1. Introduction
An outcome measure is used to analyze data from trials and is a quantitative assessment of the strength of a causal relation. The choice of outcome measure can influence one’s inferences, such as how discordant data are between trials and how strong the tested causal relation is (Sprenger and Stegenga Reference Sprenger and Stegenga2017). Here, we demonstrate that fine-grained conventional choices about a particular outcome measure can have a dramatic impact on estimates of the effectiveness of interventions.
This present article is part of a larger research program developing an innovative method that has not been widely used in the philosophy of science. We simulate patient-level data from trials on a particular class of drugs while varying features of the simulated research context, such as the real effectiveness of the drugs, the number of subjects in each trial, and in the present work, the conventional features of an outcome measure. We use simulations in the spirit of Mayo-Wilson and Zollman (Reference Mayo-Wilson and Zollman2021), who argue that simulations are like thought experiments in philosophy and indeed are often superior to thought experiments.Footnote 1
A widely used outcome measure in meta-analyses is the responder odds ratio, which is defined as the ratio of the odds of being a “responder” in the drug group divided by the odds of being a responder in the placebo group. A responder is a subject whose symptoms drop below a threshold c. In trials and meta-analyses of antidepressants, c is typically 50% of a subject’s pretrial symptom severity. Symptom severity in antidepressant trials is measured with various scales; one common scale is the Hamilton Depression Rating Scale (we will refer to this as the H scale, with corresponding H scores), a 50-point scale in which higher scores represent greater symptom severity. So, if you are a subject in a trial and your H score at the start of the trial is 26, and it is 12 at the end of the trial, you would be deemed a responder. A responder odds ratio of greater than 1 implies that a drug is effective; conversely, a responder odds ratio of close to 1 suggests that the drug is relatively ineffective.
An inference from a particular value of a responder odds ratio to a claim that a drug is effective is dubious, and although establishing this point is not our main objective, we give an argument for this as well. Our main objective is to show that this commonly used value for c falls within a range that maximizes the responder odds ratio and to explain this phenomenon. If other values for c were used in meta-analyses of antidepressants, it is very likely that lower values of the responder odds ratio would be found.
A brief thought experiment is suggestive. Based on past trial data, we know that subjects start trials with an average H score of about 24, subjects in the drug group end trials with an average H score of about 11, and subjects in the placebo group end trials with an average H score of about 14. If c were very low—say, 5%—then few subjects in either the drug group or the placebo group would be deemed responders, and thus the responder odds ratio would be close to 1, and the drug would be deemed ineffective. Conversely, if c were very high—say, 95%—then many subjects in both groups would be deemed responders, and thus again the responder odds ratio would be close to 1, and thus again the drug would be deemed ineffective. A c value of 50% looks to be roughly in a “sweet range” to maximize the responder odds ratio.
One way to test these suggestions would be to get patient-level data from trials on antidepressants and then compute the responder odds ratio while varying c. However, save few exceptions, no one has access to these data. Even the latest and biggest meta-analyses get access only to group-level summaries of trial data (Cipriani etal. Reference Cipriani, Furukawa, Georgia Salanti, Atkinson and Yusuke Ogawa2018).
Our approach gets around the problem of data access. Moreover, our approach allows us to test counterfactuals, such as scenarios in which the tested drugs are very effective or ineffective, and scenarios in which a different measurement scale is used. Tabatabaei Ghomi and Stegenga (Reference Tabatabaei Ghomi, Stegenga, Hens and De Blockforthcoming) show that setting c around 50% indeed nearly maximizes the responder odds ratio and thereby maximizes the estimate of drug effectiveness, and other values of c give lower values for the responder odds ratio. These results are consistent with previous work, such as Hadzi-Pavlovic (Reference Hadzi-Pavlovic2009). In the present article, we explore the reasons for this phenomenon.
