One-Way ANOVA and Multiple Comparisons The purpose of one-way analysis of variance is to determine if any...

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One-Way ANOVA and Multiple Comparisons
The purpose of one-way analysis of variance is to determine ifany experimental treatment, or population, means, are significantlydifferent. Multiple comparisons are used to determine which of thetreatment, or population, means are significantly different. Wewill study a statistical method for comparing more than twotreatment, or population, means and investigate several multiplecomparison methods to identify treatment differences.
-Search for a video, news item, or article (include the link inyour discussion post) that gives you a better understanding ofone-way analysis of variance and/or multiple comparison methods, oris an application in your field of study.
-Explain in your post why you chose this item and how yourlinked item corresponds to our One-Way ANOVA and MultipleComparisons course objectives.
-Then describe how you could use any of these methods in yourfuture career or a life situation.

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Previously we have tested hypotheses about two populationmeans This chapter examines methods for comparing more than twomeans Analysis of variance ANOVA is an inferential method usedto test the equality of three or more population meansH0 1 2 3 kThis method is also referred to as singlefactor ANOVA becausewe use a single property or characteristic for categorizing thepopulations This characteristic is sometimes referred to as atreatment or factorA treatment or factor is a property or characteristic thatallows us to distinguish the different populations from oneanotherThe objects of ANOVA are 1 estimate treatment means and thedifferences of treatment means 2 test hypotheses for statisticalsignificance of comparisons of treatment means where treatmentor factor is the characteristic that distinguishes thepopulationsFor example a biologist might compare the effect that threedifferent herbicides may have on seed production of an invasivespecies in a forest environment The biologist would want toestimate the mean annual seed production under the three differenttreatments while also testing to see which treatment results inthe lowest annual seed production The null and alternativehypotheses areH0 1 2 3H1 at least one of the means is significantly different fromthe othersIt would be tempting to test this null hypothesis H0 1 2 3by comparing the population means two at a time If we continuethis way we would need to test three different pairs ofhypothesesH0 1 2ANDH0 1 3ANDH0 2 3H1 1 2H1 1 3H1 2 3If we used a 5 level of significance each test would have aprobability of a Type I error rejecting the null hypothesis whenit is true of 005 Each test would have a 95 probability ofcorrectly not rejecting the null hypothesis The probability thatall three tests correctly do not reject the null hypothesis is0953 086 There is a 1 0953 014 14 probability that atleast one test will lead to an incorrect rejection of the nullhypothesis A 14 probability of a Type I error is much higher thanthe desired alpha of 5 remember is the same as Type I errorAs the number of populations increases the probability of making aType I error using multiple ttests also increases Analysis ofvariance allows us to test the null hypothesis all means areequal against the alternative hypothesis at least one mean isdifferent with a specified value of The assumptions for ANOVA are 1 observations in each treatmentgroup represents a random sample from that population 2 each ofthe populations is normally distributed 3 population variancesfor each treatment group are homogeneous ie We can easily test the normality of the samples by creating anormal probability plot however verifying homogeneous variancescan be more difficult A general rule of thumb is as followsOneway ANOVA may be used if the largest sample standarddeviation is no more than twice the smallest sample standarddeviationIn the previous chapter we used a twosample ttest to comparethe means from two independent samples with a common variance Thesample data are used to compute the test statisticwhere is the pooled estimate of the common population variance 2 Totest more than two populations we must extend this idea of pooledvariance to include all samples as shown belowwhere Sw2 represents the pooled estimate of the common variance2 and it measures the variability of the observations within thedifferent populations whether or notH0 is true This isoften referred to as the variance within samples variation due toerrorIf the null hypothesis IS true all the means are equal thenall the populations are the same with a common mean and variance2 Instead of randomly selecting different samples from differentpopulations we are actually drawing k different samplesfrom one population We know that the sampling distribution fork means based on nobservations will have meanx and variance 2 squared standard error Since wehave drawn k samples of nobservations each wecan estimate the variance of the k sample means 2 byConsequently n times the sample variance of the meansestimates 2 We designate this quantity as SB2 such thatwhere SB2 is also an unbiased estimate ofthe common variance 2 IF H0 IS TRUE This is often referred to asthe variance between samples variation due to treatmentUnder the null hypothesis that all kpopulations areidentical we have two estimates of 2 SW2 and SB2 We can usethe ratio of SB2 SW2 as a test statistic to test the nullhypothesis that H0 1 2 3 k which follows anFdistribution with degrees of freedom df1 k 1 and df2 N k where k is the number of populations and N isthe total number of observations N n1 n2 nk The numeratorof the test statistic measures the variation between sample meansThe estimate of the variance in the denominator depends only on thesample variances and is not affected by the differences among thesample meansWhen the null hypothesis is true the ratio of SB2 and SW2 willbe close to 1 When the null hypothesis is false SB2 will tend tobe larger than SW2 due See Answer

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