Regression with variables in different scale (how to encode properly and what regression to use?)

Question

I'm working here on my research proposal for my alma mattera. I'm in accountancy science field, however, I need to deal with some statistics on this occassion. Would be very grateful if you could help me.

I have an idea to built this kind of regression model:

Y= x1 + x2 + x3

The research question is: Do audit prices, non audit services provided for audit client and auditor's tenure has an impact on audit quality?

Observation - financial reporting of particular company. Let's say, we have 40 companies and their audited financial reporting.

Y (proxy for audit quality) - count of errors in financial reporting. I guess it would suitable to group the errors found in each observation. For instance, range from 1 to 5 (where 1 - from 0 to 2 errors, 2 - from 2 to 4... and so on.). Or it could be left in relative scale - just how many errors (mistakes in financial reporting) I will found.

x1 - price of audit services of financial reporting. It could be left in model in a relative scale, or coded as 1 - price above the average price in the sample, 0 - price bellow the average price in the sample

x2 - non-audit services provided for the audit client. 1 - there were other services provided, 0 - there where no other services provided.

x3 - the tenure of auditors. Like: 1 - the same auditors are auditing the financial reporting 3 or more years in the row, 0 - the same auditors are not auditing the financial reporting more or equal 3 years. Or it could be put in the model in relative scale: 2, 3, 4, 5, ...8 years in a row.

That's my assumptions how to encode the data. Could you express your opinion how the data should be encoded actually, what kind of regression should I use here and what statistical tests would be suitable for evaluating the results?

Thank you very much

Answer 1

Perhaps the safest thing to do would be to estimate ALL sensible specifications. If the results all tell the same basic story, then you can just pick your preferred specificaiton, using some logical argument as to why you think that that one best answers the question you choose to ask. If different regression specifications lead to different conclusions, you need to dig a bit deeper and think of why that might be so.

Re: your specific questions: x1: I'd probably model this as a continuous variable, or as a factor variable for prices in different bands - x11 = below(Y/N), x12 = average(Y/N), x13(Y/N). the drawback of discretizing a continuous variable is that your selected bands are essentially arbitrary. My concern about a binary above/below is that, if many people are clustered near the mean, your estimated effects will be attenuated artificially. Again, different specifications will hopefully have consistent interpretations.

x3: I'd model this as a factor. Treating it as a continuous variable involves imposing linearity, and you might not have a good justification for doing so.

Answer 2

You could also analyze your problem to answer to this question: What is the probability that audit prices, non audit services provided for audit client and auditor's tenure may have an impact on audit quality? If so, you could estimate the probability of committing k errors, given yours independent variables (modelling x1 and x3 as continuous variables and x2 as a dummy), rearranging Y and using an ordered logit/probit approach.

EDIT: With "rearranging Y", I mean that IMHO you should use the numbers of errors as Y.

Answer 3

I would drop the x2 variables. There could be multicollinearity problems between x2 and x3. I'd also leave the x1 and x3 variables on a "relative" scale, as you say.