Taking on freelance work is risky

Whenever a job is taken on like this, there is some degree of risk taken on by both parties. Who takes the most risk is dependant on the terms of the agreement, but basically:

1. The client takes the risk that you're going to deliver.
2. You take the risk that the client will pay.

These risks are mitigated by very open communication with the client about progress and a payment schedule that is reguler (milesone/weekly).

Don't underestimate yourself

I should have not taken on the contract without being confident that I could deliver.

To some extent you should be confident when taking on a job, but even the most confident and prepared person has unforseen issues. Unless you're a serious veteran in the industry with copious amounts of experience, it's pretty understandable that you're not going to see some issues coming. No one knows the future. Don't underestimate yourself!

Also, the client considered and chose you when taking this job. If you accurately represented your skills and he hired you, it means that he judged you to be good enough to do the job too.

So I wouldn't be quick to put the blame on yourself completely.

Now for what to do:

Offer solutions

I would recommend communicating your concerns to your client (#2) as suggested, however I would also make solution recommendations for the issues too.

Here's a more detailled example adressing your concerns in particular:

Addressing issue #1:

But that data is not readily available, and I do not have the resources or capability to gather that data.

Say exactly that - the data is not readily available - but then also make a recommendation. Is there a place you could get the data? What would that cost or require? Perhaps the client would pay the fee or has connections of some kind. What about other data sources?

If these things would take a long time or a lot or work to find out, perhaps tell them what you know, or do a bit of research, and then say something like "Researching the options would take some time. I can look into x further if you're interested".

Addressing issue #2:

The client wants the study done with a particular methodology. Over the course of my research, I have come to strongly suspect that the methodology is not valid and should not be used.

I would be careful how I approach this. I would stick to the facts and details precicely what's wrong with the method or why it wouldn't work. I wouldn't say that it's "wrong" or "invalid" directly. Just mention issues you see. "How is the method going to deal with x?". If this is something the client thought up, they may be attached to it and you need to be sensitive to that. Bring up your concerns in a respectful way and be open to learning more or being wrong.

Here too, make suggestions regarding your concerns. "I'm not sure how the method is going to handle x, what about doing y instead?".

If you can show your concerns about the method and suggest an alternative in a respectful way, then that's probably the best thing to do.

There is not such thing as too many questions

I have questioned the methodology in the past and have been told that I am the one who fails to understand.

I have certainly been there before.

There are two possibilities here. Either (1) the method is flawed or (2) it isn't and you misunderstand it. Asking a questiong about the potential flaw ("How will the method handle x?") will facilitate establishing which case this is.

Keep asking questions about the method until the above is resolved. Do not worry about looking stupid because you've asked (what feels like) the same question for the nth time. If you and the client have differing ideas on what this method is, deal with that ASAP.

Since everyone thinks differently, it's inevitable that there are going to be some differences in communication styles. I think in pictures, but one of my clients prefers communicating with speech. It took a while for us to figure out a way of communicating complex ideas in an efficient way. It's not that either of us are stupid (and I doubt either you or your client are too), just that people communicate very differently and it's necessary to work around that.

You're on the same side

I know that it's easy to see the client as the person who's paying and there's no vested interest there, but you both want this to work. It becomes a lot easier when you think about the client as someone who you're in a mutual relationship with, both working toward the goal of making this project suceed.