation models
Test the main and interaction effects in the equation for y2 after mvreg y1 y2 y3 = a##b
contrast a##b, equation(y2)
Reference category contrasts of estimated marginal means of y3 for levels of a
contrast r.a, equation(y3)
Test for a difference in the overall estimated marginal means of y1, y2, and y3
contrast _eqns
Contrasts of estimated marginal means of y2 and y3 with y1
contrast r._eqns
Test whether interaction effects differ across equations
contrast a#b#_eqns
Menu
Statistics > Postestimation
4 contrast — Contrasts and linear hypothesis tests after estimation
Syntax
contrast termlist
, options
where termlist is a list of factor variables or interactions that appear in the current estimation results.
The variables may be typed with or without contrast operators, and you may use any factor-variable
syntax:
See the operators (op.) table below for the list of contrast operators.
options Description
Main
overall add a joint hypothesis test for all specified contrasts
asobserved treat all factor variables as observed
lincom treat user-defined contrasts as linear combinations
Equations
equation(eqspec) perform contrasts in termlist for equation eqspec
atequations perform contrasts in termlist within each equation
Advanced
emptycells(empspec) treatment of empty cells for balanced factors
noestimcheck suppress estimability checks
Reporting
level(#) confidence level; default is level(95)
mcompare(method ) adjust for multiple comparisons; default is mcompare(noadjust)
noeffects suppress table of individual contrasts
cieffects show effects table with confidence intervals
pveffects show effects table with p-values
effects show effects table with confidence intervals and p-values
nowald suppress table of Wald tests
noatlevels report only the overall Wald test for terms that use the within @
or nested | operator
nosvyadjust compute unadjusted Wald tests for survey results
sort sort the individual contrast values in each term
post post contrasts and their VCEs as estimation results
display options control column formats, row spacing, line width, and factor-variable labeling
eform option report exponentiated contrasts
df(#) use t distribution with # degrees of freedom for computing p-values
and confidence intervals
collect is allowed; see [U] 11.1.10 Prefix commands.
df(#) does not appear in the dialog box.
contrast — Contrasts and linear hypothesis tests after estimation 5
Term Description
Main effects
A joint test of the main effects of A
r.A individual contrasts that decompose A using r.
Interaction effects
A#B joint test of the two-way interaction effects of A and B
A#B#C joint test of the three-way interaction effects of A, B, and C
r.A#g.B individual contrasts for each interaction of A and B defined by r. and g.
Par tial interaction effects
r.A#B joint tests of interactions of A and B within each contrast defined by r.A
A#r.B joint tests of interactions of A and B within each contrast defined by r.B
Simple effects
A@B joint tests of the effects of A within each level of B
A@B#C joint tests of the effects of A within each combination of the levels of B and C
r.A@B individual contrasts of A that decompose A@B using r.
r.A@B#C individual contrasts of A that decompose A@B#C using r.
Other conditional effects
A#B@C joint tests of the interaction effects of A and B within each level of C
A#B@C#D joint tests of the interaction effects of A and B within each combination of
the levels of C and D
r.A#g.B@C individual contrasts for each interaction of A and B that decompose A#B@C
using r. and g.
Nested effects
A|B joint tests of the effects of A nested in each level of B
A|B#C joint tests of the effects of A nested in each combination of the levels of B and C
A#B|C joint tests of the interaction effects of A and B nested in each level of C
A#B|C#D joint tests of the interaction effects of A and B nested in each
combination of the levels of C and D
r.A|B individual contrasts of A that decompose A|B using r.
r.A|B#C individual contrasts of A that decompose A|B#C using r.
r.A#g.B|C individual contrasts for each interaction of A and B defined by r. and g.
nested in each level of C
Slope effects
A#c.x joint test of the effects of A on the slopes of x
A#c.x#c.y joint test of the effects of A on the slopes of the product (interaction) of x and y
A#B#c.x joint test of the interaction effects of A and B on the slopes of x
A#B#c.x#c.y joint test of the interaction effects of A and B on the slopes of the product
(interaction) of x and y
r.A#c.x individual contrasts of A’s effects on the slopes of x using r.
Denominators
... / term2 use term2 as the denominator in the F tests of the preceding terms
... / use the residual as the denominator in the F tests of the preceding terms
(the default if no other /s are specified)
6 contrast — Contrasts and linear hypothesis tests after estimation
A, B, C, and D represent any factor variable in the current estimation results.
x and y represent any continuous variable in the current estimation results.
r. and g. represent any contrast operator. See the table below.
c. specifies that a variable be treated as continuous; see [U] 11.4.3 Factor variables.
Operators are allowed on any factor variable that does not appear to the right of @ or |. Operators
decompose the effects of the associated factor variable into one-degree-of-freedom effects (contrasts).
Higher-level interactions are allowed anywhere an interaction operator (#) appears in the table.
Time-series operators are allowed if they were used in the estimation.
eqns designates the equations in manova, mlogit, mprobit, and mvreg and can be specified
anywhere a factor variable appears.
/ is allowed only after anova, cnsreg, manova, mvreg, or regress.
operators (op.) Description
r. differences from the reference (base) level; the default
a. differences from the next level (adjacent contrasts)
ar. differences from the previous level (reverse adjacent contrasts)
As-balanced operators
g. differences from the balanced grand mean
h. differences from the balanced mean of subsequent levels (Helmert contrasts)
j. differences from the balanced mean of previous levels (reverse Helmert
contrasts)
p. orthogonal polynomial in the level values
q. orthogonal polynomial in the level sequence
As-observed operators
gw. differences from the observation-weighted grand mean
hw. differences from the observation-weighted mean of subsequent levels
jw. differences from the observation-weighted mean of previous levels
pw. observation-weighted orthogonal polynomial in the level values
qw. observation-weighted orthogonal polynomial in the level sequence
One or more individual contrasts may be selected by using the op#. or op(numlist). syntax. For
example, a3.A selects the adjacent contrast for level 3 of A, and p(1/2).B selects the linear and
quadratic effects of B. Also see Orthogonal polynomial contrasts and Beyond linear models.
Custom contrasts Description
{A numlist} user-defined contrast on the levels of factor A
{A#B numlist} user-defined contrast on the levels of interaction between A and B
Custom contrasts may be part of a term, such as {A numlist}#B, {A numlist}@B, {A numlist}|B, {A#B
numlist}, and {A numlist}#{B numlist}. The same is true of higher-order custom contrasts, such
as {A#B numlist}@C, {A#B numlist}#r.C, and {A#B numlist}#c.x.
Higher-order interactions with at most eight factor variables are allowed with custom contrasts.
contrast — Contrasts and linear hypothesis tests after estimation 7
method Description
noadjust do not adjust for multiple comparisons; the default
bonferroni
adjustall
Bonferroni’s method; adjust across all terms
sidak
adjustall
ˇ
Sid
´
ak’s method; adjust across all terms
scheffe Scheff
´
e’s method
Options
Main
overall specifies that a joint hypothesis test over all terms be performed.
asobserved specifies that factor covariates be evaluated using the cell frequencies observed in the
estimation sample. The default is to treat all factor covariates as though there were an equal number
of observations in each level.
lincom specifies that user-defined contrasts be treated as linear combinations. The default is to require
that all user-defined contrasts sum to zero. (Summing to zero is part of the definition of a contrast.)
Equations
equation(eqspec) specifies the equation from which contrasts are to be computed. The default is
to compute contrasts from the first equation.
atequations specifies that the contrasts be computed within each equation.
Advanced
emptycells(empspec) specifies how empty cells are handled in interactions involving factor variables
that are being treated as balanced.
emptycells(strict) is the default; it specifies that contrasts involving empty cells be treated
as not estimable.
emptycells(reweight) specifies that the effects of the observed cells be increased to accommodate
any missing cells. This makes the contrast estimable but changes its interpretation.
noestimcheck specifies that contrast not check for estimability. By default, the requested contrasts
are checked and those found not estimable are reported as such. Nonestimability is usually caused
by empty cells. If noestimcheck is specified, estimates are computed in the usual way and
reported even though the resulting estimates are manipulable, which is to say they can differ across
equivalent models having different parameterizations.
Reporting
level(#) specifies the confidence level, as a percentage, for confidence intervals. The default is
level(95) or as set by set level; see [U] 20.8 Specifying the width of confidence intervals.
mcompare(method ) specifies the method for computing p-values and confidence intervals that account
for multiple comparisons within a factor-variable term.
Most methods adjust the comparisonwise error rate, α
c
, to achieve a prespecified experimentwise
error rate, α
e
.
mcompare(noadjust) is the default; it specifies no adjustment.
α
c
= α
e
8 contrast — Contrasts and linear hypothesis tests after estimation
mcompare(bonferroni) adjusts the comparisonwise error rate based on the upper limit of the
Bonferroni inequality
α
e
≤mα
c
where m is the number of comparisons within the term.
The adjusted comparisonwise error rate is
α
c
= α
e
/m
mcompare(sidak) adjusts the comparisonwise error rate based on the upper limit of the probability
inequality
α
e
≤ 1 − (1 − α
c
)
m
where m is the number of comparisons within the term.
The adjusted comparisonwise error rate is
α
c
= 1 − (1 − α
e
)
1/m
This adjustment is exact when the m comparisons are independent.
mcompare(scheffe) controls the experimentwise error rate using the F or χ
2
distribution with
degrees of freedom equal to the rank of the term.
mcompare(method adjustall) specifies that the multiple-comparison adjustments count all
comparisons across all terms rather than performing multiple comparisons term by term. This
leads to more conservative adjustments when multiple variables or terms are specified in
marginslist. This option is compatible only with the bonferroni and sidak methods.
noeffects suppresses the table of individual contrasts with confidence intervals. This table is
produced by default when the mcompare() option is specified or when a term in termlist implies
all individual contrasts.
cieffects specifies that a table containing a confidence interval for each individual contrast be
reported.
pveffects specifies that a table containing a p-value for each individual contrast be reported.
effects specifies that a single table containing a confidence interval and p-value for each individual
contrast be reported.
nowald suppresses the table of Wald tests.
noatlevels indicates that only the overall Wald test be reported for each term containing within or
nested (@ or |) operators.
nosvyadjust is for use with svy estimation commands. It specifies that the Wald test be carried out
without the default adjustment for the design degrees of freedom. That is to say the test is carried
out as W/k ∼ F (k, d) rather than as (d − k + 1)W/(kd) ∼ F (k, d − k + 1), where k is the
dimension of the test and d is the total number of sampled PSUs minus the total number of strata.
sort specifies that the table of individual contrasts be sorted by the contrast values within each term.
post causes contrast to behave like a Stata estimation (e-class) command. contrast posts the
vector of estimated contrasts along with the estimated variance–covariance matrix to e(), so you
can treat the estimated contrasts just as you would results from any other estimation command.
For example, you could use test to perform simultaneous tests of hypotheses on the contrasts,
or you could use lincom to create linear combinations.
display options: vsquish, nofvlabel, fvwrap(#), fvwrapon(style), cformat(% fmt),
pformat(% fmt), sformat(% fmt), and nolstretch.
contrast — Contrasts and linear hypothesis tests after estimation 9
vsquish specifies that the blank space separating factor-variable terms or time-series–operated
variables from other variables in the model be suppressed.
nofvlabel displays factor-variable level values rather than attached value labels. This option
overrides the fvlabel setting; see [R] set showbaselevels.
fvwrap(#) specifies how many lines to allow when long value labels must be wrapped. Labels
requiring more than # lines are truncated. This option overrides the fvwrap setting; see [R] set
showbaselevels.
fvwrapon(style) specifies whether value labels that wrap will break at word boundaries or break
based on available space.
fvwrapon(word), the default, specifies that value labels break at word boundaries.
fvwrapon(width) specifies that value labels break based on available space.
This option overrides the fvwrapon setting; see [R] set showbaselevels.
cformat(% fmt) specifies how to format contrasts, standard errors, and confidence limits in the
table of estimated contrasts.
pformat(% fmt) specifies how to format p-values in the table of estimated contrasts.
sformat(% fmt) specifies how to format test statistics in the table of estimated contrasts.
nolstretch specifies that the width of the table of estimated contrasts not be automatically
widened to accommodate longer variable names. The default, lstretch, is to automatically
widen the table of estimated contrasts up to the width of the Results window. Specifying
lstretch or nolstretch overrides the setting given by set lstretch. If set lstretch
has not been set, the default is lstretch. nolstretch is not shown in the dialog box.
eform option specifies that the contrasts table be displayed in exponentiated form. e
contrast
is
displayed rather than contrast. Standard errors and confidence intervals are also transformed. See
[R] eform option for the list of available options.
The following option is available with contrast but is not shown in the dialog box:
df(#) specifies that the t distribution with # degrees of freedom be used for computing p-values and
confidence intervals. The default is to use e(df r) degrees of freedom or the standard normal
distribution if e(df r) is missing.
10 contrast — Contrasts and linear hypothesis tests after estimation
Remarks and examples stata.com
Remarks are presented under the following headings:
Introduction
One-way models
Estimated cell means
Testing equality of cell means
Reference category contrasts
Reverse adjacent contrasts
Orthogonal polynomial contrasts
Two-way models
Estimated interaction cell means
Simple effects
Interaction effects
Main effects
Partial interaction effects
Three-way and higher-order models
Contrast operators
Differences from a reference level (r.)
Differences from the next level (a.)
Differences from the previous level (ar.)
Differences from the grand mean (g.)
Differences from the mean of subsequent levels (h.)
Differences from the mean of previous levels (j.)
Orthogonal polynomials (p. and q.)
User-defined contrasts
Empty cells
Empty cells, ANOVA style
Nested effects
Multiple comparisons
Unbalanced data
Using observed cell frequencies
Weighted contrast operators
Testing factor effects on slopes
Chow tests
Beyond linear models
Multiple equations
Video example
Introduction
contrast performs ANOVA-style tests of main effects, interactions, simple effects, and nested
effects. It can easily decompose these tests into constituent contrasts using either named contrasts
(codings) or user-specified contrasts. Comparing levels of factor variables—whether as main effects,
interactions, or simple effects—is as easy as adding a contrast operator to the variable. The operators
can compare each level with the previous level, each level with a reference level, each level with the
mean of previous levels, and more.
contrast tests and estimates contrasts. A contrast of the parameters µ
1
, µ
2
, . . . , µ
p
is a linear
combination
P
i
c
i
µ
i
whose c
i
sum to zero. A difference of population means such as µ
1
− µ
2
is
a contrast, as are most other comparisons of population or model quantities (Coster 2005). Some
contrasts may be estimated with lincom, but contrast is much more powerful. contrast can
handle multiple contrasts simultaneously, and the command’s contrast operators make it easy to
specify complicated linear combinations.
