If you think the time is not important, you can formulate a model for dependent variable ( weight gain ) and independent variables ( types of hospital and hospitalization days ), and using this model you can find the change in weight gain by changing the hospitalization days in each hospital type.

You have two time series data: 1 week that produces weight result-1, and 2-3 weeks that produces weight result-2. Treat them as two systems that produces weight-1 and weight-2, and try to obtain system equation for each to see how can each predict system failure & success.

Try this in a different perspective: (i) list your time as t1, t2, ..., tn and your weight as y1, y2, ..., yn, then (ii) find time function to obtained Weibull regression model in a form of Yw = a + bX for the weight gain in two different time series. Using beta, eta, CDF and PDF, analysis the data in the perspective of system success and failure for weight gain.

[1]. Time Function

F(t) = (i - 0.3) / (n + 0.4)

[2]. Obtain QQx

ln(ln(A))

A = 1 / (1 - F(t))

[3]. Obtain QQy

ln(Y)

where Y = observed weight gain for particular time span.

4. With known QQx and QQy, now run regression model to obtain Yw = a + bX.

Repeat 1 - 4 for the second time series. You should have equation for Yw1 = 1 week series, and Yw2 = 2-3 weeks series. Now obtain Weibull statistics to evaluate system performance; this would lead you to conclude which, if any, time span series contribute significantly to weight gain. To obtain % target weight gain, it is a matter of going back to the two forecast functions which one would yield the most.

LINK:

https://www.weibull.com/basics/lifedata.htm

http://www.barringer1.com/pdf/Chpt1-5th-edition.pdf

http://reliawiki.org/index.php/The_Weibull_Distribution

I think that the weight gain is not (primarily) dependent on the duration in hospital, but by the environment after leaving it. If this is the case, one can, of course, run refined regressions on the base of the hospital data alone, but the results are not valid.

There seems some missing information on what is causing gain during stay...

A recurring theme in the answers already posted is that you should begin by deciding what you think is a good theoretical explanation of weight gain. This will tell you what explanatory variables to use. Your research question implies that you believe length of stay is an important determinant of weight gain. If you think that length of stay is the only relevant factor then simply regress weight gain on length of stay (the relationship might not be linear). You may think, however, that the relationship is not identical in the two hospitals. In this case you will need to either run separate regressions for each hospital or introduce a dummy variable to distinguish between hospitals and also interact this with length of stay. A potential problem is that if ALL stays in hospital Y are exactly one week (as you seem to indicate) then there is no variation in the key explanatory variable for hospital Y cases and so you cannot run a separate regression for that hospital.

If you have other relevant patient data, e.g. initial weight, type of illness, time of year(?) etc. then you might think that such variables also need to enter the regression model.

Thank you for your responses

Box-Jenkin's method can be used for forecasting