• A distribution is a mathematical function that describes the probability of observations

• It turns out that most observational data can be described by one of several common distributions

  • Continuous data often follows a Gaussian or Normal distribution
  • Count data often follows a Poisson distribution
  • When one of two events can occur (coin toss), the data often follows a binomial distribution

Defined by mean \(\mu\) and standard deviation \(\sigma\).

Probability density (relative probability) of x: \(f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\)

If you go out and measure something there is a good chance your observations can be described by a mean and standard deviation and will look more or less like this.

Defined by a single parameter \(\lambda\) that is both the mean and variance of the distribution.

Probability of observing x counts: \(p(x) = \frac{\lambda^xe^{-\lambda}}{x!}\)

Overdispersion is when there is more variance in the counts (# of trees per grid; or number of reads per gene per sample) then you would expect from Poisson.

• Statistical tests make assumptions about the underlying distribution of data.
  
• A t-test is determining the probability that your data could have come from a single Gaussian distribution (the null hypothesis) instead of from two separate Gaussian groups (with different means).
  
• If the process that generates the data isn’t Gaussian in the first place then the p-values are not valid.
• You have to use a test appropriate to you data.

• Mathematical representation of our observations
• Allows us to test whether various experimental factors (genotype, treatment) are important predictors of our observations.
  
• One way to do this is to compare the how well model predictions “fit” the observations.
  • Does a model with more predictors fit better than a simpler model?
  • Which model matches the observations better?
  • Which model explains more of the variance?

• You measure height of 500 plants grown in a dense planting (DP) or non-dense planting (NDP). You want to know if the treatment influences height.
  • Null hypothesis: treatment (DP vs NDP) does not influence height. Your observations come from a single Gaussian distribution.
  • Alternative hypothesis: treatment does influence height. Your data comes from two Gaussian distributions with different means.
• Which of these match the observed data better?

This also works with more realistic sample sizes

We need a mathematical way of describing the data. First, let’s replot it.

• You want to make the best estimates possible
• You have limited information on each individual gene
• But you have a lot of information about genes in total
• Use understanding of how genes behave in general to improve your estimate of each gene