PROC MIXED testing asymptotic vs inverted u-shaped

[emaneman]

Hello all,

I am testing a model using PROC MIXED, with one categorical and one continuous predictor.

See syntax below. I plot the results using PLM, and I am interested in testing whether the slope decreases at one level of IV but not at the other. One possibility is to use LSMEANS, and showing that at higher levels of the continuous IV (PROGRESS), that is at +1SD, there is a significant differences between the two levels of IV.

But I wonder whether there is a more sophisticated test, showing that for one level of IV there is an asymptotic shape while for the other there is a  inverted u-shape. The quadratic term (progress*progress) is significant for both levels.

Thank you for any suggestion you may have!

Eman

proc mixed data=testing ;
class id  IV ;
model DV= 	IV|progress|progress  ;
random int / subject=id;
lsmeans IV /at progress = -1 tdiff;
lsmeans IV /at progress = 0 tdiff;
lsmeans IV /at progress = 1 tdiff;

store cognizione;

proc plm source=cognizione noinfo noclprint;
effectplot slicefit (x=progress sliceby=IV  ) / clm;
run;

[Rick_SAS]

> But I wonder whether there is a more sophisticated test, showing that for one level of IV there is an asymptotic shape while for the other there is a  inverted u-shape. The quadratic term (progress*progress) is significant for both levels.

Perhaps I am misunderstanding the question. It seems to me that if the quadratic term is significant for both levels, then the shape of the model is U-shaped for both models. (An "Inverted U" if the relevant parameter estimate is negative.) I don't see how you can claim an "asymptotic shape" for either level of IV. They are both U-shaped if all coefficient estimates are nonzero, although perhaps the model for one level is monotonic when restricted to the observed range of the Progress variable?

[emaneman]

hello Rick,

I did not express myself very clearly. 

Although I am interested in the relationship overall, for theoretical reasons, it is important to test what happens in the relation between PROGRESS and DV, for levels of progress greater than 0. And, more specifically, whether this is different for the two levels of IV.  When I do that (see syntax below) I find that PROGRESS is a negative predictor of DV only IV=1 (in red in the figure); it is not for IV=-1 (blue).  In other words, for IV=-1, the slope more or less plateau after PROGRESS=0, while for IV=1 it shows a significant decrease. I attach the SOLUTION results below.

I can present these results, which indeed shows that the relationship is monotonic at one level of IV but not at the other. But I wondered whether there is any other analysis that I could run.

  

data testing2;

set testing;

where progress > 0;

proc mixed data=testing2 ;
class id IV ;
model DV= IV IV*progress /solution;
random int / subject=id;

[Rick_SAS]

Many researchers recommend that, in general, you should include main effects in the model if you are going to examine interaction effects. (Unless there is a compelling theoretical reason not to.) Having the main effects can make interpretation of the model easier. Is there a reason that the main effect  Progress is not in the model?

[emaneman]

Yes, I typically include it. The syntax below results in a significant interaction. I had then used the syntax without PROGRESS main effect to quickly see the effect of progress for the two levels of IV, separately, using / SOLUTION.

E

proc mixed;
class id IV ;
model DV= IV|progress ;
random int / subject=id;

[Rick_SAS]

Now I am more confused. Your first model has a quadratic term for PROGRESS. Your second mode (for which you show the parameter estimates) is linear in PROGRESS.  Looking at the graph you provided, those equations do not correspond to a model of the form DV = a + c*PROGRESS^2 because the curves are not symmetric about the vertical axis. As shown, each curve contains a linear term.

It sounds like you want to comment on the derivative of the model (averaged over the random effects) for the two levels of IV. So do it: Plug in the parameter estimates to get the quadratic equations for DV as a function of PROGRESS, for each level of IV. Then take the derivative and assess the values of PROGRESS for which the derivative is negative.

Let's say for IV=-1, the equation is of the form

DV = a + b*P + c*P^2,  where I am using P for the PROGRESS variable.

Then the derivative is y = b + 2*c*P.

