Dear Nam,

Multiple linear regression is quite a complicated technique. Firstly, there are different forms of stepwise methods.

Here are some additional issues to consider:

• Before running any model it is wise to investigate the nature of the relationship of Depression against each of your personality trains in turn using a scatter plot. You need to look for a "cigar shape" dispersion
• If any variable does not have the required dispersion shape you could consider transforming it. You might also consider checking for homoscedsticity (equal variance) using Koenker's method.
• Once you have decided which variables to include within the model you could try all possible models as there will only be 31 at most with 5 variables. Alternatively, you could try a model with one variable at a time and see which one gives you the best model then add one more in turn (essentially a manual forward stepwise method).
• You then need to look at the explanatory power of each model as measured, for example, by the adjusted R squared statistic as sometimes adding more variables makes the explanatory power go down - this is known as the principle of parsimony.
• The residuals also need to be checked for normality
• You should check for multicollinearity using VIF or tolerance
• You could also consider adding cross terms to the model

A good introductory worksheet is available from statstutor: http://www.statstutor.ac.uk/resources/uploaded/multiple-regression.pdf

The easiest way to find which predictors are significant is to put them all in the model and see. This presupposes, though, that they variables do not have a high level of multiollinearity between them; if they do, it's possible that none of them will show up as significant even though in reality they are all significant. If that were the case you would need to look into orthogonalizing the variables or doing some kind of dimension reduction like PCA.

Since the number of predictors are less (5), subset model selection will be a good choice to start with. Lasso and Ridge is suitable when p is considerably large though it can be used otherwise too.

Even manually it is possible to select the predictors considering the correlation ,interaction etc. with other predictors. Checking the R^2, AIC,BIC will be helpful in deciding the selection of predictors