## Example 68.6 Comparing Multiple B-Splines

This example conducts an analysis similar to Example 15 in , Examples: GLIMMIX Procedure. It uses simulated data to perform multiple comparisons among predicted values in a model with group-specific trends that are modeled through regression splines. The estimable functions are formed using nonpositional syntax with constructed effects. Consider the data in the following DATA step. Each of the 100 observations for the continuous response variable y is associated with one of two groups.

```data spline;
input group y @@;
x = _n_;
datalines;
1    -.020 1    0.199 2    -1.36 1    -.026
2    -.397 1    0.065 2    -.861 1    0.251
1    0.253 2    -.460 2    0.195 2    -.108
1    0.379 1    0.971 1    0.712 2    0.811
2    0.574 2    0.755 1    0.316 2    0.961
2    1.088 2    0.607 2    0.959 1    0.653
1    0.629 2    1.237 2    0.734 2    0.299
2    1.002 2    1.201 1    1.520 1    1.105
1    1.329 1    1.580 2    1.098 1    1.613
2    1.052 2    1.108 2    1.257 2    2.005
2    1.726 2    1.179 2    1.338 1    1.707
2    2.105 2    1.828 2    1.368 1    2.252
1    1.984 2    1.867 1    2.771 1    2.052
2    1.522 2    2.200 1    2.562 1    2.517
1    2.769 1    2.534 2    1.969 1    2.460
1    2.873 1    2.678 1    3.135 2    1.705
1    2.893 1    3.023 1    3.050 2    2.273
2    2.549 1    2.836 2    2.375 2    1.841
1    3.727 1    3.806 1    3.269 1    3.533
1    2.948 2    1.954 2    2.326 2    2.017
1    3.744 2    2.431 2    2.040 1    3.995
2    1.996 2    2.028 2    2.321 2    2.479
2    2.337 1    4.516 2    2.326 2    2.144
2    2.474 2    2.221 1    4.867 2    2.453
1    5.253 2    3.024 2    2.403 1    5.498
;
```

The following statements fit a model with separate trends for the two groups; the trends are modeled as B-splines.

```proc orthoreg data=spline;
class group;
effect spl = spline(x);
model y = group spl*group / noint;
store ortho_spline;
title 'B-splines Comparisons';
run;
```

Results from this analysis are shown in Output 68.6.1. The "Parameter Estimates" table shows the estimates for the spline coefficients in the two groups.

Output 68.6.1 Results for Group-Specific Spline Model
 B-splines Comparisons

The ORTHOREG Procedure

Dependent Variable: y

Source DF Sum of Squares Mean Square F Value Pr > F
Model 13 153.0175561 11.770581238 160.11 <.0001
Error 86 6.3223804119 0.0735160513
Corrected Total 99 159.33993651

 Root MSE 0.271138 0.960321

Parameter DF Parameter Estimate Standard Error t Value Pr > |t|
(group='1') 1 9.70265463962039 3.1341899987 3.10 0.0026
(group='2') 1 6.30619220563569 2.6299147768 2.40 0.0187
spl*group 1 1 1 -11.1786451718041 3.7008097395 -3.02 0.0033
spl*group 1 2 1 -20.1946092746139 3.9765046236 -5.08 <.0001
spl*group 2 1 1 -9.53273697995301 3.2575832048 -2.93 0.0044
spl*group 2 2 1 -5.85652496534967 2.7906116773 -2.10 0.0388
spl*group 3 1 1 -8.96118371893294 3.0717508806 -2.92 0.0045
spl*group 3 2 1 -5.55671605245205 2.5716715573 -2.16 0.0335
spl*group 4 1 1 -7.26153231478755 3.243690314 -2.24 0.0278
spl*group 4 2 1 -4.36778889738236 2.7246809593 -1.60 0.1126
spl*group 5 1 1 -6.44615256510896 2.9616955361 -2.18 0.0323
spl*group 5 2 1 -4.03801618914902 2.4588839125 -1.64 0.1042
spl*group 6 1 1 -4.63816959094139 3.7094636319 -1.25 0.2146
spl*group 6 2 1 -4.30290104395061 3.0478540171 -1.41 0.1616
spl*group 7 1 0 0 . . .
spl*group 7 2 0 0 . . .

By default, the ORTHOREG procedure constructs B-splines with seven knots. Since B-spline coefficients satisfy a sum-to-one constraint and since the model contains group-specific intercepts, the last spline coefficient for each group is redundant and estimated as 0.

The following statements make a prediction for the input data set by using the SCORE statement with PROC PLM and graph the observed and predicted values in the two groups:

```proc plm source=ortho_spline;
score data=spline out=ortho_pred predicted=p;
run;

proc sgplot data=ortho_pred;
series  y=p x=x / group=group name="fit";
scatter y=y x=x / group=group;
keylegend "fit" / title="Group";
run;
```

The prediction plot in Output 68.6.2 suggests that there is some separation of the group trends for small values of x and for values that exceed about .