You want to know where the derivative is negative. Your graph shows that c < 0, so (solving for P)

P > -b/(2c)

is the range for which the derivative is negative.  When you plug in the actual values for your model, you will be able to conclude something like 

"When IV=-1, the derivative is negative when PROGRESS > [value]".  From your discussion, it sounds like you expect value will be close to 0.

Do the same thing when IV=1 to conclude something like, "When IV=1, the derivative doesn't become negative until Progress is greater than <different value>, which is greater than the value when IV=-1."

[emaneman]

Hello Rick --

sorry, I had not seen your reply earlier. 

yes, you interpreted my confusing post correctly. 

I understand what you suggest; now I have to figure out how to actually do it -- will report back.

Eman

[StatDave]

>I am interested in testing whether the slope decreases at one level of IV but not at the other.
> ... for one level of IV there is an asymptotic shape while for the other there is a
> inverted u-shape.

If you want to allow the model to show the shape of the curves across PROGRESS and suspect that they might have quite different shapes, then you probably should not force the curves to both have a quadratic shape - particularly if you think one curve plateaus which is not something a polynomial form of model will accommodate. You could instead use flexible splines to better follow the shape of the data. You can use the EFFECT statement to define a spline over PROGRESS and then write the model to allow each IV level to have its own spline. PROC MIXED doesn't support the EFFECT statement, but you can fit the model using PROC GLIMMIX instead. This is done in the following statements.

Then, if comparing the slopes of the lines is the goal, you need to pick one or more specific values of PROGRESS at which to estimate and compare the slopes. This most easily done using the ESTIMATE statement and non-positional syntax. The first two ESTIMATE statements below estimate the slope of each curve near PROGRESS=1 by computing the change in DV mean from PROGRESS=1 to 1.25. Since the difference is computed over a range of only 0.25, each difference is multiplied by 4 to give an approximate slope. The last ESTIMATE statement then estimates and tests the difference of the two slopes.

proc glimmix data=testing;
class id IV;
effect spl=spline(progress/naturalcubic);
model DV = IV*spl / noint s;
random int / subject=id;
estimate 'IV=-1 slope from 1 to 1.25' IV*spl [4,-1 1.25] [-4,-1 1] ;
estimate 'IV=1 slope from 1 to 1.25'  IV*spl [4,1 1.25] [-4,1 1];
estimate 'slope diff IV=-1 - IV=1' IV*spl [4,-1 1.25] [-4,-1 1] [-4,-1 1.25] [4,-1 1];
run;

Another solution would be to estimate and compare the derivative of DV with respect to PROGRESS at a particular value of PROGRESS on each curve. Recall that the derivative is the slope of a line drawn tangent to the curve at the selected point. This removes the approximate estimation as above over some range of PROGRESS using the familiar rise/run computation of slope. This could be done using the MARGINS macro, but that macro would require you to switch from a random effects model to a Generalized Estimating Equations (GEE) model. It also doesn't allow for splines in the model, so you could instead use a much higher order polynomial model which might be adequate. See this note where the marginal effect is discussed (in the context of a binary response model).

[emaneman]

Thank you Dave!

I had to drop IV from the CLASS statement to make it work -- se below.

I get the estimates for both levels of IV, but not the comparison between the two estimates. I guess it's just a typo somewhere that needs to be resolved?

proc glimmix data=testing;
class id IV;
effect spl=spline(progress/naturalcubic);
model DV = IV*spl / noint s;
random int / subject=id;
estimate 'IV=-1 slope from 1 to 1.25' IV*spl [4,-1 1.25] [-4,-1 1] ;
estimate 'IV=1 slope from 1 to 1.25'  IV*spl [4,1 1.25] [-4,1 1];
estimate 'slope diff IV=-1 - IV=1' IV*spl [4,-1 1.25] [-4,-1 1] [-4,-1 1.25] [4,-1 1];
run;

[StatDave]

No, IV was intended to be a CLASS variable as it essentially provides the separate spline curves for its two levels.

[emaneman]

Then it does not work. 

I get the error message below.

I also attach the data, in case you want to give it a try -- note that I dropped the random factor (ID); it is not necessary.

