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Of course, you can conduct a multivariate regression with only one predictor variable, although that is rare in practice. The syntax for estimating a multivariate regression is similar to running a model with a single outcome, the primary difference is the use of the manova statement so that the output includes the multivariate statistics. The f- and p-values for four multivariate criterion are given, including Wilks' lambda, Lawley-Hotelling trace, Pillai's trace, and Roy's largest root. The SAS output for multivariate regression can be very long, especially if the model has many outcome variables.
The output from our example has four parts: one for each of the three outcome variables, and the fourth from the manova statement. Below we will discuss the output in sections. Above we see that the class variable prog has three levels. Just below the class level information, we see the number of observations read form the data and the number of observations used in the analysis. If the variables used in the analysis contained missing values the number of observations used would be smaller than the number of observations read. This output is shown below, but we will not discuss it further, instead we will move on to the multivariate output. SAS prints similar output for each of the predictor variables in the model in this case write , science , and prog , this output is shown below, but we will not discuss it further.
Instead we will move on to additional tests. As mentioned above, if you ran a separate 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? One of the advantages is that you can conduct tests of the coefficients across the different models. Below we show a few of the hypothesis tests you can perform.
An alternative way to state this hypothesis is that the difference between the two coefficients i. The estimate statement can be used to perform this test. The text between the apostrophes i. Next we list the variable name prog followed by a series of numbers, one for each level of prog in order, these are the values by which the coefficients will be multiplied to perform the test. The output produced by this model is similar to the output for the previous model, except that it contains additional output associated with the use of the estimate statement.
To save space, we will only show the additional output. There is separate output for each of the outcome variables. Each of the tables in the output gives the estimate in this case the difference between the coefficients , the standard error of this estimate, the t-value and associated p-value. We request this test by adding a second manova statement, where h gives the predictor variable or variables to be tested i.
Again, we will only show the portion of the output associated with the new manova statement. The first table shown below gives the matrix for the outcome variables. Because motivation isn't involved in the test, it is multiplied by zero. To perform this test we need to use both the contrast statement and the manova statement.
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