Dependent or Independent Samples

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    Ana Fernandez

    I record simultaneously several neurons while an animal performs a working memory task. He runs on a T-maze, which consist of two turns: right and left. There are two types of trials: samples and choices. In samples trials, one of the arms is blocked so the animal enters in the open arm. In choice trials, both arms are open and the animal needs to choose the opposite arm that it was forced to choose in the sample trial. If he chooses the same arm, it´s considered an error.
    Let´s take one neuron as an example. I want to know if there are differences in firing rate of one neuron between the different types of trials. Are there differences on firing rate between left and right trials. And between samples and choices?
    The data are non-normal. I checked normality of data with Shapiro test, histograms shapes and q-q plots.

    I´m not sure what type of data I´m dealing with.
    – If I compare differences on firing rate between samples and choice trials. Are these data independent?
    – And between right and left trials (it doesn´t matter if they are samples or choices)?
    – And between right sample trials with right choice trials?

    If data are independent, could use Wilcoxon rank-sum test?


    Robert Butler

    The smallest unit of independence is the individual animal. Your measures are repeated measures and the analysis must take that fact into account. If you don’t you will find all kinds of significant effects where none exist. This is due to the fact that the machine will treat each measure as independent. The repeated measure-to-measure variability will be much less than real independent measure-to-measure variation which means you are testing measurement differences using smaller estimates of variance.

    You could run a repeated measures regression analysis with firing rate being the outcome variable. The issue is going to be that of independent variable definition. Based on what you have posted I think the best be would be a 4 parameter model outcome = forced choice right, forced choice left, choice right, and choice left. I say this since your basic question is firing rate of neurons as a function of direction. The fact that an animal chooses to take the same direction as previously (?) forced to take is interesting but hardly an error and a model of this type will cover the question you are asking.

    If all you care about is mean differences of the 4 choices and if you don’t have any ready means for analyzing repeated measures data then you could build a 2 way ANOVA with the 4 choice types as one variable and the animal identification as the other variable. By including animal in the analysis you will have taken into account the repeated measures nature of the measurements. If you choose the ANOVA approach you will want a reference – the same problem with piil type across patients and a measure of fecal fat as the outcome substituted for choice type across animals with neuron firing rate can be found on pages 254-259 of Regression Methods in Biostatistics by Vitinghoff, Glidden, Shiboski, and McCulloch. That book also does a very good job of discussing the issues of repeated measures and how to deal with them in general.

    As for issues of normality of outcomes – this not an issue either with regression or with t-tests and ANOVA. In regression analysis the issue of approximate normality only applies to the residuals and both the t-test and ANOVA are robust with respect to lack of data normality. By the way, when it comes to testing for normality do yourself a favor and first plot the data – histogram and normal probability plots being the first choice. The various normality tests are so sensitive that it is possible to take generated data using the normal distribution as the underlying generator and have even that data fail one or more of the tests.

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