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The Binomial Distribution

The Binomial Distribution. Nina Gunnes October 17, 2019. Count variable. Resulting from counting the number of times an event occurs Variable only taking on the value zero and positive integer values Mean (expected value) Center of the probability distribution

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The Binomial Distribution

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  1. The Binomial Distribution Nina Gunnes October 17, 2019 Lecture 5

  2. Count variable • Resulting from counting the number of times an event occurs • Variable only taking on the value zero and positive integer values • Mean (expected value) • Center of the probability distribution • Variance and standard deviation • Measure of the dispersion, or spread, of the distribution • and Lecture 5

  3. Example • Norwegian National Advisory Unit on Women’s Health • 25 persons, of which eight are pet owners (and 17 are not) • : number of pet owners • Distribution of pet owners in a random sample of three persons Lecture 5

  4. Binomial experiment • Consisting of a sequence of repeated trials • Each trial resulting in one of two possible outcomes • (for success) • (for failure) • Probability of success (failure) being the same in each trial • Denoted by () • Trials being independent of each other • Outcome of one trial not affecting outcome of other trials Lecture 5

  5. Examples (Aalen et al., 2006) • Childbirth • Boy or girl • Throw of the die • A six or not • Treatment with penicillin • Anaphylactic shock or not • Gallup poll • Voting for a political party or not • Occurrence of disease • Suffering from allergies or not • Genetics • Sickle cell anemia or not Lecture 5

  6. Binomial distribution • Considering a binomial experiment of trials • Two possible outcomes: (success) or (failure) • : number of times occurres (i.e., number of successes) • : probability of success in each trail • A hypothetical case of successes: • Probability of this sequence: • Number of ways to arrange the order of this sequence: , where • Probability of successes: Lecture 5

  7. Binomial distribution, cont. • Mean (expected value) • Expected number of successes • Variance • Skewed distribution when is close to 0 or 1 • More symmetrical distribution for central values of • Perfect symmetry for (i.e., success and failure equally likely) Lecture 5

  8. Binomial distribution, cont. Lecture 5

  9. Example • Flipping a coin 10 times () • Two possible outcomes of each trial: head and tail • Probability of head in each trial: • : number of heads • Probability of at least nine heads? Lecture 5

  10. Hypotheses blah, blah, blah …, significance level blah, blah, blah … Hypothesis testing • Drawing conclusions from data subject to random variation • Evaluating an intervention, comparing treatments, etc. • Is the observed result indicating an effect (or difference)? • Is the observed result merely an act of chance? • Defining a null hypothesis • : no effect (or no difference) • Defining an alternative hypothesis • : effect (or difference) – possibly in a specific direction Yeah, yeah, but is the new treatment better than the old one or not?! Lecture 5

  11. Hypothesis testing, cont. • Different alternative hypotheses for one-sided and two-sided tests • One-sided : one treatment more (less) efficacious than the other • Two-sided : one treatment either more or less efficacious than the other • Direction of any potential difference usually not known in advance • Two-sided test most common in practice • Two types of possible errors • Type I error: erroneously rejecting when it is in fact true • Type II error: erroneously accepting when it is in fact false Lecture 5

  12. Hypothesis testing, cont. • Significance level expressing necessary certainty of the conclusion • Denoted by • Equal to the probability of making a type I error • Serious consequences of rejecting : choosing a low value of (e.g., 1%) • No serious consequences of rejecting : choosing a high value of (e.g., 5%) • Calculating the significance probability, or p value, of a test statistic • Probability of values at least as extreme as the observed value under • (whether one-sided or two-sided) determining what is deemed extreme Lecture 5

  13. Hypothesis testing, cont. • Comparing the p value to the significance level • Small value () • Observed result not consistent with • Rejecting in favor of (that is, accepting ) • Large value () • Failing to reject – NOT(!) the same as proving that is true • Ideally, choosing , , and before the study outcome is known Lecture 5

  14. Migraine (Aalen et al., 2006) • Testing a new medication to treat migraine • : traditional medication • : new medication • Randomized double-blind crossover study • One month of treatment with and one month of treatment with • Neither patient nor physician knowing which medication is taken when • When is the patient least affected by migraine? • Month corresponding to treatment with or ? https://pxhere.com/en/photo/1448737 Lecture 5

  15. Migraine (Aalen et al., 2006), cont. • Eight patients included in a hypothetical study • : number of patients preferring over • binomially distributed with trials and unknown probability • Seven out of the eight patients most comfortable with • Indication of being the best medication or just a coincidence? • Defining the hypotheses for a one-sided test • : (the two medications equally efficacious) • : ( more efficacious than ) Lecture 5

  16. Migraine (Aalen et al., 2006), cont. • Significance probability for a one-sided test • Rejecting at a significance level of 5% • more efficacious than • Defining the alternative hypothesis for a two-sidedtest • : (either more or less efficacious than ) Lecture 5

  17. Migraine (Aalen et al., 2006), cont. • Significance probability for a two-sided test • Not rejecting at a significance level of 5% • No basis for claiming that is more efficacious than Lecture 5

  18. Cot death (Aalen et al., 2006) • Also known as sudden infant death syndrome (SIDS) • SIDS epidemic in Norway and other Western countries in the 1980s • Related to the child’s sleeping position? • 11 children died of SIDS in Hordaland county in 1990 • 10 of the children were in a front-sleeping position (“mageleie”) • Front sleeping in 15% of children in Hordaland county on the average • : number of SIDS babies in a front-sleeping position • binomially distributed with trials and unknown probability Lecture 5

  19. Cot death (Aalen et al., 2006), cont. • Is the observed result conspicuous? • Hypotheses for a one-sided test • : (same front-sleeping proportion as the general population) • : (greater front-sleeping proportion than the general population) • Significance probability for a one-sided test • () Lecture 5

  20. Cot death (Aalen et al., 2006), cont. • Rejecting at a significance level of 5% • Strikingly high frequency of front sleeping among the 11 SIDS babies • Front-sleeping position no longer recommended for babies • Placing babies on their backs to sleep instead https://pixabay.com/vectors/baby-girl-cartoons-pink-new-baby-33252/ Lecture 5

  21. References • Aalen OO, Frigessi A, Moger TA, Scheel I, Skovlund E, Veierød MB. 2006. Statistiskemetoderimedisinoghelsefag. Oslo: Gyldendal akademisk. Lecture 5

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