Both the contrast operation and the creation of the margins for comparison can be performed as
though the data were balanced (typical for experimental designs) or using the observed frequencies
in the estimation sample (typical for observational studies). contrast can perform these analyses on
the results of almost all of Stata’s estimators, not just the linear-models estimators.
contrast — Contrasts and linear hypothesis tests after estimation 11
Most of contrast’s computations can be considered comparisons of estimated cell means from
a model fit. Tests of interactions are tests of whether the cell means for the interaction are all equal.
Tests of main effects are tests of whether the marginal cell means for the factor are all equal. More
focused comparisons of cell means (for example, is level 2 equal to level 1) are specified using
contrast operators. More formally, all of contrast’s computations are comparisons of conditional
expectations; cell means are one type of conditional expectation.
All contrasts can also easily be graphed; see [R] marginsplot.
For a discussion of contrasts and testing for linear models, see Searle and Gruber (2017) and
Searle (1997). For discussions specifically related to experimental design, see Winer, Brown, and
Michels (1991) and Milliken and Johnson (2009). Rosenthal, Rosnow, and Rubin (2000) focus on
contrasts with applications in behavioral sciences. Mitchell (2021, 2015) and Baldwin (2019) focus
on contrasts in Stata.
contrast is a flexible tool for understanding the effects of categorical covariates. If your model
contains categorical covariates, and especially if it contains interactions, you will want to use contrast.
One-way models
Suppose we have collected data on cholesterol levels for individuals from five age groups. To study
the effect of age group on cholesterol, we can begin by fitting a one-way model using regress:
. use https://www.stata-press.com/data/r18/cholesterol
(Artificial cholesterol data)
. label list ages
ages:
1 10--19
2 20--29
3 30--39
4 40--59
5 60--79
. regress chol i.agegrp
Source SS df MS Number of obs = 75
F(4, 70) = 35.02
Model 14943.3997 4 3735.84993 Prob > F = 0.0000
Residual 7468.21971 70 106.688853 R-squared = 0.6668
Adj R-squared = 0.6477
Total 22411.6194 74 302.859722 Root MSE = 10.329
chol Coefficient Std. err. t P>|t| [95% conf. interval]
agegrp
20--29 8.203575 3.771628 2.18 0.033 .6812991 15.72585
30--39 21.54105 3.771628 5.71 0.000 14.01878 29.06333
40--59 30.15067 3.771628 7.99 0.000 22.6284 37.67295
60--79 38.76221 3.771628 10.28 0.000 31.23993 46.28448
_cons 180.5198 2.666944 67.69 0.000 175.2007 185.8388
12 contrast — Contrasts and linear hypothesis tests after estimation
Estimated cell means
margins will show us the estimated cell means for each age group based on our fitted model:
. margins agegrp
Adjusted predictions Number of obs = 75
Model VCE: OLS
Expression: Linear prediction, predict()
Delta-method
Margin std. err. t P>|t| [95% conf. interval]
agegrp
10--19 180.5198 2.666944 67.69 0.000 175.2007 185.8388
20--29 188.7233 2.666944 70.76 0.000 183.4043 194.0424
30--39 202.0608 2.666944 75.76 0.000 196.7418 207.3799
40--59 210.6704 2.666944 78.99 0.000 205.3514 215.9895
60--79 219.282 2.666944 82.22 0.000 213.9629 224.601
We can graph those means with marginsplot:
. marginsplot
Variables that uniquely identify margins:
180
200
220
Linear prediction
10–19 20–29 30–39 40–59 60–79
Age group
Adjusted predictions of agegrp with 95% CIs
contrast — Contrasts and linear hypothesis tests after estimation 13
Testing equality of cell means
Are all the means equal? That is to say is there an effect of age group on cholesterol level? We can
answer that by asking contrast to test whether the means of the age groups are identical.
. contrast agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp 4 35.02 0.0000
Denominator 70
The means are clearly different. We could have obtained this same test directly had we fit our model
using anova rather than regress.
. anova chol agegrp
Number of obs = 75 R-squared = 0.6668
Root MSE = 10.329 Adj R-squared = 0.6477
Source Partial SS df MS F Prob>F
Model 14943.4 4 3735.8499 35.02 0.0000
agegrp 14943.4 4 3735.8499 35.02 0.0000
Residual 7468.2197 70 106.68885
Total 22411.619 74 302.85972
Achieving a more direct test result is why we recommend using anova instead of regress for
models where our focus is on the categorical covariates. The models fit by anova and regress are
identical; they merely parameterize the effects differently. The results of contrast will be identical
regardless of which command is used to fit the model. If, however, we were fitting models whose
responses are nonlinear functions of the covariates, such as logistic regression, then there would be
no analogue to anova, and we would appreciate contrast’s ability to quickly test main effects and
interactions.
14 contrast — Contrasts and linear hypothesis tests after estimation
Reference category contrasts
Now that we know that the overall effect of age group is statistically significant, we can explore
the effects of each age group. One way to do that is to use the reference category operator, r.:
. contrast r.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(20--29 vs 10--19) 1 4.73 0.0330
(30--39 vs 10--19) 1 32.62 0.0000
(40--59 vs 10--19) 1 63.91 0.0000
(60--79 vs 10--19) 1 105.62 0.0000
Joint 4 35.02 0.0000
Denominator 70
Contrast Std. err. [95% conf. interval]
agegrp
(20--29 vs 10--19) 8.203575 3.771628 .6812991 15.72585
(30--39 vs 10--19) 21.54105 3.771628 14.01878 29.06333
(40--59 vs 10--19) 30.15067 3.771628 22.6284 37.67295
(60--79 vs 10--19) 38.76221 3.771628 31.23993 46.28448
The cell mean of each age group is compared against the base age group (ages 10–19). The first
table shows that each difference is significant. The second table gives an estimate and confidence
interval for each contrast. These are the comparisons that linear regression gives with a factor covariate
and no interactions. The contrasts are identical to the coefficients from our linear regression.
Reverse adjacent contrasts
We have far more flexibility with contrast. Age group is ordinal, so it is interesting to compare
each age group with the preceding age group (rather than against one reference group). We specify
that analysis by using the reverse adjacent operator, ar.:
. contrast ar.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(20--29 vs 10--19) 1 4.73 0.0330
(30--39 vs 20--29) 1 12.51 0.0007
(40--59 vs 30--39) 1 5.21 0.0255
(60--79 vs 40--59) 1 5.21 0.0255
Joint 4 35.02 0.0000
Denominator 70
contrast — Contrasts and linear hypothesis tests after estimation 15
Contrast Std. err. [95% conf. interval]
agegrp
(20--29 vs 10--19) 8.203575 3.771628 .6812991 15.72585
(30--39 vs 20--29) 13.33748 3.771628 5.815204 20.85976
(40--59 vs 30--39) 8.60962 3.771628 1.087345 16.1319
(60--79 vs 40--59) 8.611533 3.771628 1.089257 16.13381
The 20–29 age group’s cholesterol level is 8.2 points higher than the 10–19 age group’s cholesterol
level; the 30–39 age group’s level is 13.3 points higher than the 20–29 age group’s level; and so on.
Each age group is statistically different from the preceding age group at the 5% level.
Orthogonal polynomial contrasts
The relationship between age group and cholesterol level looked almost linear in our graph. We
can examine that relationship further by using the orthogonal polynomial operator, p.:
. contrast p.agegrp, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(linear) 1 139.11 0.0000
(quadratic) 1 0.15 0.6962
(cubic) 1 0.37 0.5448
(quartic) 1 0.43 0.5153
Joint 4 35.02 0.0000
Denominator 70
Only the linear effect is statistically significant.
We can even perform the joint test that all effects beyond linear are zero. We do that by selecting
all polynomial contrasts above linear—that is, polynomial contrasts 2, 3, and 4.
. contrast p(2 3 4).agegrp, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(quadratic) 1 0.15 0.6962
(cubic) 1 0.37 0.5448
(quartic) 1 0.43 0.5153
Joint 3 0.32 0.8129
Denominator 70
The joint test has three degrees of freedom and is clearly insignificant. A linear effect of age group
seems adequate for this model.
16 contrast — Contrasts and linear hypothesis tests after estimation
Two-way models
Suppose we are investigating the effects of different dosages of a blood pressure medication and
believe that the effects may be different for men and women. We can fit the following ANOVA model
for bpchange, the change in diastolic blood pressure. Change is defined as the after measurement
minus the before measurement, so that negative values of bpchange correspond to decreases in blood
pressure.
. use https://www.stata-press.com/data/r18/bpchange
(Artificial blood pressure data)
. label list gender
gender:
1 Male
2 Female
. anova bpchange dose##gender
Number of obs = 30 R-squared = 0.9647
Root MSE = 1.4677 Adj R-squared = 0.9573
Source Partial SS df MS F Prob>F
Model 1411.9087 5 282.38174 131.09 0.0000
dose 963.48179 2 481.7409 223.64 0.0000
gender 355.11882 1 355.11882 164.85 0.0000
dose#gender 93.308093 2 46.654046 21.66 0.0000
Residual 51.699253 24 2.1541355
Total 1463.608 29 50.46924
Estimated interaction cell means
Everything is significant, including the interaction. So increasing dosage is effective and differs by
gender. Let’s explore the effects. First, let’s look at the estimated cell mean of blood pressure change
for each combination of gender and dosage.
. margins dose#gender
Adjusted predictions Number of obs = 30
Expression: Linear prediction, predict()
Delta-method
Margin std. err. t P>|t| [95% conf. interval]
dose#gender
250#Male -7.35384 .6563742 -11.20 0.000 -8.708529 -5.99915
250#Female 3.706567 .6563742 5.65 0.000 2.351877 5.061257
500#Male -13.73386 .6563742 -20.92 0.000 -15.08855 -12.37917
500#Female -6.584167 .6563742 -10.03 0.000 -7.938857 -5.229477
750#Male -16.82108 .6563742 -25.63 0.000 -18.17576 -15.46639
750#Female -14.38795 .6563742 -21.92 0.000 -15.74264 -13.03326
Our data are balanced, so these results will not be affected by the many different ways that
margins can compute cell means. Moreover, because our model consists of only dose and gender,
these are also the point estimates for each combination.
contrast — Contrasts and linear hypothesis tests after estimation 17
We can graph the results:
. marginsplot
Variables that uniquely identify margins:
-20
-15
-10
-5
0
5
Linear prediction
250 500 750
Dosage in milligrams per day
Male
Female
Adjusted predictions of dose#gender with 95% CIs
The lines are not parallel, which we expected because the interaction term is significant. Males
experience a greater decline in blood pressure at every dosage level, but the effect of increasing
dosage is greater for females. In fact, it is not clear if we can tell the difference between male and
female response at the maximum dosage.
Simple effects
We can contrast the male and female responses within dosage to see the simple effects of gender.
Because there are only two levels in gender, the choice of contrast operator is largely irrelevant.
Aside from orthogonal polynomials, all operators produce the same estimates, although the effects
can change signs.
. contrast r.gender@dose
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
gender@dose
(Female vs Male) 250 1 141.97 0.0000
(Female vs Male) 500 1 59.33 0.0000
(Female vs Male) 750 1 6.87 0.0150
Joint 3 69.39 0.0000
Denominator 24
Contrast Std. err. [95% conf. interval]
gender@dose
(Female vs Male) 250 11.06041 .9282533 9.144586 12.97623
(Female vs Male) 500 7.149691 .9282533 5.23387 9.065512
(Female vs Male) 750 2.433124 .9282533 .5173031 4.348944
18 contrast — Contrasts and linear hypothesis tests after estimation
The effect for females is about 11 points higher than for males at a dosage of 250, and that shrinks
to 2.4 points higher at the maximum dosage of 750.
We can form the simple effects the other way by contrasting the effect of dose at each level of
gender:
. contrast ar.dose@gender
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
dose@gender
(500 vs 250) Male 1 47.24 0.0000
(500 vs 250) Female 1 122.90 0.0000
(750 vs 500) Male 1 11.06 0.0028
(750 vs 500) Female 1 70.68 0.0000
Joint 4 122.65 0.0000
Denominator 24
Contrast Std. err. [95% conf. interval]
dose@gender
(500 vs 250) Male -6.380018 .9282533 -8.295839 -4.464198
(500 vs 250) Female
-10.29073 .9282533 -12.20655 -8.374914
(750 vs 500) Male -3.087217 .9282533 -5.003038 -1.171396
(750 vs 500) Female -7.803784 .9282533 -9.719605 -5.887963
Here we use the ar. reverse adjacent contrast operator so that first we are comparing a dosage
of 500 with a dosage of 250, and then we are comparing 750 with 500. We see that increasing the
dosage has a larger effect on females—10.3 points when going from 250 to 500 compared with 6.4
points for males, and 7.8 points when going from 500 to 750 versus 3.1 points for males.
Interaction effects
By specifying contrast operators on both factors, we can decompose the interaction effect into
separate interaction contrasts.
. contrast ar.dose#r.gender
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
dose#gender
(500 vs 250) (Female vs Male) 1 8.87 0.0065
(750 vs 500) (Female vs Male) 1 12.91 0.0015
Joint 2 21.66 0.0000
Denominator 24
contrast — Contrasts and linear hypothesis tests after estimation 19
Contrast Std. err. [95% conf. interval]
dose#gender
(500 vs 250)
(Female vs Male) -3.910716 1.312748 -6.620095 -1.201336
(750 vs 500)
(Female vs Male) -4.716567 1.312748 -7.425947 -2.007187
Look for departures from zero to indicate an interaction effect between dose and gender. Both
contrasts are significantly different from zero. Of course, we already knew the overall interaction
was significant from our ANOVA results. The effect of increasing dose from 250 to 500 is 3.9 points
greater in females than in males, and the effect of increasing dose from 500 to 750 is 4.7 points
greater in females than in males. The confidence intervals for both estimates easily exclude zero,
meaning that there is an interaction effect.
The joint test of these two interaction effects reproduces the test of interaction effects in the anova
output. We can see that the F statistic of 21.66 matches the statistic from our original ANOVA results.
Main effects
We can perform tests of the main effects by listing each variable individually in contrast.
. contrast dose gender
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
dose 2 223.64 0.0000
gender 1 164.85 0.0000
Denominator 24
The F tests are equivalent to the tests of main effects in the anova output. This is true only for
linear models. contrast provides an easy way to obtain main effects and other ANOVA-style tests
for models whose responses are not linear in the parameters—logistic, probit, glm, etc.
