Note: This example was done using Mplus version 5.2. The syntax
may not work, or may function differently, with other versions of Mplus.
As the name implies, multivariate regression is a technique that estimates a
single regression model with more than one outcome variable. When there is more
than one predictor variable in a multivariate regression model, the model is a
multivariate multiple regression.
Please note: The purpose of this page is to show how to use various data analysis commands.
It does not cover all aspects of the research process which researchers are expected to do. In
particular, it does not cover data cleaning and checking, verification of assumptions, model
diagnostics and potential follow-up analyses.
Examples of multivariate regression analysis
Example 1. A researcher has collected data on three psychological variables,
four academic variables (standardized test scores), and the type of educational
program the student is in for 600 high school students. She is interested in how
the set of psychological variables is related to the academic variables and the
type of program the student is in.
Example 2. A doctor has collected data on cholesterol, blood pressure, and
weight. She also collected data on the eating habits of the subjects
(e.g., how many ounces of red meat, fish, dairy products, and chocolate consumed
per week). She wants to investigate the relationship between the three
measures of health and eating habits.
Example 3. A researcher is interested in determining what factors influence
the health African Violet plants. She collects data on the average leaf
diameter, the mass of the root ball, and the average diameter of the blooms, as
well as how long the plant has been in its current container. For predictor variables,
she measures several elements in the soil, as well as the amount of light
and water each plant receives.
Description of the data
Let’s pursue Example 1 from above.
We have a hypothetical dataset with 600 observations on seven variables which
can be obtained by clicking on mvreg.dat.
The psychological variables are locus of control (locus), self-concept (self), and
motivation (motiv). The academic variables are standardized tests scores in
reading (read), writing (write), and science (science), as well as a categorical
variable (prog) giving the type of program the student is in; general (prog=1), academic
(prog=2), or
vocational (prog=3). In addition to the three-category variable prog, the dataset
contains a dummy variable for each level of prog (prog1, prog2,
and prog3), for example, prog1 is equal to 1 when prog=1
(general),
and 0 otherwise. You can store the data file anywhere you like, but our examples will
assume it has been stored in c:data. (Note that the names of
variables should NOT be included at the top of the data file. Instead, the
variables are named as part of the variable command.) You may want to do your
descriptive statistics in a general use statistics package, such as SAS, Stata
or SPSS, because the options for obtaining descriptive statistics are limited in
Mplus. Even if you chose to run descriptive statistics in another package, it is
a good idea to run a model with type=basic before you do anything else,
just to make sure the dataset is being read correctly. The input file below
shows such a model.
Data: File is c:datamvreg.dat ; Variable: Names are locus self motiv read write science prog prog1 prog2 prog3; Missing are all (-9999) ; analysis: type = basic;
As we mentioned above, you will want to look at the output from this command carefully to be sure that
the dataset was read into Mplus correctly. You will want to make sure that
you have the correct number of observations, and that the variables all have
means that are close to those from the descriptive statistics generated in a
general purpose statistical package. If there are missing values for some or all
of the variables, the descriptive statistics generated by Mplus will not match
those from a general purpose statistical package exactly, because by default, Mplus versions
5.0 and later use maximum likelihood based procedures for handling missing
values.
<output omitted>
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 600
<output omitted>
SAMPLE STATISTICS
Means
LOCUS SELF MOTIV READ WRITE
________ ________ ________ ________ ________
1 0.097 0.005 0.004 51.902 52.385
Means
SCIENCE PROG PROG1 PROG2 PROG3
________ ________ ________ ________ ________
1 51.763 2.088 0.230 0.452 0.318
Analysis methods you might consider
Below is a list of some analysis methods you may have encountered.
Some of the methods listed are quite reasonable while others have either
fallen out of favor or have limitations.
- Multivariate multiple regression, the focus of this page.
- Separate OLS Regressions – You could analyze these data using separate
OLS regression analyses for each outcome variable. The individual
coefficients, as well as their standard errors will be the same as those
produced by the multivariate regression. However, the OLS regressions will
not produce multivariate results, nor will they allow for testing of
coefficients across equations. - Canonical correlation analysis might be feasible if you don’t want to
consider one set of variables as outcome variables and the other set as
predictor variables.
Multivariate regression analysis
The input file for our multivariate regression in Mplus is shown below. In the variable command, the
usevariables option is used because only some of the variables in our dataset are used in the model.