ERROR: The value -1 is not a valid level for a CLASS variable.

ERROR: Invalid ESTIMATE statement; it is ignored.

ERROR: The value -1 is not a valid level for a CLASS variable.

ERROR: Invalid ESTIMATE statement; it is ignored.

id	DV	progress	IV
1	-1.25	-1.566479463	-1
1	-0.08	-1.218372915	-1
1	-1.76	-0.870266368	-1
1	-0.39	-0.522159821	-1
1	1.05	-0.174053274	-1
1	1.46	0.174053274	-1
1	0.88	0.522159821	-1
1	0.02	0.870266368	-1
1	-0.25	1.218372915	-1
1	0.33	1.566479463	-1
2	-0.72	-1.566479463	-1
2	-1.21	-1.218372915	-1
2	-0.48	-0.870266368	-1
2	1.88	-0.522159821	-1
2	0.08	-0.174053274	-1
2	-0.18	0.174053274	-1
2	0.27	0.522159821	-1
2	1.44	0.870266368	-1
2	-0.06	1.218372915	-1
2	-1.02	1.566479463	-1
3	-2.58	-1.566479463	-1
3	0.12	-1.218372915	-1
3	0.34	-0.870266368	-1
3	1.16	-0.522159821	-1
3	0.44	-0.174053274	-1
3	0.6	0.174053274	-1
3	-0.23	0.522159821	-1
3	0.14	0.870266368	-1
3	-0.35	1.218372915	-1
3	0.36	1.566479463	-1
4	-1.07	-1.566479463	1
4	0.48	-1.218372915	1
4	1.61	-0.870266368	1
4	-1.21	-0.522159821	1
4	0.72	-0.174053274	1
4	-0.55	0.174053274	1
4	-0.97	0.522159821	1
4	0.63	0.870266368	1
4	1	1.218372915	1
4	-0.64	1.566479463	1
5	-0.23	-1.566479463	-1
5	-0.39	-1.218372915	-1
5	0.28	-0.870266368	-1
5	0.18	-0.522159821	-1
5	-0.29	-0.174053274	-1
5	-0.7	0.174053274	-1
5	-0.88	0.522159821	-1
5	2.65	0.870266368	-1
5	-0.06	1.218372915	-1
5	-0.56	1.566479463	-1
6	-0.97	-1.566479463	-1
6	-0.42	-1.218372915	-1
6	-0.49	-0.870266368	-1
6	0.16	-0.522159821	-1
6	0.89	-0.174053274	-1
6	-0.41	0.174053274	-1
6	-0.03	0.522159821	-1
6	-0.63	0.870266368	-1
6	-0.54	1.218372915	-1
6	2.44	1.566479463	-1
7	1.17	-1.566479463	1
7	0.14	-1.218372915	1
7	-0.09	-0.870266368	1
7	-0.69	-0.522159821	1
7	-0.78	-0.174053274	1
7	0.64	0.174053274	1
7	0.46	0.522159821	1
7	0.46	0.870266368	1
7	0.9	1.218372915	1
7	-2.2	1.566479463	1
8	0.79	-1.566479463	1
8	0.42	-1.218372915	1
8	-1.73	-0.870266368	1
8	0.71	-0.522159821	1
8	1.34	-0.174053274	1
8	0.08	0.174053274	1
8	0.18	0.522159821	1
8	-0.81	0.870266368	1
8	0.45	1.218372915	1
8	-1.42	1.566479463	1
9	1.41	-1.566479463	1
9	-1.35	-1.218372915	1
9	-0.24	-0.870266368	1
9	-1.13	-0.522159821	1
9	0.11	-0.174053274	1
9	1.21	0.174053274	1
9	0.47	0.522159821	1
9	-0.42	0.870266368	1
9	0.97	1.218372915	1
9	-1.02	1.566479463	1
10	0.26	-1.566479463	1
10	-0.6	-1.218372915	1
10	-1.38	-0.870266368	1
10	0.09	-0.522159821	1
10	0.32	-0.174053274	1
10	-0.57	0.174053274	1
10	2.08	0.522159821	1
10	0.15	0.870266368	1
10	0.77	1.218372915	1
10	-1.12	1.566479463	1
11	0.23	-1.566479463	-1
11	0.61	-1.218372915	-1
11	0.27	-0.870266368	-1
11	0.61	-0.522159821	-1
11	0.76	-0.174053274	-1
11	1.14	0.174053274	-1
