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1Stat 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health SciencesInstructor: Ivo Dinov, Asst. Prof. of Statistics and NeurologyTeaching Assistants:Fred Phoa, Kirsten Johnson, Ming Zheng & Matilda HsiehUniversity of California, Los Angeles, Fall 2005http://www.stat.ucla.edu/~dinov/courses_students.htmlStat 13, UCLA, Ivo DinovSlide 2Sample Size Calculations & Confidence Intervals for ProportionsStat 13, UCLA, Ivo DinovSlide 3Planning a Study to Estimate µz It is important before you begin collecting data to consider whether the estimates will be sufficiently precise.z Two factors to consider: the population variability of Y sample sizeStat 13, UCLA, Ivo DinovSlide 4z First: In certain situations the variability of Y should not be controlled for (response in a medical study to treatment). However, in most studies it is important to reduce the variability of Y, by holding extraneous conditions as constant as possible. For example: study of breast cancer might want to examine only womenPlanning a Study to Estimate µStat 13, UCLA, Ivo DinovSlide 5z Second: Once the experiment is planned to reduce the variability of Y as much as possible, we consider the sample size.  For example: how many women should we sample to achieve the desired precision for our estimate?z RECALL: nsSE =Planning a Study to Estimate µStat 13, UCLA, Ivo DinovSlide 6z To decide on a proper value of n, we must specify what value of SE is desirable and have a guess of s. For SE we need to ask what value would we tolerate? For s we could use information from a pilot study or previous researchnsGuessedSEDesired =Planning a Study to Estimate µ2Stat 13, UCLA, Ivo DinovSlide 7Example: Reindeer (Cont’)= 54.78s = 8.83SE = 0.874Suppose we would like to estimate the sample size necessary for next year's round-up to keep SE < 0.6reindeernnn 21758.21672.1460.083.8≈≤≤≤yCan't have 0.6 of a reindeer, so we round (ALWAYS round up on sample size calculations) to n = 217 reindeer.Planning a Study to Estimate µStat 13, UCLA, Ivo DinovSlide 8z What happens to n as the desired precision gets smaller?Example: Reindeer (cont’) Suppose we would like to estimate the sample size necessary for next year's round-up to keep SE <0.3z When we double the precision (ie. cut SE in half) it requires 4 times as many reindeer.  This is the result of the reindeernn 86732.86683.830.0≈≥≥Planning a Study to Estimate µStat 13, UCLA, Ivo DinovSlide 9Decisions About SEz How do we make the decision of what SE we will tolerate is the estimation of µ RECALL: the + part is called the margin of error and is equivalent to t(df)0.025* SE for a 95% confidence interval If we scan the 0.025 (or 95%) column of the t table the t multipliers are roughly equal to 2.SESEdft 2)(025.0≈y⎟⎠⎞⎜⎝⎛±nsdfty2)(αSEdft025.0)(+SEdft025.0)(−Stat 13, UCLA, Ivo DinovSlide 10z So then for example, maybe we reason that we want our estimate to be within µ+ 1.2 with 95% confidence Using the logic from the previous slide thinking of the span of the CI, suppose a total span of 2.4 or +1.2 is desired, then SE would need to be <0.60y-1.2+ 1.26.02.122)(025.0==≈SESESESEdftDecisions About SEStat 13, UCLA, Ivo DinovSlide 11Conditions for Validity of Estimation Methodsz We have to be careful when making estimations computers make it easy interpretations are valid only under certain conditionsStat 13, UCLA, Ivo DinovSlide 12Conditions of validity of the SE formula z For to be an estimate of µ, we must have sampled randomly from the population If not the inference is questionable/biased z The validity of SE also requires: The population is large when compared to the sample size rare that this is a problem sample size can be as much as 5% of the population without seriously inflating SE. Observations must be independent of each other we want the n observations to give n independent pieces of information about the population.y3Stat 13, UCLA, Ivo DinovSlide 13Conditions of validity of the SE formulaz Definition: A hierarchical structure exists when observations are nested within the sampling units this is a common problem in the sciencesExample: Measure the pulse of 10 patients 3 times each. z We don't have 30 pieces of independent information. One possible naïve solution: we could use each persons averageStat 13, UCLA, Ivo DinovSlide 14Conditions of validity of a CI for µz Data must be from a random sample and observations must be independent of each other If the data is biased, the sampling distribution concepts on which the CI method is based do not hold knowing the average of a biased sample does not provide information about µStat 13, UCLA, Ivo DinovSlide 15z We also need to consider the shape of the data for Student's T distribution: If Y is normally distributed then Student's T is exactly valid  If Y is approximately normal then Student's T is approximately valid  If Y is not normal then Student's T is approximately valid only if n is large (CLT) How large? Really depends on severity of non-normality, however our rule of thumb is n >30 Page 202 has a nice summary of these conditions NOTE: If sampling distribution cannot be considered normal Student's T will not hold. Conditions of validity of a CI for µStat 13, UCLA, Ivo DinovSlide 16Verifications of Conditionsz In practice these conditions are often assumptions, but it is important to check to make sure they are reasonable  Scrutinize study design for: random sampling possible bias non-independent observations Population Normal? previous experience with other similar data histogram/normal probability plot increase sample size try a transformation and analyze on the transformed scaleStat 13, UCLA, Ivo DinovSlide 17CI for a Population Proportionz So far we have discussed a confidence interval using quantitative dataz There is also a CI for a dichotomous categorical variable when the parameter of interest is a population proportionis the sample proportionp is the population proportionpˆStat 13, UCLA, Ivo DinovSlide 18z When the sample size is large, the sampling distribution of is approximately normal Related to the CLTz When the sample size is small, the normal approximation may be inadequate To accommodate this we will modify slightlypˆpˆCI for a Population Proportion4Stat 13, UCLA, Ivo DinovSlide 19z The


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