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It's the first time I'm posting here, and I apologize in advance if the same question has been answered previously.

I am working with data of proportions, and I want to know how that proportion changes over time. For example, below are random telephone survey for what middle schooler in a particular state do with their lunch at different years. I want to know what would be the optimal statistical method to analyze the change in proportion of each option across time.

State A

2000 total: 200 Own lunch:120 Buy lunch:45 School lunch:73 Other:24

2001 total: 220 Own lunch:89 Buy lunch:39 School lunch:85 Other:17

2002 total: 270 Own lunch:70 Buy lunch:52 School lunch:121 Other:32

2003 total: 354 Own lunch:45 Buy lunch:71 School lunch:224 Other:37

2004 total: 241 Own lunch:32 Buy lunch:37 School lunch:168 Other:29

2005 total: 530 Own lunch:51 Buy lunch:94 School lunch:403 Other:14

and so on....

On the basic level, I know I can compare, say, proportion of students who bring their own lunch between 2000 and 2001 with fisher's exact or chi-square. But I want to know if there is a way to analyze changes in all options (as they are not mutually exclusive, students can pick multiple options so one option going up doesn't necessarily mean other options have to go down) and across quantitative time (instead of just two categorical time points). In addition, I have this data for many states, so it would so be nice if I can incorporate that into the same model to see if the change in proportion across time differs by states.

Please help me identify the optimal test/analysis method, ideally something I can run on R.

Alexis
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ace327
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    "what would be the optimal statistical method to analyze the change in proportion"... this is too broad.. can you narrow down and focus on the question you would like to answer? – StupidWolf Apr 28 '20 at 16:58
  • you can fit a binomial or poisson, but it's not a mega-tool box that returns you all the answers... you have to ask the right question – StupidWolf Apr 28 '20 at 16:59
  • @StupidWolf binomial or Poisson model *level* of proportions (values from $0$ to $1$), not *change* in proportions (values from $-1$ to $1$). – Alexis Apr 28 '20 at 21:56
  • How many years of data do you have? – Brash Equilibrium Apr 28 '20 at 21:59
  • @Alexis maybe i used the wrong wording, what I meant was to fit a binomial with the counts of choosing 1 category vs not choosing, for all the categories, like this https://stats.stackexchange.com/questions/259502/in-using-the-cbind-function-in-r-for-a-logistic-regression-on-a-2-times-2-t – StupidWolf Apr 28 '20 at 22:03
  • As for poisson , i was thinking of modeling the counts of each category as a rate, i.e counts of own lunch / total , buy lunch / total and so on – StupidWolf Apr 28 '20 at 22:04
  • @BrashEquilibrium I have 18 years of data, and I have the same set of data from 5 different states – ace327 Apr 28 '20 at 23:25
  • @StupidWolf Thank you for so many answers. Sorry for poorly worded question, but I'm not sure how to word it... sorry So if instead of having the numbers of people in each category, I got the age of each person who bring lunch, buy lunch, etc, then I can imaging myself running some sort of linear model with age~lunch option+year+state for main effect of each factor, and add any interactive effect I want to investigate. But with my dependent variable being a count (or proportion of a total), I'm not sure how to set up a model to test the effects of each factor on the counts/proportions – ace327 Apr 28 '20 at 23:34
  • @StupidWolf Is the difference between modeling $x_{t}$ and modeling $\Delta x_{t}$, pun intended. :) – Alexis Apr 29 '20 at 04:34

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