# How to improve the efficiency of an estimator when the estimates significantly vary among subgroups?

In a clinical trial, patients enter the hospital (hence, the study) at different time points. We have observed that patients who respond to the primary therapy early survive longer than those who respond slowly. While estimating the overall survival probability of patients, can we incorporate such information to reduce the variance of the estimator?

Say, we could take the early responders and slow responders as two strata, then we could do something like this: $\hat{S(t)}_{overall} = \sum_{i=1}^2 \dfrac{N_i}{N} \times \hat{S(t)}_i$. But here we do not know $N_1$, $N_2$ or $N$, although we can find the estimates $\hat{S(t)}_1$ and $\hat{S(t)}_2$.

Is there any way we can make the estimator more efficient when the estimates significantly vary among subgroups?

Edit: I was reading endogenous post-stratification and calibration. If I have one or more variables (e.g. doctor's perception or age) that can be related to early response or slow response (or to the responding time), then can we use any of these methods in this scenario? How will that work?

• Is my reading of your question right: there are two groups of patients (slow and fast responders), but you don't know a priori which is which. So, in statistical language, you could say that there is a latent variable (unobserved) describing patient's response -- in the simplest case a latent variable with two values (slow and fast)? – jwimberley Dec 19 '16 at 17:20
• Actually, I am talking about trials that have two stages. At the first stage, the patients respond to the primary treatment. Some respond quickly and some respond slowly. We can observe their responding times (say, $X$). Our ultimate goal is to observe their survival times (say, $T$) and we have seen that slow responders have shorter survival times than fast responders. So, can we improve the estimator $S(t)$ by using the information that survival chances are widely different between these two groups? – Blain Waan Dec 19 '16 at 23:45
• I'm a little more confused now -- in the question you say that you do not know $N_1$, $N_2$, or $N$, but in your comment you say that the patients are directly observed to respond quickly or slowly. It seems like you can simply count $N_1$, $N_2$ and $N$. Why isn't this the case? – jwimberley Dec 20 '16 at 14:44
• I may be wrong. But I said that I didn't know the population sizes $N_1$, $N_2$ and $N$ because patients are coming at random to the hospital and I don't know the "population" of fast and slow responders. I only know who are being admitted during the study period. So, I know $n_1$, $n_2$ and $n$. But I used these notations as I was trying to relate this with stratified analysis. – Blain Waan Dec 21 '16 at 3:43

I agree with @user918967 that you have wrongly formulated your problem, but I believe your misformulation involves ethical as well as technical deficiencies. In particular, your focus on an 'estimator' of overall survival in the face of two putatively distinct classes of patient seems perverse. Please consider how you might design your trial to acknowledge the ethical demands of adapting to the information stream that comes in as you perform the trial.[1] For example: if it should turn out that the 'late responders' actually do worse on the treatment (compared with some available alternative), have you designed your study to detect this as soon as possible? Or would such a finding emerge only belatedly, in retrospect, after you had reached some fixed, predetermined sample size [1]?

As to the technical shortcomings of your problem misspecification, I believe you might find it fruitful to consider your 2 putative classes of response in light of the concept of frailty; see [2] and [3]. In frailty modeling, a latent 'frailty' variable is presumed to be distributed in some way across a population, and this distribution typically morphs over time as frailer individuals 'exit' first. Your concept of 'early' and 'late' responders suggests a bimodal frailty, as if response were a monogenic phenotype. But you should consider whether you might be leaping to 'dichotomize' what is actually a more continuous phenomenon, and frailty modeling provides a formalism for such consideration.

One thing you might like to do is to preregister your study protocol on a platform like F1000Research, where you can get transparent feedback, and iteratively update your protocol (through multiple versions, if necessary) until it passes peer review.

Note 1: Your query about using "doctor's perception or [patient's] age" points very much toward frailty-type phenomena. The surgeon's 'eyeball' impression of a patient's suitability for surgery is often discussed as delivering an implicit assessment of frailty that may not be readily accessible to clinical measurement.[4]

Note 2: As with "survival", the term "frailty" does not necessarily carry a fixed positive or negative connotation in every application. Thus, if you were modeling time-to-event for an event like clinical response, then of course the 'frailty' involved would be the vulnerability of the disease process to the treatment, and in this case 'frailty' would be a good thing, and 'survival' (i.e., longer duration of the disease) would be a bad thing.

[1]: Berry, Donald A. “Bayesian Statistics and the Efficiency and Ethics of Clinical Trials.” Statistical Science 19, no. 1 (February 2004): 175–87. doi:10.1214/088342304000000044. http://projecteuclid.org/euclid.ss/1089808281

[2]: Aalen, O. O. “Effects of Frailty in Survival Analysis.” Statistical Methods in Medical Research 3, no. 3 (October 1, 1994): 227–43. doi:10.1177/096228029400300303. http://journals.sagepub.com/doi/pdf/10.1177/096228029400300303

[3]: Aalen, Odd O., Ørnulf Borgan, and S. Gjessing. Survival and Event History Analysis: A Process Point of View. Statistics for Biology and Health. New York, NY: Springer, 2008. (See especially Chapter 6, titled "Unobserved heterogeneity: The odd effects of frailty".)

[4]: Jain, Renuka, Sue Duval, and Selcuk Adabag. “How Accurate Is the Eyeball Test?: A Comparison of Physician’s Subjective Assessment versus Statistical Methods in Estimating Mortality Risk after Cardiac Surgery.” Circulation. Cardiovascular Quality and Outcomes 7, no. 1 (January 2014): 151–56. doi:10.1161/CIRCOUTCOMES.113.000329. http://circoutcomes.ahajournals.org/content/7/1/151.long

Personally I think you may be formulating the problem incorrectly.

Your data and analysis seems like more like it would be more appropriate to use Survival Analysis - of course your outcome does not literally have to be survival but could instead be Time to Event.

Once you have the model constructed you can simple log-rank tests or more sophisticated cox proportion hazards (whatever is more appropriate) to inspect differences between your groups.

Lastly, if you want a more sophisticated SA (e.g transitioning between stages of disease), you could look at multistate models.

• Thank you for the reply. I was thinking of something else. Say, my ultimate goal is to report an overall survival probability. But I can see that the estimates $\hat{S}_1(t)$ and $\hat{S}_2(t)$ computed for the sub-groups (e.g. of the fast and slow responders) are quite different. The variances of estimator among these two groups are also different. So, can I use some other method (like stratification or something else) for the estimation of $S(t)$ to improve the efficiency of estimation, instead of the scheme where I use all data at a time and find out the estimate of $S(t)$? – Blain Waan Dec 21 '16 at 3:54
• I see no need to attempt to combine the two estimates of survival into one grand value. Quite honestly, if your estimates of survival for S1 and S2 are statistically different you do not want to combine them into a unified value. – user918967 Dec 22 '16 at 5:39

Just a suggestion, try to define the following sample probability or likelihood function for the two different subgroups, $P(X|\theta)=\sum_{k=1}^{M}P(x|\theta)$. The sample of outcome is assigned by the measurements; i.e look for the conditional (a posteriori) probablity $P(\theta|X)$ of $\theta$ given the sample $X$. This can be easily obtained by the Bayes Theorem. From this you can evaluate the expected value of $\theta$ and the variance of the distribution.