How to compare response to same question asked at different time points? I asked survey respondents about two points in time, and asked participants the degree to which they "agree", a typical Likert-style question. For example, as if I had asked "Before Obama became president, life was good" and a separate question, "After Obama became President, life has been good". Normally, one would assume a t-test, and at first I thought a paired t-test, since these are two separate sets of answers I am trying to compare (sort of like time 1/time 2, right?). However, I've gotten rejected multiple times by wanna-be stats folks, none of whom can tell me what the right test is. So can anyone tell me if I ask about two points in time (in your childhood, what was your relationship like with your mother? Now, what is your relationship with your mother?), how do I compare them?
 A: It depends on several things.
One is what your scale was.  Unless it can plausibly be treated as a continuous scale, a t test could not be appropriate.  Some purists argue that you should never treat an ordinal scale as continuous for such purposes.
The other, probably more important issue, is what you are interested in.  Do you want to see if there is a perceived net improvement/decline in life; or are you interested in the correlation between the two.  For example (assuming you can treat the variable as continuous) a paired t test would show you if on average people think life is better; but a correlation coefficient might be useful if what you are interested in is whether "Obama" is a decisive factor one way or another.  
Consider a highly politically partisan sample balanced between Republicans and Democrats, where everyone who agrees with the first statement disagrees with the second and vice versa.  A paired t statistic will be exactly zero, showing no impact of Obama - but the reality is there was a big impact, in switching around who thinks life is good.  A correlation coefficient of some sort would show this up.
If you want to do a test with a correlation coefficient you should use a bootstrap method.
If you want an appropriate correlation coefficient for ordinal data you should consider a polychoric correlation.
If in fact you are interested in the net increase (up or down), then a paired t test would be ok if the data is sufficiently "close" to being continuous that you can pretend it is; but a non-parametric paired test as @psj suggests might be better.
A: One approach is to visualize the changes by individuals. I can't recall the name of this chart type offhand, but this would be a chart in which each individual response to the first question is plotted in its y-axis position on the left side, and each response is plotted for the second question on the right side, with a line connecting the two datapoints for each individual. By the slopes of the lines, we can visually interpret the changes that individuals make in their responses. Because there are only 5 possible values for each axis, many of the lines will overlap. In order to make the density of the lines apparent, you can fake this by making the axes 1-500 instead of 1-5, and apply a random jitter to the exact location of each plot. Then they won't be precisely overlapping, and you will be able to distinguish areas of high and low density.
A numerical approach would be to compute a variable for each respondent showing the direction and extent of the shift in their answer between the two questions. If somebody responded with a 1 on the first question and  3 on the second, you would calculate this change variable to be +2. You can then report on this many different ways; for example, x, y, and z% of respondents responded higher, lower, or the same between the two questions; you can give yourself a nice stacked bar chart going from +4 to -4, etc. 
These approaches I think make sense in light of Peter Ellis's comment that the overall group behavior obscures the shifts that individuals make, and the interesting thing is the individual shifts. You can look at the groups that have positive values, 0, or negative values and see how they differ in demographics or political affiliation.
My main concern is that the data is being framed as "time 1" and "time 2", and it's really not that. It's "question 1" and "question 2". That's not to say that it is without value, it's just an entirely different kind of data collection which we would be expected to tell us more about their current positive and negative opinions, and their current perception of any shift. That's still valuable, it's just different.
A: Your outcomes are paired ordered categories, so the t-test is not appropriate, but the generalizations of McNemar's test would be.  Read the linked page to get the general idea, then follow the "Related Tests" links to find which is best for your situation.
