SPEAKER: One of the most successful applications
of statistical models in high-throughput experiments
is related to estimating variance.
In this video, we're going to introduce parametric models that
can be used to model variance.
This picture here, we're showing you an example
where we took a data set with 12 biological replicates for around
10,000 genes.
And we computed the standard deviation across samples for each gene.
So this gives us around 10,000 estimates of the standard deviation or sample
standard deviations.
So on the picture on the left, we see an empirical distribution
of these sample standard deviations.
And on the right, we see a Q-Q plot of these values
plotted against the normal distribution.
We immediately see that the normal distribution is not appropriate.
The right tail is too large.
And also the normal distribution has negative numbers,
and variances are strictly positive.
So we have several choices for models of variance, for modeling variance.
The F-distribution is an example.
I'm showing you the equation for this parametric distribution.
You don't have to memorize it or know this distribution by heart.
But I do want to show you the formula, so you
can see that we have certain parameters that
permit us to change the shape of this distribution,
namely through the degrees of freedom, d1 and d2.
And another variable that isn't in this formula is called a scale parameter.
You're going to see how we use it in the next slide.
But basically, it lets us account for changes in units.
Imagine that we change the units from centimeters
to meters or from grams to kilograms.
We would have to multiply the standard deviation
by a constant, which means that this distribution will
have to change by a constant.
In this case, s0 is that constant.
So here's an example for the data that we've been working with.
Here we see three versions of the F-distribution
with two different values for s0 and three different values for d.
In black, you see the histogram of the sample standard deviations.
And in green, we see the fitted parametric distribution to the data.
And you can see how it changes as we change s0
and how it changes as we change d.
s0 changes the location.
And d changes the shape.
We see that one of those cases, the fitted parametric model
fits the data quite well.
And here's a close-up.
Here we see a Q-Q plot of the observed values
versus the theoretical predictive values from our model fit.
We see that the fit is much better than, for example, the normal distribution
that we saw earlier.
And on the right, we see the histogram along
with the fitted parametric distribution, in green, for the values
that we obtained by finding the best fits s0 and d.
You can see them in the title of the figure.
So this figure shows us how we can use parametric models
to model standard deviations.
And it works pretty well.
Later, we're going to see how we use this
for a downstream improvement of results from a high-throughput experiment.