# Can I use “left eye” and “right eye” in my sample as two different subjects?

My data is as follows. I have two groups of patients. Patients in each group had a different type of eye surgery. 5 variables were measured on patients in each group. I want to compare those variables between the two groups using a permutation test or MANOVA. The eye on which the surgery was done does not really matter in the analysis. However, Patient 2 in group A for example, had the surgery on both eyes and therefore has those 5 variables measured twice, one time on each eye. Can I consider patient 2 Left and patient 2 Right as two different observations? Same for patient 31 in group B. $$\begin{array} \hline \text{Patient} & \text{Surgery type} & \text{Side} & \text{V1}& \ldots & V5\\ 1 & \text{A} & \text{Left} & 91 & \ldots & 22\\ 2 & \text{A} & \text{Left} & 87 & \ldots & 19\\ 2 & \text{A} & \text{Right} & 90 & \ldots & 23\\ . & . & . &\\ 31 & \text{B} & \text{Left} & 90 & \ldots & 17\\ 31 & \text{B} & \text{Right} & 88 & \ldots & 19\\ 32 & \text{B} & \text{Right} & 91 & \ldots & 24\\ . & . & . &\\ \hline \end{array}$$

• Your test may be accomplished by something similar to a matched pair test, unbalanced randomized block design. But before I guess further, could you please elaborate on your data, what is it what is looks like etc.? – suncoolsu Aug 4 '11 at 7:43
• Thanks. I am trying to present my data in a nice table format here on the blog but I still didn't figure out how. I will make sure to present my data in details in my next question. I would like to reiterate that both eyes had the same type of surgery, so are in the same group. – sara Aug 8 '11 at 3:38
• I have made a sample table, you can now edit it to show your data. – suncoolsu Aug 8 '11 at 6:22
• @suncoolsu, the question is answerable without the data. What is your intent in having the OP post data? – Iterator Aug 8 '11 at 11:21
• @Iterator I agree and you have already answered it (and I have upvoted it :-)). I was just curious to see the data and what kind of models could be fit on the data. – suncoolsu Aug 8 '11 at 12:25

I would not recommend it. Not being a domain expert, I can still identify three things that would reduce the independence of the outcomes:

1. Both eyes were treated at (almost) the same time. While this is not necessarily a problem, it affects the other assumptions of independence. What's more, the surgical team may have opted to treat both in the same way or may make a decision about one eye with consideration of aspects of the other eye.
2. Both eyes were treated by the same surgical team (surgeon and everyone else involved)
3. Both eyes are subject to the same patient "factors", i.e. anything that would be intrinsic to the patient that could affect outcomes, such as compliance with other treatments, overall health, etc.

If anything about the outcome could be attributed to the surgical team or the patient, then there's a problem.

Because all the answers so far are negative (in terms of advocating using less than the full data set or in suggesting limited uses for the two-eye cases), let's see what can be done. For that, we need a probability model.

Consider a single response variable, $Y$ (one of V1 through V5, apparently). As a point of departure, suppose the response depends on several factors, including

• An average or "typical" response $\mu$.

• A random patient-specific factor, $\varepsilon$, with zero mean.

• Perhaps an indicator that both eyes were involved, $X_2$.

• A surgery type factor, $X_s$, which ought to be an attribute of the eye, but which appears to be constant within each patient (thereby limiting our ability to identify this factor).

• A factor for any systematic difference between the right and the left, $X_e$.

• For each eye, a random variation from the expected response in that eye, $\delta$, with zero mean and independent of the patient factor $\varepsilon$.

It is implicit here that the experiment was designed in certain standard ways: namely, that patients were randomly selected from a specified population; that the determination to treat the left eye, right eye, or both, was either randomized or can be assumed independent of other factors; etc. Changes to these assumptions would require concomitant changes in the model.

According to this model, the response expected of eye $j$ ($j \in {\text{right}, \text{left}}$) within patient $i$ is

$$Y(i,j) = \mu + \beta_2 X_2(i,j) + \beta_s X_s(i,j) + \beta_e X_e(j) + \varepsilon(i) + \delta(j).$$

This looks like a somewhat complex partially nested mixed model. Fitting the parameters $\mu$, $\beta_2$ and $\beta_s$ can be done with maximum likelihood (or possibly generalized least squares regression).

I offer this purely as an illustration, to show how one might profitably think about this problem and arrive at a way to exploit the dataset to the fullest. Some of my assumptions may be incorrect and should be modified; additional interactions may be needed; some thought may be required about how best to handle potential differences between eyes. (It's unlikely there's a universal difference between left and right, but maybe there's a difference related to the patient's dominant eye, for instance.)

The point is that there does not appear to be any reason either to limit the analysis to one eye per patient or to use ad hoc analytical methods. Standard methodology appears to be applicable and a good way to use it begins by modeling the experiment.

• I think it's important to add that the assumption of independence can and should be tested, if using the samples with binocular treatments. Such testing for dependence should be done before running ahead with a model, for two reasons: 1: If there is some dependence, then this could be very interesting. 2: Understanding the dependence could lead to a better model. – Iterator Aug 8 '11 at 17:47
• @Iterator Your good suggestion is exactly what I was hoping this discussion would prompt: when we contemplate how to model our data, we often get insight into what assumptions are being made and need to be tested. – whuber Aug 8 '11 at 19:10
• @whuber Good start. As always, you are right about "mixed-model-ish" treatment! I agree with you that we should not " throw away" any data. – suncoolsu Aug 9 '11 at 3:13

I agree with the others that two eyes of the same patient are not independent. However, I do not per se agree on using only one sample. After all that is throwing away precious samples.

In a somewhat similar situation (some of my patients were operated again on the same tumour) I do use their samples.

• For the (iterated/repeated cross) validation I make sure the splitting is done patient-wise.
• I cannot state the effective (statistical) sample size. For me that is anyways no problem due to more samples of some patients. I have hundreds of spectra for each sample, and they are neither repeated (they are taken from different places) nor independent. So I don't loose anything here.
• I sometimes use the number of patients as conservative bound for the effective (statistical) sample size: at least the patients are independent
• You may weigh the samples so that each patient enters the analysis with the same weight.

I concur with @iterator. If a large proportion had surgery on both eyes, I'd do some sort of matched pairs. If only a small proportion had surgery on both eyes, I'd probably just not use either eye for those people, but certainly not both.

• Peter's right. Actually, the set could be quite interesting in itself: conditioned on needing to operate on both eyes, were outcomes any worse? The reason we advocate not assuming independence is that there are a lot of reasons why this could be wrong. If there's a large enough sample, test for independence. The insight could be very interesting & practically useful. – Iterator Aug 4 '11 at 19:08

One point to add to the comments of iterator and peter. When analyzing the overall data set, you should use only the data from one eye for patients who were operated on both (because the outcome for the two eyes is unlikely to be independent). Which eye? Use a randomization method, so you don't choose the one with the better (or worse) outcome, which would influence (bias) the results.

As part of a separate study, you might want to look at only patients with good outcome in one eye and not in the other, and try to see is there are any hints about what causes the difference.