11	0.34	0.522159821	-1
11	-1.33	0.870266368	-1
11	-0.65	1.218372915	-1
11	-1.98	1.566479463	-1
12	-1.74	-1.566479463	1
12	0.5	-1.218372915	1
12	-0.31	-0.870266368	1
12	0.89	-0.522159821	1
12	-0.68	-0.174053274	1
12	0.92	0.174053274	1
12	-0.31	0.522159821	1
12	0.08	0.870266368	1
12	1.59	1.218372915	1
12	-0.93	1.566479463	1
13	-1.45	-1.566479463	-1
13	-0.31	-1.218372915	-1
13	-0.69	-0.870266368	-1
13	-0.78	-0.522159821	-1
13	-0.5	-0.174053274	-1
13	0.3	0.174053274	-1
13	0.16	0.522159821	-1
13	0.47	0.870266368	-1
13	0.63	1.218372915	-1
13	2.17	1.566479463	-1
14	1.8	-1.566479463	-1
14	-0.62	-1.218372915	-1
14	-0.53	-0.870266368	-1
14	-0.87	-0.522159821	-1
14	-0.48	-0.174053274	-1
14	-0.45	0.174053274	-1
14	-0.06	0.522159821	-1
14	-1.09	0.870266368	-1
14	0.97	1.218372915	-1
14	1.33	1.566479463	-1
15	-0.41	-1.566479463	1
15	0.23	-1.218372915	1
15	1.11	-0.870266368	1
15	0.24	-0.522159821	1
15	-1.7	-0.174053274	1
15	-0.37	0.174053274	1
15	0.05	0.522159821	1
15	1.11	0.870266368	1
15	1.14	1.218372915	1
15	-1.39	1.566479463	1
16	-1.07	-1.566479463	-1
16	-0.52	-1.218372915	-1
16	1.4	-0.870266368	-1
16	-0.46	-0.522159821	-1
16	-1.6	-0.174053274	-1
16	0.83	0.174053274	-1
16	-0.71	0.522159821	-1
16	0.98	0.870266368	-1
16	0.45	1.218372915	-1
16	0.7	1.566479463	-1
17	0.57	-1.566479463	1
17	1.39	-1.218372915	1
17	0.14	-0.870266368	1
17	-1.55	-0.522159821	1
17	-1.68	-0.174053274	1
17	-0.1	0.174053274	1
17	-0.41	0.522159821	1
17	1.05	0.870266368	1
17	0.22	1.218372915	1
17	0.38	1.566479463	1
18	-1.57	-1.566479463	-1
18	-0.74	-1.218372915	-1
18	0.11	-0.870266368	-1
18	0.71	-0.522159821	-1
18	1.22	-0.174053274	-1
18	1.59	0.174053274	-1
18	-1	0.522159821	-1
18	-0.11	0.870266368	-1
18	-0.57	1.218372915	-1
18	0.34	1.566479463	-1
19	-1.63	-1.566479463	-1
19	0.53	-1.218372915	-1
19	-0.01	-0.870266368	-1
19	-1.33	-0.522159821	-1
19	-0.13	-0.174053274	-1
19	0.38	0.174053274	-1
19	0.47	0.522159821	-1
19	1.94	0.870266368	-1
19	0.17	1.218372915	-1
19	-0.37	1.566479463	-1
20	-0.96	-1.566479463	1
20	-0.69	-1.218372915	1
20	-1.1	-0.870266368	1
20	-0.64	-0.522159821	1
20	0.35	-0.174053274	1
20	0.9	0.174053274	1
20	1.26	0.522159821	1
20	1.35	0.870266368	1
20	-1.1	1.218372915	1
20	0.63	1.566479463	1
21	-1.76	-1.566479463	1
21	-0.28	-1.218372915	1
21	-0.33	-0.870266368	1
21	0.85	-0.522159821	1
21	0.4	-0.174053274	1
21	-0.01	0.174053274	1
21	-0.52	0.522159821	1
21	1.61	0.870266368	1
21	1.01	1.218372915	1
21	-0.98	1.566479463	1
22	1.14	-1.566479463	-1
22	1.75	-1.218372915	-1
22	-1.02	-0.870266368	-1
22	0.25	-0.522159821	-1
22	-0.35	-0.174053274	-1
22	-1.48	0.174053274	-1
22	-0.24	0.522159821	-1
22	-0.87	0.870266368	-1
22	0.6	1.218372915	-1
22	0.22	1.566479463	-1
