# Am I introducing bias by assuming birthdate is middle of month?

I have a dataset containing dichotomous disease measures as well as some continuous anthropometric measures on a cohort of patients, and includes their month and year of birth as well as the exact date of data collection.

In an effort to reduce noise in my statistical analyses, I want to replace the default age variable with a more accurate one, as the provided age variable is rounded down to the nearest year. As we only have the date of birth to the nearest month (owing to data sensitivity), I made the assumption that patients were born in the middle of the month (16th for 31 day months, 15.5th for 30 days months, etc...) then calculated the difference between their assumed D.O.B. and their "data collection" date, giving me an approximate age in days, accurate to the closest half month.

A colleague informs me that I'm introducing bias into my derived age variable and they have attempted to explain to me why, but I'm simply unable to understand their explanation. They also tell me that what I am doing is no more accurate than simply counting in months (i.e. rounding the data collection date to the nearest month then finding the difference in number of months), but surely by using the exact date of collection, I am incorporating more information?

My questions are:

1. Is my derived age variable biased? If so, why?
2. Is my derived age in days more accurate than calculating age in whole months?

I apologise if this seems incredibly obvious to you and I appreciate any help you can give! Thank you.

• The material related to stats.stackexchange.com/search?q=sheppard may provide you some theoretical support. – whuber Jul 1 '16 at 15:56
• Perhaps a different question you could ask why is your method better that differencing months? Over what time scale does your phenomenon vary? Additionally, do you have enough data to make predictions that are accurate to within a fortnight or week or day? Because if you don't, then why bother? Also, you could probably use a measurement error model anyway (eg EM algorithm) – probabilityislogic Jul 2 '16 at 14:17
• @probabilityislogic This data set is not longitudinal - we have a single measure of many phenotypes but there are several years between the first and last patients being recruited, and so each patient has a different "data collection" date. The reason I wish to have a more accurate measure of age is that age is a risk factor in many of our phenotypes and so I wish to better control for age in my regression models. – tiit_helimut Jul 4 '16 at 10:44
• Fair enough - but the EM algorithm could be a useful framework. So you have "reported age" following a distribution around "actual age". You then have your model based on "actual age" which you replace with an expected value given the "reported age" that you observed. Taking the midpoint sounds like a good option in this framework. – probabilityislogic Jul 5 '16 at 11:14

Is my derived age in days more accurate than calculating age in whole months?

Obviously not. You cannot make measurement more precise then it was actually measured. Imagine that besides assuming middle of month, you assumed also that the patients were born at 12:30, 30 seconds, 30 milliseconds, etc. - would it make your measurement super precise? It is impossible to de-aggregate aggregated data. Notice that in long run your procedure would yield the same results as picking uniformly random day for each patient - would such procedure make measurement any more accurate?

• By calculating age in months, don't I need to throw away the day of the month at data collection (which we have accurate to the day)? Wouldn't I therefore have lost useful information? – tiit_helimut Jul 3 '16 at 15:57
• @tiit_helimut I'm confused: you wrote that you don't have daily data..? If you had it there wouldn't be any reason to delete it. But you don't have it... – Tim Jul 3 '16 at 18:11
• We have date of birth only to nearest month (e.g. Jan 1967, Mar 1952, etc) but we have the exact "data collection" date (this is different every patient). The age variable that I'm deriving is simply the age at data collection for each person (difference between data collection date and date of birth). – tiit_helimut Jul 4 '16 at 9:35
• I should point out that the data is not longitudinal - we just have single measures of all phenotypes at a specific point (i.e. the day the patient joined) which is different for each patient and there are several years between the first and last patient being recruited. – tiit_helimut Jul 4 '16 at 9:48
• @tiit_helimut sorry but I don't understand your comments. Do you have data about date of birth in days or in months? Do you have data about age in days or months? If you do not have data about day of birth than you don't have it. Pretending that you have it doesn't make anything more precise. – Tim Jul 4 '16 at 9:56

I'm just going to state the obvious and say that I agree with you and I can't evaluate your colleague's explanation without seeing it. For (1), I can't even tell what direction the bias should be expected to be in. Are people substantially more likely to be born near the beginning or end of a month? Not that I know of. For (2), it's obvious that using your method will give you slightly more accurate estimates than computing in whole months. The increase in accuracy may be so slight as to make no difference, but certainly your method won't hurt.

• Thanks for the response. My colleague tells me that by assuming that all patients were born in the middle of the month, that patients born at the beginning or end of the month (we don't know who these are as we don't have access to exact D.O.B.) would systematically have higher errors in my derived age variable. Surely this wouldn't be a problem unless in our regression models, the dependent phenotype variable is somehow dependent on when in the month you were born? – tiit_helimut Jul 4 '16 at 10:42
• I agree with you. And if the day of month of birth was in fact important, whole-month arithmetic as your colleague suggested wouldn't help with that a bit. – Kodiologist Jul 4 '16 at 12:28
• As far as I understand that's kind of their point - if day of the month was an important risk factor, using whole months avoids creating a "biased" age variable by us not having to assume people were born in the middle of the month. However, we lose some accuracy in the process. I guess I'm still not satisfied by my colleague's argument in this! – tiit_helimut Jul 4 '16 at 13:27