Nonparametric Statistics STAT 514
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Mr. Alex Berge
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This 6 page Class Notes was uploaded by Mr. Alex Berge on Friday October 23, 2015. The Class Notes belongs to STAT 514 at University of Idaho taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/227940/stat-514-university-of-idaho in Statistics at University of Idaho.
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Date Created: 10/23/15
AOV continued The AOV table Source df Sum of Mean squares square F test Model t 1 SSB SSBt 1 f PVal Error nT t SSW SSW7LT t Corrected nT SST total The AOV model 3 N normal m 02 where 239 l 2 t andj l 2 m Different parameterization M u at where m a2 at 0 oz is the effect ofthe ith treatment level There are still t parameters u a1 a2 at1 and at a1 a2 ozt1 Note that M M1M2Mttandozm M Another way of writing the model Yij 2 M Eij where eij N norma10 02 Model evaluation Statistics is the study of how to draw conclusions from counts and measurements Statistical analysis proceeds by building probabilistic stochastic models of the processes that produce the variability in the counts and measurements You should think of statistics as focused on models rather than methods Fitting a model to data E estimating the unknown model parameters with data Model parameters in 1way AOV M1 M2 pt 02 or u a1 a2 0154 02 Estimates usually denoted with hats for instance 2 pi and so on Evaluating the tted AOV model centers around the key ingredients of the model 1 The variance 02 is constant among the t populations homoskedastic 2 The Yijs within each population 239 have a normal distribution 3 The st are independent random variables Evaluation uses the following quantities Predicted value for yij under the model 31739 yi Residual for yij under the model 61739 yij 31739 yij Recall the model Yij m eij Residual can be thought of as an estimate of the noise amount Eij for that observation If the AOV model is adequate the residuals should be similar to a random sample from a normal distribution with constant variance Note the residuals are in fact dependent but the amount of dependence is small in adequatesized samples For example if there is just one normal population N normalu 02 with randonl sample Y1 Y2 Yn where n 2 2 estimate l7 residual Ej Y one can showthat CorrEk El 2 ln 1 1 Evaluating constant variance assumption important Sidebyside box plots of residuals from each group Scatter plot of residuals vertical vs predicted values Hartley test very sensitive to normality departures or Levine test text section 74 pp 365371 2 Evaluating normality not quite as important Normal probability plot of residuals Test of normality for residuals KolmogorovSmirnov etc Note can obtain both of these from PROC UNIVARIATE with the PLOT and NORMAL options 3 Independence assured by design of sampling or experiment Fixes for nonnormal data 1 Nonparametric model free statistical methods Stat 514 These approaches use weaker assumptions such as assuming that the distributions are symmetric one sample Wilcoxon test for a median two sample MannWhitneyWilcoxon test for comparing two distributions 1way AOV KruskalWallis test These are all examples of methods based on linear rank statistics In fact one can perform the equivalent of all these tests by calculating the ranks of the data ties get averaged ranks and performing the usual normalbased methods on the ranks 2 Transformations Data can often be transformed so that the transformed observations are approximately normal andor have their variances stabilized Multiplicative processes nancesincome biological growth are often normalized by the logarithmic transformation Yij logXlj Poisson observations can be normalized and variance stabilized with the square root transformation Yij xXij 38 Binomial observations can be normalized and variance stabilized with the arcsin inverse sin transformation Kj arcsinXlj The BoxCox transformation is a general family of transformations one for each value of A Xij 1 10gltXija O 3 Nonnormal models
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