I have a dataset containing one response variable, and 3 independent variables. There are 6 number of observations. I want to see, in AICc framework, which of these independent variables best explain my response variable. I have chosen AICc because my number of observation is low.
I have 8 models in my model set (all possible models, without interactions, plus a null model). FYI, those three independent variables are chosen from a set of more than 20 variables based on literature (working hypotheses), and after assessing collinearity.
AIC results seem to match with my thoughts and hypotheses. One of the models (with two independent variables) is the best model (has the lowest AIC). This model has Adj-R2 of ~0.60, which is among the highest in my model set. And for the fully saturated model (with all three independent variables), Adj.R2 and AIC do not improve, which matches with AIC penalty etc. But according to AICc, the null model is the best one (the lowest AICc), and the more variable is added to the model, the higher the AICc.
How it is possible to have completely different interpretation for AIC and AICc? I am not sure if it is reasonable to have the best AICc for the null model (which has Adj-R2=0), while another model that has Adj.R2 = 0.60 having one of the highest AICc. Any thoughts?
I should add that the parameter estimates (slope) are not high in none of the models (usually less than 0.02) and the intercept remains high and close to what it is in the null model (i.e., the intercept for null model is around 1.2, and even for the best model according to AIC, it remains around 0.8. This may suggest, I think, the variability in the response variable is not that much, and is not being explained by the independent variables. But as mentioned, Adj.R2 is 0.6 for the best model according to AIC, which shows there are some variation in the response variable (although not that much) and this variation is explained by the two variable.
Any input and thoughts will be appreciated. Mehdi