What Difference Representation: Inconclusive Evidence
Congratulations to the authors on an excellent study that promotes and explores the importance of random assignment.
My comment supports the article’s emphasis on caution and not overgeneralizing. My focus is on the article’s Question 2: Did an offer of HLAB representation increase the probability that the claimant would prevail? My analysis of the simple frequencies (I have not delved into the regressions and ignore weights) suggests that HLAB attorneys should view the results as modest, but inconclusive, evidence that an offer of representation improves outcomes.
Based on Table 1, page 24, there are 129 No offer observations and 78 Offer observations. Ignoring weights, which I think are said not to make a huge difference, page 26 reports that .76 of claimants who received an offer prevailed in their first level appeals, and that .72 of claimants who did not receive an offer prevailed in their first level appeal.
So, those who were offered representation fared better; one measure of which is they did .04/.72 x 100, or 5.6% better. Given the high background (no offer condition) rate of prevailing, the maximum improvement (to 1.00 success rate) is .28/.72 x 100 or 38.9%. Another measure could be the proportionate reduction in defeat. The no offer group was “defeated” 28% of the time. The offer group was defeated 24% of the time. The reduction in defeat is .04/.28 x 100 is 14.3%. This measure has the sometimes attractive feature that it can range from 0% to 100%. So by this measure the offeree group did 14% better than the non-offeree group, a modest improvement for the offer condition.
A concern expressed in the paper is that the result is not statistically significant. This raises the question: given the sample size, how likely was it that a statistically significant effect would be detected? Assessing this requires hypothesizing what size effect of an offer would be of societal interest. Suppose we say that lawyers should do about 10% better and move the win rate from .72 for non offerees to .80 for offerees. This is an 11.1% improvement by the first measure and a 28.6% improvement by the second measure. Both strike me as socially meaningful but others might specify different numbers.
We can now pose the question: given the sample size and the effect of specified size, what is the probability of observing a statistically significant effect if one exists? I use the following Stata command to explore the statistical power of the study:
sampsi .72 .80,n1(129) n2(78), which yields the following output:
Estimated power for two-sample comparison of proportions
Test Ho: p1 = p2, where p1 is the proportion in population 1 and p2 is the proportion in population 2
alpha = 0.0500 (two-sided)
p1 = 0.7200
p2 = 0.8000
sample size n1 = 129
n2 = 78
n2/n1 = 0.60
power = 0.1936
A power of 0.19 is too low to conclude that the study was large enough to detect an effect of the specified size at a statistically significant level. If one concluded that an offer of representation did not make a significant difference from this study, there is a good chance the conclusion would be incorrect. To achieve power of about 0.70, one would need a sample four times as large as that in the study. If one thought that smaller effects were meaningful, the sample would be even more undersized.
I think my analysis so far underestimates the benefit of an offer by HLAB attorneys. Perhaps we can take .72 as a reasonable lower bound on success. Even folks without an offer succeeded at that rate. The realistic upper bound on success is likely not 1.00. Some cases simply cannot be won, even by the best lawyer in the world. Perhaps not more than 90% of cases are ever winnable, with the real winnable rate likely somewhere between .8 and .9. If the winnable rate was .8, then the offer got clients halfway there, from .72 to .76. If the real rate was higher, the offer was less effective but not trivial in size. At .9, the offer got the clients 22% closer to the ideal. The study just was not large enough to detect much of an effect at a statistically significant level.
So while I agree that the study provides no significant evidence that an offer increases success, my analysis (obviously incomplete) suggests that the study provides no persuasive evidence that an offer does not increase success. The study is inconclusive on this issue because of sample size.
HLAB lawyers should not feel that they have to explain away these results; the results modestly, but inconclusively, support the positive effect of an offer because they are in the right direction in a small study.