Here is another way of thinking about our thought experiment mentioned earlier. For a causal relation evaluated in a trial with data generated by a particular measuring instrument, there is a sweet range for c which minimizes the proportion of subjects deemed responders in the placebo group and maximizes the proportion of subjects deemed responders in the drug group, thereby maximizing the responder odds ratio (for similar suggestions, see Ragland Reference Ragland1992).
Now consider another thought experiment. Suppose we modify the H scale by adding questions that are irrelevant to the causal relation under investigation and that add the same score to all subjects, such as “Are you human?” and “Is your body composed of at least five atoms?” This increases the total possible H score without changing the absolute difference in mean scores between the drug and placebo groups. The sweet range for c that maximizes the responder odds ratio will correspondingly increase. We evaluate this via our simulation approach, and we confirm our speculation.
Another objective of this article is to enter the recent debate between some philosophers, who argue that absolute outcome measures are informative while relative outcome measures are misleading, and other philosophers, who argue that both absolute and relative outcome measures can be informative. The responder odds ratio is an example of a relative outcome measure that is misleading and uninformative about the real effectiveness of interventions.
2. Methods
Our simulation approach was initially presented in another article (Tabatabaei Ghomi and Stegenga, Reference Tabatabaei Ghomi, Stegenga, Hens and De Blockforthcoming). We briefly reiterate it here, adding details particular to this article.
2.1 Modeling responders
OR is the odds of being a “responder” in the drug group (
${O_d}$
) divided by the odds of being a responder in the placebo group (
${O_p}$
). The odds of being a responder in the placebo or the drug group equals the number of responders divided by the number of nonresponders in each group:
$$OR = {{{O_d}} \over {{O_p}}}$$
$${O_g} = {{\left| {{R_g}} \right|} \over {\left| g \right| - \left| {{R_g}} \right|}}\;,\;g \in \left\{ {d,\;p} \right\},$$
where
${R_g}$
is the set of responders in the drug (
$g = d$
) or the placebo (
$g = p$
) group, and
$\left| {{R_g}} \right|$
and
$\left| g \right|$
are the sizes of
${R_g}$
and g respectively.
A subject i in group g counts as a responder (
$i \in {R_g}$
) if their H score after treatment (
$H_i^a$
) is less than or equal to a fraction c of their H score before treatment (
$H_i^b$
):
$$\;{R_g} = \left\{ \forall i \in g\;|\;{{H_i^a} \over {H_i^b}}\; \le c\right\} .$$
A drug is deemed effective if the lower 95% confidence interval of OR is higher than 1 (Bland Reference Bland2015).
2.2 Simulating patient data
We used a random generator to generate patient data before and after treatment with drug or placebo based on the parameters that we calculated from published meta-analyses of trials on antidepressants. Our parameter estimates were based on one of the most recent and most comprehensive meta-analyses of antidepressants (Cipriani etal. Reference Cipriani, Furukawa, Georgia Salanti, Atkinson and Yusuke Ogawa2018). Because the absolute values of these parameters are central to this article, in this section we provide some details of how we estimated them.
The most frequently used scale was the 17-question H scale, used in 537 out of 1,199 parameters reported in the supplementary material of Cipriani etal. (Reference Cipriani, Furukawa, Georgia Salanti, Atkinson and Yusuke Ogawa2018). After excluding reports with missing data, we ended up with 105 parameter values for the placebo and 337 values for various drugs (drug values were combined, regardless of drug identity). We calculated both the simple and the weighted averages (by the number of participants in each study) of the parameter values (weighting had a negligible impact). Because Cipriani etal. (Reference Cipriani, Furukawa, Georgia Salanti, Atkinson and Yusuke Ogawa2018) report standard deviations only after treatment, we used Hieronymus etal. (Reference Hieronymus, Fredrik Emilsson, Nilsson and Eriksson2016) to obtain the standard deviations of the placebo and drug groups before treatment. The other distribution parameter values reported by Hieronymus etal. (Reference Hieronymus, Fredrik Emilsson, Nilsson and Eriksson2016) were close to the values we calculated and so cross-validated our results. The calculated values are reported in table 1.
