Steady state trials: another valid substitution of counterfactual ideal to measure causal effects
 Okujou Iwami^{1}Email author and
 Masayuki Ikeda^{2}
https://doi.org/10.1007/s1219901203128
© The Author(s) 2012
Received: 18 July 2012
Accepted: 9 October 2012
Published: 31 October 2012
Abstract
Objectives
Many traditionally established medical interventions are not examined with randomized trials especially in emergency medicine. We researched what is the scientific basis of the measurement of the causal effect in these interventions and proposed another trial to measure causal effects.
Methods
We deduced steady state trials from the counterfactual model and used Bayesian approaches to estimate causal effects statistically.
Results
When the state of the observed person is fairly steady before an exposure, the ratio of the afterperiod to the beforeperiod of the exposure is sufficiently small, and changes are obtained in relatively short time, it is possible to postulate that the state of the counterfactual person to be compared is almost equal to the state of the real person before the exposure. Bayesian approaches show that the causal effect of the exposure is estimated even in only oneperson steady state trials, when large changes are observed.
Conclusions
Steady state trials are valid methods to measure causal effects and can measure causal effects even in oneperson trials. When we can measure the causal effect of interventions with steady state trials, these interventions should be regarded as scientific without use of randomized trials.
Keywords
Crossover trials Counterfactual model Steady state Period ratio Individual causal effectIntroduction
Evidencebased medicine (EBM) appeared as a handy tool kit for clinicians who had not understood the basic thinking of epidemiology [1]. After the advocates of EBM succeeded in nominating randomized trials to be paramount [2], the socalled “Hierarchy of Strength of Evidence” towered in medical practice and many clinical guidelines prostrated themselves in front of the pyramid [3, 4]. Many traditionally established medical interventions were stripped of their rank for reasons having to do with observational studies. Under these circumstances, Smith and Pell [5] asked a sarcastic question why protagonists of EBM did not participate in a randomized trial of parachute use.
In epidemiological studies, the counterfactual or potentialoutcome model has become increasingly standard for causal inference [6–8]. However, the theoretical ideal to measure causal effects of exposure is impossible. To achieve a valid substitution for the counterfactual experience, we resort to various design methods that promote comparability. One approach is a crossover study and another is a randomized trial. Other approaches might involve choosing unexposed study subjects who have the same or similar riskfactor profiles for disease as the exposed subjects [9]. Casecrossover design was introduced for estimating a short term, transient effect of intermittent exposures on acuteonset diseases [10, 11]. For each case, one or more predisease or postdisease time periods are selected as matched control periods for the case. The exposure status of the case at the time of the disease onset is compared with the distribution of exposure status for the same person in the control periods. The key feature of the casecrossover design is that each case serves as its own control. In this paper, we expand this key feature and propose another valid substitution of the counterfactual ideal to measure causal effects and show that parachute use and many interventions in emergency medicine have the scientific basis of the causal inference without randomized trials.
Materials and methods
Results
Steady state trials
 t :

time
 T _{0} :

the time when the observation starts
 T _{1} :

the time when the exposure is done
 T _{2} :

the time when the outcome is observed
 B = (T _{1} − T _{0}):

the period before the exposure
 A = (T _{2} − T _{1}):

the period after the exposure
 n :

the integer which gives the ratio of A to B, A:B = 1:n
 S :

the state of the observed person which is a function of time
 X :

the state S just before the time T _{1}
 Y :

the state S at the time T _{2}
 Z :

the state of the counterfactual ideal of the unexposed person which is a function of time
 W :

