Paired t-test when each data point was repeatedly measured different number of times? I have an existing data set that comes from the same group of people before and after they received a treatment.
The data set comes from when participants tested their blood sugar values over a 30 day period before receiving an insulin pump and a 30 day period after receiving an insulin pump. This data was obtained from user logs (archival data) and was not controlled to ensure that they were tested at fixed intervals. Participants tested themselves when they needed to test throughout the day for a period of 30 days.
My goal is to determine whether the average blood sugar across 30 days was different for the before vs after group. 
Normally this would be a paired samples t-test but unfortunately the before and after group have an unequal number of data points with the after group having significantly more. People are testing more often after receiving an insulin pump.
What is the correct way to handle this?
I can collapse the data to find a mean for each participant before treatment and after treatment (across all participants) so that the data matches up and then run a paired samples t-test on this data but I think this is not the ideal solution. 
Would a one way within subjects ANOVA be the appropriate test to run in this case?
 A: For the purposes of hypothesis testing, I often find that the simpler approach is the best. 
In this case, I would do exactly what you considered yourself: average all the pre-treatment values and all the post-treatment values for each participant, obtaining two values per participant. Then you can run a paired t-test on the resulting averages.
There is nothing wrong with this approach. If you do that and you get $p$-value sufficiently low for your purposes, you can call it a day. I guess there are much more complicated mixed models that one can set up here, but I would be skeptical that they produce much lower $p$-values (and if not, then there is no gain). Whereas two big advantages of the simple t-test on averages are: (1) it takes five minutes to perform; (2) it takes two lines to explain in a paper.

PS. If I am not mistaken, then a simple repeated measures ANOVA (that you asked about) cannot be applied in your case.
PPS. Note that without a control group you will not be able to say if the difference between post and pre (in case you observe any) is due to treatment or due to some time passing.

Update. What I wrote above was under assumption that either you have no information about times of individual measurements, or you are happy to assume that the time is irrelevant. @psarka argued (+1) that time of the day is very relevant and, worse, it is unlikely that measurement pre- and post-treatment were distributed along the day in the same way. So if you have the information about measurement times, then you should better take it into account, and the exercise becomes more complicated then. If not, then well, not.
In addition, @robin argued that the day number is important as well, see discussion in the comments.
A: I'm skeptical about possibility to say anything meaningful in this situation without taking daytime into account. The reason is the process that generates your data.
Blood sugar levels have a daily pattern (more erratic or less erratic, depending on the patient), partly related to carbohydrate consumption. In theory, after eating your sugar levels rise and then fall as insulin does its job.
Blood sugar measurements also have a daily pattern: patients usually measure their blood before eating, as they need to know if they have to adjust the amount of insulin. This should be especially true, if patient is used to his/her therapy (that is, before switching to pump).
If we agree that the true average blood sugar level is the area under curve which we could get if we measured blood glucose continuously, then


*

*it seems likely that average of measurements before switching to pump generate biased estimate of true average 

*and unlikely that the average of more frequent measurements generate the estimate which is biased in the same way.


Even if these statements are not true in your case, you'll have to convince everyone (at least me) that they indeed aren't.
Now if you are willing to take daytime into account, then you can infer so called modal day before switch and after switch, or compare corresponding measurements, say before breakfast.
A: If I'm understanding you correctly, you should be running a repeated-measures model, sort of like an interrupted time series model except you only have 1 (in some cases, none) observations before the treatment. Longitudinal regression models can handle imbalance, if you treat it as a repeated-measure hierarchical or multilevel model. In this case, the level 2 model is the person, and the level 1 model is for the observation within the person. 
Something like this:
Level 1: $y_{it}=\hat{\beta_{0i}}+\hat{\beta_1}treat+\hat{\beta_2}time_t + \epsilon_{it}$
Level 2: 
$\hat{\beta_{0i}}=\alpha+\Sigma\hat{\gamma_i}X_i + \mu_i$
Where indices $i$ and $t$ represent person and time, respectively. $\epsilon_{it}$ is the level 1 error term, for the observations within persons, and $\mu_i$ is the error term for the persons. The variable $treat$ is a binary variable, 1 if the person was treated at time $t$, 0 otherwise. I have here time as a linear term, but you could add polynomials as well. This specification should tell you if on average, independent of when the measurement was taken, if it was taken after the treatment was it higher than before. You can then also see if there is a time effect independent of treatment.
$\gamma_i$ is the coefficient for time-invariant person characteristics $X$, assuming you have that. You could also include time-varying person characteristics, if you tracked them. I've also only assumed you are interested in a random intercept - which means the intercept for each person is a normally distributed, random variable, centered around the mean. 
This is a basic model - you can make it far more complicated. But it can get your started. 
Regarding individuals with no pre-treatment observation, that can still be included in your model - they will not contribute any pre-treatment information, but they will provide additional comparison for those with pre-treatment observations. If you have additional covariates this would make it much stronger. 
A: Your stated goal is to compare the means from the two samples, each of which has different sample size (i.e. n1 does not equal n2). The classic way to handle this problem is to use Welch's t test ( http://en.wikipedia.org/wiki/Welch's_t_test). 
This test is exactly appropriate in your case because it relaxes the requirement of the paired t-test, namely that n1 = n2.
This approach will of course require you to properly construct your hypothesis test, but that shouldn't be a problem since you were already thinking of applying a paired t-test. In short, the rest of the hypothesis test should proceed exactly as expected with the only difference being the application of Welch's test.
Good luck.
