### Stata Data Analysis Examples Multivariate Regression Analysis

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

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. he psychological variables are locus of control (locus_of_control), self-concept (self_concept), and motivation (motivation). 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, academic, or vocational).

Let's look at the data (note that there are no missing values in this data set).

use http://www.ats.ucla.edu/stat/stata/dae/mvreg, clear

summarize locus_of_control self_concept motivation read write science

Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
locus_of_c~l |       600    .0965333    .6702799  -1.995957   2.205511
self_concept |       600    .0049167    .7055125   -2.53275   2.093563
motivation |       600    .0038979       .8224  -2.746669   2.583752
read |       600    51.90183    10.10298   24.62001   80.58649
write |       600    52.38483    9.726455   20.06888   83.93482
-------------+--------------------------------------------------------
science |       600    51.76333    9.706179   21.98953   80.36942

tabulate prog

program |
type |      Freq.     Percent        Cum.
------------+-----------------------------------
general |        138       23.00       23.00
vocational |        191       31.83      100.00
------------+-----------------------------------
Total |        600      100.00

correlate locus_of_control self_concept motivation
(obs=600)

| locus_~l self_c~t motiva~n
-------------+---------------------------
locus_of_c~l |   1.0000
self_concept |   0.1712   1.0000
motivation |   0.2451   0.2886   1.0000

(obs=600)

-------------+---------------------------
write |   0.6286   1.0000
science |   0.6907   0.5691   1.0000

#### 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.

• 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

To conduct a multivariate regression in Stata, we need to use two commands, manova and mvreg. The manova command will indicate if all of the equations, taken together, are statistically significant. The F-ratios and p-values for four multivariate criterion are given, including Wilks' lambda, Lawley-Hotelling trace, Pillai's trace, and Roy's largest root. Next, we use the mvreg command to obtain the coefficients, standard errors, etc., for each of the predictors in each part of the model.  We will also show the use of the test command after the mvreg command. The use of the test command is one of the compelling reasons for conducting a multivariate regression analysis.

Below we run the manova command. Note the use of c. in front of the names of the continuous predictor variables -- this is part of the factor variable syntax introduced in Stata 11. It is necessary to use the c. to identify the continuous variables, because, by default, the manova command assumes all predictor variables are categorical.

manova locus_of_control self_concept motivation = c.read c.write c.science prog

Number of obs =     600

W = Wilks' lambda      L = Lawley-Hotelling trace
P = Pillai's trace     R = Roy's largest root

Source |  Statistic     df   F(df1,    df2) =   F   Prob>F
-----------+--------------------------------------------------
Model | W   0.6992      5    15.0  1634.7    15.08 0.0000 a
| P   0.3196           15.0  1782.0    14.17 0.0000 a
| L   0.4035           15.0  1772.0    15.89 0.0000 a
| R   0.3271            5.0   594.0    38.86 0.0000 u
|--------------------------------------------------
Residual |               594
-----------+--------------------------------------------------
read | W   0.9764      1     3.0   592.0     4.76 0.0027 e
| P   0.0236            3.0   592.0     4.76 0.0027 e
| L   0.0241            3.0   592.0     4.76 0.0027 e
| R   0.0241            3.0   592.0     4.76 0.0027 e
|--------------------------------------------------
write | W   0.9474      1     3.0   592.0    10.96 0.0000 e
| P   0.0526            3.0   592.0    10.96 0.0000 e
| L   0.0555            3.0   592.0    10.96 0.0000 e
| R   0.0555            3.0   592.0    10.96 0.0000 e
|--------------------------------------------------
science | W   0.9834      1     3.0   592.0     3.33 0.0193 e
| P   0.0166            3.0   592.0     3.33 0.0193 e
| L   0.0169            3.0   592.0     3.33 0.0193 e
| R   0.0169            3.0   592.0     3.33 0.0193 e
|--------------------------------------------------
prog | W   0.8914      2     6.0  1184.0    11.67 0.0000 e
| P   0.1086            6.0  1186.0    11.35 0.0000 a
| L   0.1217            6.0  1182.0    11.99 0.0000 a
| R   0.1209            3.0   593.0    23.89 0.0000 u
|--------------------------------------------------
Residual |               594
-----------+--------------------------------------------------
Total |               599
--------------------------------------------------------------
e = exact, a = approximate, u = upper bound on F
• The tests for the overall mode, shown in the section labeled Model (under Source), indicate that the model is statistically significant, regardless of the type of multivariate criteria that is used (i.e. all of the p-values are less than 0.0001). If the overall model was not statistically significant, you might want to modify it before running mvreg.
• Below the overall model tests, are the multivariate tests for each of the predictor variables. Each of the predictors is statistically significant overall, regardless of which test is used.