If we include contrast operators on the variables, we can also decompose the main effects into
individual contrasts:
. contrast ar.dose r.gender
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
dose
(500 vs 250) 1 161.27 0.0000
(750 vs 500) 1 68.83 0.0000
Joint 2 223.64 0.0000
gender 1 164.85 0.0000
Denominator 24
20 contrast — Contrasts and linear hypothesis tests after estimation
Contrast Std. err. [95% conf. interval]
dose
(500 vs 250) -8.335376 .6563742 -9.690066 -6.980687
(750 vs 500) -5.4455 .6563742 -6.80019 -4.090811
gender
(Female vs Male) 6.881074 .5359273 5.774974 7.987173
By specifying the ar. operator on dose, we decompose the main effect for dose into two one-degree-
of-freedom contrasts, comparing the marginal mean of blood pressure change for each dosage level
with that of the previous level. Because gender has only two levels, we cannot decompose this main
effect any further. However, specifying a contrast operator on gender allowed us to calculate the
difference in the marginal means for women and men.
Partial interaction effects
At this point, we have looked at the total interaction effects and at the main effects of each variable.
The partial interaction effects are a midpoint between these two types of effects where we collect the
individual interaction effects along the levels of one of the variables and perform a joint test of those
interactions. If we think of the interaction effects as forming a table, with the levels of one factor
variable forming the rows and the levels of the other forming the columns, partial interaction effects
are joint tests of the interactions in a row or a column. To perform these tests, we specify a contrast
operator on only one of the variables in our interaction. For this particular model, these are not very
interesting because our variables have only two and three levels. Therefore, the tests of the partial
interaction effects reproduce the tests that we obtained for the total interaction effects. We specify a
contrast operator only on dose to decompose the overall test for interaction effects into joint tests
for each ar.dose contrast:
. contrast ar.dose#gender
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
dose#gender
(500 vs 250) (joint) 1 8.87 0.0065
(750 vs 500) (joint) 1 12.91 0.0015
Joint 2 21.66 0.0000
Denominator 24
The first row is a joint test of all the interaction effects involving the (500 vs 250) comparison
of dosages. The second row is a joint test of all the interaction effects involving the (750 vs 500)
comparison. If we look back at our output in Interaction effects, we can see that there was only one of
each of these interaction effects. Therefore, each test labeled (joint) has only one degree-of-freedom.
We could have instead included a contrast operator on gender to compute the partial interaction
effects along the other dimension:
contrast — Contrasts and linear hypothesis tests after estimation 21
. contrast dose#r.gender
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
dose#gender 2 21.66 0.0000
Denominator 24
Here we obtain a joint test of all the interaction effects involving the (Female vs Male) comparison
for gender. Because gender has only two levels, the (Female vs Male) contrast is the only reference
category contrast possible. Therefore, we obtain a single joint test of all the interaction effects.
Clearly, the partial interaction effects are not interesting for this particular model. However, if our
factors had more levels, the partial interaction effects would produce tests that are not available in
the total interaction effects. For example, if our model included factors for four dosage levels and
three races, then typing
. contrast ar.dose#race
would produce three joint tests, one for each of the reverse adjacent contrasts for dosage. Each of
these tests would be a two-degree-of-freedom test because race has three levels.
Three-way and higher-order models
All the contrasts and tests that we reviewed above for two-way models can be used with models
that have more terms. For instance, we could fit a three-way full factorial model by using the anova
command:
. use https://www.stata-press.com/data/r18/cont3way
. anova y race##sex##group
We could then test the simple effects of race within each level of the interaction between sex
and group:
. contrast race@sex#group
To see the reference category contrasts that decompose these simple effects, type
. contrast r.race@sex#group
We could test the three-way interaction effects by typing
. contrast race#sex#group
or the interaction effects for the interaction of race and sex by typing
. contrast race#sex
To see the individual reference category contrasts that decompose this interaction effect, type
. contrast r.race#r.sex
We could even obtain joint tests for the interaction of race and sex within each level of group
by typing
. contrast race#sex@group
22 contrast — Contrasts and linear hypothesis tests after estimation
For tests of the main effects of each factor, we can type
. contrast race sex group
We can calculate the individual reference category contrasts that decompose these main effects:
. contrast r.race r.sex r.group
For the partial interaction effects, we could type
. contrast r.race#group
to obtain a joint test of the two-way interaction effects of race and group for each of the individual
r.race contrasts.
We could type
. contrast r.race#sex#group
to obtain a joint test of all the three-way interaction terms for each of the individual r.race contrasts.
Contrast operators
contrast recognizes a set of contrast operators that are used to specify commonly used contrasts.
When these operators are used, contrast will report a test for each individual contrast in addition
to the joint test for the term. We have already seen a few of these, like r. and ar., in the previous
examples. Here we will take a closer look at each of the unweighted operators.
Here we use the cholesterol dataset and the one-way ANOVA model from the example in One-way
models:
. use https://www.stata-press.com/data/r18/cholesterol
(Artificial cholesterol data)
. anova chol agegrp
(output omitted )
The margins command reports the estimated cell means, bµ
1
, . . . , bµ
5
, for each of the five age
groups.
. margins agegrp
Adjusted predictions Number of obs = 75
Expression: Linear prediction, predict()
Delta-method
Margin std. err. t P>|t| [95% conf. interval]
agegrp
10--19 180.5198 2.666944 67.69 0.000 175.2007 185.8388
20--29 188.7233 2.666944 70.76 0.000 183.4043 194.0424
30--39 202.0608 2.666944 75.76 0.000 196.7418 207.3799
40--59 210.6704 2.666944 78.99 0.000 205.3514 215.9895
60--79 219.282 2.666944 82.22 0.000 213.9629 224.601
contrast — Contrasts and linear hypothesis tests after estimation 23
Contrast operators provide an easy way to make certain types of comparisons of these cell means.
We use the ordinal factor agegrp to demonstrate these operators because some types of contrasts are
meaningful only when the levels of the factor have a natural ordering. We demonstrate these contrast
operators using a one-way model; however, they are equally applicable to main effects, simple effects,
and interactions for more complicated models.
Differences from a reference level (r.)
The r. operator specifies that each level of the attached factor variable be compared with a
reference level. These are referred to as reference-level or reference-category contrasts (or effects),
and r. is the reference-level operator.
In the following, we use the r. operator to test the effect of each category of age group when
that category is compared with a reference category.
. contrast r.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(20--29 vs 10--19) 1 4.73 0.0330
(30--39 vs 10--19) 1 32.62 0.0000
(40--59 vs 10--19) 1 63.91 0.0000
(60--79 vs 10--19) 1 105.62 0.0000
Joint 4 35.02 0.0000
Denominator 70
Contrast Std. err. [95% conf. interval]
agegrp
(20--29 vs 10--19) 8.203575 3.771628 .6812991 15.72585
(30--39 vs 10--19) 21.54105 3.771628 14.01878 29.06333
(40--59 vs 10--19) 30.15067 3.771628 22.6284 37.67295
(60--79 vs 10--19) 38.76221 3.771628 31.23993 46.28448
In the first table, the row labeled (20--29 vs 10--19) is a test of µ
2
= µ
1
, a test that the mean
cholesterol levels for the 10–19 age group and the 20–29 age group are equal. The tests in the
next three rows are defined similarly. The row labeled Joint provides the joint test for these four
hypotheses, which is just the test of the main effects of age group.
The second table provides the contrasts of each category with the reference category along with
confidence intervals. The contrast in the row labeled (20--29 vs 10--19) is the difference in the cell
means of the second age group and the first age group, bµ
2
− bµ
1
.
The first level of a factor is the default reference level, but we can specify a different reference
level by using the b. operator; see [U] 11.4.3.2 Base levels. Here we use the last age group, (60-79),
instead of the first as the reference category. We also include the nowald option so that only the
table of contrasts and their confidence intervals is produced.
24 contrast — Contrasts and linear hypothesis tests after estimation
. contrast rb5.agegrp, nowald
Contrasts of marginal linear predictions
Margins: asbalanced
Contrast Std. err. [95% conf. interval]
agegrp
(10--19 vs 60--79) -38.76221 3.771628 -46.28448 -31.23993
(20--29 vs 60--79) -30.55863 3.771628 -38.08091 -23.03636
(30--39 vs 60--79) -17.22115 3.771628 -24.74343 -9.698877
(40--59 vs 60--79) -8.611533 3.771628 -16.13381 -1.089257
Now, the first row is labeled (10--19 vs 60--79) and is the difference in the cell means of the first
and fifth age groups.
Differences from the next level (a.)
The a. operator specifies that each level of the attached factor variable be compared with the next
level. These are referred to as adjacent contrasts (or effects), and a. is the adjacent operator. This
operator is meaningful only with factor variables that have a natural ordering in the levels.
We can use the a. operator to perform tests that each level of age group differs from the next
adjacent level.
. contrast a.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(10--19 vs 20--29) 1 4.73 0.0330
(20--29 vs 30--39) 1 12.51 0.0007
(30--39 vs 40--59) 1 5.21 0.0255
(40--59 vs 60--79) 1 5.21 0.0255
Joint 4 35.02 0.0000
Denominator 70
Contrast Std. err. [95% conf. interval]
agegrp
(10--19 vs 20--29) -8.203575 3.771628 -15.72585 -.6812991
(20--29 vs 30--39) -13.33748 3.771628 -20.85976 -5.815204
(30--39 vs 40--59) -8.60962 3.771628 -16.1319 -1.087345
(40--59 vs 60--79) -8.611533 3.771628 -16.13381 -1.089257
In the first table, the row labeled (10--19 vs 20--29) tests the effect of belonging to the 10–19 age
group instead of the 20–29 age group. Likewise, the rows labeled (20--29 vs 30--39), (30--39 vs 40--
59), and (40--59 vs 60--79) are tests for the effects of being in the younger of the two age groups
instead of the older one.
In the second table, the contrast in the row labeled (10--19 vs 20--29) is the difference in the
cell means of the first and second age groups, bµ
1
− bµ
2
. The contrasts in the other rows are defined
similarly.
contrast — Contrasts and linear hypothesis tests after estimation 25
Differences from the previous level (ar.)
The ar. operator specifies that each level of the attached factor variable be compared with the
previous level. These are referred to as reverse adjacent contrasts (or effects), and ar. is the reverse
adjacent operator. As with the a. operator, this operator is meaningful only with factor variables that
have a natural ordering in the levels.
In the following, we use the ar. operator to report tests for the individual reverse adjacent effects
of agegrp.
. contrast ar.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(20--29 vs 10--19) 1 4.73 0.0330
(30--39 vs 20--29) 1 12.51 0.0007
(40--59 vs 30--39) 1 5.21 0.0255
(60--79 vs 40--59) 1 5.21 0.0255
Joint 4 35.02 0.0000
Denominator 70
Contrast Std. err. [95% conf. interval]
agegrp
(20--29 vs 10--19) 8.203575 3.771628 .6812991 15.72585
(30--39 vs 20--29) 13.33748 3.771628 5.815204 20.85976
(40--59 vs 30--39) 8.60962 3.771628 1.087345 16.1319
(60--79 vs 40--59) 8.611533 3.771628 1.089257 16.13381
Here the Wald tests in the first table for the individual reverse adjacent effects are equivalent to the
tests for the adjacent effects in the previous example. However, if we compare values of the contrasts
in the bottom tables, we see the difference between the r. and the ar. operators. This time, the
contrast in the first row is labeled (20--29 vs 10--19) and is the difference in the cell means of
the second and first age groups, bµ
2
− bµ
1
. This is the estimated effect of belonging to the 20–29
age group instead of the 10–19 age group. The remaining rows make similar comparisons with the
previous level.
Differences from the grand mean (g.)
The g. operator specifies that each level of a factor variable be compared with the grand mean of
all levels. For this operator, the grand mean is computed using a simple average of the cell means.
26 contrast — Contrasts and linear hypothesis tests after estimation
Here are the grand mean effects of agegrp:
. contrast g.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(10--19 vs mean) 1 68.42 0.0000
(20--29 vs mean) 1 23.36 0.0000
(30--39 vs mean) 1 0.58 0.4506
(40--59 vs mean) 1 19.08 0.0000
(60--79 vs mean) 1 63.65 0.0000
Joint 4 35.02 0.0000
Denominator 70
Contrast Std. err. [95% conf. interval]
agegrp
(10--19 vs mean) -19.7315 2.385387 -24.48901 -14.974
(20--29 vs mean) -11.52793 2.385387 -16.28543 -6.770423
(30--39 vs mean) 1.809552 2.385387 -2.947953 6.567057
(40--59 vs mean) 10.41917 2.385387 5.661668 15.17668
(60--79 vs mean) 19.0307 2.385387 14.2732 23.78821
There are five age groups in our estimation sample. Thus, the row labeled (10--19 vs mean) tests µ
1
=
(µ
1
+µ
2
+µ
3
+µ
4
+µ
5
)/5. The row labeled (20--29 vs mean) tests µ
2
= (µ
1
+µ
2
+µ
3
+µ
4
+µ
5
)/5.
The remaining rows perform similar tests for the third, fourth, and fifth age groups. In our example,
the means for all age groups except the 30–39 age group are statistically different from the grand
mean.
Differences from the mean of subsequent levels (h.)
The h. operator specifies that each level of the attached factor variable be compared with the mean
of subsequent levels. These are referred to as Helmert contrasts (or effects), and h. is the Helmert
operator. For this operator, the mean is computed using a simple average of the cell means. This
operator is meaningful only with factor variables that have a natural ordering in the levels.
contrast — Contrasts and linear hypothesis tests after estimation 27
Here are the Helmert contrasts for agegrp:
. contrast h.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(10--19 vs >10--19) 1 68.42 0.0000
(20--29 vs >20--29) 1 50.79 0.0000
(30--39 vs >30--39) 1 15.63 0.0002
(40--59 vs 60--79) 1 5.21 0.0255
Joint 4 35.02 0.0000
Denominator 70
Contrast Std. err. [95% conf. interval]
agegrp
(10--19 vs >10--19) -24.66438 2.981734 -30.61126 -18.7175
(20--29 vs >20--29) -21.94774 3.079522 -28.08965 -15.80583
(30--39 vs >30--39) -12.91539 3.266326 -19.42987 -6.400905
(40--59 vs 60--79) -8.611533 3.771628 -16.13381 -1.089257
The row labeled (10--19 vs >10--19) tests µ
1
= (µ
2
+ µ
3
+ µ
4
+ µ
5
)/4, that is, that the cell mean
for the youngest age group is equal to the average of the cell means for the older age groups. The row
labeled (20--29 vs >20--29) tests µ
2
= (µ
3
+ µ
4
+ µ
5
)/3. The tests in the other rows are defined
similarly.
Differences from the mean of previous levels (j.)
