Age-Matching Groups Who Differ on Age

#1
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.
 
#2
One Approach

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 :)
 
#3
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.
 
#4
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.
 
#5
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.