We can use mvreg to obtain estimates of the coefficients in our model. Normally mvreg requires the user to specify both outcome and predictor variables, however, because we have just run the manova command, we can use the mvreg command, without additional input, to run a multivariate regression corresponding to the model just estimated by maova (note that this feature was introduced in Stata 11, if you are using an earlier version of Stata, you'll need to use the full syntax for mvreg).

mvreg

Equation          Obs  Parms        RMSE    "R-sq"          F        P
----------------------------------------------------------------------
locus_of_c~l      600      6    .6069966    0.1868   27.28199   0.0000
self_concept      600      6    .6890684    0.0540   6.786094   0.0000
motivation        600      6     .761402    0.1500   20.96388   0.0000

------------------------------------------------------------------------------
|      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
locus_of_c~l |
read |   .0125046   .0037178     3.36   0.001      .005203    .0198062
write |    .012145   .0033914     3.58   0.000     .0054845    .0188056
science |   .0057615   .0036412     1.58   0.114    -.0013896    .0129126
|
prog |
2  |   .1277951    .063955     2.00   0.046     .0021896    .2534005
3  |   .2516705   .0684699     3.68   0.000      .117198     .386143
|
_cons |  -1.624765   .1570053   -10.35   0.000    -1.933118   -1.316412
-------------+----------------------------------------------------------------
self_concept |
read |   .0013076   .0042205     0.31   0.757    -.0069812    .0095965
write |  -.0042934   .0038499    -1.12   0.265    -.0118545    .0032676
science |   .0053059   .0041335     1.28   0.200    -.0028121     .013424
|
prog |
2  |   .2764834   .0726023     3.81   0.000     .1338949    .4190719
3  |   .4233592   .0777277     5.45   0.000     .2707047    .5760137
|
_cons |  -.3723412   .1782339    -2.09   0.037    -.7223865   -.0222958
-------------+----------------------------------------------------------------
motivation   |
read |   .0096735   .0046635     2.07   0.038     .0005146    .0188325
write |   .0175354    .004254     4.12   0.000     .0091807    .0258902
science |  -.0090015   .0045674    -1.97   0.049    -.0179716   -.0000313
|
prog |
2  |   .3603294   .0802236     4.49   0.000     .2027729    .5178859
3  |    .619696    .085887     7.22   0.000     .4510169    .7883751
|
_cons |  -1.310842   .1969437    -6.66   0.000    -1.697633   -.9240513
------------------------------------------------------------------------------
• The output from the mvreg command looks much like the output from the regress command, except that the output is for three equations (one for each outcome measure) instead of one. In addition to looking like the output from an OLS regression, the output is interpreted much like the output from an OLS regression.
• The first table gives the number of observations, number of parameters, RMSE, R-squared, F-ratio, and p-value for each of the three models.
• Looking at the column labeled P, we see that each of the three univariate models are statistically significant.
• In the column labeled R-sq, we see that the five predictor variables explain 19%, 5%, and 15% of the variance in the outcome variables locus_of_control, self_concept, and motivation, respectively. (Note this value is a standard R-squared, not an adjusted R-squared.)
• The second table contains the coefficients, their standard errors, test statistic (t), p-values, and 95% confidence interval, for each predictor variable in the model, grouped by outcome. As mentioned above, the coefficients are interpreted in the same way coefficients from an OLS regression are interpreted. For example, looking at the top of the table, a one unit change in read is associated with a 0.013 unit change in the predicted value of locus_of_control.

If you ran a separate OLS regression for each outcome variable, you would get exactly the same coefficients, standard errors, t- and p-values, and confidence intervals as shown above.  So why conduct a multivariate regression? As we mentioned earlier, one of the advantages of using mvreg is that you can conduct tests of the coefficients across the different outcome variables. (Please note that many of these tests can be preformed after the manova command, although the process can be more difficult because a series of contrasts needs to be created.)  In the examples below, we test four different hypotheses.

For the first test, the null hypothesis is that the coefficients for the variable read are equal to 0 in all three equations.  (Note that this duplicates the test for the variable read in the manova output above.)

test read

F(  3,   594) =    4.78
Prob > F =    0.0027

The results of this test 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.

Second, we can test the null hypothesis that the coefficients for prog=2 (identified as 2.prog) and prog=3 (identified as 3.prog) 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). Note that the variable name in brackets (i.e. locus_of_control) indicates which equation the coefficient being tested belongs to, with the equation identified by the name of the outcome variable.

test [locus_of_control]2.prog [locus_of_control]3.prog

( 1)  [locus_of_control]2.prog = 0
( 2)  [locus_of_control]3.prog = 0

F(  2,   594) =    6.83
Prob > F =    0.0012

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. The null hypothesis printed by the test command is that the difference in the coefficients is 0, which is another way of saying two coefficients are equal. Another way of stating this null hypothesis is that, that the effect of write on locus_of_control is equal to the effect of write on self_concept.

test [locus_of_control]write = [self_concept]write

( 1)  [locus_of_control]write - [self_concept]write = 0

F(  1,   594) =   11.89
Prob > F =    0.0006

The results of this test indicate that the difference between the coefficients for write with locus_of_control and self_concept as the outcome is significantly different from 0, in other words, the coefficients are significantly different.

For the final example, 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. We tested the difference in the coefficients for write in the last example, so we can use the accum option to add the test of the difference in coefficients for science, allowing us to test both sets of coefficients at the same time.

test [locus_of_control]science = [self_concept]science, accum

( 1)  [locus_of_control]write - [self_concept]write = 0
( 2)  [locus_of_control]science - [self_concept]science = 0

F(  2,   594) =    6.39
Prob > F =    0.0018

The results of the above test indicate that taken together the differences in the two sets of coefficients is statistically significant.

#### 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.
• If the outcome variables are dichotomous, then you will want to use either mvprobit or biprobit.

Stata Online Manual

#### References

• Afifi, A., Clark, V. and May, S. (2004). Computer-Aided Multivariate Analysis. 4th ed. Boca Raton, Fl: Chapman & Hall/CRC.

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