Table 1. Parameters Used in Simulations
We simulated trials of various sizes, from 100 to 500 subjects (equally distributed between the drug and placebo groups; the size of each group is denoted by n). In our previous study, we varied drug effectiveness, indicated by the after-treatment mean H score in the drug group (
$m_d^a$
). For each combination of values for
$m_d^a$
(seven values, from 9 to 15) and n (nine values, from 50 to 250), we repeated the simulation 5,000 times (thereby generating data for 315,000 trials in total). In the present study, we fixed the drug effectiveness to
$m_d^a = 11$
(estimated from meta-analyses), but we varied the H scale as described in the following section, resulting in different absolute values for mean H scores in the drug and placebo groups before and after treatment.Footnote
2
We repeated the simulation 5,000 times for each combination of n and H scale (thereby generating data for 225,000 trials in total).
2.3 Adjusting measurement scale
We simulated data using multiple versions of the H scale. The base scale is H50, the 17-question scale commonly used in real trials, with a total of 50 points. The mean values reported in table 1 are based on this scale. We adjusted this scale by adding a number of irrelevant questions for which all subjects would answer affirmatively, generating a scale with a total of 57 points (H57), 67 points (H67), 88 points (H88), and 151 points (H151). Section 3 explains why we used these particular scales. These additional questions shift
$m_d^b$
,
$m_d^a$
,
$m_p^b$
and
$m_p^a$
up by a fixed value equivalent to the number of added questions without changing the absolute difference between the means of the placebo and drug groups. For each scale, we assessed the effect of varying c on the difference in the number of responders in the placebo and drug groups, the lower 95% confidence interval of OR, and the probability of concluding that a drug is effective.
3. Results and discussion
3.1 The sweet range
The probability of concluding that a drug is effective under various combinations of n,
$m_d^a$
and c is displayed in figure 1; this was one of the main results of Tabatabaei Ghomi and Stegenga (Reference Tabatabaei Ghomi, Stegenga, Hens and De Blockforthcoming), and we reproduce it here to lay the grounds for the remainder of our results. We begin by noting three features of figure 1. First, as the effectiveness of the drug decreases from
$m_d^a = 9$
to
$m_d^a = 15$
, the probability of concluding that a drug is effective decreases, as expected. Second, the probability of finding a drug to be effective increases by increasing the number of participants, which is a result of the increased power of larger trials. Third, and most relevant to our present interest, the influence of the (arbitrary and conventional) value of c on inferences of effectiveness is abundantly clear.
Figure 1. The probability of concluding that a drug is effective under various combinations of sample size (
$n$
), real drug effectiveness (
$m_d^a$
), and conventional threshold for definition of responder (
$c$
).
We observe three key features of the influence of c on the probability of concluding that a drug is effective. First, as mentioned earlier,
$c \approx 50\%$
nearly maximizes the probability of concluding that a drug is effective. Second, the range of c with a high probability of concluding that a drug is effective narrows as we go toward less effective drugs (higher
$m_d^a$
). And third, the optimum range of c for concluding that a drug is effective shifts slightly to the right as we go toward less effective drugs. In Tabatabaei Ghomi and Stegenga (Reference Tabatabaei Ghomi, Stegenga, Hens and De Blockforthcoming), we noted these phenomena but did not explain them. Here, we provide an explanation for these phenomena, thereby articulating the source of a systematic bias in some trials and meta-analyses.