the state Z at the time T _{2.}
However, we cannot really observe \( \frac{{{\text{d}}Z}}{{{\text{d}}t}} \), so the value of (k + δz) is replaced with the observed value of (k + δ_{ is }). In order to estimate the difference between (Y –X)/A and (k + δ_{ is }), we postulate that the distribution of (Y – X)/A follows the normal distribution with the same variance as σ^{2} which is estimated by the sample variance of (k + δ_{ is }). Then the difference between (Y – X)/A and (k + δ_{ is }) can be statistically estimated with the t distribution. When the outcome Y has the quality different from the state X, the nominal scale is applied.
Statistical inference of causal effects
Suppose when we observe the outcome Y which belongs to the category different from the state X and the change is of practical importance, or when the difference between (Y – X)/A and (k + δ_{ is }) is statistically significant and large enough to be of practical importance. We now discuss the causation of the incidence of such an important outcome Y, which we designate Y _{imp} in the following discussion.
As the state is steady and the period A is small relative to the period B, we can postulate that the counterfactual condition of the unexposed state in the period A is equivalent to the real condition in the period B, and the probability that Y _{imp} happens within the time span of the period A during the period B is equal to θu. Then the period B has a sequence of ntimes repetitions of a trial with constant probability θu.
When we are uncertain about the prior distribution, we adopt Beta(0.5, 0.5) or Beta(1, 1) as the reference prior distribution for θe and θu. The posterior distribution of θu shifts to zero as n becomes larger. With the reference prior Beta(0.5, 0.5) for θe and θu, the lower limit of the 95 % credible interval of Δ is over zero when n is equal to or more than four. With the reference prior Beta(1, 1) for θe and θu, the lower limit of the 95 % credible interval of Δ is over zero when n is equal to or more than seven. Classical statistical approaches also show similar results [15, 16]. The larger the number of n is, the more credibility we can gain in the inference of the causal effect. However, the lower limit of the 95 % credible interval of Δ cannot be over 0.15 with the prior Beta(0.5, 0.5) and not over 0.16 with the prior Beta(1, 1), no matter how n may become large. This is the limitation of oneperson trials. Population studies with many persons can show larger lower limits of the credible interval, if the success proportion is high.
Steady state implies that the previous observations of the same condition showed no incidence of Y _{imp} without the exposure. When we believe the previous evidence for the noincidence of Y _{imp} under the nonexposure, we can adopt, for example, Beta(1, 1000000) as the prior distribution for θu. Adopting almost null prior distribution Beta(1, 1000000) for θu means that p(Δ) is practically equal to p(θeye).
Relation to crossover trials
Period 1  Period 2  

Group I (FG)  μ + τ_{F} + π_{1}  μ + τ_{G} + π_{2} + γ_{FG} 
Group II (GF)  μ + τ_{G} + π_{1}  μ + τ_{F} + π_{2} + γ_{GF} 
Here, μ is a general mean, the τ terms represent treatment effects, the π terms represent period effects, and the γ terms represent the treatment × period interaction.
Period 1  Period 2  

Group I (FG)  μ + π_{1}  μ + τ_{G} + π_{2} 
Suppose the ratio of the period 2 to the period 1 is 1:n. The constancy of \( \frac{{{\text{d}}S}}{{{\text{d}}t}} \) means that the period effect π_{1} is constant during the period 1. Under the condition of state steadiness which is confirmed by the (n + 1) times observations during the period 1, when the period 2 follows successively the period 1 and n is sufficiently large, we can postulate that the π_{2} is almost equal to (π_{1} + π_{1}/n) in the group I. The larger n is, the more we can believe the steadiness of the state and the approximation of π_{2}. Then the difference of the response between the two periods is (τ_{G} + π_{1}/n) and we can measure τ_{G} with the repeated observations of group I. Thus steady state trials are considered as variants of crossover trials.
Prerequisite for steady state trials
How many figures should we adopt for n? In the above model, we postulate that \( \frac{{{\text{d}}Z}}{{{\text{d}}t}} \) is equal to k, or π_{2} is equal to (π_{1} + π_{1}/n). When n is infinitely large, this postulation is reasonable. However, when n is moderately large, the postulation receives criticism. There are many biological parameters which show cyclical or periodic variations, for example folliclestimulating hormone or luteinizing hormone levels in female blood plasma. Another criticism is that the observed variable might reach the critical point after the steady state and change drastically without exposures. Before executing steady state trials in medicine, we have to examine biologically the trial condition for the possibility of cyclical or drastic state change. If some period ratio is thought to be critical, we have to avoid using such n for steady state trials.
Discussion
We have deduced steady state trials (SSTs) from the counterfactual model, from which randomized controlled trials (RCTs) were also deduced. Although RCTs are thought to be paramount trials in recent clinical research, STTs can also offer the valid method to measure causal effects, when the state before the exposure is steady and large changes are immediately observed. The smaller the ratio of the afterperiod to the beforeperiod is, the better we can rely on the measurement of the causal effect. When the afterperiod is relatively long, the measurements of SSTs may be confounded and RCTs should be considered in such situations. RCTs are also necessary when outcomes long after the exposure are important, even if SSTs show causal effects immediately.
Individual causal effects are defined as a contrast of the counterfactual outcomes. Because only one of those values is observed, it has been proposed that individual causal effects cannot be identified in epidemiological research [18, 19]. The epidemiologic principle is that a person may be exposed to an agent and then develop disease without there being any causal connection between exposure and disease [9]. SSTs show that we can measure individual causal effects in the condition where repeated observations are performed, the state before the exposure is steady, and large changes are immediately obtained after the exposure. This approach could open the door to the individual causal inference and other conditions for the individual causation that should be researched in epidemiology.
We adopt Beta(0.5, 0.5) as the prior distributions for θe and θu. However, the huge sample size fixes practically the same posterior distribution as when we believe the prior probability of θu is almost null. The 95 % credible interval of Δ is computed as 0.9995–0.9996 with WinBUGS. Classical statistics show that the 95 % confidence interval of the safe skydiving proportion is 0.99954–0.99959.
SSTs are practicable in the situation where immediate clinical responses are important, such as in the emergency room, where confounders are under the control of practitioners. Many treatments in emergency medicine have a long good history of SSTs in innumerable persons and can be regarded as scientific medical interventions without RCTs, such as intravenous injection of glucose for patients in hypoglycemic coma, injection of adrenalin (epinephrine) for patients with anaphylactic shock, a tourniquet for bleeding patients, and so on.
Systolic blood pressure in an imaginary adult patient with anaphylactic shock
Time (min)  S (mmHg)  dS/dt (mmHg/min) 