The j. operator specifies that each level of the attached factor variable be compared with the
mean of the previous levels. These are referred to as reverse Helmert contrasts (or effects), and j.
is the reverse Helmert operator. For this operator, the mean is computed using a simple average of
the cell means. This operator is meaningful only with factor variables that have a natural ordering in
the levels.
Here are the reverse Helmert contrasts of agegrp:
. contrast j.agegrp
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(20--29 vs 10--19) 1 4.73 0.0330
(30--39 vs <30--39) 1 28.51 0.0000
(40--59 vs <40--59) 1 43.18 0.0000
(60--79 vs <60--79) 1 63.65 0.0000
Joint 4 35.02 0.0000
Denominator 70
28 contrast — Contrasts and linear hypothesis tests after estimation
Contrast Std. err. [95% conf. interval]
agegrp
(20--29 vs 10--19) 8.203575 3.771628 .6812991 15.72585
(30--39 vs <30--39) 17.43927 3.266326 10.92479 23.95375
(40--59 vs <40--59) 20.2358 3.079522 14.09389 26.37771
(60--79 vs <60--79) 23.78838 2.981734 17.8415 29.73526
The row labeled (20--29 vs 10--19) tests µ
2
= µ
1
, that is, that the cell means for the 20–29 and the
10–19 age groups are equal. The row labeled (30--39 vs <30--29) tests µ
3
= (µ
1
+ µ
2
)/2, that is,
that the cell mean for the 30–39 age group is equal to the average of the cell means for the 10–19
and 20–29 age groups. The tests in the remaining rows are defined similarly.
Orthogonal polynomials (p. and q.)
The p. and q. operators specify that orthogonal polynomials be applied to the attached factor
variable. Orthogonal polynomial contrasts allow us to partition the effects of a factor variable into
linear, quadratic, cubic, and higher-order polynomial components. The p. operator applies orthogonal
polynomials using the values of the factor variable. The q. operator applies orthogonal polynomials
using the level indices. If the level values of the factor variable are equally spaced, as with our
agegrp variable, then the p. and q. operators yield the same result. These operators are meaningful
only with factor variables that have a natural ordering in the levels.
Because agegrp has five levels, contrast can test the linear, quadratic, cubic, and quartic effects
of agegrp.
. contrast p.agegrp, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp
(linear) 1 139.11 0.0000
(quadratic) 1 0.15 0.6962
(cubic) 1 0.37 0.5448
(quartic) 1 0.43 0.5153
Joint 4 35.02 0.0000
Denominator 70
The row labeled (linear) tests the linear effect of agegrp, the only effect that appears to be
significant in this case.
The labels for our agegrp variable show the age ranges that correspond to each level.
. label list ages
ages:
1 10--19
2 20--29
3 30--39
4 40--59
5 60--79
Notice that these groups do not have equal widths. Now, let’s refit our model using the agemidpt
variable. The values of agemidpt indicate the midpoint of each age group that was defined by the
agegrp variable and are, therefore, not equally spaced.
contrast — Contrasts and linear hypothesis tests after estimation 29
. anova chol agemidpt
Number of obs = 75 R-squared = 0.6668
Root MSE = 10.329 Adj R-squared = 0.6477
Source Partial SS df MS F Prob>F
Model 14943.4 4 3735.8499 35.02 0.0000
agemidpt 14943.4 4 3735.8499 35.02 0.0000
Residual 7468.2197 70 106.68885
Total 22411.619 74 302.85972
Now if we use the q. operator, we will obtain the same results as above because the level indices
of agemidpt are equivalent to the values of agegrp.
. contrast q.agemidpt, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agemidpt
(linear) 1 139.11 0.0000
(quadratic) 1 0.15 0.6962
(cubic) 1 0.37 0.5448
(quartic)
1 0.43 0.5153
Joint 4 35.02 0.0000
Denominator 70
However, if we use the p. operator, we will instead fit an orthogonal polynomial to the midpoint
values.
. contrast p.agemidpt, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agemidpt
(linear) 1 133.45 0.0000
(quadratic) 1 5.40 0.0230
(cubic)
1 0.05 0.8198
(quartic) 1 1.16 0.2850
Joint 4 35.02 0.0000
Denominator 70
Using the values of the midpoints, the quadratic effect is also significant at the 5% level.
Technical note
We used the noeffects option when working with orthogonal polynomial contrasts. Apart from
perhaps the sign of the contrast, the values of the individual contrasts are not meaningful for orthogonal
polynomial contrasts. In addition, many textbooks provide tables with contrast coefficients that can be
used to compute orthogonal polynomial contrasts where the levels of a factor are equally spaced. If
30 contrast — Contrasts and linear hypothesis tests after estimation
we use these coefficients and calculate the contrasts manually with user-defined contrasts, as described
below, the Wald tests for the polynomial terms will be equivalent, but the values of the individual
contrasts will not necessarily match those that we obtain when using the polynomial contrast operator.
When we use one of these contrast operators, an algorithm is used to calculate the coefficients of the
polynomial contrast that will allow for unequal spacing in the levels of the factor as well as in the
weights for the cell frequencies (when using pw. or qw.), as described in Methods and formulas.
User-defined contrasts
In the previous examples, we performed tests using contrast operators. When there is not a contrast
operator available to calculate the contrast in which we are interested, we can specify custom contrasts.
Here we fit a one-way model for cholesterol on the factor race, which has three levels:
. label list race
race:
1 Black
2 White
3 Other
. anova chol race
Number of obs = 75 R-squared = 0.0299
Root MSE = 17.3775 Adj R-squared = 0.0029
Source
Partial SS df MS F Prob>F
Model 669.27823 2 334.63912 1.11 0.3357
race 669.27823 2 334.63912 1.11 0.3357
Residual 21742.341 72 301.97696
Total 22411.619 74 302.85972
margins calculates the estimated cell mean cholesterol level for each race:
. margins race
Adjusted predictions Number of obs = 75
Expression: Linear prediction, predict()
Delta-method
Margin std. err. t P>|t| [95% conf. interval]
race
Black 204.4279 3.475497 58.82 0.000 197.4996 211.3562
White 197.6132 3.475497 56.86 0.000 190.6849 204.5415
Other 198.7127 3.475497 57.18 0.000 191.7844 205.6409
Suppose we want to test the following linear combination:
3
X
i=1
c
i
µ
i
where µ
i
is the cell mean of chol when race is equal to its ith level (the means estimated using
margins above). Assuming the c
i
elements sum to zero, this linear combination is a contrast. We
can specify this type of custom contrast by using the following syntax:
contrast — Contrasts and linear hypothesis tests after estimation 31
{race c
1
c
2
c
3
}
The null hypothesis for the test of the main effects of race is
H
0
race
: µ
1
= µ
2
= µ
3
Although H
0
race
can be tested using any of several different contrasts on the cell means, we will test
it by comparing the second and third cell means with the first. To test that the cell means for blacks
and whites are equal, µ
1
= µ
2
, we can specify the contrast
{race -1 1 0}
To test that the cell means for blacks and other races are equal, µ
1
= µ
3
, we can specify the contrast
{race -1 0 1}
We can use both in a single call to contrast.
. contrast {race -1 1 0} {race -1 0 1}
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race
(1) 1 1.92 0.1699
(2) 1 1.35 0.2488
Joint
2 1.11 0.3357
Denominator 72
Contrast Std. err. [95% conf. interval]
race
(1) -6.814717 4.915095 -16.61278 2.983345
(2) -5.715261 4.915095 -15.51332 4.082801
The row labeled (1) is the test for µ
1
= µ
2
, the first specified contrast. The row labeled (2) is the
test for µ
1
= µ
3
, the second specified contrast. The row labeled Joint is the overall test for the
main effects of race.
32 contrast — Contrasts and linear hypothesis tests after estimation
Now, let’s fit a model with two factors, race and age group:
. anova chol race##agegrp
Number of obs = 75 R-squared = 0.7524
Root MSE = 9.61785 Adj R-squared = 0.6946
Source Partial SS df MS F Prob>F
Model 16861.438 14 1204.3884 13.02 0.0000
race 669.27823 2 334.63912 3.62 0.0329
agegrp 14943.4 4 3735.8499 40.39 0.0000
race#agegrp 1248.7601 8 156.09501 1.69 0.1201
Residual 5550.1814 60 92.503024
Total 22411.619 74 302.85972
The null hypothesis for the test of the main effects of race is now
H
0
race
: µ
1·
= µ
2·
= µ
3·
where µ
i·
is the marginal mean of chol when race is equal to its ith level.
We can use the same syntax as above to perform this test by specifying contrasts on the marginal
means of race:
. contrast {race -1 1 0} {race -1 0 1}
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race
(1) 1 6.28 0.0150
(2) 1 4.41 0.0399
Joint 2 3.62 0.0329
Denominator 60
Contrast Std. err. [95% conf. interval]
race
(1) -6.814717 2.720339 -12.2562 -1.37323
(2) -5.715261 2.720339 -11.15675 -.2737739
contrast — Contrasts and linear hypothesis tests after estimation 33
Custom contrasts may be specified on the cell means of interactions, too. Here we use margins
to calculate the mean of chol for each cell in the interaction of race and agegrp:
. margins race#agegrp
Adjusted predictions Number of obs = 75
Expression: Linear prediction, predict()
Delta-method
Margin std. err. t P>|t| [95% conf. interval]
race#agegrp
Black#10--19 179.2309 4.301233 41.67 0.000 170.6271 187.8346
Black#20--29 196.4777 4.301233 45.68 0.000 187.874 205.0814
Black#30--39 210.6694 4.301233 48.98 0.000 202.0656 219.2731
Black#40--59 214.097 4.301233 49.78 0.000 205.4933 222.7008
Black#60--79 221.6646 4.301233 51.54 0.000 213.0609 230.2684
White#10--19 186.0727 4.301233 43.26 0.000 177.469 194.6765
White#20--29 184.6714 4.301233 42.93 0.000 176.0676 193.2751
White#30--39 196.2633 4.301233 45.63 0.000 187.6595 204.867
White#40--59 209.9953 4.301233 48.82 0.000 201.3916 218.5991
White#60--79 211.0633 4.301233 49.07 0.000 202.4595 219.667
Other#10--19 176.2556 4.301233 40.98 0.000 167.6519 184.8594
Other#20--29 185.0209 4.301233 43.02 0.000 176.4172 193.6247
Other#30--39 199.2498 4.301233 46.32 0.000 190.646 207.8535
Other#40--59 207.9189 4.301233 48.34 0.000 199.3152 216.5227
Other#60--79
225.118 4.301233 52.34 0.000 216.5143 233.7218
Now, we are interested in testing the following linear combination of these cell means:
3
X
i=1
5
X
j=1
c
ij
µ
ij
We can specify this type of custom contrast using the following syntax:
{race#agegrp c
11
c
12
. . . c
15
c
21
c
22
. . . c
25
c
31
c
32
. . . c
35
}
Because the marginal means of chol for each level of race are linear combinations of the cell
means, we can compose the test for the main effects of race in terms of the cell means directly.
The constraint that the marginal means for blacks and whites are equal, µ
1·
= µ
2·
, translates to the
following constraint on the cell means:
1
5
(µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
) =
1
5
(µ
21
+ µ
22
+ µ
23
+ µ
24
+ µ
25
)
Ignoring the common factor, we can specify this contrast as
{race#agegrp -1 -1 -1 -1 -1 1 1 1 1 1 0 0 0 0 0}
contrast will fill in the trailing zeros for us if we neglect to specify them, so
{race#agegrp -1 -1 -1 -1 -1 1 1 1 1 1}
is also allowed. The other constraint, µ
1·
= µ
3·
, translates to
1
5
(µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
) =
1
5
(µ
31
+ µ
32
+ µ
33
+ µ
34
+ µ
35
)
34 contrast — Contrasts and linear hypothesis tests after estimation
This can be specified to contrast as
{race#agegrp -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1}
The following call to contrast yields the same test results as above.
. contrast {race#agegrp -1 -1 -1 -1 -1 1 1 1 1 1}
> {race#agegrp -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1}, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race#agegrp
(1) (1) 1 6.28 0.0150
(2) (2) 1 4.41 0.0399
Joint 2 3.62 0.0329
Denominator 60
The row labeled (1) (1) is the test for
µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
= µ
21
+ µ
22
+ µ
23
+ µ
24
+ µ
25
It was the first specified contrast. The row labeled (2) (2) is the test for
µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
= µ
31
+ µ
32
+ µ
33
+ µ
34
+ µ
35
It was the second specified contrast. The row labeled Joint tests (1) (1) and (2) (2) simultaneously.
We used the noeffects option above to suppress the table of contrasts. We can omit the 1/5
from the equations for µ
1·
= µ
2·
and µ
1·
= µ
3·
and still obtain the appropriate tests. However, if
we want to calculate the differences in the marginal means, we must include the 1/5 = 0.2 on each
of the contrast coefficients as follows:
. contrast {race#agegrp -0.2 -0.2 -0.2 -0.2 -0.2 ///
0.2 0.2 0.2 0.2 0.2} ///
{race#agegrp -0.2 -0.2 -0.2 -0.2 -0.2 ///
0 0 0 0 0 ///
0.2 0.2 0.2 0.2 0.2}
So far, we have reproduced the reference category contrasts by specifying user-defined contrasts
on the marginal means and then on the cell means. For this test, it would have been easier to use the
r. contrast operator:
. contrast r.race, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race
(White vs Black) 1 6.28 0.0150
(Other vs Black) 1 4.41 0.0399
Joint 2 3.62 0.0329
Denominator 60
contrast — Contrasts and linear hypothesis tests after estimation 35
In most cases, we can use contrast operators to perform tests. However, if we want to compare,
for instance, the second and third age groups with the fourth and fifth age groups with the test
1
2
(µ
·2
+ µ
·3
) =
1
2
(µ
·4
+ µ
·5
)
there is not a contrast operator that corresponds to this particular contrast. A custom contrast is
necessary.
. contrast {agegrp 0 -0.5 -0.5 0.5 0.5}
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
agegrp 1 62.19 0.0000
Denominator 60
Contrast Std. err. [95% conf. interval]
agegrp
(1) 19.58413 2.483318 14.61675 24.5515
Empty cells
An empty cell is a combination of the levels of factor variables that is not observed in the estimation
sample. In the previous examples, we have seen data with three levels of race, five levels of agegrp,
and all level combinations of race and agegrp present. Suppose there are no observations for white
individuals in the second age group (ages 20–29).
. use https://www.stata-press.com/data/r18/cholesterol2
(Artificial cholesterol data, empty cells)
. label list
race:
1 Black
2 White
3 Other
ages:
1 10--19
2 20--29
3 30--39
4 40--59
5 60--79
36 contrast — Contrasts and linear hypothesis tests after estimation
. regress chol race##agegrp
note: identifies no observations in the sample.