In the model command, each of the outcome variables (i.e., locus, self, and
motiv) is predicted by the four predictor variables using the keyword
on. In the output command we have requested fully standardized output
(in addition to the unstandardized coefficients) using the stdyx option, this will produce standardized estimates of the
coefficients, which you may find useful, but it also requests that Mplus produce
the R-square statistic for each of the outcome variables.
data: file is C:datamvreg.dat ; variable: names are locus self motiv read write science prog prog1 prog2 prog3; missing are all (-9999) ; usevariables are locus self motiv read write science prog2 prog3; model: locus on read write science prog2 prog3; self on read write science prog2 prog3; motiv on read write science prog2 prog3; output: stdyx;
SUMMARY OF ANALYSIS Number of groups 1 Number of observations 600 Number of dependent variables 3 Number of independent variables 5 Number of continuous latent variables 0 Observed dependent variables Continuous LOCUS SELF MOTIV Observed independent variables READ WRITE SCIENCE PROG2 PROG3 Estimator ML Information matrix OBSERVED Maximum number of iterations 1000 Convergence criterion 0.500D-04 Maximum number of steepest descent iterations 20 Maximum number of iterations for H1 2000 Convergence criterion for H1 0.100D-03
- At the top of the output we see that 600 observations were used.
- From the output we see that the model includes 3 continuous dependent (outcome)
variable and 5 independent (predictor) variables. - The analysis summary is followed by a block of technical information about the model,
we won’t discuss most of this information, but we will note that the default estimator is maximum
likelihood (i.e., ML).
<output omitted>
THE MODEL ESTIMATION TERMINATED NORMALLY
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 0.000
Degrees of Freedom 0
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value 311.076
Degrees of Freedom 18
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.000
Loglikelihood
H0 Value -8807.819
H1 Value -8807.819
Information Criteria
Number of Free Parameters 24
Akaike (AIC) 17663.637
Bayesian (BIC) 17769.164
Sample-Size Adjusted BIC 17692.970
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.000
90 Percent C.I. 0.000 0.000
Probability RMSEA <= .05 0.000
SRMR (Standardized Root Mean Square Residual)
Value 0.000
- Several measures of model fit are included in the output. The log
likelihood (-8807.819) can be used in comparisons of nested models, but we
won't show an example of that here. - The Akaike information criterion (AIC) and the Bayesian information
criterion (BIC, sometimes also called the Schwarz criterion), can also
be used to compare models. - Because the model is saturated, the
chi-square test of model fit (for the current model, not the baseline
model), as well as the CFI, TLI, RMSEA, and SRMR, all show perfect fit. This does not necessarily
mean that the model fits well in the sense that the model does a good job of
predicting the outcomes, just that these measures are not the appropriate
measures of fit for a saturated model.
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
LOCUS ON
READ 0.013 0.004 3.380 0.001
WRITE 0.012 0.003 3.599 0.000
SCIENCE 0.006 0.004 1.590 0.112
PROG2 0.128 0.064 2.008 0.045
PROG3 0.252 0.068 3.694 0.000
SELF ON
READ 0.001 0.004 0.311 0.755
WRITE -0.004 0.004 -1.121 0.262
SCIENCE 0.005 0.004 1.290 0.197
PROG2 0.276 0.072 3.827 0.000
PROG3 0.423 0.077 5.474 0.000
MOTIV ON
READ 0.010 0.005 2.085 0.037
WRITE 0.018 0.004 4.143 0.000
SCIENCE -0.009 0.005 -1.981 0.048
PROG2 0.360 0.080 4.514 0.000
PROG3 0.620 0.085 7.252 0.000
SELF WITH
LOCUS 0.057 0.017 3.335 0.001
MOTIV WITH
LOCUS 0.060 0.019 3.179 0.001
SELF 0.130 0.022 5.935 0.000
Intercepts
LOCUS -1.625 0.156 -10.401 0.000
SELF -0.372 0.177 -2.100 0.036
MOTIV -1.311 0.196 -6.689 0.000
Residual Variances
LOCUS 0.365 0.021 17.321 0.000
SELF 0.470 0.027 17.320 0.000
MOTIV 0.574 0.033 17.321 0.000
- Under the heading MODEL RESULTS we first see the regression coefficients for
the three outcome variables (locus, self, and motiv)
regressed on the predictor variables, denoted ON.
The output includes the estimate (i.e., the coefficient), its standard
error, the est./s.e. (i.e., the z-value), and the associated p-value. The
regression coefficients are followed by estimates of the covariances among
the residuals of the outcome variables, denoted WITH. Finally, we see the
intercepts for the three regression equations and the residual variances of the
outcome variables. - The coefficients are interpreted in the same way coefficients from an
OLS regression are interpreted. For example, looking at the top of the
MODEL RESULTS, a one unit change in read is associated with a 0.013 unit
change in the predicted value of locus_of_control.