23	1.36	-1.566479463	1
23	-0.05	-1.218372915	1
23	-0.23	-0.870266368	1
23	0.38	-0.522159821	1
23	0.07	-0.174053274	1
23	1.8	0.174053274	1
23	-0.22	0.522159821	1
23	-0.53	0.870266368	1
23	-1.46	1.218372915	1
23	-1.12	1.566479463	1
24	-1.19	-1.566479463	-1
24	-1.14	-1.218372915	-1
24	-1.45	-0.870266368	-1
24	-0.39	-0.522159821	-1
24	0.75	-0.174053274	-1
24	0.83	0.174053274	-1
24	0.77	0.522159821	-1
24	-0.2	0.870266368	-1
24	1.21	1.218372915	-1
24	0.81	1.566479463	-1
25	-1.07	-1.566479463	1
25	1.69	-1.218372915	1
25	-0.6	-0.870266368	1
25	-0.07	-0.522159821	1
25	0.97	-0.174053274	1
25	0.78	0.174053274	1
25	0.73	0.522159821	1
25	-0.63	0.870266368	1
25	-1.42	1.218372915	1
25	-0.38	1.566479463	1
26	0.39	-1.566479463	1
26	-0.07	-1.218372915	1
26	-0.09	-0.870266368	1
26	0.2	-0.522159821	1
26	0.74	-0.174053274	1
26	-1.43	0.174053274	1
26	-0.09	0.522159821	1
26	2.14	0.870266368	1
26	-0.9	1.218372915	1
26	-0.9	1.566479463	1
27	-1.17	-1.566479463	1
27	-1.04	-1.218372915	1
27	-0.9	-0.870266368	1
27	0.04	-0.522159821	1
27	0.46	-0.174053274	1
27	0.88	0.174053274	1
27	-0.55	0.522159821	1
27	1.34	0.870266368	1
27	-0.58	1.218372915	1
27	1.52	1.566479463	1
28	-1.12	-1.566479463	1
28	1.11	-1.218372915	1
28	0.69	-0.870266368	1
28	-0.52	-0.522159821	1
28	1.31	-0.174053274	1
28	0.21	0.174053274	1
28	1.09	0.522159821	1
28	-1.01	0.870266368	1
28	-1.23	1.218372915	1
28	-0.52	1.566479463	1
29	-1.18	-1.566479463	1
29	1.48	-1.218372915	1
29	-0.28	-0.870266368	1
29	0.91	-0.522159821	1
29	-1.18	-0.174053274	1
29	1.26	0.174053274	1
29	0.45	0.522159821	1
29	-0.21	0.870266368	1
29	-0.1	1.218372915	1
29	-1.13	1.566479463	1
30	-1.97	-1.566479463	1
30	0.02	-1.218372915	1
30	-0.5	-0.870266368	1
30	-0.05	-0.522159821	1
30	1.47	-0.174053274	1
30	0.36	0.174053274	1
30	0.43	0.522159821	1
30	-0.74	0.870266368	1
30	1.31	1.218372915	1
30	-0.31	1.566479463	1
31	-1.25	-1.566479463	1
31	-0.62	-1.218372915	1
31	0.48	-0.870266368	1
31	-0.08	-0.522159821	1
31	1.33	-0.174053274	1
31	0.57	0.174053274	1
31	0.59	0.522159821	1
31	-1.86	0.870266368	1
31	0.95	1.218372915	1
31	-0.11	1.566479463	1
32	0.39	-1.566479463	1
32	1.26	-1.218372915	1
32	-0.81	-0.870266368	1
32	-1.49	-0.522159821	1
32	0.25	-0.174053274	1
32	-0.06	0.174053274	1
32	-0.69	0.522159821	1
32	1.87	0.870266368	1
32	-0.15	1.218372915	1
32	-0.55	1.566479463	1
33	0.18	-1.566479463	1
33	-0.8	-1.218372915	1
33	-0.35	-0.870266368	1
33	-0.62	-0.522159821	1
33	-1	-0.174053274	1
33	1.28	0.174053274	1
33	1.24	0.522159821	1
33	-0.62	0.870266368	1
33	-0.89	1.218372915	1
33	1.57	1.566479463	1
34	-1.47	-1.566479463	-1
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[StatDave · Accepted solution]