Source SS df MS Number of obs = 70
F(13, 56) = 13.51
Model 15751.6113 13 1211.66241 Prob > F = 0.0000
Residual 5022.71559 56 89.6913498 R-squared = 0.7582
Adj R-squared = 0.7021
Total 20774.3269 69 301.077201 Root MSE = 9.4706
chol Coefficient Std. err. t P>|t| [95% conf. interval]
race
White 12.84185 5.989703 2.14 0.036 .8430383 24.84067
Other -.167627 5.989703 -0.03 0.978 -12.16644 11.83119
agegrp
20--29 17.24681 5.989703 2.88 0.006 5.247991 29.24562
30--39 31.43847 5.989703 5.25 0.000 19.43966 43.43729
40--59 34.86613 5.989703 5.82 0.000 22.86732 46.86495
60--79 44.43374 5.989703 7.42 0.000 32.43492 56.43256
race#agegrp
White#20--29 0 (empty)
White#30--39 -22.83983 8.470719 -2.70 0.009 -39.80872 -5.870939
White#40--59 -14.67558 8.470719 -1.73 0.089 -31.64447 2.293306
White#60--79 -10.51115 8.470719 -1.24 0.220 -27.48004 6.457735
Other#20--29 -6.054425 8.470719 -0.71 0.478 -23.02331 10.91446
Other#30--39 -11.48083 8.470719 -1.36 0.181 -28.44971 5.488063
Other#40--59 -.6796112 8.470719 -0.08 0.936 -17.6485 16.28928
Other#60--79 -1.578052 8.470719 -0.19 0.853 -18.54694 15.39084
_cons 175.2309 4.235359 41.37 0.000 166.7464 183.7153
Now, let’s use contrast to test the main effects of race:
. contrast race
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race (not testable)
Denominator 56
By “not testable”, contrast means that it cannot form a test for the main effects of race based
on estimable functions of the model coefficients. agegrp has five levels, so contrast constructs an
estimate of the ith margin for race as
bµ
i·
=
1
5
5
X
j=1
bµ
ij
= bµ
0
+ bα
i
+
1
5
5
X
j=1
n
b
β
j
+ (
c
αβ)
ij
o
but (
c
αβ)
22
was constrained to zero because of the empty cell, so bµ
2·
is not an estimable function
of the model coefficients.
contrast — Contrasts and linear hypothesis tests after estimation 37
See Estimable functions in Methods and formulas of [R] margins for a technical description of
estimable functions. The emptycells(reweight) option causes contrast to estimate µ
2·
by
bµ
2·
=
bµ
21
+ bµ
23
+ bµ
24
+ bµ
25
4
which is an estimable function of the model coefficients.
. contrast race, emptycells(reweight)
Contrasts of marginal linear predictions
Margins: asbalanced
Empty cells: reweight
df F P>F
race 2 3.17 0.0498
Denominator 56
We can reconstruct the effect of the emptycells(reweight) option by using custom contrasts.
. contrast {race#agegrp -4 -4 -4 -4 -4 5 0 5 5 5}
> {race#agegrp -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1}, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race#agegrp
(1) (1) 1 1.06 0.3080
(2) (2) 1 2.37 0.1291
Joint 2 3.17 0.0498
Denominator 56
The row labeled (1) (1) is the test for
1
5
(µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
) =
1
4
(µ
21
+ µ
23
+ µ
24
+ µ
25
)
It was the first specified contrast. The row labeled (2) (2) is the test for
µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
= µ
31
+ µ
32
+ µ
33
+ µ
34
+ µ
35
It was the second specified contrast. The row labeled Joint is the overall test of the main effects of
race.
38 contrast — Contrasts and linear hypothesis tests after estimation
Empty cells, ANOVA style
Let’s refit the linear model from the previous example with anova to compare with contrast’s
test for the main effects of race.
. anova chol race##agegrp
Number of obs = 70 R-squared = 0.7582
Root MSE = 9.47055 Adj R-squared = 0.7021
Source Partial SS df MS F Prob>F
Model 15751.611 13 1211.6624 13.51 0.0000
race 305.49046 2 152.74523 1.70 0.1914
agegrp 14387.856 4 3596.964 40.10 0.0000
race#agegrp 795.80757 7 113.6868 1.27 0.2831
Residual 5022.7156 56 89.69135
Total 20774.327 69 301.0772
contrast and anova handled the empty cell differently; the F statistic reported by contrast
was 3.17, but anova reported 1.70. To see how they differ, consider the following table of the cell
means and margins for our situation.
agegrp
1 2 3 4 5
1 µ
11
µ
12
µ
13
µ
14
µ
15
µ
1·
race 2 µ
21
µ
23
µ
24
µ
25
3 µ
31
µ
32
µ
33
µ
34
µ
35
µ
3·
µ
·1
µ
·3
µ
·4
µ
·5
For testing the main effects of race, we know that we will be testing the equality of the marginal
means for rows 1 and 3, that is, µ
1·
= µ
3·
. This translates into the following constraint:
µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
= µ
31
+ µ
32
+ µ
33
+ µ
34
+ µ
35
Because row 2 contains an empty cell in column 2, anova dropped column 2 and tested the equality
of the marginal mean for row 2 with the average of the marginal means from rows 1 and 3, using
only the remaining cell means. This translates into the following constraint:
2(µ
21
+ µ
23
+ µ
24
+ µ
25
) = µ
11
+ µ
13
+ µ
14
+ µ
15
+ µ
31
+ µ
33
+ µ
34
+ µ
35
(1)
Now that we know the constraints that anova used to test for the main effects of race, we can use
custom contrasts to reproduce the anova test result.
contrast — Contrasts and linear hypothesis tests after estimation 39
. contrast {race#agegrp -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1}
> {race#agegrp 1 0 1 1 1 -2 0 -2 -2 -2 1 0 1 1 1}, noeffects
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race#agegrp
(1) (1) 1 2.37 0.1291
(2) (2) 1 1.03 0.3138
Joint 2 1.70 0.1914
Denominator 56
The row labeled (1) (1) is the test for µ
1·
= µ
3·
; it was the first specified contrast. The row labeled
(2) (2) is the test for the constraint in (1); it was the second specified contrast. The row labeled
Joint is an overall test for the main effects of race.
Nested effects
contrast has the | operator for computing simple effects when the levels of one factor are nested
within the levels of another. Here is a fictional example where we are interested in the effect of
five methods of teaching algebra on students’ scores for the math portion of the SAT. Suppose three
algebra classes are randomly sampled from classes using each of the five methods so that class is
nested in method as demonstrated in the following tabulation.
. use https://www.stata-press.com/data/r18/sat
(Fictional SAT data)
. tabulate class method
Five methods of teaching algebra
Class ID
1 2 3 4 5 Total
1 5 0 0 0 0 5
2 5 0 0 0 0 5
3 5 0 0 0 0 5
4 0 5 0 0 0 5
5 0 5 0 0 0 5
6 0 5 0 0 0 5
7 0 0 5 0 0 5
8 0 0 5 0 0 5
9 0 0 5 0 0 5
10 0 0 0 5 0 5
11 0 0 0 5 0 5
12 0 0 0 5 0 5
13 0 0 0 0 5 5
14 0 0 0 0 5 5
15 0 0 0 0 5 5
Total 15 15 15 15 15 75
40 contrast — Contrasts and linear hypothesis tests after estimation
We will consider method as fixed and class nested in method as random. To use class nested
in method as the error term for method, we can specify the following anova model:
. anova score method / class|method /
Number of obs = 75 R-squared = 0.7599
Root MSE = 71.8517 Adj R-squared = 0.7039
Source Partial SS df MS F Prob>F
Model 980312 14 70022.286 13.56 0.0000
method 905872 4 226468 30.42 0.0000
class|method 74440 10 7444
class|method 74440 10 7444 1.44 0.1845
Residual 309760 60 5162.6667
Total 1290072 74 17433.405
Like anova, contrast allows the | operator, which specifies that one variable is nested in the
levels of another. We can use contrast to test the main effects of method and the simple effects
of class within method.
. contrast method class|method
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
method (not testable)
class|method
1 2 2.80 0.0687
2 2 0.91 0.4089
3 2 1.10 0.3390
4 2 0.22 0.8025
5 2 2.18 0.1221
Joint 10 1.44 0.1845
Denominator 60
Although contrast was able to perform the individual tests for the simple effects of class within
method, empty cells in the interaction between method and class prevented contrast from testing
for a main effect of method. Here we add the emptycells(reweight) option so that contrast
can take the empty cells into account when computing the marginal means for method.
contrast — Contrasts and linear hypothesis tests after estimation 41
. contrast method class|method, emptycells(reweight)
Contrasts of marginal linear predictions
Margins: asbalanced
Empty cells: reweight
df F P>F
method 4 43.87 0.0000
class|method
1 2 2.80 0.0687
2 2 0.91 0.4089
3 2 1.10 0.3390
4 2 0.22 0.8025
5 2 2.18 0.1221
Joint 10 1.44 0.1845
Denominator 60
Now, contrast does report a test for the main effects of method. However, if we compare this with
the anova results, we will see that the results are different. They are different because contrast
uses the residual error term to compute the F test by default. Using notation similar to anova, we
can use the / operator to specify a different error term for the test. Therefore, we can reproduce the
test of main effects from our anova command by typing
. contrast method / class|method /, emptycells(reweight)
Contrasts of marginal linear predictions
Margins: asbalanced
Empty cells: reweight
df F P>F
method 4 30.42 0.0000
class|method 10 (denominator)
class|method
1 2 2.80 0.0687
2 2 0.91 0.4089
3 2 1.10 0.3390
4 2 0.22 0.8025
5 2 2.18 0.1221
Joint 10 1.44 0.1845
Denominator 60
Multiple comparisons
We have seen that contrast can report the individual linear combinations that make up the
requested effects. Depending upon the specified option, contrast will report confidence intervals,
p-values, or both in the effects table. By default, the reported confidence intervals and p-values are
not adjusted for multiple comparisons. Use the mcompare() option to adjust the confidence intervals
and p-values for multiple comparisons of the individual effects.
42 contrast — Contrasts and linear hypothesis tests after estimation
Let’s compute the grand mean effects of race using the g. operator. We also specify the mcom-
pare(bonferroni) option to compute p-values and confidence intervals using Bonferroni’s adjust-
ment.
. use https://www.stata-press.com/data/r18/cholesterol
(Artificial cholesterol data)
. anova chol race##agegrp
(output omitted )
. contrast g.race, mcompare(bonferroni)
Contrasts of marginal linear predictions
Margins: asbalanced
Bonferroni
df F P>F P>F
race
(Black vs mean) 1 7.07 0.0100 0.0301
(White vs mean) 1 2.82 0.0982 0.2947
(Other vs mean) 1 0.96 0.3312 0.9936
Joint 2 3.62 0.0329
Denominator 60
Note: Bonferroni-adjusted p-values are reported for tests on
individual contrasts only.
Number of
comparisons
race 3
Bonferroni
Contrast Std. err. [95% conf. interval]
race
(Black vs mean) 4.17666 1.570588 .3083743 8.044945
(White vs mean) -2.638058 1.570588 -6.506343 1.230227
(Other vs mean) -1.538602 1.570588 -5.406887 2.329684
The last table reports a Bonferroni-adjusted confidence interval for each individual contrast. (Use
the effects option to add p-values to the last table.) The first table includes a Bonferroni-adjusted
p-value for each test that is not a joint test.
contrast — Contrasts and linear hypothesis tests after estimation 43
Joint tests are never adjusted for multiple comparisons. For example,
. contrast race@agegrp, mcompare(bonferroni)
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race@agegrp
10--19 2 1.37 0.2620
20--29 2 2.44 0.0958
30--39
2 3.12 0.0512
40--59 2 0.53 0.5889
60--79 2 2.90 0.0628
Joint 10 2.07 0.0409
Denominator 60
Note: Bonferroni-adjusted p-values are reported
for tests on individual contrasts only.
Number of
comparisons
race@agegrp 10
Bonferroni
Contrast Std. err. [95% conf. interval]
race@agegrp
(White vs base) 10--19 6.841855 6.082862 -10.88697 24.57068
(White vs base) 20--29 -11.80631 6.082862 -29.53513 5.922513
(White vs base) 30--39 -14.40607 6.082862 -32.13489 3.322751
(White vs base) 40--59 -4.101691 6.082862 -21.83051 13.62713
(White vs base) 60--79 -10.60137 6.082862 -28.33019 7.127448
(Other vs base) 10--19 -2.975244 6.082862 -20.70407 14.75358
(Other vs base) 20--29 -11.45679 6.082862 -29.18561 6.272031
(Other vs base) 30--39 -11.41958 6.082862 -29.1484 6.309244
(Other vs base) 40--59 -6.17807 6.082862 -23.90689 11.55075
(Other vs base) 60--79 3.453375 6.082862 -14.27545 21.1822
Here we have five tests of simple effects with two degrees of freedom each. No Bonferroni-adjusted
p-values are available for these tests, but the confidence intervals for the individual contrasts are
adjusted.
Unbalanced data
By default, contrast treats all factors as balanced when computing marginal means. By balanced,
we mean that contrast assumes an equal number of observations in each level of each factor and
an equal number of observations in each cell of each interaction. If our data are balanced, there
is no issue. If, however, our data are not balanced, we might prefer that contrast use the actual
cell frequencies from our data in computing marginal means. We instruct contrast to use observed
frequencies by adding the asobserved option.
Even if our data are unbalanced, we might still want contrast to compute balanced marginal
means. It depends on what we want to test and what our data represent. If we have data from a designed
44 contrast — Contrasts and linear hypothesis tests after estimation
experiment that started with an equal number of males and females but the data became unbalanced
because the data from a few males were unusable, we might still want our margins computed as
though the data were balanced. If, however, we have a representative sample of individuals from Los
Angeles with 40% of European descent, 34% African-American, 25% Hispanic, and 1% Australian,
we probably want our margins computed using these representative frequencies. We do not want
Australians receiving the same weight as Europeans.
The following examples will use an unbalanced version of our dataset.
. use https://www.stata-press.com/data/r18/cholesterol3
(Artificial cholesterol data, unbalanced)
. tab race agegrp
Age group
Race 10--19 20--29 30--39 40--59 60--79 Total
Black 1 5 5 4 3 18
White 4 5 7 4 4 24
Other 3 7 6 5 4 25
Total 8 17 18 13 11 67
The row labeled Total gives observed cell frequencies for age group. These can be obtained
by summing frequencies from the cells in the corresponding column. In this respect, we can also
refer to them as marginal frequencies. We use the terms marginal frequencies and cell frequencies
interchangeably below.
