Did you run the normality tests on the predictors or the residual of your model? Most parametric techniques like regression ask for normality of residuals, not inputs. The degree of non-normality is important too, many factors can result in a significant normality test that don't amount to much. 

Can you ask your question differently? I'm having trouble understanding exactly what you're asking.

Thank you Mr Rafael, please I have run the normality test but the result shows that the data is not normally distributed. But, I want to investigate whether sets of dependent factors variable can influence an independent variable. So, I want to know which test of significance I can use for this.

You should run a standard regression model and look at the model residual. If that is approximately normal, you should be fine . See Gelman's discussion here: 

http://andrewgelman.com/2013/08/04/19470/

Tests of normality must be believed with caution as a large enough sample will invariably yield significance. Look at the frequency distribution of the residuals. Are they approximately normal?

if the rule for regression analysis is that the data must be normally distributed. What do you think would the case for this data that were not normally distributed. 

The rule for regression is that the ERRORS (residuals) must be normally distributed, not "the data". Did test the residuals? Even a normality test is significant, it may be a trivial difference and not worth sacrificing power by using non parametric approaches. 

If you must, you can often use bootstrapping methods to overcome limitations of normality. Assuming that it is meaningful...

Exact-based approaches are one way to handle these sorts of problems. I think, however, that there is something to be said for the "good enough" principle in certain contexts.

In a field like psychology (where we frankly do not know enough about the relevant confounding variables) there is such a thing as fitting the sample data too well (thus destroying our generalizability). 

Even the "best methods" have weakness. I find it better to get converging evidence from various approaches, sources, and techniques than to use essentially the same method for everything. Put another way, I think it's "best" to practice critical multiplism in all aspects of research. 

I do not know anything about ODA methods (and therefore cannot speak to it specifically), but will read some of the papers from the journal linked above. I appreciate the references.

Back to Paulinus's original question: I suggest trying the non-parametric ODA regression techniques that Paul is advocating if you are uncomfortable with using traditional regression techniques. I do think that any publication that came from this would be better with both analyses included.

Raf

I think this question goes back to what "non-normal" actually means here. Are we talking left or right skewed or bi/multi-modal data? Is it leptokurtic or power law or what? All of these things imply different things about your data. For instance, bi-modal data may suggest that you have one or more additional categorical variables in your data, making it possible to looking at some form of multiple regression. Conversely, a log or other transformation may solve your problem but with a cost to interpretive power. For instance, I tend to work with network data, so it's not uncommon to use the log of both the explanatory variable(s) and their associated responses (Power Law).

Rafael thank you so much on your point. Remember I am not a statistician and that is why I have ask this question. When a data is not normal, it means that it is either positively or negatively skewed and does not have the bell shape of a normal distribution curve. Again, I have run a correlation analysis using Spearman's procedure but the results shows that all the dependent factors were within themselves correlated and none did show a correlation with the independent variable. Please I need a more clear answer in this regard. Thank you all 

Paulinus - can you plot the distributions for us or offer some normalized statistics?  eg. can you offer the mean, median and standard deviations? That said - you can transform the data using the natural log of the values and this will tend to give you normal distributions. The sacrifice is that you lose interpretive value.

@Ryan: " The sacrifice is that you lose interpretive value." - No. It is just that you seek to explain relative changes rather than absolute changes. This is often particularily sensible.

@Paul,

Any response I have (I think) would be better framed if I read up more on ODA first. 

Rafael ("Not-quite-yet-a-doctor") Garcia