While there may be examples where more complex analyses provide identical results to a more simple, perhaps less accurate method, you never know until you have tried! While simple models may provide a good proxy of what is happening in reality, this is not always the case. On the other hand, a more complex model does not always provide more predictive or explanatory power than a simple one. A number of statistical methods allow us to test a model's fit and performance and might provide one way of finding the optimal solution for a given problem and parameter space. There is no rule of thumb that simple methods or complex methods are more accurate or useful, that is why we have to keep at the science :)

Examples abound! Take a look at the growing dependence on numerical software to solve simple thermodynamic and electrodynamic problems, It's pathetic! Nothing is added, in fact, it's just the opposite - there is a reduction in understanding, because these apparent scientists look at the output and cannot understand what it means!

It is normally the case with quantitative studies, which I don't think myself an expert of; but in case of qualitative research, complex methods normally provide us ways to have deeper insights into the matter. 

I think it is difficult to answer this question without additional details.  For example, how do you define complex? If by complex you mean additional variables then the value of collecting and analyzing additional variables on the same phenomenon of interest seems straightforward.  Second, you seem to be using methodology and methods to mean the same thing.  Is this intentional? I don't think that they are the same.  Clarification on this point would be help.  Also, it seems that at least one of your assumptions about the inquiry process is that the outcomes of the study are accurate or "true".  Is that "true"?

One additional question, what do you mean by "same science-based conclusions"?

You are right , keeping it simple is key now making it complex is trend.

Value addition is important than making things complex , Keeping everytime simple wont be possible but in case if it is then it should be.

Problem we are facing is we still not clear with fundamentals and hence we require complex stuff to explain things, once fundamentals are clear , we might get simple and clear view.

You are supposed to define your methodology when you design the experiment.  Assuming this to be the case, you've chosen a "complex" method because you hypothesize that it will control variation/bias that you think may affect the results of a "simpler" method that may be available to you.

If the simple method agrees with the complex method after the fact, that doesn't mean the complex method was the wrong choice; and it may be a valuable discussion point if they do.  You simply must justify your choice of the complex method---- or the simple method.

While there may be examples where more complex analyses provide identical results to a more simple, perhaps less accurate method, you never know until you have tried! While simple models may provide a good proxy of what is happening in reality, this is not always the case. On the other hand, a more complex model does not always provide more predictive or explanatory power than a simple one. A number of statistical methods allow us to test a model's fit and performance and might provide one way of finding the optimal solution for a given problem and parameter space. There is no rule of thumb that simple methods or complex methods are more accurate or useful, that is why we have to keep at the science :)

Dear @Marcel. The question is the other way round! In fact, simple (numerical) methods using huge storage (and maybe long computational time too) to solve complex problems (sky building, design of airplane, ...). Only for simple cases the old fashion thinking works!

Probably just to keep our computers busy.

If the "complex method" does not add any novel and relevent result, there is no reason to use it. But, if necessary to reach further objectives, it is wellcome.

In normal circumstances simplicity always has the advantage over complexity. Simple happiness seems so much more productive than “complex” happiness. Newton has had the advantage over Berkeley, Descartes over Pascal, Hegel over Nietschze ...

In order to utilize a "complex" methods successfully, the experimental design and data acquisition usually needs to be developed with the complex analysis in mind.  Complex methods applied to simple data aren't likely to perform any better.

For me, one thing is very sure: if complex methods are used to result in the exactly same science-based conclusions as simple methods, the “complex” one will be replaced by the “simple” one sooner or later------ natural metabolism in science.

Complex methods are infinitely better when simple methods fail to delineate fungal species. In earlier work  fungi are identified by simple morphological methods resulting in creating several names for one fungus. Later, workers, on the basis of perfect state formation found  one valid name for several fungi. At present, DNA and molecular methods are used in delineating fungal species and several other microbes.  These complex methods resulted in clubbing two or more genera of fungi into one genus.  An experienced mycologist can detect the presence of any fungus in tissues by simple KOH mount examination. An inexperienced worker who is not aware of the nature of fungi may follow complex methods to detect the same.

When simple methods do not yield good results, it it is infinitely better to go for complex methods to achieve the desired result.

Sure it's wiki, but it's a classic.

http://en.wikipedia.org/wiki/All_models_are_wrong

"Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity."

George Box

The limit of our knowledge is bounded By The limits of our language. Whenever empírical data is in contradiction with The prediction of our theoreticall models we go on And propose a new theoreticall model. Complex?,that's how

science progress.

Very interesting question...

