Two problems: The problem mentioned is correct, nothing can be deduced from the null result. Second, the participant set is the same pre- and posttest, which needs to be taken into account when testing.

The research questions seems to be whether the correlation increased (no matter if it has been present before). I suggest to set up the test difference score model for depression and estimate the prediction of the trait for the difference score (or equivalently the correlation of the trait with the difference score).

This can be done using a correlation of depression change with trait, but for completeness and expansionability, I'd suggest to set it up as a simple SEM (for something this small, the SEM program Onyx can be used, it's very intuitive and easy to use).

Two problems: The problem mentioned is correct, nothing can be deduced from the null result. Second, the participant set is the same pre- and posttest, which needs to be taken into account when testing.

The research questions seems to be whether the correlation increased (no matter if it has been present before). I suggest to set up the test difference score model for depression and estimate the prediction of the trait for the difference score (or equivalently the correlation of the trait with the difference score).

This can be done using a correlation of depression change with trait, but for completeness and expansionability, I'd suggest to set it up as a simple SEM (for something this small, the SEM program Onyx can be used, it's very intuitive and easy to use).

you can annotate the second error of the correlation test to understand better the lack of correlation. In addition, it could be necessary increase the sample size, or to add new independent variables that can improve the correlation.

If the question is really about whether the post-test correlation is larger than the pre-test correlation, I think Timo's advice is good, but I would also estimate confidence intervals around your two correlations. It'll give you some sense of how large the two correlations actually are (and therefore how different from one another), as opposed to whether either is demonstrably different from 0. It's an approach more common to public health, as well as periodically emphasized in social sciences by people like Cohen!

The conceptual interest is probably more around what the treatment is doing to the distribution of depression scores - does high depression level merely mask a population-level correlation between depression and the trait? Or does treatment affect trait X scores as well?

Thanks for the pointer to Onyx, Timo. In fact, the study authors did do something like that. Effect sizes (Cohen’s d) were calculated for the mean changes scores between baseline and follow-up. The results were 0.78 for trait X and 3.14 for depression (i.e. four times more).

The purpose of the overall experiment was to test the relative stability of each measure following treatment. Due to the experimental design, a zero or weak correlation in the pre-treatment group would be fine for establishing the baseline; in fact, a non-significant weak correlation could be anticipated on the basis of theory and previous experiments. The fact they didn’t find a significant correlation at treatment initiation doesn’t invalidate their experiment, which wasn’t aimed at showing interactions. It was aimed at assessing the stability of each, relative to the other, in order to show that X is a stable personality trait only weakly linked to depression (to settle an ongoing state-trait controversy).

The subjects were selected on the basis of a major depressive disorder – i.e. a high score on depression scales. When the selection criteria are narrowed like this, the prospect of finding a significant correlation with trait X narrows too. In a properly randomised study, with an unbiased population sample, you have a much greater chance of finding a significant positive correlation (as many other studies have done with trait X). But if you select only participants who all have exactly the same depression score, there is nothing for trait X to correlate with. Between these poles, with a narrow selection band, you are likely to find some signs of a correlation (if one exists), but not at a significant level.

Why select only those with major depressive disorder? Well, there's little point in giving antidepressants to people who aren't depressed. (However, there was, as expected, a wide distribution of trait X scores amongst the group.)

There are various permutations of the mantra that nothing can be deduced from a non-significant result: but of course this actually means that the specific null hypothesis can’t be accepted or rejected on that basis (according to the .05 convention, anyway – but let’s not open that can of worms!) What about other logical implications of the values? How likely is it that the non-significant result cloaks a perfect correlation of 1? Logically, it’s not possible – even taking sampling errors into account. So, can we compute the likelihood of these (non-significant) results occurring if there was actually a correlation of .38 amongst the wider population of patients with major depressive disorder?

I do not know the causal order in conception of these traits and correlation is not causation - you can investigate further relations in structural linear models, e.g. interaction between trait X and depression through treatment.

Sam, if I'm reading the last part of your comment of 4 days ago correctly, you are considering whether your sample at baseline has a correlation between trait X and depression that is significantly different from a population correlation of 0.38. to be honest, I think I'd still generate 95% confidence intervals around the 0.09 estimate and base my conclusions on whether the CIs included 0.38 with your N. The Cohen's d would work just as well as an effect size estimate.

There are more complicated formal tests in path analysis/SEM. You can estimate model fit constraining the correlation between X and depression to 0.38 (or 0.09!) and see if it damages fit significantly compared to a model in which that correlation is estimated freely from the observed data. That kind of model can be run in any basic program like LISREL or AMOS. I am not sure it adds value to the simple approach, though.

If the question is not for the population value, but for the sample, the CIs go around the 0.38 estimate and see if 0.09 is included. I take it this is a single group, pre-post sample.

Adding to all of the above esp re CIs, since this sounds like alexithymia, how about just dichotomizing the TAS-20 scale into