Earlier today I was discussing statistical analysis software with a colleague of mine. My colleague had primarily used SPSS in previous work for performing t-tests, anovas, manovas, and other statistical tests similar to those. My colleague also mentioned that SPSS does not handle regressions well. I have no idea whether or not SPSS handles regressions well, but I did respond that a t-test can be formulated as a regression, so SPSS must not be all that bad with regressions. We got to talking about t-tests, regression, and causality, and it came up that "you cannot prove causality with regression, while t-tests are able to prove causality." I've always thought that causality could be established given an appropriate experimental design, regardless of whether you use a t-test or regression to estimate something or perform a hypothesis test. Is it possible to establish a causal relationship using a t-test and not regression? Is it possible to establish a causal relationship without knowing about the experimental design?
@John is correct, but, in addition you cannot prove causation with any experimental design: You can only have weaker or stronger evidence of causality.
In any study, but especially in an observational study, evidence for causality is increased by including relevant covariates, giving a scientifically plausible causal path, replicating results and so on.
However, even in the best experimental design, you don't prove causality.
As for t-tests vs. regression - your friend does not know what he/she is talking about. T-tests results can be duplicated exactly with regression procedures: Just use a single independent variable that is dichotomous.
Causal relationships are established by experimental design, not a particular statistical test. You could use a correlation as your statistical test and demonstrate that the true experiment you conducted showed causation. You could perform a t-test as your statistic and show a relationship in your quasi or observational study that does not motivate a causal explanation.