-> https://en.wikipedia.org/wiki/Multinomial_logistic_regression

Logit Models, Logistic Regression

You can using a generalized linear model and a logit link function (logit models). A binary variable can have only two outcomes, for example Yes and No, or 0 and 1. The probability distribution of the binary variable y has the Bernoulli distribution:

In SAS software program the following procedure may be helped.

DATA a;

INPUT @ ## ;

DATALINES;

infile

;

PROC GENMOD DATA=a;

MODEL ## = @/ DIST = BIN

LINK = LOGIT

TYPE1

TYPE3;

RUN;

Thanks Jochen, I'll check that out!

Fausto, I'd rather not send my data! However, it looks the data table in the attachment. The picture in the attachment is simplified, however the structure is the same. In my table I have 19 response variables as behavior!

Waleed, thanks for your answer!

However you say "A binary variable can have only two outcomes, for example Yes and No, or 0 and 1 ", but this is not the case for me right? I have percentages that go from 0 to 100 (or data from 0 to 1 if you don't multiply by 100). Could you perhaps clarify?

Thank you!

Mybe I can try to clarify your question to Waleed:

Your percentages are nothing but relative frequencies, that is, you have a "number of events" (e.g.: how often does the animal show feeding bahaviour) divided by the "number of observations" (how often was the animal observed, how often could the event have the chance to happen / to be recorded).

Given one of your variables (e.g. feeding behaviour), your response would be binominal (dicotomous): either "yes" or "no". Such variables are analyzed by logisic regression. That's essentially about the log odds of showing that behaviour.

You have a not a two possibilities (feeding or not) but a group of several possibilities (feeding or standing or sleeping or ...). You would still analyze the log odds of these behaviours, so it's still a logistic regression, but the response is now multinominal (polytomous): either "feeding" or "sleeping" or ... . Therefore I suggested a multinominal logistic regression.

Hope that clarified a bit.

Hi Jochen, I think I get it now!

However, when I read your answer I get the idea that I'd better use the ''raw'' data set of mine. At the observation point (let's say the first observation) the behavior that is observed gets a ''1'', while the other behaviors get a ''0''! And this of course for every observation point. (Indeed, the percentages are total numbers of ''1'' for each behavior over all the observation points divided by the total number of observation points.)

Should I use this dataset rather then the literal percentages?

Dear Robbert

You can also use percentages after transformed by the arcsine transformation prior to analysis.

@ Robbert, yes. Although some software accept percentages plus the total number of observations. Using only the percentages is not enough. Eventually, the analysis needs to know the counts.

@ Waleed: I see that this would be a possible work-around for a simple (binomial) logistic regression. But does this also work for a multinomial logistic regression? I don't think so, because the percentages add to 100%, so the data of the groups is not independent (but a linear regression using the transformed precentages assumes this).

Thanks both of you! I will try it out in SPSS 24 today... Thank you guys for the help!

Still wondering though if the arcsine transformation also could work! This because I have a lot of observational points (1000-2000) per individual so handeling the percentages is more clear..

@Jochen: I've been reading about multinominal logistic regression, and I don't 100% get how my dataset should be constructed. Should it be one column and then for each observation point the number (=0,1,2,3,4,....,17,18) of the recorded behavior, or should it be 19 columns (because I have 19 different behaviors) with in only the behavior observed a ''1'' and in the other behaviors (not observed) a ''0''? The things I've read online gives me the impression that the first dataset contruction with one column is the correct one! What do you think?

That depends on the software you use. For instance, the R package nnet provides a function multinom() to fit a multinominal regression to a factor (like: sleep, feed, feed, stand, feed, ....). The function from the R package mlogit does require some reshaping. I don't know how SPSS or SAS handle this.

I'm using SPSS, and I think that I need one column with 19 possible outcomes I think!

Thanks!

Hi Jochen, I'm starting to doubt if multinominal logistic regression is the right thing to do!

I've checked some datasets online of people performing a multinominal log reg, but there, one row is one individual. However in my dataset, the behavior of each individual is checked each second and this for a longer period of time. So for example if I log the behavior each second for 5 minutes, I have 300 rows of the same individual at a certain hour and each row represents a second and a corresponding behavior at that second...

I've performed the multinom log reg in SPSS but I don't think the output makes much sense..

Any ideas?

You have a closed number set that adds across the individual to 1. ( You have just multiplied this by 100 to get a percentage.)

You need to think about couple of questions - what is the denominator (there must be one eg the total time the individual was observed ) and does it differ for each individual? So you need to take account of this denominator and the closed nature of the set which generates inbuilt correlation - as one proportion goes up the others must go down.

I think this data set is actually repeated measures of an individual where you recorded whether an individual is doing one of 19 mutually exclusive behaviors in a time interval. Say you observed every 5 minutes for 1 hour and the individual only did two activities A on four occasions and D on 8 occasions so the denominator is 12 and the observed proportion is 4/12 and 8/12; all other responses are 0/12. Essentially you have aggregated from the time interval data to the individual (collapsed over time) and this is fine if the predictor variable in not time varying that is it is about the individual. These aggregated data can be analysed by a multinomial model - different software uses different set ups but you need software where you can give the denominator column, and the software does not assume incorrectly it is 1. This denominator is sometimes called the number of trials or is entered by setting it up as frequency

see https://support.minitab.com/en-us/minitab/18/help-and-how-to/modeling-statistics/regression/how-to/nominal-logistic-regression/perform-the-analysis/enter-your-data/

I looked to see if I could find this in SPSS but could not.

The arcsine is transform is a bit of a bodge for two reasons. The logit transform models the underlying log-odds of the activity not the observed values and the residual distribution is multinomial not say Normal ( the variance depends on the mean and the co-variance is expected to be negative as one proportion goes up towards one, another proportion must must go down). Not getting this approximately correct will affect the standard error .

This model assumes that all individuals are subjected to the same situations before expressing their choice; if not consider the conditional multinomial logit introduced by Nobel Laureate McFadden. This sort of stuff is highly developed as discrete choice theory in transport economics as you make decisions about mode of transport. it is known as the red bus blue bus problem - or more prosaically as the independence of irrelevant alternatives assumption of the multinomial logit .

Dear Kelvyn,

thank you for your answer! I will keep trying then!

Robbert

Hi Kelvyn,

I saw minitab is providing a free trail, so I could use that probably. Do you work with minitab?

So, according to you, I have to use the total sum of how much each response occurs right? My denominator is 1800 (quite large) and this is the same for almost all the observed individuals. However, when I check the website you posted in your answer: Do I have to sum all the response variables of the 10 observed individuals of 9h, all the response variables of 13h, and all the response variables of 17h and use that as dataset? Or do I have to put in one row per observed individual? I ask this because of the example dataset construction of the minitab website you posted. Subjects would be my behaviors, ''Model'' would be the hours (9h,13h,17h), and frequency the number that that behaviors was observed at all 10 individuals at the corresponding hour.

(Each hour, 10 individuals were observed)

Thanks!

You really have repeated measures with 2 factors, viz the individual and the time . You need to do what is called a fixed effects analysis when a dummy is entered in to the model for each individual. That would take out anything idiosysncratic about the individual and then you can see the time effects net of that.

The data needs to be in the long form

Numerator Denominator Behaviour Individual Time

3 12 Feeding One Early

0 12 Fighting One Early

9 12 Mating One Early

3 12 Mating Ten Late