There is a “sweet range” for c that minimizes the proportion of subjects deemed responders in the placebo group and maximizes the proportion of subjects deemed responders in the drug group, thereby maximizing the lower confidence interval of the responder odds ratio; this can be formally represented as follows:
$${{m_d^a} \over {m_d^b}} \lt c \lt {{m_p^a} \over {m_p^b}}{\kern 1pt} {\kern 1pt} \left( {{\rm{sweet\ range}}} \right).$$
Here is why the sweet range is the case. Recall that an individual patient in the drug or placebo group is deemed a responder if the ratio of her H score after treatment over her H score before treatment (
$H_i^a/H_i^b$
) falls below c. A c value that on average satisfies this criterion for patients in the drug group but not for the patients in the placebo group increases the difference between the number of responders in the two groups and thus increases the OR. c values within the sweet range have this property.Footnote
3
When using H17, on average the mean H score before treatment is 24, and the mean H score after treatment is 14 and 11 for the placebo and drug groups, respectively. Coincidently, these values are such that
$c = 50\%$
falls within the sweet range:
$${{11} \over {24}} = 0.45 \lt c = 0.5 \lt {{14} \over {24}} = 0.58.$$
This is why we see, in figure 1, the probability of deeming a drug to be effective clustered in a range bounding c = 0.5.Footnote 4
As drugs become less effective,
$m_d^a$
increases, and consequently, the lower bound of the sweet range approaches the upper bound. This explains the dome shape of figure 1, in which a progressively smaller range of c values has a high chance of finding less effective drugs to be effective. Also, as the effectiveness of drugs decreases to that of placebo (
$m_d^a \to m_p^a$
), the lower bound of the sweet range increases so much that
$c = 0.5$
falls out of the sweet range, and higher values of c are within the range. This explains the slight shift toward higher values of c at the peak of the inverse dome.
These are, in part, numerical consequences of the arbitrary range of scores of H17. Were the H scale designed differently, assigning different absolute values to patients,
$c = 0.5$
would not have been within the sweet range. To show this, we repeated the simulations, this time with our hypothetical H scales with additional irrelevant questions. Although the drug effectiveness is fixed in these simulations (and consequently, the absolute mean difference between drug and placebo H scores is fixed), the absolute value that each H scale assigns to patients shifts, and consequently, the absolute values of mean H scores in both the drug and placebo groups differ between scales.
We chose the particular adjustments to the H scale such that the following values of c fall within the sweet range:
-
H50 ⇒ c ≈ 50%
-
H57 ⇒ c ≈ 60%
-
H67 ⇒ c ≈ 70%
-
H88 ⇒ c ≈ 80%
-
H151 ⇒ c ≈ 90%
Figure 2 shows the average difference between the number of responders in the placebo and drug groups in the simulations of trials with 500 subjects. The results in other trial sizes are similar (save small stochastic fluctuations) and so are not shown. The average difference between the number of responders in the placebo and the drug groups maximizes approximately at 50%, 60%, 70%, 80%, and 90% for H17, H57, H67, H88, and H151, respectively. This maximum difference, in turn, maximizes the lower 95% confidence interval of OR (figure 3) and, consequently, the probability of concluding that the drug is effective (figure 4). Figures 3 and 4 also clearly visualize the concept of the sweet range (the spread of the darkest red tone). This result shows that were the H scales designed differently, some drugs that were deemed effective by the OR would have been deemed ineffective based on exactly the same data.
Figure 2. Difference in the number of responders between the drug and placebo groups.
Figure 3. Average lower 95% confidence interval of
$OR$
under combinations of various H scales and
$c$
values.
Figure 4. Probability of finding the same drug to be effective under combinations of various H scales and
$c$
values.
3.2 Superiority of absolute outcome measures
Our results emphasize the superiority of absolute measures, such as the difference in mean H score reduction between the drug and placebo groups, compared with relative measures, such as OR. The analyses with the newly created H scales show that the value of the OR is sensitive to arbitrary changes to the H scales, whereas the absolute mean H score reduction is not sensitive to such changes. Choose a particular c on the x-axis of figure 3, and move up the corresponding column; as the H scale changes, so does the probability of concluding that a drug is effective based on OR. Yet this is an undesirable feature of an outcome measure. Recall that the questions that were added to the newly formed H scales are, by stipulation, totally irrelevant to that which is being measured (namely, severity of depression symptoms).
