# Looking for something similar to a three way anova analysis for non-independent response samples

I have devised an experiment that consists of measuring how quickly a person can complete a set of tasks. I have three variables that I wish to investigate whether or not they affect the performance of the person as well as any interaction that exists between the variables. First I will give a bit of background about the experiment followed by the issue I face.

The person is only informed intermittently of the tasks he must complete. I begin measuring the time it takes to complete each individual task from the point in time the person was informed about it. Each task does not take the same amount of time to complete. Furthermore, because these tasks may arrive while the person is currently working on a given task, he may have more than one task to complete when he has finished his current one. Therefore, the time lapse between when he is informed of any given task and when he completes it (other than the first) depends on how quickly the previous tasks have been completed.

I have done some reading and from what I understand it would not be valid for me to apply the three way ANOVA in this instance as it violates the assumption that the samples (ie time to complete a task) are independent. Is there any other statistical method that I may use to obtain similar insight that the three way ANOVA analysis would?

I am familiar with MATLAB and have the statistics toolbox installed so anything that can easily be done on this would be preferred.

I would appreciate any suggestions you may have.

You should be able to design this experiment in such a way the responses are independent. could you set things up so that you have, say

• condition 1: person is informed of task 1 and completes the task. Observe time.
• condition 2: person is informed of task 1 and is interrupted with arrival of task 2. Observe time of task 1.
• condition 3: person is informed of task 1, is interrupted with arrival of task 2. Observe time of task 2.

And so on. Each condition is one level of a design category that you enter into the model. You also have other design criteria, so the actual experimental design could be somewhat complex, with several factors, but you would have independence in the response, conditional on the subject. Since each subject has their own basic response time, this will be a repeated measures (or random effecs) design,

If you really need to analyse the response times together .. say task 1, task 2, task 3 ... then you can do a multivariate repeated measures. But the task itself could be just another experimental category.

I think the key here is to have a balanced and careful design of the tasks. If you just hit people with a random sequence of tasks, it's going to get tricky.

If I might make a suggestion, experimental design takes more than having "just done some reading on ANOVA", so now might be a good time to consult a statistician professionally.