Example  Test path for 1995-16Valve failing code 51:

Click above too zoom.  (1.8L Sport is far different )

If you removed the CAT Converter  (or defeated it) the above red air path will fail. 
I have no solution other than putting it back in.
The mod valve (modulator) was renamed to Transducer in 1996, but it still works the same exact way.

There are a number of tests.
in order of difficulty .
1: cold engine running, reach behind EGR, see skeleton frame, see diaphragm exposed, now gently press diagphragm with finger, it must move.
if it moves then It may be good.  If stuck it is bad.
While engine is running push it, the engine must die or almost and it will at least stumble badly.
if this test passes then most likely the problem as at step 2 and beyond.

If above test fails, the EGR main path is bad. (assuming diaphragm moves freely)
If diaphram is not moving , then EGR is bad.
The path from Exhaust all the way to the intake must be cleaned (Iuse a flex cable 1/8 th inch ,chucked to a drill and ream it out)
if diaphragm is stuck, you remove EGR and  Clean out the carbon and spray "Penetrating Oil"  or the like into holes you just cleaned and try to free up this diagphram.  if you can't free it , buy a new one at Rockauto.com  (best prices not affiliated to them and they ship to Canada and elsewhere).

2:  The Test above passes but you get 51 and the EGR and its pathes,  are clean.
Apply vacuum tothe EGR main  valve directly , and test it .
This is easy with a mityvac tool or just a spare length of hose.
Put spare hose to EGR vsv hose input hose (not nipple), then to EGR valve  , again the EGR must open "LOOK on back to see it" and engine again must die or stumble. Start engine to make vacuum , pull manifold side of VSV valve hose, feel it to be sure there is vacuum , if not that is a BIG Problem, fix that.
now extend this hose to the Main EGR valve. The Valve must open , look at the diaphragm with a small flashlight.
if not the EGR diaphram is bad.  ( one can use the MityVac tool to check for a leaking diaphram) [ no leak down allowed at all, is spec]

  Pull the hose on top of Main EGR, this a larger hose. (going to bottom of mod. valve, check hose for cracks).
Look at this main EGR valve nipple , it LOVES to be  clogged, and will produce as code 51.  ( not the diaphragm nipple)
This hose goes to EGR MOD above.
Mod is modulator,

The above, are the most likely failure modes., there are more more tests.

MOD test and VSV test.

There is a different test for the modulator and the VSV
They are tricky to test, the  mod and vsv.

Here is a short way:
engine  On.
apply 12vdc to EGR VSV (vacuum solenoid)
the valve should open and apply vacuum to the
the P side of the MOD valve.  inplug VSV elect. conn. first. 
Just feel it with finger tip, valve opens and you feel vacuum on the other side.  Yes, then the VSV is GOOD !

The MOD Q side will be closed.  ( because exhaust back pressure is too low idlng ).
Next , while 12vdc is applied to VSV , apply low shop
air 5psi to the bottom of the MOD valve.
I use a paint spray regulator to do this.  (bicycle tire pump? or inflatable toy pump, < no jokes, ok?) [improvise here ]
this pressure , simulates exhaust backpressure and
opens the MOD valve, at the moment the EGR will open .  ( engine makes vacuum and it gets applied to the MOD valve and the bottom of the MOD valve
moves its internal diaphragm and the mod valve opens and this applies vacuum to the main EGR valve.
just watch EGR main open , and the engine stall or almost stall.

if all the above passes, the EGR system is working.

5: Acid test, FSM:  ( cars old than 95 , have less rules for this test )
the last test is the FSM factory test.
It must be done with rear wheels raised.                           No Kids, No observers, no one near the wheels. Please !
The EPA is trying to kill us , as you can see, it is a dangerous test.

The valve will not open unless a bunch of conditions ,  are true.  a giant "AND function".
RPM, Speed, backpressure, hot engine. etc.
See my link for that.
Here are the rule sets: (from my main EGR page)

1-Less Than 8200 feet altitude  ( you are not living at Breckenridge, CO.) [ I guess you must move ?]
2- RPM Greater-Than 3500 (engine RPM, not wheels)
3 -Engine Temperature Greater-Than  136 degrees  with years 89-93 or GT 149 deg. years 94-96     { 149d F, in this example 95'}
4- Not idling, not W.O.T., yes cruising fast.   (not wide open throttle).  see step 2 .
5- The wheels (rear) must turn on 94-96 yr (on jack stands)
6- EGR Mod., must find some exhaust backpressure . (with out proper back pressure the EGR MOD WILL FAIL to open properly)
( good luck if you run straight pipes and a cherrybomb or other heavy mods, or a rodded out CAT or gutted CAT )
7 - EGR Parts all good, passages must be clear, on 96-97 passages must be SUPER clean. (fails on minor restrictions, on newer cars). 95 year is easy.
8- Below 94 only Cal cars produce code 51. Thermo probe bad or EGR not opening. Fed cars do not have the EGR thermo probe.

Code 51 can be created , with nothing more bad , besides a bad EGR thermo probe (Calif car only or if your state/ country uses Calif cars).
If you see this wire and probe entering your EGR valve than it is a Calif car. It is called the REGTS.
It is easy to test, this sensor.
repeat of my main page:
Data Points for the REGTS:  source Pg, 6E99 FSM.  ( has a graph plot and test procedure)
212 F. = 11.2 to 13.6k ohms   or about 12,400 ohms
140 F = 42.1-55.5k ohms.
68 F = 214-313.8k ohms
The REGTS  in my Garage , right now, which is 45F , reads 500k ohms.  It is just a Termistor ( with a reverse temperature coefficient.  NTC)

These rules are dictated by the coding in side the ECU.

end EGR 95.

rough draft. FYI