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I am confused with applying student's t-test with my biological experiment (even after checking you-tube videos and blogs). I have microarray data from control and hormone-treated plants with 2 replicates each and looks like this,

ctrl-rep1   ctrl-rep2   treated-rep1  treated-rep2
6.11        6.44        7.26          7.08
8.46        9.07        8.99          8.81
12.74       13.98       12.22         13.13
7.22        6.12        7.08          6.51
15.5        16.48       16.67         16.89

Since gene expression can be induced OR reduced upon hormone treatment, I assume two-tail t-test is fit in this case. However, I am not sure about the type of the test to be applied. I assume it is unpaired t-test based on a discussion here. If I decide to use, unpaired t-test, how can I determine it is unpaired with equal variance or unpaired with unequal variance?

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    $\begingroup$ If this is a standard microarray analysis involving thousands of genes at a time, you also have to take into account your multiple statistical comparisons. For example, at p < 0.05 for each individual test you will come up with about 500 false-positive results if you test 10,000 genes. The limma package, noted in the helpful answer from @SmallChess, not only takes that into account but also uses estimates of variance based on results from all the genes to provide more power than a set of repeated t-tests. Use software designed for microarray analysis, like limma. $\endgroup$ – EdM Jul 23 '17 at 14:40
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A nice background reading on microarray t-test is the limma Bioconductor package. It's a very popular package for what you're doing, and I personally take it like a bible. Give it a read when you have an opportunity.

1. Paired vs independent

Let's take a look what a typical paired samples should be:

https://www.bioconductor.org/packages/devel/bioc/vignettes/limma/inst/doc/usersguide.pdf

Paired samples occur when we compare two treatments and each sample given one treatment is naturally paired with a particular sample given the other treatment. This is a special case of blocking with blocks of size two. The classical test associated with this situation is the paired t-test. Suppose an experiment is conducted to compare a new treatment (T) with a control (C). Six dogs are used from three sib-ships. For each sib-pair, one dog is given the treatment while the other dog is a control.

What it really mean is that, if you apply both control and treatment to the same biology source, it's a repeated measurement and therefore you should use the paired t-test.

There is no information on how you conducted your experiment in your question, so I assume your samples are independent and therefore you should use the independent t-test. But please double check.

2. Equal or unequal variance

This is normally not a problem in my works because I just run the limma package and let it figure out the variance... (it uses a Bayesian denominator, but let's not get into that).

There's no right or wrong answer on how you assume your variance. From what I see your data, the variance looks quite close in your control samples and treatment samples. You might want make box-plots for your samples and decide if the equal variance assumption is reasonable. There're some formal ways to do that, but a simple box-plot visualization should be sufficient.

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