Best way to compare changes in proportions I am reviewing a manuscript so I cannot be specific but the authors had a treatment that was supposed to improve access to care. They had a treated group and a control group. Access to care was measured in three different areas. Now, with made up numbers, suppose, in the treated group:
Area 1 went from 10% to 20%
Area 2 went from 50% to 60%
Area 3 went from 80% to 90%
while there were very minor changes in the control group.
(The "areas" are things like "prenatal care", "wellness visits" and such, where the ideal is 100%)
The authors called area 1 a 100% improvement, area 2 a 20% improvement and area 3 a 12% improvement. This obviously isn't wrong but it strikes me as misleading.
What would be better measures?
 A: Phrases like "percent differences" or "percent improvements" are a big problem. It's often not clear whether those who use that phrase are talking about percent differences from a pre-treatment condition, as in your example, or percentage-point differences. And the dependence of "percent differences" on the value of the denominator (e.g., pre-treatment value) can lead to all sorts of misleading interpretations.
Consider asking the authors to specify in the text both the magnitude of difference in the original scale and the multiplicative factor representing post/pre-treatment access in each of the Areas. In your made-up example that could be something like:

Treatment was associated with a 10-percentage-point increase in access for all 3 Areas. Given the differences in baseline access among Areas, this represented a doubling of access in Area 1, an increase to 1.2 times baseline access in Area 2, and an increase to 1.12 times baseline access in Area 3.

That's a bit awkward but it is at least fair to the actual findings, avoids the ambiguity (and potentially misleading nature) of "percent improvement," and explicitly notes how baseline access affected the "percent improvement."
A: Comment continued:
If the counts are 40 vs. 50 out of 100 individuals, then the following printout from Minitab's 2-proportion procedure, shows a non-significant P-value of 15% > 5%.
Test and CI for Two Proportions 

Sample   X    N  Sample p
1       40  100  0.400000
2       50  100  0.500000

Difference = p (1) - p (2)
Estimate for difference:  -0.1
95% CI for difference:  (-0.237197, 0.0371975)
Test for difference = 0 (vs ≠ 0):  
  Z = -1.43  P-Value = 0.153

By contrast, the difference between the same two percentages based on larger samples of 1000, is highly 
significant with P-value < 0.0005 (denoted as 0.000 in the output).
Test and CI for Two Proportions 

Sample    X     N  Sample p
1       400  1000  0.400000
2       500  1000  0.500000

Difference = p (1) - p (2)
Estimate for difference:  -0.1
95% CI for difference:  (-0.143386, -0.0566143)
Test for difference = 0 (vs ≠ 0):  
  Z = -4.52  P-Value = 0.000

A: "Access to care" sounds like a semi-quantitative metric, using rubrics on health literacy, transporation, etc. 
It sounds like they measured a "percent" difference using a log-transform of the outcome, interpreting the exponentiated regression coefficient as the percent change. E.g. $\exp(\beta_1)=1.05 \implies$ AtC improved 5% from before. In this case it is correct, although misleading to report the "percentage change" (being a change-as-a-percent) as 100% meaning a doubling. You could ask that they report raw values alongside every "percent change" reported, further clarifying that a raw value should have units of "points". Text should read "Area 1 improved by 100% (10 points to 20 points)". It's not many more words.
You should also ask whether a log transformed outcome is the correct metric to measure change. A point-difference could also be considered. Or even more complex changes of variable. Considerations are: what is the residual error? Also to what extent does the baseline value predict growth? Log transformed outcomes predict smallest growth among worst performing clusters at baseline, this is usually an inadequate assumption; commonly the opposite is true: that best performing clusters are usually already implementing many of the successful strategies targeted by the intervention.