We begin by fitting the two-factor model with an interaction.
. anova chol race##agegrp
Number of obs = 67 R-squared = 0.8179
Root MSE = 8.37496 Adj R-squared = 0.7689
Source Partial SS df MS F Prob>F
Model 16379.993 14 1169.9995 16.68 0.0000
race 230.7544 2 115.3772 1.64 0.2029
agegrp 13857.988 4 3464.4969 49.39 0.0000
race#agegrp 857.81521 8 107.2269 1.53 0.1701
Residual 3647.2774 52 70.13995
Total 20027.27 66 303.44349
Using observed cell frequencies
Recall that the marginal means are computed from the cell means. Treating the factors as balanced
yields the following marginal means for race:
η
1·
=
1
5
(µ
11
+ µ
12
+ µ
13
+ µ
14
+ µ
15
)
η
2·
=
1
5
(µ
21
+ µ
22
+ µ
23
+ µ
24
+ µ
25
)
η
3·
=
1
5
(µ
31
+ µ
32
+ µ
33
+ µ
34
+ µ
35
)
contrast — Contrasts and linear hypothesis tests after estimation 45
If we have a fixed population and unbalanced cells, then the η
i·
do not represent population means. If,
however, our data are representative of the population, we can use the frequencies from our estimation
sample to estimate the population marginal means, denoted µ
i·
.
Here are the results of testing for a main effect of race, treating all the factors as balanced.
. contrast r.race
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race
(White vs Black) 1 3.28 0.0757
(Other vs Black) 1 1.50 0.2263
Joint 2 1.64 0.2029
Denominator 52
Contrast Std. err. [95% conf. interval]
race
(White vs Black) -5.324254 2.93778 -11.21934 .5708338
(Other vs Black) -3.596867 2.93778 -9.491955 2.298221
The row labeled (White vs Black) is the test for η
2·
= η
1·
. The row labeled (Other vs Black)
is the test for η
3·
= η
1·
.
If the observed marginal frequencies are representative of the distribution of the levels of agegrp,
we can use them to form the marginal means of chol for each of the levels of race from the cell
means.
µ
1·
=
1
67
(8µ
11
+ 17µ
12
+ 18µ
13
+ 13µ
14
+ 11µ
15
)
µ
2·
=
1
67
(8µ
21
+ 17µ
22
+ 18µ
23
+ 13µ
24
+ 11µ
25
)
µ
3·
=
1
67
(8µ
31
+ 17µ
32
+ 18µ
33
+ 13µ
34
+ 11µ
35
)
46 contrast — Contrasts and linear hypothesis tests after estimation
Here are the results of testing for the main effects of race, using the observed marginal frequencies:
. contrast r.race, asobserved
Contrasts of marginal linear predictions
Margins: asobserved
df F P>F
race
(White vs Black) 1 7.25 0.0095
(Other vs Black) 1 3.89 0.0538
Joint 2 3.74 0.0304
Denominator 52
Contrast Std. err. [95% conf. interval]
race
(White vs Black) -7.232433 2.686089 -12.62246 -1.842402
(Other vs Black) -5.231198 2.651203 -10.55123 .0888295
The row labeled (White vs Black) is the test for µ
2·
= µ
1·
. The row labeled (Other vs Black)
is the test for µ
3·
= µ
1·
. Both tests were insignificant when we tested the cell means resulting from
balanced frequencies; however, when we tested the cell means from observed frequencies, the first
test is significant beyond the 5% level (and the second test is nearly so).
Here we reproduce the results of the asobserved option with custom contrasts. Because we are
modifying the way that the marginal means are constructed from the cell means, we will specify the
contrasts on the predicted cell means. We use macro expansion, =exp, to evaluate the fractions instead
of approximating them with decimals. Macro expansion guarantees that the contrast coefficients sum
to zero. For more information, see Macro expansion operators and function in [P] macro.
contrast — Contrasts and linear hypothesis tests after estimation 47
. contrast {race#agegrp -‘=8/67’ -‘=17/67’ -‘=18/67’ -‘=13/67’ -‘=11/67’
> ‘=8/67’ ‘=17/67’ ‘=18/67’ ‘=13/67’ ‘=11/67’}
> {race#agegrp -‘=8/67’ -‘=17/67’ -‘=18/67’ -‘=13/67’ -‘=11/67’
> 0 0 0 0 0
> ‘=8/67’ ‘=17/67’ ‘=18/67’ ‘=13/67’ ‘=11/67’}
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race#agegrp
(1) (1) 1 7.25 0.0095
(2) (2) 1 3.89 0.0538
Joint 2 3.74 0.0304
Denominator 52
Contrast Std. err. [95% conf. interval]
race#agegrp
(1) (1) -7.232433 2.686089 -12.62246 -1.842402
(2) (2) -5.231198 2.651203 -10.55123 .0888295
Weighted contrast operators
contrast provides observation-weighted versions of five of the contrast operators—gw., hw.,
jw., pw., and qw.. The first three of these operators perform comparisons of means across cells, and
like the marginal means just discussed, these means can be computed in two ways: 1) as though the
cell frequencies were equal or 2) using the observed cell frequencies from the estimation sample. The
weighted operators provide versions of the standard (as balanced) operators that weight these means
by their cell frequencies. The two orthogonal polynomial operators involve similar adjustments for
weighting.
Let’s examine what this means by using the gw. operator. The gw. operator is a weighted version
of the g. operator. The gw. operator computes the grand mean using the cell frequencies for race
obtained from the model fit.
Here we test the effects of race, comparing each level with the weighted grand mean but otherwise
treating the factors as balanced in the marginal mean calculations.
. contrast gw.race
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race
(Black vs mean) 1 2.78 0.1014
(White vs mean) 1 2.06 0.1573
(Other vs mean) 1 0.06 0.8068
Joint 2 1.64 0.2029
Denominator 52
48 contrast — Contrasts and linear hypothesis tests after estimation
Contrast Std. err. [95% conf. interval]
race
(Black vs mean) 3.24931 1.948468 -.6605779 7.159198
(White vs mean) -2.074944 1.44618 -4.976915 .8270276
(Other vs mean) -.347557 1.414182 -3.18532 2.490206
The observed marginal frequencies of race are 18, 24, and 25. Thus, the row labeled (Black vs
Mean) tests η
1·
= (18η
1·
+24η
2·
+25η
3·
)/67; the row labeled (White vs Mean) tests η
2·
= (18η
1·
+
24η
2·
+ 25η
3·
)/67; and the row labeled (Other vs Mean) tests η
3·
= (18η
1·
+ 24η
2·
+ 25η
3·
)/67.
Now, we reproduce the above results using custom contrasts. We are weighting the calculation
of the grand mean from the marginal means for each of the races, but we are not weighting the
calculation of the marginal means themselves. Therefore, we can specify the custom contrast on the
marginal means for race instead of on the cell means.
. contrast {race ‘=49/67’ -‘=24/67’ -‘=25/67’}
> {race -‘=18/67’ ‘=43/67’ -‘=25/67’}
> {race -‘=18/67’ -‘=24/67’ ‘=42/67’}
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
race
(1) 1 2.78 0.1014
(2) 1 2.06 0.1573
(3) 1 0.06 0.8068
Joint 2 1.64 0.2029
Denominator 52
Contrast Std. err. [95% conf. interval]
race
(1) 3.24931 1.948468 -.6605779 7.159198
(2) -2.074944 1.44618 -4.976915 .8270276
(3) -.347557 1.414182 -3.18532 2.490206
Now, we will test for each race the difference between the marginal mean and the weighted grand
mean, treating the factors as observed in the marginal mean calculations.
contrast — Contrasts and linear hypothesis tests after estimation 49
. contrast gw.race, asobserved wald ci
Contrasts of marginal linear predictions
Margins: asobserved
df F P>F
race
(Black vs mean) 1 6.81 0.0118
(White vs mean) 1 3.74 0.0587
(Other vs mean) 1 0.26 0.6099
Joint 2 3.74 0.0304
Denominator 52
Contrast Std. err. [95% conf. interval]
race
(Black vs mean) 4.542662 1.740331 1.050432 8.034891
(White vs mean) -2.689771 1.39142 -5.481859 .1023172
(Other vs mean) -.6885363 1.341261 -3.379973 2.002901
The row labeled (Black vs Mean) tests µ
1·
= (18µ
1·
+ 24µ
2·
+ 25µ
3·
)/67; the row labeled (White
vs Mean) tests µ
2·
= (18µ
1·
+ 24µ
2·
+ 25µ
3·
)/67; and the row labeled (Other vs Mean) tests
µ
3·
= (18µ
1·
+ 24µ
2·
+ 25µ
3·
)/67.
Here we use a custom contrast to reproduce the above result testing µ
1·
= (18µ
1·
+ 24µ
2·
+
25µ
3·
)/67. Because both the calculation of the marginal means and the calculation of the grand mean
are adjusted, we specify the custom contrast on the cell means.
. contrast {race#agegrp ‘=49/67*8/67’ ‘=49/67*17/67’ ‘=49/67*18/67’
> ‘=49/67*13/67’ ‘=49/67*11/67’
> -‘=24/67*8/67’ -‘=24/67*17/67’ -‘=24/67*18/67’
> -‘=24/67*13/67’ -‘=24/67*11/67’
> -‘=25/67*8/67’ -‘=25/67*17/67’ -‘=25/67*18/67’
> -‘=25/67*13/67’ -‘=25/67*11/67’}, nowald
Contrasts of marginal linear predictions
Margins: asbalanced
Contrast Std. err. [95% conf. interval]
race#agegrp
(1) (1) 4.542662 1.740331 1.050432 8.034891
The Helmert and reverse Helmert contrasts also involve calculating averages of the marginal means;
therefore, weighted versions of these parameters are available as well. The hw. operator is a weighted
version of the h. operator that computes the mean of the subsequent levels using the cell frequencies
obtained from the model fit. The jw. operator is a weighted version of the j. operator that computes
the mean of the previous levels using the cell frequencies obtained from the model fit.
For orthogonal polynomials, we can use the pw. and qw. operators, which are the weighted
versions of the p. and q. operators. In this case, the cell frequencies from the model fit are used in
the calculation of the orthogonal polynomial contrast coefficients.
50 contrast — Contrasts and linear hypothesis tests after estimation
Testing factor effects on slopes
For linear models where the independent variables are all factor variables, the linear prediction
at fixed levels of the factor variables turns out to be a cell mean. With these models, contrast
computes and tests the effects of the factor variables on the expected mean of the dependent variable.
When factor variables are interacted with continuous variables, contrast distinguishes factor effects
on the intercept from factor effects on the slope.
Here we have 1980 census data including information on the birthrate (brate), the median age
(medage), and the region of the country (region) for each of the 50 states. We can fit an ANCOVA
model for brate using main effects of the factor variable region and the continuous variable medage.
. use https://www.stata-press.com/data/r18/census3
(1980 Census data by state)
. label list cenreg
cenreg:
1 NE
2 NCentral
3 South
4 West
. anova brate i.region c.medage
Number of obs = 50 R-squared = 0.8264
Root MSE = 12.7575 Adj R-squared = 0.8110
Source Partial SS df MS F Prob>F
Model 34872.859 4 8718.2147 53.57 0.0000
region 2197.7545 3 732.58484 4.50 0.0076
medage 15327.423 1 15327.423 94.18 0.0000
Residual 7323.9611 45 162.75469
Total 42196.82 49 861.15959
For those more comfortable with linear regression, this is equivalent to the regression model
. regress brate i.region c.medage
You may use either.
We can use contrast to compute reference category effects for region. These contrasts compare
the adjusted means of NCentral, South, and West regions with the adjusted mean of the NE region.
. contrast r.region
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
region
(NCentral vs NE) 1 2.24 0.1417
(South vs NE) 1 0.78 0.3805
(West vs NE) 1 10.33 0.0024
Joint 3 4.50 0.0076
Denominator 45
contrast — Contrasts and linear hypothesis tests after estimation 51
Contrast Std. err. [95% conf. interval]
region
(NCentral vs NE) 9.061063 6.057484 -3.139337 21.26146
(South vs NE) 5.06991 5.72396 -6.458738 16.59856
(West vs NE) 21.71328 6.755616 8.106774 35.31979
Let’s add the interaction between region and medage to the model.
. anova brate region##c.medage
Number of obs = 50 R-squared = 0.9000
Root MSE = 10.0244 Adj R-squared = 0.8833
Source Partial SS df MS F Prob>F
Model 37976.315 7 5425.1878 53.99 0.0000
region 3405.0704 3 1135.0235 11.30 0.0000
medage 5279.7145 1 5279.7145 52.54 0.0000
region#medage 3103.456 3 1034.4853 10.29 0.0000
Residual 4220.5051 42 100.48822
Total 42196.82 49 861.15959
The parameterization for the expected value of brate as a function of region and medage is given
by
E(brate|region = i, medage) = α
0
+ α
i
+ β
0
medage + β
i
medage
where α
0
is the intercept and β
0
is the slope of medage. We are modeling the effects of region
in two different ways. The α
i
parameters measure the effect of region on the intercept, and the β
i
parameters measure the effect of region on the slope of medage.
contrast computes and tests effects on slopes separately from effects on intercepts. First, we
will compute the reference category effects of region on the intercept:
. contrast r.region
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
region
(NCentral vs NE) 1 0.09 0.7691
(South vs NE) 1 0.01 0.9389
(West vs NE) 1 8.50 0.0057
Joint 3 11.30 0.0000
Denominator 42
Contrast Std. err. [95% conf. interval]
region
(NCentral vs NE) -49.38396 167.1281 -386.6622 287.8942
(South vs NE) -9.058983 117.424 -246.0302 227.9123
(West vs NE) 343.0024 117.6547 105.5656 580.4393
52 contrast — Contrasts and linear hypothesis tests after estimation
Now, we will compute the reference category effects of region on the slope of medage:
. contrast r.region#c.medage
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
region#c.medage
(NCentral vs NE) 1 0.16 0.6917
(South vs NE) 1 0.03 0.8558
(West vs NE) 1 8.18 0.0066
Joint 3 10.29 0.0000
Denominator 42
Contrast Std. err. [95% conf. interval]
region#c.medage
(NCentral vs NE) 2.208539 5.530981 -8.953432 13.37051
(South vs NE) .6928008 3.788735 -6.953175 8.338777
(West vs NE) -10.94649 3.827357 -18.67041 -3.22257
At the 5% level, the slope of medage for the West region differs from that of the NE region, but at
that level of significance, we cannot say that the slope for the NCentral or the South region differs
from that of the NE region.
