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1 Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to comparng two means or proportons to each other to determne whether they were dfferent ( ) or whether the dfference was some specfed value ( some number other than ) Now n chapter, we turn to comparng more than two populatons usng an analyss of varance, or ANOVA A sngle-factor ANOVA focuses on a comparson of more than two populaton or treatment means The usual scenaros are: (a) dfferent populatons: sample data collected to compare a shared characterstc (b) same populaton: sample groups gven dfferent treatments and compared Notaton: the number of populatons or treatments beng compared µ the mean of populaton or the true average response when treatment s appled M µ the mean of populaton or the true average response when treatment s appled The letters and Y were used n two-sample problems (chapter 9) to dfferentate the observatons n one sample from those n the other Because ths s cumbersome for three or more samples, the usual practce s to use a sngle letter wth two subscrpts The frst subscrpt dentfes the sample number, correspondng to the populaton or treatment beng sampled, and the second subscrpt denotes the poston of the observaton wthn that sample, j the random varable that denotes the jth measurement taken from the th populaton, or the measurement taken on the jth expermental unt that receves the th treatment x, j the observed value of the jth measurement taken from the th populaton/treatment when the experment s performed Whle the text uses j and x j when both and j are sngle-dgt numbers, for the Lectures ll always have a comma For the purposes of secton, all sample szes wll be equal, e n n K n The ndvdual sample means wll be denoted by random varables, K,, where Note: The text uses a notaton j, j, so that calculated, x x The dot n place of the second subscrpt sgnfes that we have added over all values of that second subscrpt (j,,, ) whle holdng the frst subscrpt value () fxed (A parallel s the means of and Y n the jont probablty dstrbutons of chapter 5) n a smlar fashon, we can calculate the varance of each sample: ( S ) The average of all tmes observatons, called the grand mean, s denoted by Note: The text uses a notaton x j, j, so that calculated ( ), j j x x j, j j, j
2 For a sngle-factor ANOVA, the hypotheses wll be H : µ µ K µ versus H a : at least two the of the µ s are dfferent For example, f we had, we would want to fal to reject H only when µ µ µ, and would want to reject H when µ or µ µ or µ µ or µ µ A test of these hypotheses requres that we have avalable a random sample from each populaton or treatment The observed data wll be dsplayed n a rectangular table, n whch samples from the dfferent populatons appear n dfferent rows of the table, and x, j s the jth number n the th row Basc assumptons: The x, j s wthn any partcular sample are ndependent (e we have a random sample taken from the th populaton or treatment dstrbuton) Dfferent samples are ndependent of one another Each of the populaton or treatment dstrbutons s normal, and each has the same varance σ Note that the sample standard devatons wll generally dffer somewhat, even when the correspondng populaton/treatment σ s are dentcal A rough rule of thumb s that f the largest s s not much more than two tmes the smallest, t s reasonable to assume equal σ s f ether the normalty assumpton or the assumpton of equal varances s judged mplausble, a method of analyss other than an ANOVA must be used Prelmnary calculatons sums of squares: The total sum of squares (SST) ( x ) ( ) ( ), j x x, j x j j The treatment sum of squares (SSTr) ( x ) ( ) ( ) x x x j The error sum of squares (SSE) ( j x ) Recall from earler, j j,, j, and x j The shortcut expressons for SST and SSTr are convenent f ANOVA calculatons are to be done by hand n practce, the wde avalablty of statstcal software wll usually make ths unnecessary The fundamental dentty relatng the three sums of squares s SST SSTr + SSE The proof s algebrac, and s gven n outlne form n the text n practce, we ll calculate SST and SSTr and then use them to fnd SSE Theory: f H s true, the observatons n each sample come from a normal populaton dstrbuton wth the same mean value µ, n whch case the sample means x, x, K, x should be reasonably close to one another The test procedure s based on comparng a measure of dfferences among the x s ( between-samples varaton) to a measure of varaton calculated from wthn each of the samples SST s a measure of the total varaton n the data the sum of all squared devatons about the grand mean The dentty SST SSTr + SSE says that ths total varaton can be parttoned nto two peces SSE measures varaton that would be present (wthn rows) whether H s true or false, and s thus the part of total varaton that s unexplaned by the status of H SSTr s the amount of varaton (between rows) that can be explaned by possble dfferences n the µ s H s rejected f the explaned varaton s large relatve to unexplaned varaton, j
3 Method: Once SSTr and SSE are computed, each s dvded by ts assocated degrees of freedom to obtan a mean square (mean n the sense of average) SSTr SSE The mean square for treatments (MSTr) The mean square for error (MSE) ( ) Proposton: When H s true, E(MSTr) E(MSE) σ, whereas when H s false, E(MSTr) > E(MSE) σ That s, both statstcs are unbased for estmatng the common populaton varance σ when H s true, but MSTr tends to overestmate σ when H s false The unbased nature of MSE, E(MSE) σ, s a consequence of the fact that E ( ) σ S, whether or not H s true When H s true, each has the same mean value µ and varance σ /, so the sample varance of the s, of σ ( ) estmates σ unbasedly Then, multplyng by gves MSTr as an unbased estmator The s tend to spread out more when H s false than when t s true, tendng to nflate the value of MSTr when H s false, and we would thus want to reject t Test statstc: Our test statstc, called F, s defned as the rato of the two mean squares MSTr and MSE: MSTr F MSE An F dstrbuton arses n connecton wth any rato n whch there s one number of degrees of freedom assocated wth the numerator, and another number of degrees of freedom assocated wth the denomnator We ll use ν and ν ( ) to denote the number of numerator and denomnator degrees of freedom, respectvely, for our F statstc A value of F that greatly exceeds, correspondng to an MSTr much larger than MSE, gves us statstcal reason to queston the assumpton that H s true The approprate form of the rejecton regon s therefore f a boundary value c The value c should be chosen to gve P(F c when H s true) α, the desred sgnfcance level Ths necesstates knowng the dstrbuton of F when H s true n practce, when we have statstcal software we wll use t to calculate the P-value When calculatng by hand, we ll use Appendx Table A9 to determne the crtcal value c for α, 5,, and Values of ν are dentfed wth dfferent columns of the table, and the rows are labeled wth varous values of ν A calculated test statstc f c mples p α, n whch case we wll reject the null hypothess Our computatons wll be summarzed n a tabular format, called an ANOVA table (See Example A below)
4 Example A A paper n Measurement and Evaluaton n Counselng and Development (Oct 9, pp 7) dscussed a survey nstrument called the Mathematcs Anxety Scale for Chldren (MASC) Suppose the MASC was admnstered to three groups of fve sxth graders, wth each group havng been taught usng a dfferent method Test whether the results of the three methods dffer (α 5) Data are as follows Group x x Group x x Group x x hypotheses: x x ν ν ( x, j ) SST SSTr SSE MSTr MSE f Source of Varaton Treatments Degrees of Freedom Sum of Squares Mean Square f Error Total crtcal value c (from Table A9) concluson:
5 Appendx Table A9 Crtcal Values for F Dstrbutons ν numerator df ν α ,84 5, 54,79 56,5 576,45 585,97 59,87 598,44 6,
6 Appendx Table A9 Crtcal Values for F Dstrbutons ν numerator df ν α ,6 6,668 65,764 6,98 64,7 66,99 68,7 6,85 6,
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