Randomized controlled trials constitute what are generally considered to be the gold standard design for evaluating the effects of some intervention or treatment of interest. The fact that participants are randomized to the two (sometimes more) groups ensures that, at least in expectation, the two treatment groups are balanced in respect of both measured, and importantly, unmeasured factors which may influence the outcome. As a consequence, differences in outcomes between the two groups can be attributed to the effect of being randomized to the treatment rather than the control (which often would be another treatment).
R squared and goodness of fit in linear regression
I’ve been teaching a modelling course recently, and have been reading and thinking about the notion of goodness of fit. R squared, the proportion of variation in the outcome Y, explained by the covariates X, is commonly described as a measure of goodness of fit. This of course seems very reasonable, since R squared measures how close the observed Y values are to the predicted (fitted) values from the model.
R squared and adjusted R squared
One quantity people often report when fitting linear regression models is the R squared value. This measures what proportion of the variation in the outcome Y can be explained by the covariates/predictors. If R squared is close to 1 (unusual in my line of work), it means that the covariates can jointly explain the variation in the outcome Y. This means Y can be accurately predicted (in some sense) using the covariates. Conversely, a low R squared means Y is poorly predicted by the covariates. Of course, an effect can be substantively important but not necessarily explain a large amount of variance – blood pressure affects the risk of cardiovascular disease, but it is not a strong enough predictor to explain a large amount of variation in outcomes. Put another way, knowing someone’s blood pressure can’t tell you with much certainty whether a particular individual will suffer from cardiovascular disease.