I have two groups of children, A and B. Group A is the treatment group and has an age range of 5-7 years old. Group B is the control group and has an age range of 2-5 years old.

Is there a statistical way in which I can age-match these groups even though children in Group B are systematically younger than children in Group A?

I've been introduced very briefly to propensity scores, but I'm just not sure.

Suppose your ages are continuous variables (this might be appropriate if you have a lot of birthdays), a one-to-one correspondence can be created easily:

x - age G1, 5 <= x <= 7
y - age G2, 2 <= y <= 5.

Let f(x) = y. We seek f(5) = 2, and f(7) = 5, as well as f(x) being linear (i.e. injective). f(x) = ax + b. Solving for a, b:

f(5) = 2 = a*5 + b -> b = 2 - a*5
f(7) = 5 = a*7 + b -> b = 5 - a*7

2 - a*5 = 5 - a*7 -> a = 2*a = 3 -> a = 3/2 -> b = 2 - (3/2)*5 = -11/2

Now f(x) = 3/2x - 11/2, so that f(5) = (3/2)*5 - 11/2 = 2, and f(7) = (3/2)*7 - 11/2 = 5. f(x) effectively maps the interval [5, 7] to [2, 5]. If you want to map [2, 5] to [5, 7], use h(y) = (y + 11/2)*(2/3).

I'd only use this approach if you can accurately pair ages in group 1 with ages in group 2.

Hope it provides some insight :)

Suppose your ages are continuous variables (this might be appropriate if you have a lot of birthdays), a one-to-one correspondence can be created easily:

x - age G1, 5 <= x <= 7
y - age G2, 2 <= y <= 5.

Let f(x) = y. We seek f(5) = 2, and f(7) = 5, as well as f(x) being linear (i.e. injective). f(x) = ax + b. Solving for a, b:

f(5) = 2 = a*5 + b -> b = 2 - a*5
f(7) = 5 = a*7 + b -> b = 5 - a*7

2 - a*5 = 5 - a*7 -> a = 2*a = 3 -> a = 3/2 -> b = 2 - (3/2)*5 = -11/2

Now f(x) = 3/2x - 11/2, so that f(5) = (3/2)*5 - 11/2 = 2, and f(7) = (3/2)*7 - 11/2 = 5. f(x) effectively maps the interval [5, 7] to [2, 5]. If you want to map [2, 5] to [5, 7], use h(y) = (y + 11/2)*(2/3).

I'd only use this approach if you can accurately pair ages in group 1 with ages in group 2.

Hope it provides some insight :)

Thank you for this. I'm not sure, however, that it's getting at what I want to do. I don't really want a one-to-one correspondance between the two groups since they obviously differ on chronological age.

I think at the bottom line, age difference is serious issue in most situations, and there might not be much to do to make it "as if there isn't difference." But I might try to standardise the dependent variables in each group, and use the group-wise standardised numbers to do the rest of analysis.

I think at the bottom line, age difference is serious issue in most situations, and there might not be much to do to make it "as if there isn't difference." But I might try to standardise the dependent variables in each group, and use the group-wise standardised numbers to do the rest of analysis.

I tend to agree, having looked at other options the last couple of days. I'm meeting with a statistican tomorrow though to discuss this. Perhaps I'll provide an update if we come up with anything useful.