**An example: the ViDIO-trial case**

ViDIO is an acronym for Vitamin D supplementation by an Intensive Outreach programme for alcoholics. Effectiveness of oral vitamin D supplementation on vitamin D status and muscle performance is tested in a randomized two-arm trial including 150 community-dwelling adults with a history of alcohol abuse and vitamin D deficiencies of lower than 50 nmol/L serum 25(OH)D. The intervention is intended to enhance treatment compliance by means of a simple medication regime in one-on-one patient contacts. Prevalence of vitamin D and B1 deficiencies will be described. Primary outcomes are serum 25(OH)D concentrations at 6 and 12 months follow-up. The trial protocol includes a comprehensive statement of the statistical methodology to be employed. The main research question will be analyzed using a generalized linear model with vitamin D concentrations after 6 month follow-up, intervention condition as factor, and vitamin D level at baseline as covariate.

Conventional analyses

Key analysis in this study is fitting a so-called covariance regression model using maximum likelihood estimation. In formula: Y2i = B0 + B1(Y1i) + B2(Condition) + error. The term B1(Y1i) is negligible as only patients with serious vitamin D deficiency will be included. Thus, the overall intervention effect can be measured using difference scores: Y(2-1)i = B0 + B1(Condition) + error. This is a T-test for the null-hypothesis that the intervention has no effect, H0: B1=0. No conclusion is reached in case this hypothesis is not rejected (absence of proof is no proof of absence). Most likely, the outcome will reject the null-hypothesis – it would be foolish if we spent all the time and effort on this study expecting no effect of vitamin D supplementation. But then the only thing that we will have learned from this trial is that the intervention has a statistically significant effect. The P-value (< 5%) and 95% confidence interval signal that when H0 is true the outcome or a more extreme result is very unlikely. This is a statement about the probability of the data given that H0 is true (Data|H0), not to be misinterpreted as the probability of finding the null by chance (H0|Data). These probabilities can be very different. The probability that someone died by hanging (D|H) does not equal the probability of suicide by hanging as cause of death (H|D).

Alternative hypothesis

What we really want to know is the probability that the intensive outreach programme restores vitamin D deficiencies below 50 nmol to above the norm level of 80 nmol (“tachtig is prachtig”). Expecting an average vitamin D level of 30 nmol at baseline, we hope to see a 50 points increase on average in 6 months in the intervention group. So a more meaningful hypothesis would be H0: B1=50. This hypothesis will probably not be rejected so that in the framework of significance or hypothesis testing we get stuck.

Bayesian perspective

But the P-value can be interpreted in a Bayesian perspective as the lower bound of the probability that the observed association is in the wrong direction (Greenland & Poole, 2013). So, finding a 60 point difference and P=0.30 would indicate a 15% chance that the true difference does not favour the intervention. Better yet, we could forget the intermediate P-value step and employ Bayesian statistics (Gelman, 2013). Or use both conventional and Bayesian analyses. As Hays (1973) in the second edition of his well-known statistical handbook puts it: “there is no law against the inclusion of both kinds of analysis in a research report”.