I am confused by some of your description, so maybe you can clarify:

"each device may be used once, twice or thrice (ranks) as a treatment in an individual"

• So some patients only get Device 1 (D1) used once, some get it used twice in their treatment, etc.? Is that correct?
• If the above is true, how or why does "number of times the device was used" become characterized as a "rank"? Usually the term "rank" refers to something like patients or doctors indicating a preference for the device, or it's the order in which the device was used. This variable seems (from the description) to be something quite different.
• I assume your outcome measures are (or can validly be considered) interval-level measures, and that the device "ranks" are not among the outcome measures--rather, they constitute predictors.
• How is the number of times each device is used per treatment or per patient (or rank, if I've misunderstood) decided? Who makes the decision? Is it randomly assigned? Is it left up to the treatment provider or the patient? Is it because of equipment failure or limited availability? A description of this might help.

Your data setup sounds really interesting. I can't promise anything truly helpful, but I will try, if I can learn enough about the study to feel I have a grasp of the most important characteristics.

It seems like a Latin Square design. You should be more explicit.

As far as I understand, each experimental unit will receive treatment in a different order. For example unit1

Thank you, Darrin Rogers PhD, let me do some clarifications:

For every patient who undergo cardiac imaging , we use 4 disposables devices. However, lately in medicine, many hospitals in the world including US and Canada are resterilizing those devices as a measure of cost containment. No much of worldwide clear data on the safety of this practice. I am studying the effect of repeated devices resterilization on patients health (the four devices), whether devices are sterilized once, twice or thrice,and used in the same patient; one device may have the first resterilization, the other may be first or second or third ..etc.. Now you can see that each device of the four used in a patient may have similar/different rank sterilization once, twice or thrice.

I am trying to find a formula to measure the resterilization rank of the four devices as total and separate effects on patient safety (outcome measure), for all patients.

Thank you again.

Thank you, Mehmet Sinan Iyisoy

It could be interpreted as 4C3 ,each time the four devices are used in a patient (order is not important)

Dear Darrin Rogers

1- point one: that is correct, but remember that a patient must use at random 4 devices with different rank of sterilization.

2 - here the rank of sterilization is not preffered by the doctor but randomly assigned.

3- how many a device is sterilized is supposed to be a predictor of outcome.

4- Devices are randomly assigned.

Thank you again

I think I am starting to get a grasp of the situation. It sounds both complicated and interesting as a "natural experiment" setup. 

I'm going to make an assumption I am not sure about, so please correct me if I'm wrong: Each patient always gets four devices used in his/her treatment, and those four devices are not interchangeable--they are different kinds of devices. Since I don't know about the devices, I'll imagine it in this silly way, which should help you see if I'm making mistakes in my understanding:

If we replace "medical treatment" with "preparing a breakfast of frozen waffles and grape juice for children" we might say there are four necessary devices always used for each meal:

And we can imagine that each one has been sterilized between one and four times. So we might end up with something like this:

Child 1:

• D1 (knife): sterilized 1 time
• D2 (fork): sterilized 3 times
• D3 (plate): sterilized 2 times
• D4 (glass): sterilized 4 times

Child 2:

• D1 (knife): sterilized 1 time
• D2 (fork): sterilized 1 time
• D3 (plate): sterilized 2 times
• D4 (glass): sterilized 1 time

... and so forth. At this point, I hope my silly example has made it clear that I either do or do not understand your situation. But assuming things are as I have described them so far, this could be a fairly simple analysis. If there are no other complications, you might try a multiple regression analysis for each outcome (this assumes the outcome variable is numerical, like the results of a psychometric questionnaire, or a physical measurement of some kind--or binary, so logistic regression could be used):

• Response/Outcome variable: [whatever the variable is]
• Predictor 1: Number of sterilizations of Device 1
• Predictor 2: Number of sterilizations of Device 2
• Predictor 3: Number of sterilizations of Device 3
• Predictor 4: Number of sterilizations of Device 4

Let's assume (since I don't know) that your outcome variable is a physician-completed rating scale of patient health one month after surgery, on some numerical scale where higher numbers indicate better health (or whatever). Your data might look like this, in the file:

OV  P1  P2  P3  P4

21    1     3     2     4

53    1     1     1     1

43    3     2     3     4

17    4     3     3     2

and so on (sorry if that doesn't look like a nice matrix table; formatting options are limited, here).

If the conditions of OLS ("regular type") regression are met, you would analyze using the standard multiple regression procedure your software has. The results (if conditions are met) will tell you:

• How much variability all four predictors together can account for in the outcome variable (a.k.a. "model fit", multiple-R2, etc.)
• How much each individual predictor is associated with the outcome after accounting for the association of all the other predictors with the outcome.

The second point is a good one, because the regression coefficients of P1 through P4 will tell you how uniquely important the sterilization rank of each instrument is in affecting the outcome. Put together with overall model fit (which can be thought of as how much combined predictive ability the four predictors have), you can get a good idea of the situation.

Ordinal versus Numerical: I note that you have described "number of sterilizations" as "ranks," which is a special keyword in statistics (some might say it's a "trigger" word... heh heh). This normally means that the variable is ordinal, not interval/ratio (i.e., "numerical") and thus must not be used in parametric procedures (like OLS regression). However, in this case:

A. I think "number of sterilizations" is actually numerical (ratio, in fact), so I don't think you should worry about its ordinal-ness.

B. If someone decides it's actually ordinal, use regression procedures designed for ordinal predictors.

A final issue: Those predictors (the sterilization numbers of D1 through D4) are very unlikely to be normally distributed. They will be discrete (1, 2, 3, 4), and almost certainly strongly positively skewed--in fact, my guess is that they will follow the Poisson distribution. This is not automatically a problem; the biggest issue is the distribution of your outcome variable, followed by the distribution of the residuals from the regression analysis. If those are OK, then don't worry about your positively-skewed discrete predictors.

However, if the positive skew of the predictors causes problems you now have a variety of options:

• Reverting to thinking of the predictors as ordinal and then using regression procedures that correctly model ordinal predictors (such as using lavaan or Mplus to do the modeling and specifying which predictors are "ordered")
• Transforming your predictors (e.g., square root or log transformation)
• Using a Generalized Linear Model and hunting around  until you find the correct linking function that makes for a good model of your outcome as a function of those skewed predictors
• Probably other things I'm not smart enough to know.

Multiple outcomes: There are regression procedures that can model multiple outcomes, sort of like MANOVA. Those could be useful. However, if you don't care much about the relationship between your outcomes/DVs, then you could just do one analysis per outcome variable, then make sense of them separately.

There are certainly other ways to approach this problem. I don't claim this is the best.