A plausible desideratum for an outcome measure is that if an outcome measure is used to summarize the same data from a trial generated from two measurement scales that are identical in every respect relevant to the causal relation under investigation and differ only in ways causally unrelated to the causal relation, then the outcome measure should report the same value. For example, if you measure the temperature today at lunchtime and then at dinnertime using a blue Celsius thermometer and a red Celsius thermometer, you should expect to compute the same lunch–dinner temperature difference with both thermometers because the color of the thermometers is causally irrelevant to measuring temperature. The difference in the mean reduction in symptom severity between the drug and placebo groups satisfies this desideratum (because the mean H scores for both the drug group and the placebo group scale up identically when changing the H scale, the mean difference between the groups remains the same). As we have seen in figure 3, the responder odds ratio does not satisfy this desideratum.
In general, the responder odds ratio involves a loss of information because the definition of responder does not include the magnitude of change in symptom severity for a subject. Consider a hypothetical weight-loss drug and a trial in which a responder is defined as anyone who loses at least a nonzero amount of weight. Suppose that in the trial of the drug, there were 100 subjects in each of the drug and placebo groups, and in the drug group, 75 subjects lost 10 grams, yet 25 subjects gained 5 kilograms, and in the placebo group, 50 subjects lost 10 grams, and 50 subjects gained 10 grams; this drug would have a responder odds ratio of 3 in favor of weight loss, even though it caused virtually no weight loss in most subjects and caused substantial weight gain in a large proportion of subjects. For reasons such as this, statistician Stephen Senn claims that measures such as the responder odds ratio are “liable to be extravagantly interpreted” (Senn Reference Senn2003, 239).
Although some philosophers have argued that both absolute and relative measures should be reported on the presumed grounds that both are informative (e.g., Hoefer and Krauss Reference Hoefer and Krauss2021), our analysis here supports those who have argued that reporting relative measures can be misleading. Stegenga and Kenna (Reference Stegenga, Kenna and McClimans2017) and Sprenger and Stegenga (Reference Sprenger and Stegenga2017) argue for the superiority of absolute outcome measures for binary outcomes, and our argument here extends this to continuous outcome measures.
4. Conclusion
Our results show that the conventional choice of
$c = 50\%$
in the responder odds ratio is in a range that maximizes estimates of the effectiveness of antidepressants. The dependence of the probability of concluding that a drug is effective on the choice of c indicates the problem with this arbitrary measure. We further offer an explanation for where this “sweet range” comes from, along with other nuances related to this sweet range.
In what is probably the most significant meta-analysis of antidepressants to date, the 95% lower confidence interval of the odds ratio for antidepressants was estimated to be between 1.37 and 2.13, with many dugs showing values around 1.6 (Cipriani etal. Reference Cipriani, Furukawa, Georgia Salanti, Atkinson and Yusuke Ogawa2018). This might strike you as rather modest. Yet, we hope to have demonstrated how uninformative such a measure is about the real effectiveness of the drugs. Moreover, we have shown that this value depends on a conventional choice for c, and other choices for c would entail even lower values for the responder odds ratio. It is a happy coincidence for those who wish to substantiate the putative effectiveness of antidepressants that the commonly used value of c in meta-analyses of antidepressants is tuned to the commonly used measurement scale and actual facts about subjects in trials of antidepressants such that the responder odds ratio is nearly maximized.
Our method deployed here is an example of a possible way to evaluate speculative hypotheses about research practices and can be extended to other domains.
Acknowledgments
For commentary and discussion, we are grateful to Cristian Larroulet Philippi, Adria Segarra, Sophia Crüwell, Adrian Erasmus, Oliver Holdsworth, Ina Jantgen, Charlotte Zemmel, Zinhle Mncube, and two anonymous referees.
Open access