This model is simple enough that the reference category contrasts reproduce the coefficients for
region and for the interactions in an equivalent model fit by regress.
. regress brate region##c.medage
Source SS df MS Number of obs = 50
F(7, 42) = 53.99
Model 37976.3149 7 5425.18784 Prob > F = 0.0000
Residual 4220.5051 42 100.488217 R-squared = 0.9000
Adj R-squared = 0.8833
Total 42196.82 49 861.159592 Root MSE = 10.024
brate Coefficient Std. err. t P>|t| [95% conf. interval]
region
NCentral -49.38396 167.1281 -0.30 0.769 -386.6622 287.8942
South -9.058983 117.424 -0.08 0.939 -246.0302 227.9123
West 343.0024 117.6547 2.92 0.006 105.5656 580.4393
medage -8.802707 3.462865 -2.54 0.015 -15.79105 -1.814362
region#
c.medage
NCentral 2.208539 5.530981 0.40 0.692 -8.953432 13.37051
South .6928008 3.788735 0.18 0.856 -6.953175 8.338777
West -10.94649 3.827357 -2.86 0.007 -18.67041 -3.22257
_cons 411.8268 108.2084 3.81 0.000 193.4533 630.2002
This will not be the case for models that are more complicated.
contrast — Contrasts and linear hypothesis tests after estimation 53
Chow tests
Now, let’s suppose we are fitting a model for birthrates on median age and marriage rate. We are
also interested in whether the regression coefficients differ for states in the east versus states in the
west. We use census divisions to create a new variable, west, that indicates which states are in the
western half of the United States.
. generate west = inlist(division, 4, 7, 8, 9)
We fit a model that includes a separate intercept for west as well as an interaction between west
and each of the other variables in our model.
. regress brate i.west##c.medage i.west##c.mrgrate
Source SS df MS Number of obs = 50
F(5, 44) = 92.09
Model 38516.2172 5 7703.24344 Prob > F = 0.0000
Residual 3680.60281 44 83.6500639 R-squared = 0.9128
Adj R-squared = 0.9029
Total 42196.82 49 861.159592 Root MSE = 9.146
brate Coefficient Std. err. t P>|t| [95% conf. interval]
1.west 327.8733 58.71793 5.58 0.000 209.5351 446.2115
medage -7.532304 1.387624 -5.43 0.000 -10.32888 -4.735731
west#
c.medage
1 -10.11443 1.849103 -5.47 0.000 -13.84105 -6.387808
mrgrate 828.6813 643.3443 1.29 0.204 -467.8939 2125.257
west#
c.mrgrate
1 -800.8036 645.488 -1.24 0.221 -2101.699 500.092
_cons 366.5325 47.08904 7.78 0.000 271.6308 461.4343
We can test the effects of west on the intercept and on the slopes of medage and mrgrate. We will
specify all of these effects in a single contrast command and include the overall option to obtain
a joint test of effects, that is, a test that the coefficients for eastern states and for western states are
equal.
. contrast west west#c.medage west#c.mrgrate, overall
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
west 1 31.18 0.0000
west#c.medage 1 29.92 0.0000
west#c.mrgrate 1 1.54 0.2213
Overall 3 22.82 0.0000
Denominator 44
This overall test is referred to as a Chow test in econometrics (Chow 1960).
54 contrast — Contrasts and linear hypothesis tests after estimation
Beyond linear models
contrast may be used after almost any estimation command, with the added benefit that contrast
provides direct support for testing main and interaction effects that is not available in most estimation
commands. To illustrate, we will use contrast with results from a logistic regression. Stata’s logit
command fits logistic regression models, reporting the fitted regression coefficients. The logistic
command fits the same models but reports odds ratios. Although contrast can report odds ratios for
the computed effects, the tests are all computed from linear combinations of the model coefficients
regardless of which estimation command we used.
Suppose we have data on patient satisfaction for three hospitals in a city. Let’s begin by fitting a
model for satisfied, whether the patient was satisfied with his or her treatment, using the main
effects of hospital:
. use https://www.stata-press.com/data/r18/hospital, clear
(Artificial hospital satisfaction data)
. logit satisfied i.hospital
Iteration 0: Log likelihood = -393.72216
Iteration 1: Log likelihood = -387.55736
Iteration 2: Log likelihood = -387.4768
Iteration 3: Log likelihood = -387.47679
Logistic regression Number of obs = 802
LR chi2(2) = 12.49
Prob > chi2 = 0.0019
Log likelihood = -387.47679 Pseudo R2 = 0.0159
satisfied Coefficient Std. err. z P>|z| [95% conf. interval]
hospital
2 .5348129 .2136021 2.50 0.012 .1161604 .9534654
3 .7354519 .2221929 3.31 0.001 .2999618 1.170942
_cons 1.034708 .1391469 7.44 0.000 .7619855 1.307431
Because there are no other independent variables in this model, the reference category effects of
hospital computed by contrast will match the fitted model coefficients, assuming a common
reference level.
. contrast r.hospital
Contrasts of marginal linear predictions
Margins: asbalanced
df chi2 P>chi2
hospital
(2 vs 1) 1 6.27 0.0123
(3 vs 1) 1 10.96 0.0009
Joint 2 12.55 0.0019
Contrast Std. err. [95% conf. interval]
hospital
(2 vs 1) .5348129 .2136021 .1161604 .9534654
(3 vs 1) .7354519 .2221929 .2999618 1.170942
contrast — Contrasts and linear hypothesis tests after estimation 55
We see that the reference category effects are equal to the fitted coefficients. They also have the same
interpretation, the difference in log odds from the reference category. The top table also provides a
joint test of these effects, a test of the main effects of hospital.
We also have information on the condition for which each patient is being treated in the variable
illness. Here we fit a logistic regression using a two-way crossed model of hospital and illness.
. label list illness
illness:
1 Heart attack
2 Stroke
3 Pneumonia
4 Lung disease
5 Kidney failure
. logistic satisfied hospital##illness
Logistic regression Number of obs = 802
LR chi2(14) = 38.51
Prob > chi2 = 0.0004
Log likelihood = -374.46865 Pseudo R2 = 0.0489
satisfied Odds ratio Std. err. z P>|z| [95% conf. interval]
hospital
2 1.226496 .5492177 0.46 0.648 .509921 2.950049
3 1.711111 .8061016 1.14 0.254 .6796395 4.308021
illness
Stroke 1.328704 .6044214 0.62 0.532 .544779 3.240678
Pneumonia .7993827 .3408305 -0.53 0.599 .3466015 1.843653
Lung dise.. 1.231481 .5627958 0.46 0.649 .5028318 3.016012
Kidney fa.. 1.25 .5489438 0.51 0.611 .5285676 2.956102
hospital#
illness
2#Stroke 2.434061 1.768427 1.22 0.221 .5860099 10.11016
2#Pneumonia 4.045805 2.868559 1.97 0.049 1.008058 16.23769
2 #
Lung dise.. .54713 .3469342 -0.95 0.342 .1578866 1.89599
2 #
Kidney fa.. 1.594425 1.081104 0.69 0.491 .4221288 6.022312
3#Stroke
.5416535 .3590089 -0.93 0.355 .1477555 1.985635
3#Pneumonia 1.579502 1.042504 0.69 0.489 .4332209 5.758783
3 #
Lung dise.. 3.137388 2.595748 1.38 0.167 .6198955 15.87881
3 #
Kidney fa.. 1.672727 1.226149 0.70 0.483 .3976256 7.036812
_cons 2.571429 .8099239 3.00 0.003 1.386983 4.767358
Note: estimates baseline odds.
Using contrast, we can obtain an ANOVA-style table of tests for the main effects and interaction
effects of hospital and illness.
56 contrast — Contrasts and linear hypothesis tests after estimation
. contrast hospital##illness
Contrasts of marginal linear predictions
Margins: asbalanced
df chi2 P>chi2
hospital 2 14.92 0.0006
illness 4 4.09 0.3937
hospital#illness 8 20.45 0.0088
Our interaction effect is significant, so we decide to evaluate the simple reference category effects of
hospital within illness. We are particularly interested in patient satisfaction when being treated
for a heart attack or stroke, so we will use the i. operator to limit our output to simple effects within
the first two illnesses.
. contrast r.hospital@i(1 2).illness, nowald
Contrasts of marginal linear predictions
Margins: asbalanced
Contrast Std. err. [95% conf. interval]
hospital@illness
(2 vs 1) Heart attack .2041611 .4477942 -.6734995 1.081822
(2 vs 1) Stroke 1.093722 .5721288 -.0276296 2.215074
(3 vs 1) Heart attack .5371429 .4710983 -.3861928 1.460479
(3 vs 1) Stroke -.0759859 .4662325 -.9897847 .8378129
The row labeled (2 vs 1) heart attack estimates simple effects on the log odds when comparing
hospital 2 with hospital 1 for patients having heart attacks. These effects are differences in the cell
means of the linear predictions.
We can add the or option to report an odds ratio for each of these simple effects:
. contrast r.hospital@i(1 2).illness, nowald or
Contrasts of marginal linear predictions
Margins: asbalanced
Odds ratio Std. err. [95% conf. interval]
hospital@illness
(2 vs 1) Heart attack 1.226496 .5492177 .509921 2.950049
(2 vs 1) Stroke 2.985366 1.708014 .9727486 9.162089
(3 vs 1) Heart attack 1.711111 .8061016 .6796395 4.308021
(3 vs 1) Stroke .9268293 .4321179 .3716567 2.311306
These odds ratios are just the exponentiated version of the contrasts in the previous table.
For contrasts of the margins of nonlinear predictions, such as predicted probabilities, see [R] margins,
contrast.
contrast — Contrasts and linear hypothesis tests after estimation 57
Multiple equations
contrast works with models containing multiple equations. Commands such as intreg and
gnbreg allow their ancillary parameters to be modeled as functions of independent variables, and
contrast can compute and test effects within these equations. In addition, contrast allows a special
pseudofactor for equation—called eqns—when working with results from manova, mvreg, mlogit,
and mprobit.
In example 4 of [MV] manova, we fit a two-way MANOVA model using data from Woodard (1931).
Here we will fit this model using mvreg. The data represent patients with jaw fractures. y1 is the
patient’s age, y2 is blood lymphocytes, and y3 is blood polymorphonuclears. Two factor variables,
gender and fracture, are used as independent variables.
. use https://www.stata-press.com/data/r18/jaw
(Table 4.6. Two-way unbalanced data for fractures of the jaw, Rencher (1998))
. mvreg y1 y2 y3 = gender##fracture, vsquish nofvlabel
Equation Obs Parms RMSE "R-sq" F P>F
y1 27 6 10.21777 0.4086 2.902124 0.0382
y2 27 6 5.268768 0.4743 3.78967 0.0133
y3 27 6 4.993647 0.4518 3.460938 0.0195
Coefficient Std. err. t P>|t| [95% conf. interval]
y1
2.gender -17.5 11.03645 -1.59 0.128 -40.45156 5.451555
fracture
2 -12.625 5.518225 -2.29 0.033 -24.10078 -1.149222
3 5.666667 5.899231 0.96 0.348 -6.601456 17.93479
gender#
fracture
2 2 21.375 12.68678 1.68 0.107 -5.008595 47.75859
2 3 8.833333 13.83492 0.64 0.530 -19.93796 37.60463
_cons 39.5 4.171386 9.47 0.000 30.82513 48.17487
y2
2.gender 20.5 5.69092 3.60 0.002 8.665083 32.33492
fracture
2 -3.125 2.84546 -1.10 0.285 -9.042458 2.792458
3 .6666667 3.041925 0.22 0.829 -5.659362 6.992696
gender#
fracture
2 2 -19.625 6.541907 -3.00 0.007 -33.22964 -6.02036
2 3 -23.66667 7.133946 -3.32 0.003 -38.50252 -8.830813
_cons 35.5 2.150966 16.50 0.000 31.02682 39.97318
y3
2.gender -18.16667 5.393755 -3.37 0.003 -29.38359 -6.949739
fracture
2 1.083333 2.696877 0.40 0.692 -4.52513 6.691797
3 -3 2.883083 -1.04 0.310 -8.9957 2.9957
gender#
fracture
2 2 19.91667 6.200305 3.21 0.004 7.022426 32.81091
2 3 23.5 6.76143 3.48 0.002 9.438837 37.56116
_cons 61.16667 2.038648 30.00 0.000 56.92707 65.40627
contrast computes Wald tests using the coefficients from the first equation by default.