Output 68.6.2 Observed Data and Predicted Values by Group

In order to determine the range on which the trends separate significantly, the PLM procedure is executed in the following statements with an ESTIMATE statement that applies group comparisons at a number of values for the spline variable x:

```%macro GroupDiff;
%do x=0 %to 75 %by 5;
"Diff at x=&x" group 1 -1 group*spl [1,1  &x] [-1,2  &x],
%end;
'Diff at x=80' group 1 -1 group*spl [1,1 80] [-1,2 80]
%mend;

proc plm source=ortho_spline;
show effects;
estimate %GroupDiff / adjust=simulate seed=1 stepdown;
run;
```

For example, the following ESTIMATE statement compares the trends between the two groups at :

```estimate 'Diff at x=25' group 1 -1 group*spl [1,1 25] [-1,2 25];
```

The nonpositional syntax is used for the group*spl effect. For example, the specification requests that the spline be computed at for the second level of variable group. The resulting coefficients are added to the vector for the estimate after being multiplied with .

Because comparisons are made at a large number of values for x, a multiplicity correction is in order to adjust the p-values to reflect familywise error control. Simulated p-values with step-down adjustment are used here.

Output 68.6.3 displays the "Store Information" for the item store and information about the spline effect (the result of the SHOW statement).

Output 68.6.3 Spline Details
 B-splines Comparisons

The PLM Procedure

Store Information
Item Store WORK.ORTHO_SPLINE
Data Set Created From WORK.SPLINE
Created By PROC ORTHOREG
Date Created 18FEB11:10:49:41
Response Variable y
Class Variable group
Constructed Effect spl
Model Effects group spl*group

 B-splines Comparisons

The PLM Procedure

Knots for Spline Effect spl
Knot Number Boundary x
1 * -48.50000
2 * -23.75000
3 * 1.00000
4   25.75000
5   50.50000
6   75.25000
7 * 100.00000
8 * 124.75000
9 * 149.50000

 B-splines Comparisons

The PLM Procedure

Basis Details for Spline Effect spl
Column Support Support Knots
1 -48.50000 25.75000 1-4
2 -48.50000 50.50000 1-5
3 -23.75000 75.25000 2-6
4 1.00000 100.00000 3-7
5 25.75000 124.75000 4-8
6 50.50000 149.50000 5-9
7 75.25000 149.50000 6-9

Output 68.6.4 displays the results from the ESTIMATE statement.

Output 68.6.4 Estimate Results with Multiplicity Correction
Estimates
Label Estimate Standard Error DF t Value Pr > |t| Adj P
Diff at x=0 12.4124 4.2130 86 2.95 0.0041 0.0206
Diff at x=5 1.0376 0.1759 86 5.90 <.0001 <.0001
Diff at x=10 0.3778 0.1540 86 2.45 0.0162 0.0545
Diff at x=15 0.05822 0.1481 86 0.39 0.6952 0.9101
Diff at x=20 -0.02602 0.1243 86 -0.21 0.8346 0.9565
Diff at x=25 0.02014 0.1312 86 0.15 0.8783 0.9565
Diff at x=30 0.1023 0.1378 86 0.74 0.4600 0.7418
Diff at x=35 0.1924 0.1236 86 1.56 0.1231 0.2925
Diff at x=40 0.2883 0.1114 86 2.59 0.0113 0.0450
Diff at x=45 0.3877 0.1195 86 3.24 0.0017 0.0096
Diff at x=50 0.4885 0.1308 86 3.74 0.0003 0.0024
Diff at x=55 0.5903 0.1231 86 4.79 <.0001 <.0001
Diff at x=60 0.7031 0.1125 86 6.25 <.0001 <.0001
Diff at x=65 0.8401 0.1203 86 6.99 <.0001 <.0001
Diff at x=70 1.0147 0.1348 86 7.52 <.0001 <.0001
Diff at x=75 1.2400 0.1326 86 9.35 <.0001 <.0001
Diff at x=80 1.5237 0.1281 86 11.89 <.0001 <.0001

Notice that the "Store Information" in Output 68.6.3 displays the classification variables (from the CLASS statement in PROC ORTHOREG), the constructed effects (from the EFFECT statement in PROC ORTHOREG), and the model effects (from the MODEL statement in PROC ORTHOREG). Output 68.6.4 shows that at the 5% significance level the trends are significantly different for and for . Between those values you cannot reject the hypothesis of trend congruity.

To see this effect more clearly, you can filter the results by adding the following filtering statement to the previous PROC PLM run:

```filter adjp > 0.05;
```

This produces Output 68.6.5, which displays the subset of the results in Output 68.6.4 that meets the condition in the FILTER expression.

Output 68.6.5 Filtered Estimate Results
 B-splines Comparisons

The PLM Procedure

Estimates