Statistics in Bioinformatics • May 12, 2005 • Quiz 3-on May 12 • Learning objectives-Understand equally likely outcomes, counting techniques (Example, genetic code, PCR primer), E value • Homework 10-due May 14, 2005
Statistical analysis of results underlies bioinformatics • When you run a program the computer will always give an answer. • The bioinformaticist will analyze the data from two points of view: • 1) Statistical • 2) Biological • Assessment through these filters will determine if the result is reasonable
Two big questions you need to ask yourself • Does the result fit with what is currently • known about biology (protein structure, • evolution, function, etc.)? • Could the results have been obtained • by random chance? Part of this comes from • scientific intuition but another part comes • from statistics.
Types of statistics typically used in bioinformatics • Yes-Likelihood methods • No-ANOVA, regression analysis, hypothesis testing • When one performs a sequence comparison search one must ask what is the likelihood that one would obtain a match based on random chance. This depends on the sequence you are searching for and the amount of data within the database you are mining.
Equally likely outcomes sample space S= set of all possible outcomes. Assumption: all outcomes are equally likely. Then, for any event A (=set of outcomes) P(A)=number of elements in A = |A| number of elements in S |S| For an experiment consisting of k parts, each of which can have ni outcomes |S|=n1n2 . . .nk
Multiplication Rule n things taken k at a time with repetition is nk Familiar example: the genetic code. Given that there are 4 nucleotides (A,T,G,C) how many different triplet codons are possible? This is the same as saying 4 items taken 3 at a time with repetition. Answer: 43= 64 4 4 4 Position: 1 2 3
Multiplication rule n things taken k at a time with repetition is nk Second example: the PCR primer design. How many different PCR primers of 16 nucleotides in length are possible? This is the same as saying 4 items taken 16 at a time with repetition. Answer: 416= 4.29 x 109 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Position:1 2 3 4 5 6 7 8 9 10111213141516 Any 16mer pattern can be expected to appear approximately once in the human genome by chance alone because the human genome contains 3 x 109 bases
1 43 = 0.0156 One may convert the previous calculations to probabilities • What is the probability that the codon CCC will occur assuming all codons are represented equally?
1 416 = 2.32 x 10-10 • What is the probability that the sequence ATAGCGTACTGCATCA will occur given equal probability of nucleotides at each position?
1 46 = 2.44 x 10-4 Restriction Enzymes • What is the probability that you would expect an EcoRI site in a six nucleotide sequence assuming equal representation of all nucleotides? The sequence is : GAATTC
The E value (false positive expectation value) The Expect value (E) is a parameter that describes the number of “hits” one can "expect" to see just by chance when searching a database of a particular size. It decreases exponentially as the Similarity Score (S) increases (inverse relationship). The higher the Similarity Score, the lower the E value. Essentially, the E value describes the random background noise that exists for matches between two sequences. The E value is used as a convenient way to create a significance threshold for reporting results. When the E value is increased from the default value of 10 prior to a sequence search, a larger list with more low-similarity scoring hits can be reported. An E value of 1 assigned to a hit can be interpreted as meaning that in a database of the current size you might expect to see 1 match with a similar score simply by chance.
E value E = K•m•n•e-λS Where K is constant, m is the length of the query sequence, n is the length of the database sequence, λ is the decay constant, S is the similarity score. If S increases, E decreases exponentially. If the decay constant increases, E decreases exponentially If m•n increases the “search space” increases and there is a greater chance for a random “hit”, E increases. Larger database will increase E.