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Lecture 3: Statistics Review I Date: 9/3/02 Distributions Likelihood Hypothesis tests Sources of Variation Definition: Sampling variation results because we only sample a fraction of the full population (e.g. the mapping population). Definition: There is often substantial experimental error in the laboratory procedures used to make measurements. Sometimes this error is systematic. Parameters vs. Estimates Definition: The population is the complete collection of all individuals or things you wish to make inferences about it. Statistics calculated on populations are parameters. Definition: The sample is a subset of the population on which you make measurements. Statistics calculated on samples are estimates. Types of Data Definition: Usually the data is discrete, meaning it can take on one of countably many different values. Definition: Many complex and economically valuable traits are continuous. Such traits are quantitative and the random variables associated with them are continuous (QTL). Random We are concerned with the outcome of random experiments. production of gametes union of gametes (fertilization) formation of chiasmata and recombination events Set Theory I Set theory underlies probability. Definition: A set is a collection of objects. Definition: An element is an object in a set. Notation: sS “s is an element in S” Definition: If A and B are sets, then A is a subset of B if and only if sA implies sB. Notation: AB “A is a subset of B” Set Theory II Definition: Two sets A and B are equal if and only if AB and BA. We write A=B. Definition: The universal set is the superset of all other sets, i.e. all other sets are included within it. Often represented as. Definition: The empty set contains no elements and is denoted as. Sample Space & Event Definition: The sample space for a random experiment is the set that includes all possible outcomes of the experiment. Definition: An event is a set of possible outcomes of the experiment. An event E is said to happen if any one of the outcomes in E occurs. Example: Mendel I Mendel took inbred lines of smooth AA and wrinkled BB peas and crossed them to make the F1 generation and again to make the F2 generation. Smooth A is dominant to B. The random experiment is the random production of gametes and fertilization to produce peas. The sample space of genotypes for F2 is AA, BB, AB. Random Variable Definition: A function from set S to set T is a rule assigning to each sS, an element tT. Definition: Given a random experiment on sample space , a function from to T is a random variable. We often write X, Y, or Z. If we were very careful, we’d write X(s). Simply, X is a measurement of interest on the outcome of a random experiment. Example: Mendel II Let X be the number of A alleles in a randomly chosen genotype. X is a random variable. Sample space is = {0, 1, 2}. Discrete Probability Distribution Suppose X is a random variable with possible outcomes {x1, x2, …, xm}. Define the discrete probability distribution for random variable X as with P X xi x xi p X x 0 x xi p x 1 i 1 X i Example: Mendel III p X 0 0.25 p X 1 0.50 p X 2 0.25 p X otherwise 0 Cumulative Distribution The discrete cumulative distribution function is defined as P X xi F xi P X x x xi The continuous cumulative distribution function is defined as F x P X x f u du x Continuous Probability Distribution dF ( x) F ' ( x) f ( x) dx If exists, then f(x) is the continuous probability distribution. As in the discrete case, f u du 1 Expectation and Variance xi p x xi for discrete random variable xi E X uf u du for continuous random variable xi Exi 2 p x xi for discrete random variable xi Var X u Eu 2 f u du for continuous random variable Moments and MGF Definition: The rth moment of X is E(Xr). Definition: The moment generating function is defined as E(etX). mgf X E etX etxi p x xi for discrete random variable xi etu f u du for continuous random variable Example: Mendel IV Define the random variable Z as follows: 0 if seed is smooth Z 1 if seed is wrinkled If we hypothesize that smooth dominates wrinkled in a single-locus model, then the corresponding probability model is given by: Example: Mendel V PZ 0 3 4 pZ z 1 P Z 1 4 EZ 3 0 1 1 1 4 4 4 Var Z 0 1 3 1 1 1 3 4 4 4 4 16 2 2 Joint and Marginal Cumulative Distributions Definition: Let X and Y be two random variables. Then the joint cumulative distribution is F x, y P X x, Y y Definition: The marginal cumulative distribution is P X x, Y y for discrete random variables y FX x F x x f u, v dvdu for continuous random variables Joint Distribution Definition: The joint distribution is px, y P X x, Y y As before, the sum or integral over the sample space sums to 1. Conditional Distribution Definition: The conditional distribution of X given that Y=y is p x, y P X x, Y y px y P X x Y y p y PY y Lemma: If X and Y are independent, then p(x|y)=p(x), p(y|x)=p(y), and p(x,y)=p(x)p(y). Example: Mendel VI P(homozygous | smooth seed) = P X 2, Z 1 1 4 PX 2 Z 1 3 PZ 1 3 4 1 Binomial Distribution Suppose