One may even do some jokes on it, as for example...  that we need to show how complex could be the methods that we use to ... simply do nothing of meaningful.

Here a serious cite by an excellent physicist:

"If all our common-sense notions about the universe were correct, then science would have solved the secrets of the universe thousands of years ago. The purpose of science is to peel back the layer of the appearance of objects to reveal their underlying nature.

In fact, if appearance and essence were the same thing, there would be no need for science."

by Michio Kaku, HYPERSPACE, A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension.

Perhaps complex methods are as like ...trying to imitate some essence and simple methods as as like... giving a short solution concerning the appearance?

Excellent point, I think, Radu Leca. Yes, business, indeed.

And... I know several Real medical doctors that say that medicine is not a science. They, of course, read everything that comes as novel research and practical knowledge in their field, but they decide what shell they do in each particular case - "sugar cube, dispensed at street corners" or the use of some "heavy-duty instrumentation"... Sure, they are not at all good in making business.

So, to the initial question, perhaps the old-fashioned dualistic approach to this "what is better" dilemma is the best one?

Sometimes the people start to invent a bicycle ... Everything is developing in spiral mode, but everything excellent is the simpliest one

Complex methods? No body would use them unless it becomes a necessity or the formalism has been developed for a complicated general problem. Use of computers as mentioned by @ Luisiana Cundin is not a complicated method but a brute force unintelligent method where you want accurate numbers without going into the depth.Most specific problems have simple solutions but when you want to add on various other factors making them general, the complication arises. Further whether something is complex or not is a matter of individual perception. Solving an integral by Complex Analysis using Cauchy's residue theorem may be very simple for some but very complicated for others.

One of my least favorite aspects of "complexity science" is the false sense of "complex" that haunts the "complex methods."  Whatever it is we call "complex" is often a function of our interest and our habits.

There is nothing really "simple" or straightforward about a t-test or an average or an ANOVA. All of these statistical measures carry the burdens of rich, history-laden, bias-suffering premises that inspire them--just as much as any newer measures/summaries carry them. If the measures of linear modeling seem simple it is only because of history and habit.  I propose that the desired simplicity is the logical validity of the measure relative to the underlying hypothesis. Statistics for atheoretical fun bores me, but the number of computational steps or corners cut is no guarantee of the "simplicity." A close cousin of the hand-wringing over "complex" measures seems also to be an impatience that we just "show the real data." Raw data are what they are. The way scientists choose to collect them and summarize them have failed so far to reveal any sort of universally right way to find what's "the truth."

Like your answer Damian.

I'll just add, while there may be no universally right way... there are abundant universally wrong ways to find "the truth".

What statistics and other computational methods, simple or complex, have going for them is the history, or at least the consistency in their application.

I prefer Howard Skipper's quotation just a little bit more than George Box's: "A model is a lie that helps you see the truth."

I do not think that there are complex and simple methods.What is complex for some students may be simple to others due to the individual differences.Some methods may seem to be difficult to be used by the teacher because using them require the availability of some educational technologies.This does not mean that the method itself is difficult or complex.

Simple Method Vs. Complex Method is Really a Complex Topic.

When Microprocessors just came into the Market,it became a Trend to use Microprocessors for Problems that could be Solved through Simple Hardware Circuits that give Instant Output compared to Writing  Programs using Complex Logic and the time taken to execute the program.A Good Example is  Calculating Time Delay wherein one has to consider the Time Taken in executing each Statement of the Delay Program. But it was Fascination and the urge to Explore New Technology.

Simple Solutions save Time and Cost.

But in certain cases simple closed Solutions like Solving a set of Linear Algebraic Equations even single equation of a higher degree, leave alone  Non Linear Equations or a set of Differential Equations ,Partial Differential Equations etc. cannot be done without Resorting to Numerical Methods.

For certain problems there are Empirical Formulas obtained from Experience.

For Certain types of Design Problems  Prototypes can be prepared and used but are Costly.

In such Cases Computer Simulation aids in Design. Simulation also avoids destructive  Testing.

Thus each Method has its Own Advantages and Disadvantages.

Hence a Judicial Selection of Simple or Complex or a Combination of Both has to be made according to the Needs and Convenience.

P.S.

Dear Marcel: In my opinion, the best things are always the simpler (I agree with Amir Answer).

Thank you!

Accepting that your statistical results will depend on the complexity of the statistical tool used (e.g. SAS vs R), e.g. to discover subtle effects in peculiar conditions, what will be the probability to find the same results in a new study using the same or another more or less complex tool?