If the relationship between x and y is "rather random" then it's rather random; transformations are not likely to help that. Transformations aren't meant to create a correlation between two variables where there was none; transformations are meant for a single variable that's non-normally distributed. i.e., the decision to transform or not transform should be based on looking at that one variable, rather than the relationship between the two variables. In fact, trying a bunch of different transformations to try to get one that's significant is essentially p-hacking (also known as "fishing" and several other things) and is very bad practice (see e.g. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1850704, http://www.stat.columbia.edu/~gelman/research/unpublished/p_hacking.pdf).  If you collected data expecting there to be a relationship between X and Y but there wasn't one, then you report the data honestly and acknowledge that the predicted effect wasn't there; you don't try to go create it.

Also note that transforming variables to look more normal is not really recommended practice anymore.

OK thanks a lot for your answer. It really helps.

hi Stephen

I have one more question. Can you suggest any articles which report random relationships between dependent and independent variables

You mean "no relationship"? There are a lot. I don't know your field, but surely you can find some articles. In an article testing a lot of variables, probably several of the variables have no relationship.

If you were looking to find a relationship between two variables and you didn't find it, then this is called a "null result". There are lots of places where null results can potentially be published; in fact, this issue has been discussed very much recently in the statistics world. If you search online for discussions about null results you will find a lot.

Thank you so much Stephen. My field in accounting by the way and yes when I search for 'null results' I can find many articles. Thanks a lot

Dear Stephen

I am confused about another thing. I have searched online for the articles which give null results in my field. I am particularly trying to figure out how to report null results formally in a thesis orarticle. What they are doing is they are putting the variables as independent variables in the form of a linear regression first. Then they are reporting the p values and saying that these variables are not significantly related to the dependent variable. None of them is reporting the scatter plot.

My question is why are they putting these variables as independent variables in a linear regression in the first place? When the scatter plot predicts that there is no relation shouldn’t those variables be dropped there and then (and not be put in the regression at all)? shouldnt only those variables be placed in a linear regression which are linearly related to y?

Dear Saadia,

What data you want to transform? This is the data from the file (non linear data.xlsx) in the attachmeht to your quation ("Can i run an OLS regression?"), is not it? If so, which of the independent variables are random variables?

Dear Saadia,

Scatter plots are just supporting evidences to desciptive statistics of your variables. You need to include numerical statistics like Kolmogorov-Smirnov, Shapiro-Wilks, etc. depending on your data sets. If you have tried ladder-of-power transformation, you can try box-cox transformation which is normally being carried out after all the ladder powers failed.

Hi Sergiy,

Thanks for your reply. For example if you look at the independent variable 'board size' it has a random relationship with the dependent variable 'amount of fraud'... how to transform this into linear to be able to run ols?

Like I said above, you can't use transformations to create a correlation where there was none.

Also, out of all your variables, "board size" is one of the few that does seem to show an effect on amount of fraud. Looking at the scatterplot, there appears to be a slight negative trend.

thanks Stephen. Can you please also answer the following. I am really grateful to you for your help.

I am confused about another thing. I have searched online for the articles which give null results in my field. I am particularly trying to figure out how to report null results formally in a thesis orarticle. What they are doing is they are putting the variables as independent variables in the form of a linear regression first. Then they are reporting the p values and saying that these variables are not significantly related to the dependent variable. None of them is reporting the scatter plot.

My question is why are they putting these variables as independent variables in a linear regression in the first place? When the scatter plot predicts that there is no relation shouldn’t those variables be dropped there and then (and not be put in the regression at all)? shouldnt only those variables be placed in a linear regression which are linearly related to y?

Hi Saadia,

For example I built the linear regression with the normalized dependent variable 'amount of fraud' and the normalized independent variable 'board size' by SB family of Johnson system and OLS (see fig.1 in attachment).After that the non-linear regression (black line), 95% confidence interval (green lines) and 95% prediction interval (red lines) for non-linear regression were built based on SB family of Johnson system and the linear regression (see fig.2 in attachment).

Saadia, what should be reported and what should be put in a linear regression always depends on your hypotheses and what you are trying to test (and on whether the purpose of research is only to predict outcomes, or is to identify effects of individual variables). There is never one hard-and-fast answer to what should be analyzed.

But, if some variable is of theoretical interest and a person needs to report whether or not it had a significant effect, then of course it needs to be put into the regression; if it's not in the regression then there's no way to report whether or not it is significant.

Thanks Stephen, this really helps. In fact you have clarified a lot of ambiguities.

One last thing. I have highlighted my approach in the attached file. If you have time please go through it and let me know if I am in the right direction. 

There are different types of non-linear functions in regression analysis, sometimes we apply log on one variable, sometimes on both variables. Logarithmic curve, exponential curve or hyperbolic curve may be useful.

Saadia, I don't see a need for the first step you performed (conducting separate regressions on each variable to choose which to include). You can put all variables in the same regression and use that to determine which variables you think have a significant effect. (However, you will need to check for multicollinearity. You have a large number of variables and many of them are probably correlated; if you have high multicollinearity a set of variables, then it's possible to have a good regression fit with none of the variables individually significant; in short, the regression coefficients for the individual variables will be meaningless. There are lots of resources you can find online describing how to detect multicollinearity and how to handle it, so I won't go into it in this post.)

I also still find it hard to understand what you're doing here. Calling all the variables "X1", "X2", etc., makes everything hard to follow; it's easiest to use descriptive names for all your variables. I don't understand what this "ln2" variable you created is. And I don't understand why the number of observations in the final model is 97 while in the earlier models it was 111-120.