Refereeing Croquet Strokes and Ethics of refereeing:
Clearly an understanding of the Laws of Croquet, and experience in their application. make one less likely to make incorrect calls. when required to do so. having being summoned to the lawn, to adjudicate a situation by a player or their adversary.
The referee initially should understand his role. Initial request of what state the game is in, why you have been called, and what stroke is requested to observe and judge stroke play and fairness. The time limit of claims and knowledge past events and play may be required. Mostly however a stroke is about to be played, and where a fault is deemed possible the call is made by the striker.
The typeof stroke or intent of the player is crucial, governing where best you safely stand, having marked both relevant balls. Once the correct and best position is gained it is clearly helpful to anticopate what MAY occur. The purpose of these lawn-craft chapters dresses just that aspect of Refereeing. Many of the strokes require knowledge of the static position of the balls, for example when either ball is in the jaws, just touching another ball  or 2-5 mm apart. Most players simply require you to
Firstly SEE  
Secondly HEAR  
OR Finally ANTICIPATE - since currently, that is all you can rely upon.
It is probably appropriate to comment here upon the role of Video cameras, especially solution cameras. I believe they have no current place as a tool to assist a Croquet Referee today with a decision. You simply have to rule on what you observe.
While the excellent YouTube clips, and detailed solution strokes assist a referee to be prepared (hence anticipate and for what to watch), solution video cannot practically be used in International or National events. Having said this, new technologies are being digitally developed all the time, and future referees may well have cameras that at 4 yards, well away from play, can be cited at 1000fps to a 1050p 24 inch monitor for review instantly. 
Hence I agree with others who state;
 " I take the view that you have to have witnessed a fault, by sight or sound. I lie at an extreme here - I also seem to be called upon to do a lot of refereeing?
Plummer 1999  
He adds:

 

 

 

Sight: you have to adopt the best position to observe the stroke. You should remove any court accessories which confuse what you are seeing. You have to analyse which the likely fault will be, e.g. you need to be side-on to the balls and low down if you expect a beveled edge with the bottom of the mallet face, or standing beside the striker's shoulder if you expect a side beveled edge. It is popular to crouch behind a player close to a hoop to watch for a crush shot. Why? You can see the direction that the mallet is traveling? I consider this inadequate. If the mallet is swung with the mallet face at an angle to the direction of the swing the ball will happily glance off cleanly through the hoop but from the back it looks like they are driving it full into the wire! Do not just do it one way just because everyone else does.

Sound: Sound is the best indication of crushes and double taps. Before refereeing a shot you should remove all of the clips from the hoop and check how firmly it is in the ground - gently try to flex it with thumb and forefinger. If it is not solid then you would expect a different sound.

What does a crush sound like? Difficult to express, so you should experiment and note the sound. If there are two separate clean sounds; ball on mallet and ball on hoop, then no problem. If there is a rattle, then it may be due to a loose hoop. If the hoop was not loose then a crush would be a likely cause. If the sound is a blur or a buzz then a crush is suspected. Note that if you hold a mallet loosely you will get a cleaner impact sound than if you grip it tightly. Experience is your main measure.

Expectation: I would give one example, you are asked to watch if a ball hits another ball lying against a wire, then a marker placed in a line through the contact point of the target ball on the wire and its centre would define the ball's path if the hoop alone was struck. If contact is made with the ball it is unlikely that it could travel down that line. This may weight your decision"

In the final analysis we all will make mistakes, but players and referees would clearly benefit from careful reflection on these matters. Remember ethically the player must accept what you decide (no matter what he may inwardly reflect or even say), right or wrong. They elected to hand over their responsibility to the referee having been called upon. It would be very rare in my experience, for a striker to having been judged to play a clean stroke, to then fault himself! There may indeed be some strikers request as to what fault has been detected, however you are under no obligation, to immediately say anything. Insisting on an answer would be badgering the referee, no decision can be reversed once decisively made, and if this argument continues the TR should be summonsed. Law 55, is not just an idle Law.