Because we used the stdyx option of the output command the
output includes standardized coefficients. We did this primarily to obtain the
R-square values for the outcome variables, so we have omitted the standardized
output to save space.
<output omitted>
R-SQUARE
Observed Two-Tailed
Variable Estimate S.E. Est./S.E. P-Value
LOCUS 0.187 0.029 6.508 0.000
SELF 0.054 0.018 3.010 0.003
MOTIV 0.150 0.027 5.580 0.000
- The output above gives the R-square value for each of the outcome variables, as well
as the standard error, z-value, and associated p-value for each. Based on
the output, the model explains 18% of variance in locus of control (locus).
If you ran a separate OLS regression
for each outcome variable, you would get exactly the same coefficients and
standard errors as shown above. So why conduct a
multivariate regression? One advantage of estimating the series of
equations as a single model is that you can conduct tests of the coefficients
across the different outcome variables. For example, the input file below uses
the model test command to test
the null hypothesis that the coefficients for the variable read are equal to
0 in all three equations. Notice that in the model command each of the
terms we wish to test (i.e., each instance of read) is followed by a label
in parentheses (e.g., "(r1)"). These parameter labels are then used to refer to
the associated coefficients in the model test command. There are a few important
things to note about parameter labels. First, the labels must always appear at
the end of a line (but not necessarily the end of the command). Second, the
labels apply to all
parameters listed on the line (meaning all of the parameters on the line are
constrained to equality). This is why read is the only predictor variable
on the line with the label on it. In the model test command, we give the null
hypotheses we wish to test together, in this case, that each of the parameters
for read (identified as r1, r2, and r3) are simultaneously
equal to zero.
data:
file is C:datamvreg.dat ;
variable:
names are locus self motiv read write science prog prog1 prog2 prog3;
missing are all (-9999) ;
usevariables are locus self motiv read write science prog2 prog3;
model:
locus on read (r1)
write science prog2 prog3;
self on read (r2)
write science prog2 prog3;
motiv on read (r3)
write science prog2 prog3;
model test:
r1 = 0;
r2 = 0;
r3 = 0;
output:
stdyx;
The output generated by this syntax will be identical to the output shown above, except that
it will include the additional output generated by the model test
command, the additional
output is shown below (all other output is omitted).
Wald Test of Parameter Constraints
Value 14.486
Degrees of Freedom 3
P-Value 0.0023
The Wald test statistic of 14.486 with 3 degrees of freedom has an associated
p-value of 0.0023. These results reject the null hypothesis that the coefficients for
read
across the three equations are simultaneously equal to 0, in other words, the coefficients
for read, taken for all three outcomes together, are statistically significant.
We can also test the null hypothesis that the coefficients for prog=2 (prog2) and prog=3
(prog3) are simultaneously equal to 0 in the equation for locus_of_control.
When used to test the coefficients for dummy variables that form a single
categorical predictor, this type of test is sometimes called an overall
test for the effect of the categorical predictor (i.e., prog).
data:
file is C:datamvreg.dat ;
variable:
names are locus self motiv read write science prog prog1 prog2 prog3;
missing are all (-9999) ;
usevariables are locus self motiv read write science prog2 prog3;
model:
locus on read write science
prog2 (p1)
prog3 (p2);
self on read write science prog2 prog3;
motiv on read write science prog2 prog3;
model test:
p1 = 0;
p2 = 0;
output:
stdyx;
Wald Test of Parameter Constraints
Value 13.788
Degrees of Freedom 2
P-Value 0.0010
The results of the above test indicate that the two coefficients
together are significantly different from 0, in other words, the overall
effect of prog on locus_of_control is statistically significant.
The next example tests the null hypothesis that the coefficient for the variable
write in the equation with locus_of_control as the outcome is equal to the coefficient
for write in the equation with self_concept as the outcome. Another way
of stating this null hypothesis is that the effect of write on locus_of_control is equal to the effect of
write on self_concept.
Data:
File is mvreg.dat ;
Variable:
Names are locus self motiv read write science prog prog1 prog2 prog3;
Missing are all (-9999) ;
usevariables are locus self motiv read write science prog2 prog3;
model:
locus on read
write (wl)
science prog2 prog3;
self on read
write (ws)
science prog2 prog3;
motiv on read write science prog2 prog3;
model test:
wl = ws;
Wald Test of Parameter Constraints
Value 12.006
Degrees of Freedom 1
P-Value 0.0005
The results of this test indicate that the
coefficients for write with locus_of_control and
self_concept as the outcome are significantly different.