there is a random experiment with two possible outcomes, we call them “success” and “failure”. Suppose there is a constant probability p of success for each experiment and multiple experiments of this type are independent. Let X be the random variable that counts the total number of successes. Then XBin(n,p). Properties of Binomial Distribution n x n x f x; n, p P X x n, p p 1 p x n x tx n x t mgf X e p 1 p 1 p pe x 0 x E X np Var X np1 p n Examples: Binomial Distribution recombinant fraction between two loci: count the number of recombinant gametes in n sampled. phenotype in Mendel’s F2 cross: count the number of smooth peas in F2. Multinomial Distribution Mn, p1 , p2 ,, pm Suppose you consider genotype in Mendel’s F2 cross, or a 3-point cross. Definition: Suppose there are m possible outcomes and the random variables X1, X2, …, Xm count the number of times each outcome is observed. Then, n! P X 1 x1 , X 2 x2 , , X m xm p1x1 p2x2 pmxm x1! x2 ! xm ! Poisson Distribution Consider the Binomial distribution when p is small and n is large, but np= is constant. Then, x n x e n x p 1 p x! x The distribution obtained is the Poisson Distribution. Properties of Poisson Distribution e f x x! x e e t 1 mgf X e e x! E X Var X tx x Normal Distribution Confidence intervals for recombinant fraction can be estimated using the Normal distribution. N , 2 Properties of Normal Distribution 1 f x e 2 t mgf X e E X Var X 2 2t 2 2 x 2 2 2 Chi-Square Distribution Many hypotheses tests in statistical genetics use the chi-square distribution. k 1 1 2 k 2 1 x 2 f x x e k 2 2 1 mgf X for t 0.5 k 1 2t 2 E X k Var X 2k Likelihood I Likelihoods are used frequently in genetic data because they handle the complexities of genetic models well. Let be a parameter or vector of parameters that effect the random variable X. e.g. =(,) for the normal distribution. Likelihood II Then, we can write a likelihood L L x1 , xn Pxi n i 1 where we have observed an independent sample of size n, namely x1,x2,…,xn, and conditioned on the parameter . Normally, is not known to us. To find the that best fits the data, we maximize L() over all . Example: Likelihood of Binomial n x n x L Ln, p P X x n, p 1 x n l log L log x log p n x log 1 p x l x n x p p 1 p x pˆ n The Score Definition: The first derivative of the log likelihood with respect to the parameter is the score. For example, the score for the binomial parameter p is l x n x p p 1 p Information Content Definition: The information content is 2 I E l x 2 E 2 l x If evaluated at maximum likelihood estimate ˆ, then it is called expected information. Hypothesis Testing Most experiments begin with a hypothesis. This hypothesis must be converted into statistical hypothesis. Statistical hypotheses consist of null hypothesis H0 and alternative hypothesis HA. Statistics are used to reject H0 and accept HA. Sometimes we cannot reject H0 and accept it instead. Rejection Region I Definition: Given a cumulative probability distribution function for the test statistic X, F(X), the critical region for a hypothesis test is the region of rejection, the area under the probability distribution where the observed test statistic X is unlikely to fall if H0 is true. The rejection region may or may not be symmetric. Rejection Region II Distribution under H0 1-F(xc) 1-F(xl) or 1-F(xu) Acceptance Region Region where H0 cannot be rejected. One-Tailed vs. Two-Tailed Use a one-tailed test when the H0 is unidirectional, e.g. H0: 0.5. Use a two-tailed test when the H0 is bidirectional, e.g. H0: =0.5. Critical Values Definition: Critical values are those values corresponding to the cut-off point between rejection and acceptance regions. P-Value Definition: The p-value is the probability of observing a sample outcome, assuming H0 is true. p - value 1 F xˆ Reject H0 when the p-value. The significance value of the test is . Chi-Square Test: Goodness-ofFit a oi ei 2 i 1 ei 2 Calculate ei under H0. 2 is distributed as Chi-Square with a-1 degrees of freedom. When expected values depend on k unknown parameters, then df=a1-k. Chi-Square Test: Test of Independence a b 2 i 1 j 1 o ij eij 2 eij eij = np0ip0j degrees of freedom = (a-1)(b-1) Example: test for linkage Likelihood Ratio Test LR L ˆ X L0 X G=2log(LR) G ~ 2 with degrees of freedom equal to the difference in number of parameters. LR: goodness-of-fit & independence test goodness-of-fit n oi G 2 oi log ei i 1 independence test a b G 2 oij log i 1 j 1 oij eij Compare 2 and Likelihood Ratio Both give similar results. LR is more powerful when there are unknown parameters involved. LOD Score LOD stands for log of odds. It is commonly denoted by Z. L ˆ X Z log 10 L0 X The interpretation is that HA is 10Z times more likely than H0. The p-values obtained by the LR statistic for LOD score Z are approximately 10-Z. Nonparametric Hypothesis Testing What do you do when the test statistic does not follow some standard probability distribution? Use an empirical distribution. Assume H0 and resample (bootstrap or jackknife or permutation) to generate: empirical CDF( X ) P X x