Recode IV. Assuming the data you provided is in a data set named A, then this fits the model and compares the slopes from 0 to 0.25.

data b; set a; 
iv2=iv; if iv=-1 then iv2=2; 
run;
proc glimmix data=b;
class id iv2;
effect spl=spline(progress/naturalcubic);
model DV = iv2*spl / noint s;
random int / subject=id;
estimate 'IV2=1 slope from 0 to .25' iv2*spl [4,1 .25] [-4,1 0] ;
estimate 'IV2=2 slope from 0 to .25'  iv2*spl [4,2 .25] [-4,2 0];
estimate 'slope diff IV=1 - IV=2' iv2*spl [4,1 .25] [-4,1 0] [-4,2 .25] [4,2 0];
store mod;
run;
proc plm source=mod;
effectplot slicefit(x=progress sliceby=iv2);
run;

[StatDave]

Since the plotted data suggest both curves are quite linear beyond about 0.5, perhaps it is the comparison of those slopes that is of most interest to you. Is so, then you could use this in the slope diff ESTIMATE statement:
iv2*spl [1,1 1.5] [-1,1 0.5] [-1,2 1.5] [1,2 0.5]

[emaneman]

Terrific, Dave, thank you very much.

The spline strategy is excellent, for it truly allows me to compare slopes at the ranges of PROGRESS that are the most theoretically meaningful.

All the best.

[StatDave]

Just to follow up on my previous remark about an alternative approach based on derivatives: This can be easily done using the Margins macro (as I mentioned earlier) but that macro uses GENMOD to fit the model and does not accommodate splines. Since GENMOD doesn't fit random effects models, a  GEE model is used. So, the statements below fit higher-order (cubic in this case) curves over PROGRESS for each IV level. Note that using the name PROGRESS results in the higher-order effect names being too long, so a copy of PROGRESS is created named P to be used instead.

The Margins macro fits the GEE model as specified and computes the marginal effect (the derivative of DV with respect to PROGRESS) at each of four values of PROGRESS. These are in the output section labeled "P Average Marginal Effects." Note that these are the slopes of lines drawn tangent to the fitted curves at each of those four values. It also estimates the difference in the slopes at each of the four points. These are in the output section labeled "Differences Of P Average Marginal Effects." Tests and confidence intervals are provide for the individual slopes and differences. The "IV2 Predictive Margins" and "Differences Of IV2 Margins" sections might not be of as much interest, but these provide the estimated means on each curve at each of the four points and the differences at each point.

data atdat; 
   do p=0,0.5,1,1.5; output; end; 
   run;
%margins(data=b, class=id iv2, response=dv, 
     model=iv2|p|p|p, geesubject=id, 
     margins=iv2, effect=p,
     at=p, atdata=atdat, diff=all, options=cl)

You can fit the same model as fit by the Margins macro and plot the fitted curves as below. The fitted curves are not too different from the spline models. 

proc genmod;
class id iv2;
model DV = iv2|progress|progress|progress;
repeated subject=id;
effectplot slicefit(x=progress sliceby=iv2);
run;