This page shows several possible uses of the **predict** statement in **proc
nlmixed**. For more information about fitting models using **proc nlmixed**
see our FAQ pages listed at the bottom of the page. Below we show how to generate predicted values for each
case holding some values constant (e.g., some variables held at their
mean), and finally how predict can be used with a random intercept model to get
predictions that do or do not contain the random effect.

## Predicted values for each case

You can download the data used in this example by clicking here: hsb2.sas7bdat.

In this example we use scores on standardized tests in writing (**write**),
and science (**science**), as well as student’s gender (**female**) to
predict scores on a standardized test of reading (**read**). This is a normal
multiple regression (also known as OLS regression). It is not necessary to use
**proc nlmixed** to fit this model, but it works well to demonstrate the **predict**
statement. The SAS code to
run this model using **proc nlmixed** appears below. After the **proc nlmixed** statement, we define the parameters in the
model. Here **xb** is the linear combination of the variables used to predict **read**,
and **b0**, **b1**, **b2**
and **b3** are coefficients. SAS will recognize variable names and mathematical
operators in these statements. Any letter, series of letters, or letter(s)
followed by number(s) that are not variable names are assumed to be parameter
names. Although this model only requires one line to define all the parameters,
more complex models may involve more than one statement. Next we use the **
predict**
statement to record the predicted value for each case. The statement starts
with **predict**, then gives the expression to record, in this case the
predicted value of **read**, that is **xb** from the previous line. The
**out=output1** gives the name of the dataset (output1) in which we wish to place the
predicted values. Next is the **model**
statement. Here we define **read** as distributed (~) normally with mean equal
to **xb**, and variance **s2**, both of which we wish to estimate (**xb**
is estimated through the coefficients; **s2** is estimated based on the model residuals).

proc nlmixed data="d:datahsb2"; xb = b0 + b1*female + b2*write + b3*science; predict xb out=output1; model read ~ normal(xb,s2); run;

We have omitted the output from **nlmixed**, since our primary interest
in this FAQ is the new dataset **output1**. After we run the above model we
can run **proc contents** on the dataset **output1**.
We use the **varnum** option to list the variables by their order in the
dataset, rather than in alphabetical order. The first eight variables (i.e., **ID** to
**SOCST**) are from the dataset **hsb2** that we used to estimate the above model; the
rest of the variables were created by the **predict** statement.

proc contents data=output1 varnum; run;<output omitted> Variables in Creation Order # Variable Type Len Label 1 ID Num 4 2 FEMALE Num 3 3 RACE Num 3 4 SES Num 3 5 SCHTYP Num 3 type of school 6 PROG Num 3 type of program 7 READ Num 3 reading score 8 WRITE Num 3 writing score 9 MATH Num 3 math score 10 SCIENCE Num 3 science score 11 SOCST Num 3 social studies score 12 Pred Num 8 Predicted Value 13 StdErrPred Num 8 Standard Error of Prediction 14 DF Num 8 Degrees of Freedom 15 tValue Num 8 t Value 16 Probt Num 8 Pr > |t| 17 Alpha Num 8 Alpha 18 Lower Num 8 Lower Confidence Limit 19 Upper Num 8 Upper Confidence Limit

Let’s take a closer look at the new variables. The code below uses **proc print** to print the values of the 8 new variables for the
first 10 observations. The variable **Pred**
contains the predicted value for each case, **StdErrPred **contains the
standard error of that prediction. The variables **tValue**, and **Probt**
contain the t-value and p-value for a test that the predicted value (**Pred**)
is equal to zero, the degrees of freedom for this test are shown in the variable
**DF**. The variables Lower and Upper contain the upper and lower bounds of
the confidence interval for the predicted value. Alpha contains the alpha level
for the confidence interval, and alpha of 0.05 corresponds to a 95%
confidence interval.

proc print data=output1 (obs=10); var pred stderrpred df tvalue probt alpha lower upper; run;StdErr Obs Pred Pred DF tValue Probt Alpha Lower Upper 1 63.2338 1.13727 200 55.6013 1.3424E-123 0.05 60.9912 65.4763 2 45.1094 0.96035 200 46.9717 5.4564E-110 0.05 43.2157 47.0032 3 37.5781 1.35648 200 27.7026 2.19543E-70 0.05 34.9032 40.2529 4 51.0225 0.70175 200 72.7070 8.5867E-146 0.05 49.6387 52.4063 5 44.2521 0.93671 200 47.2423 1.9008E-110 0.05 42.4050 46.0992 6 45.1268 0.86562 200 52.1325 2.3098E-118 0.05 43.4199 46.8337 7 41.3422 1.05910 200 39.0351 1.71071E-95 0.05 39.2537 43.4306 8 42.2548 0.97955 200 43.1368 2.9686E-103 0.05 40.3233 44.1864 9 44.8790 0.94246 200 47.6190 4.417E-111 0.05 43.0206 46.7375 10 49.0661 1.21092 200 40.5197 2.25591E-98 0.05 46.6782 51.4539

## Predicted values holding the values of some variables constant

This example uses the dataset used in the previous section.

You can also request predicted values holding one or more
of the variables in the model constant, and using the actual value from
each case for other variables. In the example below we hold **write** at its
mean (52.775), **female**
at one, and use each student’s **science** score to calculate the predicted
values. This allows us to examine how predicted values of **read** change as
the student’s **science** score changes, holding writing scores and gender
constant. Further below we show some ways to use this information.

