So if I wanted to measure the effect of a drug on a group of people, I think I would use a paired T test based on my research. But what if something caused a change in my group that I did not account for? Could I randomly sample my group into two groups and not give the drug to some people, then do a T test on both groups? What if the T test measures an increase in both groups? So if my control group had an increase and my treatment had an increase, can I conclude the treatment was effective because the treatment group increased much more than the control group?
I think what you are stating is directionally right. But, to clarify you can conduct a paired t test with your one single group and measure how they fare before and after the treatment. You can also conduct an unpaired t test by dividing patients into two separate groups with a Control group to whom you just give a placebo and a Test group to whom you give the drug. You measure how they compare before being given the drug, and after being the drug or placebo using the unpaired t test. The test before is to check how well they are matched (Control vs Test group). The second test after being given the drug is to check the drug performance vs the placebo. Actually, another and better way to do that is to measure the difference of after minus before for the Test group vs the Control Group. And, you use the unpaired t test to check if the Test group that received the drug had a difference that showed an improvement vs the Control group that just received a placebo. I think this latter unpaired t test structure focused on the difference of after minus before is how clinical trials are conducted. If you do this test structure, you don't have to do any of the other test structures mentioned (except maybe a unpaired t test to check how well the Test group and Control group are matched in terms of the relevant physiological characteristics).
Seconding Gaetans answer, I'd say the following setup would be ok, depending on what you're measuring and that you've checked the assumptions for a t-test (normality, independence etc.).
- Treatment Group (TG) vs. Control Group (CG) -> unpaired t-test
- TG/CG pre vs. post treatment -> paired t-test
For TC/CG with multiple longitudinal measurements (intra-group comparisons) some people would suggest random-effects modelling to rule out false-effects as well. But that depends a little bit on how likely you think your data will be affected by such things.
Regarding the overall study design (randomly assigning no patients any treatment and hence analyzing the effect of confounders such as "recruitment/enrollment", "time" etc.), that's a totally different ball game and actually plays into your overall study design and plays into my comment about random-effects modelling.
If you want to start matching patients from your TG and CG, then things may become a bit more complicated, but as far as I understand your question, the answers above would suffice.