Comparing two, or more, independent paired t-tests 
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*Can we compare the results from two, or more, independent paired t-tests?

For example: I want to test if drug 1 and drug 2 are effective to reduce weight. I have a control group (that will consume a placebo drug), one test group for drug 1, and another test group for drug2, all of the same size.
Instead of doing ANOVA, I want to make three independent paired t-tests, i.e, one paired t-test for each one of the groups (measuring weight before and weight after). Say that for the paired t-test for the control group I see a mean weight loss of 0.6kg, for the test group (drug 1) there is a mean weight loss of 1.3kg and for the other test group (drug 2) there is a mean weight loss of 1.4kg.
Can I compare the outputs from the three paired t-tests? From these independent tests can I already say that drug 1 and drug 2 are more effective than the placebo drug?
Edit: The problem with repeated measures ANOVA, is that if I added a variable time with $t_1$ (before) and $t_2$ (after), the model would assume that $t_1$ was already depending on the group. For example, if my data looks like this:
id   drug    t1-weight t2-weight
 1     1        3.4       3.1
 2   placebo    3.8       3.7
 3     2        4.0       3.7
 4   placebo    3.3       3.2
 5     1        4.4       4.4
...   ...       ...       ...
 

Doesn't the ANOVA think that the value measured at $t_1$ is already being affected by the drug type? This is a problem because at $t_1$ none of the drugs are affecting because that measure is BEFORE the drug administration. I think the only way here is to do paired t-tests. However, how can I compare them?
This problem would also be more complex if I needed to add another variable, for example a variable for gender (male and female) to check if gender also affects the treatment?
 A: Using the difference between t2 and t1 as feature should work; one can use ANOVA to test the equality of the means.
Note that there are post-hoc tests that allow you to get more insight into the differences, if any, found. This allows you to identify which means deviate. These post-hoc tests take into account that multiple test are being made; i.e. 'Bonferroni test' included.
You can also use a two way ANOVA if you want to add gender as second variable.
Would you want to add more variables, you could try to setup the tests as a hierarchical linear regression problem with dummy variables. Starting out with a model with a single mean, and comparing this model to one that has two dummy variables for the experimental conditions would give you the answer you want. The smaller and the larger model can be compared because the models are considered hierarchical; i.e. the larger model extends the smaller model.
Using an extra dummy for gender, and potentially dummies for interaction effects for gender x treatment, one can squeeze a lot of information out of the data. Note that one has to apply some Bonferroni type correction in this case. One would like to do some kind of power analysis upfront to see if the sample size sustains multiple testing without loosing to much power.
Happy testing!
A: There is no reason you cannot just apply ANOVA with a standard regression model in this case.  Taking three seperate T-tests and combining them into a single inference is inferior to conducting a single ANOVA with a model that includes data from all the groups.
In regard to your concerns about ANOVA, it is important to note that ANOVA is a method of analysis, not a model (see e.g., here).  It can be applied to a linear regression model irrespective of the particular terms and interactions you include or exclude from your model.
The simplest way to undertake this kind of analysis with your data would be to form a new variable for the weight difference (i.e., weight at the later time minus weight at the earlier time).  You can easily form a linear regression model with the drug as your sole explanatory variable and the weight difference as your response variable.  The model formula (in R notation) would be:
weight.diff ~ factor(drug)

If you form a model like this then you can easily apply ANOVA to test whether or not there is a statistical relationship between the drug variable and the weight.diff variable.  It is best to do a holistic ANOVA for all groups first, and then proceed down to individual tests afterward, with proper consideration of multiple comparisons.  Note also that one thing that is important in this kind of work is for your drug allocations to be randomised so that you are doing an Randomised Controlled Trial (RCT).  Randomisation of your drug allocations, and use of placebos for the control group, should ensure that your drug variable is not statistically dependent with any possible confounding factors, which allows you to make causal inferences from your statistical inferences.
A: Yes, you can compare multiple independent t-tests, but note that by doing so you increase the probability of making a Type-I error. Assuming an $\alpha$ of .05, the probability of a Type-I error is 14.3% if comparing between three groups.
Instead of comparing the start and end weights as a pair, compare the difference between the two. I.e, for every participant, subtract the end from the start, and run an ANOVA on that value.
A: If you compare multiple independent t-test(s) you should counteract the multiple comparison problem.
If you want to go down this path, one way to tackle the multi comparison problem is to use theBonferroni correction but others are available. you can find a bunch of them here.