0  50  
1  50  0 
2  49  −1 
3  50  1 
4  53  3 
5  53  0 
6  54  1 
7  56  2 
8  57  1 
9  58  1 
10  60  2 
Exposure  
11  92  32 
[(−1)^{2} + (−2)^{2} + 0^{2} + 2^{2} + (−1)^{2} + 0^{2} + 1^{2} + 0^{2} + 0^{2} + 1^{2}]/(10 − 1) = 1.33.
The standard error of [(Y − X)/A − (k + δ_{ is })] is √[1.33(1/1 + 1/10)] = 1.21.
Twosample t statistic is (32 − 1)/1.21 = 25.6,
which follows the t distribution on (1 + 10 − 2) = 9 degrees of freedom. The P value is computed as <10^{−8}. In this statistical estimation, we postulate that the distribution of (Y – X)/A follows the normal distribution with the same variance as σ^{2} which is estimated by the sample variance of (k + δ_{ is }). However, we do not really have data for the estimation of the variance of (Y – X)/A. We propose the above P value as an informal index to be used as a measure of discrepancy between (Y – X)/A and (k + δ_{ is }). We recommend that this P value should be smaller than 0.001 to show discrepancy. SBP over 90 mmHg is a clinically important value in shock, and we can consider that \( \frac{{{\text{d}}S}}{{{\text{d}}t}} \) of 32 mmHg/min for 1 min after the exposure is Y _{imp}. With prior Beta(0.5, 0.5) for θe and θu, the 95 % credible interval of the posterior distribution Δ is computed as 0.09–0.99. When we have the prior information that Y _{imp} have not been observed in the past unexposed states for 1000 min in total, we can adopt Beta(0.5, 1000.5) as the prior distribution for θu and n can be smaller to infer causal effects, if we confirm that unexposed state in the period B is in the same condition as the previously reported unexposed states. After observing one success of Y _{imp} after the exposure, with prior Beta(0.5, 0.5) for θe, prior Beta(0.5, 1000.5) for θu and n = 1, the 95 % credible interval of the posterior distribution Δ is computed as 0.15–1.00. Practically, n will be more than two in order to confirm the equal conditions, even when we adopt almost null prior for θu. When we execute SSTs in population studies, if some period ratio is thought to be critical, the participants are divided into two or three groups to which different values of n are allocated. RCTs between the groups are also possible in this setting.
SSTs are also possible in treatment trials of neurodegenerative diseases whose patients almost always show progressive deteriorations. For example, although RCTs has not been performed in the levodopa therapy of Parkinson’s disease [21], neurologists will admit that symptoms lasting for several months of early Parkinson’s patients almost always improve within a few days after receiving levodopa.
In preventive medicine, oral rehydration therapy is effective against diarrhea [22]. RCTs have compared oral rehydration with intravenous hydration [23]. SSTs can offer the measurement of the effect difference between the treated and the nontreated in acute stage diarrhea. Because of the necessity of controlling confounders, SSTs may be restricted within a narrow set of research topics in preventive medicine. However, if we have sufficient past databases of disease incidence and the derivative of the incidence rate with respect to time is constant, we may use SSTs for the measurement of effects of exposures which have a latent period of several years.
Notes
Declarations
Acknowledgments
We are grateful to Prof. Akio Koizumi for useful comments.
Conflict of interest
The authors declare that they have no conflict of interest.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
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