The code below is the same as above except for the **predict** statement.
This time, instead of simply specifying that we want predicted values of **xb**
we write out the regression equation, this is necessary in order to fix the values of some
of the variables. In the **predict** statement, the variable name
**female** has been replaced with a 1, and the variable name **write** has
been replaced with the value 52.774 (its mean). This fixes these values to a
constant for the purpose of calculating predicted values. Only the variable name
**science** remains in the equation, so that each case’s value on this variable will be used
to calculate the predicted values. The results of this **predict** statement
are placed in the dataset **output2**.

proc nlmixed data="d:datahsb2"; xb = b0 + b1*female + b2*write + b3*science; predict b0 + b1*1 + b2*52.775 + b3*science out=output2; model read ~ normal(xb,sd); run;

Using the predicted values in **output2** we can calculate the predicted
mean of **read** for female students with average writing scores, and the
sample values for **science**. The average predicted reading score,
holding writing score and gender constant,
while
allowing science to vary is 51.23, with a standard deviation of 3.95.

proc means data=output2; var pred; run;The MEANS Procedure Analysis Variable : Pred Predicted Value N Mean Std Dev Minimum Maximum 200 51.2252437 3.9549304 40.8994103 60.0731049

The predicted values can also be used to graph the relationship between
**science** and predicted values of **read**, holding the other variables constant (if
the other variables in the model aren’t held constant, the result will not be a
single regression line, but a scattering of points). Below we use **proc gplot**
to generate a line graph showing this relationship. The first line of code below
clears previous graph setting, the second instructs SAS to connect the data
points to form a line, rather than graphing a series of unconnected points. Next
is the **proc gplot** statement, on which we tell SAS that we want to create a graph
using the dataset **output2**. The **plot** statement instructs SAS to
graph **pred** against **science**. Finally, **run;** executes the command and
**quit;** instructs SAS that we will not be working with this graph further.
(For more information on **proc gplot**, see our SAS Learning Module:
Graphing data in SAS.)

goptions reset=all border; symbol1 interpol=join; proc gplot data=output2; plot pred*science; run; quit;

Below we show the resulting graph. This graph shows the predicted values of read across the observed values of science, holding write at its mean and female at one. Since this is a linear model, with no interaction terms, if we used other values of write and/or female, the predicted values would be different, but the slope of the line would be the same.

## Predict in a model with a random intercept

You can download the data used in this example by clicking here: https://stats.idre.ucla.edu/wp-content/uploads/2016/02/hsb-1.sas7bdat.

In this example we show how the **predict** statement can be used
in a model with random effects. The dataset for this example includes
information on 7185 students in 160 schools. (Note: this dataset comes from the HLM
manual.) The variable **mathach** is a measure of each student’s math
achievement, the variable **female** is a binary variable equal to one if the
student is female and zero otherwise, and the variable **pracad** is a school
level variable, the proportion of students at each school who are on an academic
track. The variable **id** identifies which school a student attends. In the model we use **female** and **pracad** to predict **mathach**,
and include a random intercept for schools.

The first line of code below begins the **proc nlmixed** command.
The second line specifies the conditional mean of **mathach **for student *i* in classroom *j* as a linear combination of the fixed and random effects, and calls this value **mu_ij**. The
third line specifies that the random term be called **u_j**. The **model**
statement specifies that **mathach** is distributed (~) normally with a mean
of **mu_ij **and residual variance **sigma2**. The **random** statement defines the random
effect **u_j** as normally distributed with mean zero and a variance term, **
****tau2, **to be estimated. The level 2 units, schools, are identified by
**subject=id;**. The last two lines of the command are **predict** statements.
The first **predict** statement gives us the
predicted means for each observation based on the fixed and random effects, identified by **
mu_ij**, and outputs a dataset called **output_fixed_and_random**. The second **predict**
statement generates predicted values that include the estimate of the random intercept
in addition to the fixed portion of the model. To get the predictions using only the fixed effects, subtract the predictions in **output_random **from the predictions in **output_fixed_and_random.**

proc nlmixed data="d:datahsb"; mu_ij = b0 + u_j + female*b1 + pracad*b2; rand = u_j; model mathach ~ normal(mu_ij,sigma2); random u_j ~ normal(0,tau2) subject=id; predict mu_ij out=output_fixed_and_random; predict rand out=output_random; run;

## Using predicted values to calculate and graph residuals

We can use the predicted value is to calculate residuals.
Using the dataset **output_random** from the previous example, the data step below calculates the residuals by subtracting each student’s
predicted mathach score (**xb**) from their actual score (**mathach**),
creating a new variable **resid**. This example calculates residuals for the dataset
where the predicted values include the random intercept (**output_random**),
however, depending on our purpose, we also could have used the dataset with only
the fixed effects (**output_fixed**). After calculating the residuals we use **proc sgplot** to plot
the residuals (**resid**) versus fitted values (**pred**) in a scatterplot.

data output_random; set output_random ; resid=mathach-pred; run; proc sgplot data=output_random; scatter x=pred y=resid; run;

## See Also

- SAS FAQ: How can I run simple linear and nonlinear models using nlmixed?
- SAS FAQ: From an OLS model to full mixed models using proc nlmixed