Below we test the null hypothesis that the
coefficient of science in the equation for
locus_of_control is equal to the coefficient for science in the
equation for self_concept, and that the coefficient for the variable
write in the equation with the outcome variable
locus_of_control equals the coefficient for write in the
equation with the outcome variable self_concept.
data:
file is mvreg.dat ;
variable:
names are locus self motiv read write science prog prog1 prog2 prog3;
missing are all (-9999) ;
usevariables are locus self motiv read write science prog2 prog3;
model:
locus on read
write (w1)
science (s1)
prog2 prog3;
self on read
write (w2)
science (s2)
prog2 prog3;
motiv on read write science prog2 prog3;
model test:
w1 = w2;
s1 = s2;
output:
stdyx;
Wald Test of Parameter Constraints
Value 12.902
Degrees of Freedom 2
P-Value 0.0016
The results of the above test indicate that taken together the
two sets of coefficients are significantly different. Note that
the degrees of freedom is now 2, reflecting the fact that we are comparing two sets
of coefficients, rather than 1.
Unlike some other packages, Mplus does not automatically provide a test for the
overall model. However, we can produce an equivalent test by constraining the
regression coefficients to 0 in our model and comparing the fit of that model to
the current saturated model. To constrain all of the regression coefficients to
0, we first constrain all of the coefficients by giving them the label n
(recall from above that the label applies to all coefficients on the line. Below
that, we use the model constraint command to fix n to 0.
Data: File is mvreg.dat ; Variable: Names are locus self motiv read write science prog prog1 prog2 prog3; Missing are all (-9999) ; usevariables are locus self motiv read write science prog1 prog2 ; model: locus on read write science prog1 prog2 (n) ; self on read write science prog1 prog2 (n); motiv on read write science prog1 prog2 (n); model constraint: n = 0;
Because this isn't the model we want to interpret, we have omitted most of the output.
Shown below are the chi-square test of model fit (which provides the overall test) and the
MODEL RESULTS section so that we can check to see we have estimated the desired model.
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 214.658
Degrees of Freedom 15
P-Value 0.0000
The chi-square test of model fit compares the fit of the current model to a saturated model. In the
models we estimated above (i.e., the unconstrained or saturated models), this value was 0, because the model was saturated (i.e., has 0 degrees of freedom). By adding
constraints to the model we have freed up 15 parameters, so now we get a
positive
value. The chi-square value of 214.658 with 15 degrees of freedom with an associated
p-value of less than 0.0001, indicates that the constrained model fits
significantly worse than the saturated model. In other words, the saturated
model shown above fits significantly better than the model with the regression
coefficients constrained to 0.
The MODEL RESULTS are shown below. It can be a good idea to check this section
to make sure the model estimated was the desired model. Note that all of the regression
coefficients (denoted ON) are constrained to 0, while the residual covariances
(denoted WITH) and variances, as well as the intercepts have been estimated.
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
LOCUS ON
READ 0.000 0.000 999.000 999.000
WRITE 0.000 0.000 999.000 999.000
SCIENCE 0.000 0.000 999.000 999.000
PROG1 0.000 0.000 999.000 999.000
PROG2 0.000 0.000 999.000 999.000
SELF ON
READ 0.000 0.000 999.000 999.000
WRITE 0.000 0.000 999.000 999.000
SCIENCE 0.000 0.000 999.000 999.000
PROG1 0.000 0.000 999.000 999.000
PROG2 0.000 0.000 999.000 999.000
MOTIV ON
READ 0.000 0.000 999.000 999.000
WRITE 0.000 0.000 999.000 999.000
SCIENCE 0.000 0.000 999.000 999.000
PROG1 0.000 0.000 999.000 999.000
PROG2 0.000 0.000 999.000 999.000
SELF WITH
LOCUS 0.081 0.020 4.133 0.000
MOTIV WITH
LOCUS 0.135 0.023 5.832 0.000
SELF 0.167 0.025 6.791 0.000
Intercepts
LOCUS 0.097 0.027 3.531 0.000
SELF 0.005 0.029 0.171 0.864
MOTIV 0.004 0.034 0.116 0.907
Residual Variances
LOCUS 0.449 0.026 17.321 0.000
SELF 0.497 0.029 17.321 0.000
MOTIV 0.675 0.039 17.321 0.000
Things to consider
- The residuals from multivariate regression models are assumed to be multivariate normal.
This is analogous to the assumption of normally distributed errors in univariate linear
regression (i.e., OLS regression). - Multivariate regression analysis is not recommended for small samples.
- The outcome variables should be at least moderately correlated for the
multivariate regression analysis to make sense.
References
Afifi, A., Clark, V. and May, S. 2004. Computer-Aided Multivariate Analysis. 4th ed. Boca Raton, Fl: Chapman & Hall/CRC.