Stata Technical Bulletin
13
overlap! is an immediate version of overlap. For the case of equal variances, the syntax is
overlap! mean1 mean2 sd
where sd is the common standard deviation of the two variables. For the case of unequal variances, the syntax is
overlap! mean1 mean2 sd1 sd2 (sd1 ≠ sd2)
For the general case, the syntax is
overlap! mean1 mean2 sd1 sd2 n1 n2
where n1 and n2 are the numbers of observations used to estimate the means and standard deviations of the two variables. This
syntax displays both the equal and unequal variance versions of the overlapping coefficient.
Example
As an example, we compare the distributions of fuel efficiency of foreign and domestic cars in the automobile data set
supplied with Stata. We begin by calculating the t test of the null hypothesis that the average miles per gallon is the same for
both sets of cars.
. use ∖stata∖auto, clear
|
(1978 Automobile Data) | |||
|
. ttest mpg, |
by(foreign) |
Mean |
Std. Dev. |
|
---------+. 0 I |
52 22 |
19.82692 24.77273 |
— 4.743297 6.611187 |
|
---------+- Hc |
74 »: mean(x) t Pr > ∣tI |
21.2973 5.785503 = mean(y) (assuming equal variances) = -3.63 with 72 d.f. = 0.0005 | |
The t test reveals a highly significant difference between the average mileage for foreign cars (24.8 MPG) and the average
mileage for domestic cars (19.8 MPG). However, the variance of MPG seems a bit higher for the foreign cars. Since the default
t test assumes equal variances across groups, we investigate further to see whether this assumption is justified.
. sdtest mpg, by(foreign)
|
Variable |
I |
Obs |
Mean |
Std. Dev. |
|
— 0 |
I |
52 22 |
19.82692 24.77273 |
— 4.743297 6.611187 |
|
— combined |
I Ho: |
74 F(21,51) |
= sd(y) = 1.94 = 0.0549 |
5.35582 (two-tailed test) |
The evidence is mixed: the p value of the test statistic is 5.5 percent. Since we are not sure whether the variances are equal,
we calculate the unequal variance version of the t test, to see if the apparent significance of the mileage difference is sensitive
to the assumption of equal variances.
. ttest mpg, by(foreign) unequal
Variable I Obs Mean Std. Dev.
---------+---------------------------------
0 I 52 19.82692 4.743297
1 I 22 24.77273 6.611187
---------+---------------------------------
combined I 74 21.2973
Ho: mean(x) = mean(y) (assuming unequal variances)
t = -3.18 with 31 d.f.
Pr > ∣t∣ = 0.0033