58 contrast — Contrasts and linear hypothesis tests after estimation
. contrast gender##fracture
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
y1
gender 1 2.16 0.1569
fracture 2 2.74 0.0880
gender#fracture 2 1.69 0.2085
Denominator 21
Here we use the equation() option to compute the Wald tests in the y2 equation:
. contrast gender##fracture, equation(y2)
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
y2
gender 1 5.41 0.0301
fracture 2 7.97 0.0027
gender#fracture 2 5.97 0.0088
Denominator 21
Here we use the equation index to compute the Wald tests in the third equation:
. contrast gender##fracture, equation(#3)
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
y3
gender 1 2.23 0.1502
fracture 2 6.36 0.0069
gender#fracture 2 6.66 0.0058
Denominator 21
Here we use the atequations option to compute Wald tests for each equation in the model. We
also use the vsquish option to suppress the extra blank lines between terms.
contrast — Contrasts and linear hypothesis tests after estimation 59
. contrast gender##fracture, atequations vsquish
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
y1
gender 1 2.16 0.1569
fracture 2 2.74 0.0880
gender#fracture 2 1.69 0.2085
y2
gender 1 5.41 0.0301
fracture 2 7.97 0.0027
gender#fracture 2 5.97 0.0088
y3
gender 1 2.23 0.1502
fracture 2 6.36 0.0069
gender#fracture 2 6.66 0.0058
Denominator 21
Because we are investigating the results from mvreg, we can use the special eqns factor to test
for a marginal effect on the means among the dependent variables:
. contrast _eqns
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
_eqns 2 49.19 0.0000
Denominator 21
Here we test whether the main effects of gender differ among the dependent variables:
. contrast gender#_eqns
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
gender#_eqns 2 3.61 0.0448
Denominator 21
60 contrast — Contrasts and linear hypothesis tests after estimation
Although it is not terribly interesting in this case, we can even calculate contrasts across equations:
. contrast gender#r._eqns
Contrasts of marginal linear predictions
Margins: asbalanced
df F P>F
gender#_eqns
(joint) (2 vs 1) 1 5.82 0.0251
(joint) (3 vs 1) 1 0.40 0.5352
Joint 2 3.61 0.0448
Denominator 21
Video example
Introduction to contrasts in Stata: One-way ANOVA
Stored results
contrast stores the following in r():
Scalars
r(df r) variance degrees of freedom
r(k terms) number of terms in termlist
r(level) confidence level of confidence intervals
Macros
r(cmd) contrast
r(cmdline) command as typed
r(est cmd) e(cmd) from original estimation results
r(est cmdline) e(cmdline) from original estimation results
r(title) title in output
r(overall) overall or empty
r(emptycells) empspec from emptycells()
r(mcmethod) method from mcompare()
r(mctitle) title for method from mcompare()
r(mcadjustall) adjustall or empty
r(margin method) asbalanced or asobserved
Matrices
r(b) contrast estimates
r(V) variance–covariance matrix of the contrast estimates
r(error) contrast estimability codes;
0 means estimable,
8 means not estimable
r(L) matrix of contrasts applied to the model coefficients
r(table) matrix containing the contrasts with their standard errors,
test statistics, p-values, and confidence intervals
r(F) vector of F statistics; r(df r) present
r(chi2) vector of χ
2
statistics; r(df r) not present
r(p) vector of p-values corresponding to r(F) or r(chi2)
r(df) vector of degrees of freedom corresponding to r(p)
r(df2) vector of denominator degrees of freedom corresponding to r(F)
contrast — Contrasts and linear hypothesis tests after estimation 61
contrast with the post option stores the following in e():
Scalars
e(df r) variance degrees of freedom
e(k terms) number of terms in termlist
Macros
e(cmd) contrast
e(cmdline) command as typed
e(properties) b V
e(est cmd) e(cmd) from original estimation results
e(est cmdline) e(cmdline) from original estimation results
e(title) title in output
e(overall) overall or empty
e(emptycells) empspec from emptycells()
e(margin method) asbalanced or asobserved
e(asbalanced) factor variables fvset as asbalanced
e(asobserved) factor variables fvset as asobserved
Matrices
e(b) contrast estimates
e(V) variance–covariance matrix of the contrast estimates
e(error) contrast estimability codes;
0 means estimable,
8 means not estimable
e(L) matrix of contrasts applied to the model coefficients
e(F) vector of unadjusted F statistics; e(df
r) present
e(chi2) vector of χ
2
statistics; e(df r) not present
e(p) vector of unadjusted p-values corresponding to e(F) or e(chi2)
e(df) vector of degrees of freedom corresponding to e(p)
e(df2) vector of denominator degrees of freedom corresponding to e(F)
Methods and formulas
Methods and formulas are presented under the following headings:
Marginal linear predictions
Contrast operators
Reference level contrasts
Adjacent contrasts
Grand mean contrasts
Helmert contrasts
Reverse Helmert contrasts
Orthogonal polynomial contrasts
Contrasts within interactions
Multiple comparisons
Marginal linear predictions
contrast treats intercept effects separately from slope effects. To illustrate, consider the following
parameterization for a quadratic regression of y on x that also models the effects of two factor variables
A and B, where the levels of A are indexed by i = 1, . . . , k
a
and the levels of B are indexed by
j = 1, . . . , k
b
.
62 contrast — Contrasts and linear hypothesis tests after estimation
E(y|A = i, B = j, x) = η
0ij
+ η
1ij
x + η
2ij
x
2
η
0ij
= η
0
+ α
0i
+ β
0j
+ (αβ)
0ij
η
1ij
= η
1
+ α
1i
+ β
1j
+ (αβ)
1ij
η
2ij
= η
2
+ α
2i
+ β
2j
+ (αβ)
2ij
We have partitioned the coefficients into three groups of parameters: η
0ij
is a cell prediction for the
intercept, η
1ij
is a cell prediction for the slope on x, and η
2ij
is a cell prediction for the slope on
x
2
. For the intercept parameters, η
0
is the intercept, α
0i
represents a main effect for factor A at its
ith level, β
0j
represents a main effect for factor B at its jth level, and (αβ)
0ij
represents an effect
for the interaction of A and B at the ijth level. The individual coefficients in η
1ij
and η
2ij
have
similar interpretations, but the effects are on the slopes of x and x
2
, respectively.
The marginal intercepts for A are given by
η
0i.
=
k
b
X
j=1
f
ij
η
0ij
where f
ij
is a marginal relative frequency of the jth level of B and is controlled by the asobserved
and emptycells(reweight) options according to
f
ij
=
1/k
b
, default
w
.j
/w
..
, asobserved
1/(k
b
− e
i.
), emptycells(reweight)
w
ij
/w
i.
, emptycells(reweight) and asobserved
Above, w
ij
is the number of individuals with A at its ith level and B at its jth,
w
i.
=
k
b
X
j=1
w
ij
w
.j
=
k
a
X
i=1
w
ij
w
..
=
k
a
X
i=1
k
b
X
j=1
w
ij
and e
i.
is the number of empty cells where A is at its ith level. The marginal intercepts for B and
marginal slopes on x and x
2
are similarly defined.
Estimates for the cell intercepts and slopes are computed using the corresponding linear combination
of the coefficients from the fitted model. For example, the estimated cell intercepts are computed
using
bη
0ij
= bη
0
+ bα
0i
+
b
β
0j
+ (
c
αβ)
0ij
contrast — Contrasts and linear hypothesis tests after estimation 63
and the estimated marginal intercepts for A are computed as
bη
0i.
=
k
b
X
j=1
f
ij
bη
0ij
Contrast operators
contrast performs Wald tests using linear combinations of marginal linear predictions. For
example, the following linear combination can be used to test for a specific effect of factor A on the
marginal intercepts.
k
a
X
i=1
c
i
η
0i.
If the c
i
elements sum to zero, the linear combination is called a contrast. If the factor A is represented
by a variable named A, then we specify this contrast using the following syntax:
{A c
1
c
2
... c
k
a
}
Similarly, the following linear combination can be used to test for a specific interaction effect of
factors A and B on the marginal slope of x.
k
a
X
i=1
k
b
X
j=1
c
ij
η
1ij
If the factor B is represented by a variable named B, then we specify this contrast using the following
syntax:
{A#B c
11
c
12
... c
1k
b
c
21
... c
k
a
k
b
}
contrast has variable operators for several commonly used contrasts. Each contrast operator
specifies a matrix of linear combinations that yield the requested set of contrasts to be applied to the
marginal linear predictions associated with the attached factor variable.
Reference level contrasts
The r. operator compares each level with a reference level. Let R be the corresponding contrast
matrix for factor A, and then R is a (k
a
− 1) × k
a
matrix with elements
R
ij
=
−1, if j is the reference level
1, if i = j and j is less than the reference level
1, if i + 1 = j and j is greater than the reference level
0, otherwise
64 contrast — Contrasts and linear hypothesis tests after estimation
If k
a
= 5 and the reference level is the third level of A (specified as rb(#3).A), then
R =
1 0 −1 0 0
0 1 −1 0 0
0 0 −1 1 0
0 0 −1 0 1
Adjacent contrasts
The a. operator compares each level with the next level. Let A be the corresponding contrast
matrix for factor A, and then A is a (k
a
− 1) × k
a
matrix with elements
A
ij
=
(
1, if i = j
−1, if i + 1 = j
0, otherwise
If k
a
= 5, then
A =
1 −1 0 0 0
0 1 −1 0 0
0 0 1 −1 0
0 0 0 1 −1
The ar. operator compares each level with the previous level. If A is the contrast matrix for the
a. operator, then −A is the corresponding contrast matrix for the ar. operator.
Grand mean contrasts
The g. operator compares each level with the mean of all the levels. Let G be the corresponding
contrast matrix for factor A, and then G is a k
a
× k
a
matrix with elements
G
ij
=
1 − 1/k
a
, if i = j
− 1/k
a
, if i 6= j
If k
a
= 5, then
G =
4/5 −1/5 −1/5 −1/5 −1/5
−1/5 4/5 −1/5 −1/5 −1/5
−1/5 −1/5 4/5 −1/5 −1/5
−1/5 −1/5 −1/5 4/5 −1/5
−1/5 −1/5 −1/5 −1/5 4/5
The gw. operator compares each level with the weighted mean of all the levels. The weights are
taken from the observed weighted cell frequencies in the estimation sample of the fitted model. Let
G
w
be the corresponding contrast matrix for factor A, and then G
w
is a k
a
×k
a
matrix with elements
contrast — Contrasts and linear hypothesis tests after estimation 65
G
ij
=
1 − w
i
/w
·
, if i = j
− w
j
/w
·
, if i 6= j
where w
i
is a marginal weight representing the number of individuals with A at its ith level and
w
·
=
P
i
w
i
.
Helmert contrasts
The h. operator compares each level with the mean of the subsequent levels. Let H be the
corresponding contrast matrix for factor A, and then H is a (k
a
− 1) × k
a
matrix with elements
H
ij
=
(
1, if i = j
−1/(k
a
− i), if i < j
0, otherwise
If k
a
= 5, then
H =
1 −1/4 −1/4 −1/4 −1/4
0 1 −1/3 −1/3 −1/3
0 0 1 −1/2 −1/2
0 0 0 1 −1
The hw. operator compares each level with the weighted mean of the subsequent levels. Let H
w
be the corresponding contrast matrix for factor A, and then H
w
is a (k
a
− 1) × k
a
matrix with
elements
H
wij
=
(
1, if i = j
−w
j
/
P
k
a
l=j
w
l
, if i < j
0, otherwise
Reverse Helmert contrasts
The j. operator compares each level with the mean of the previous levels. Let J be the corresponding
contrast matrix for factor A, and then J is a (k
a
− 1) × k
a
matrix with elements
J
ij
=
(
1, if i + 1 = j
−1/i, if j ≤ i
0, otherwise
If k
a
= 5, then
H =
−1 1 0 0 0
−1/2 −1/2 1 0 0
−1/3 −1/3 −1/3 1 0
−1/4 −1/4 −1/4 −1/4 1
66 contrast — Contrasts and linear hypothesis tests after estimation
The jw. operator compares each level with the weighted mean of the previous levels. Let J
w
be
the corresponding contrast matrix for factor A, and then J
w
is a (k
a
− 1) × k
a
matrix with elements
J
wij
=
(
1, if i + 1 = j
−w
j
/
P
i
l=1
w
l
, if i ≤ j
0, otherwise
Orthogonal polynomial contrasts
The p. operator applies orthogonal polynomial contrasts using the level values of the attached
factor variable. The q. operator applies orthogonal polynomial contrasts using the level indices of
the attached factor variable. These two operators are equivalent when the level values of the attached
factor are equally spaced. The pw. and qw. operators are weighted versions of p. and q., where
the weights are taken from the observed weighted cell frequencies in the estimation sample of the
fitted model. contrast uses the Christoffel–Darboux recurrence formula for computing orthogonal
polynomial contrasts (Abramowitz and Stegun 1964). The elements of the contrasts are normalized
such that
Q
0
WQ =
1
w
·
I
where W is a diagonal matrix of the marginal cell weights w
1
, w
2
, . . . , w
k
of the attached factor
variable (all 1 for p. and q.), and w
·
is the sum of the weights (the number of levels k for p. and
q.).
Contrasts within interactions
Contrast operators are allowed to be specified on factor variables participating in interactions. In
such cases, contrast applies the proper matrix product of the contrast matrices to the cell margins
of the interacted factor variables.
For example, consider the contrasts implied by specifying r.A#h.B. Let M be the matrix of
estimated cell margins for the levels of A and B, where the rows of M are indexed by the levels of
A and the columns are indexed by the levels of B. contrast puts the estimated cell margins in the
following vector form:
v = vec(M
0
) =
M
11
M
12
.
.
.
M
1k
b
M
21
M
22
.
.
.
M
2k
b
.
.
.
M
k
a
k
b
contrast — Contrasts and linear hypothesis tests after estimation 67
The individual contrasts are then given by the elements of
(R ⊗ H)v
where ⊗ denotes the Kronecker direct product.
Multiple comparisons
See [R] pwcompare for details on the methods and formulas used to adjust p-values and confidence
intervals for multiple comparisons. The formulas for Bonferroni’s method and
ˇ
Sid
´
ak’s method are
presented with m = k (k − 1)/2, the number of pairwise comparisons for a factor term with k
levels. For contrasts, m is instead the number of contrasts being performed on the factor term; often,
m = k − 1 for a term with k levels.
References
Abramowitz, M., and I. A. Stegun, ed. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables. Washington, DC: National Bureau of Standards.
Baldwin, S. 2019. Psychological Statistics and Psychometrics Using Stata. College Station, TX: Stata Press.
Chow, G. C. 1960. Tests of equality between sets of coefficients in two linear regressions. Econometrica 28: 591–605.
https://doi.org/10.2307/1910133.
Coster, D. 2005. Contrasts. In Vol. 2 of Encyclopedia of Biostatistics, ed. P. Armitage and T. Colton, 1153–1157.
Chichester, UK: Wiley.
Freese, J., and S. Johfre. 2022. Binary contrasts for unordered polytomous regressors. Stata Journal 22: 125–133.
Milliken, G. A., and D. E. Johnson. 2009. Analysis of Messy Data, Volume 1: Designed Experiments. 2nd ed. Boca
Raton, FL: CRC Press.
Mitchell, M. N. 2015. Stata for the Behavioral Sciences. College Station, TX: Stata Press.
. 2021. Interpreting and Visualizing Regression Models Using Stata. 2nd ed. College Station, TX: Stata Press.
Rosenthal, R., R. L. Rosnow, and D. B. Rubin. 2000. Contrasts and Effect Sizes in Behavioral Research: A Correlational
Approach. Cambridge: Cambridge University Press.
Searle, S. R. 1997. Linear Models for Unbalanced Data. New York: Wiley.
Searle, S. R., and M. H. J. Gruber. 2017. Linear Models. 2nd ed. Hoboken, NJ: Wiley.
Winer, B. J., D. R. Brown, and K. M. Michels. 1991. Statistical Principles in Experimental Design. 3rd ed. New
York: McGraw–Hill.
Woodard, D. E. 1931. Healing time of fractures of the jaw in relation to delay before reduction, infection,
syphilis and blood calcium and phosphorus content. Journal of the American Dental Association 18: 419–442.
https://doi.org/10.14219/jada.archive.1931.0096.
Also see
[R] contrast postestimation — Postestimation tools for contrast
[R] lincom — Linear combinations of parameters
[R] margins — Marginal means, predictive margins, and marginal effects
[R] margins, contrast — Contrasts of margins
[R] pwcompare — Pairwise comparisons
[R] test — Test linear hypotheses after estimation
68 contrast — Contrasts and linear hypothesis tests after estimation
[U] 20 Estimation and postestimation commands
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