Currently I am investigating whether or not auditor tenure (AT) has a non-linear relationship with audit quality (AQ). I expected a non-linear or parabolic relationship.
AT is measured in years, financial restatements (R) are used as a proxy for AQ.
Currently I have a set of restatements. I want to determine AT for each R so I can see how many R's there are for each AT year and can plot this. However I don't know how to obtain statistical significance for this matter.
I would be most grateful if you could help me out!
Boudewijn, It sounds to me that you want to fit a regression model to your data, which will yield estimates for an intercept (i.e., the average AT at R = 0) and a slope (i.e., change in AT for a given change in R) as well as their statistical significance. How AT is distributed will determine what distribution to assume for the errors about the regression model. For example, your standard regression assumes normally distributed error, which can be fit in almost any spreadsheet application (e.g., Excel). If your residuals (errors) about this type of model are not homogeneous about the line you fit to your data, then the distribution (or model expression) is likely inappropriate. Under such circumstances you will have to either change the model structure (eg from linear to non-linear), or use a generalized linear model with a link function to some other distribution type (e.g., Poisson, etc.).
To compare a non-linear trend to a linear trend is called model selection, you can do this by calculating an information criterion (AIC, DIC, or BIC) for the model when AT is expressed in linear terms (say y=mx+b) or non-linear terms (say quadratic - y=m1x+m2x^2+b). Generally the model with the lower criterion is the model that is favored by the data. If the models are nested, then you can also use a likelihood ratio test to compare two models. Hope this helps.
A simple way, as suggested above, is to do some transformation of the independent variable or dependent variable, such as natural log, inverse, inverse squared, square, square root, etc, and choose the model based on AIC/AICC or BIC. Don't forget to check the significance of the coefficients of the (transformed) independent variables.
Others:
In sas, there are some other procedure for non-linear regression such as PROC NLIN (http://www.ats.ucla.edu/stat/sas/library/SASNLin_os.htm) and PROC NLMIXED (if panel data) (http://www.ats.ucla.edu/STAT/sas/library/nlmixedsugi.pdf)
If you have sufficient data, make histogram(or kernel density) of AT for each AQ to see if AT is normally distributed, plot scatter plot of AT and AQ. This may help you catch the non-linear relationship and choose the model.
Your dependent variable is " whether or not auditor tenure (AT) " so it seems to me that this is a binary response. A logistic regression might be a good choice if you define AT as a binary instead of # of years.
If your DV is a binary variable, then you can use logistic regression. If you have a set of characteristics for financial restatement, you can convert it to index/ continuous variable and run a multiple regression. The other solution is to transform your IV or DV .
As I read it, the response variable AT is measured in number of years? So, the response may be log-normal, normal or Poisson, etc. but not really binary. The 'whether or not' seems to refer to statistical significance, and not a 0 - 1 response? Either way, Boudewijn, the type of distribution for your response will determine which 'flavor' of regression you should use.
I envision your model being structured something like:
AT = B0 + B1*R + error
where AT is the expected mean number of tenure years for a given R, and R is the number of restatements. If you assume the error is normally distributed about the line that is formed by B0+B1*R then you can use sum-of-squares in any standard stats software to obtain values for B0 and B1, as well as their p-values for 'significance' . If the error (aka residuals) are not homogeneous about the fitted line, then you can change either the model's structure (for example quadratic) to improve model fit (If this is the case, then this is evidence that a non-linear trend is more appropriate to describe your data), and if this doesn't work then I would suggest trying a different flavor of regression to accommodate a different error structure about the model.
Other than that, we're going to need more information about your data.
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics