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1、Chapter4In troduct ion toAn alysis-of-Varia nee Procedures Chapter T able of Conten ts52Chapter4 .In troductio n to An alysis-of-Varia nee Procedures SASOnlin eDoc?:Versio n8Chapter4In troduct ion toAn alysis-of-Varia nee Procedures54Chapter4 .In troductio n to An alysis-of-Varia nee ProceduresThe f

2、ollowi ng secti on prese nts an overview of some of the fun dame ntal features of an alysis of varia nce.Subseque nt sect ions describe how this an alysis is performed with procedures in SAS/STAT software.For more detail,see the chapters for the in dividual procedures.Additional sources are describe

3、d in the“ References ” section on page61.De?n iti onsAn alysis of varia nce(ANOV Ais a tech nique for an alyz ing experime ntal data in which one or more resp on se(or depe ndent or simply Yvariables are measured un-der various con diti ons ide nti?ed by one or more classi?cati on variables.The coi-

4、b in ati ons of levels for the classi?cati on variables form the cells of the experime ntal desig n for the data.For example,a n experime nt may measure weight cha nge(the depe ndent variablefor men and wome n who participated in three differe nt weight-loss programs.The six cells of the desig n are

5、 formed by the six comb in ati ons of sex (me n, wome nand program(A,B,C.In an analysis of variance,the variation in the response is separated into variation attributable to differe nces betwee n the classi?catio n variables and variati on attributable to ran dom error.A n an alysis of varia nee con

6、 structs tests to determ ine the sig ni ?ca nee of the classi?cati on effects.A typical goal in an an alyss of varia nee is to compare means of the resp onse variable for various comb in ati ons of the classi?cati on varlbles.An an alysis of varia nee may be writte n as a lin ear model.A nalysis of

7、varia nee procedures in SAS/STAT software use the model to predict the resp onse for each ob- servati on. The differe nee betwee n the actual and predicted resp onse is the residual error.Most of the procedures?t model parameters that mini mize the sum of squares of residual errors.Thus,the method i

8、s called least squares regressio n.The varia nee due to the ran dom error,is estimated by the mea n squared error(MSE or.Gen eral Lin ear Models55 In repeated-measures experime nts with people or ani mals as subjects,subjectsare declared ran dom because they are selected from the larger populatio n

9、towhich you want to gen eralize.A typical assumpti on is that ran dom effects have values draw n from a n ormally distributed ran dom process with mean zero and com mon varia nce.Effects are declared ran dom whe n the levels are ran domly selected from a large populati on of possible levels .Inferen

10、 ces are made using only a few levels but can be gen eralized across thewhole populati on of ran dom effects levels.The eon seque nee of hav ing ran dom effects in your model is that some observati onsare no Ion ger un correlated but in stead have a covaria nee that depe nds on the varia neeof the r

11、an dom effectn fact,a more gen eral approach to ran dom effect models is to model the covaria nee betwee n observati ons.MS(Ehas the distribution under the null hypothesis.When the null hypothesis is false,the nu merator term has a larger expected value,but the expected value of the denomin ator rem

12、ai ns the same.Thus,large values lead to rejectio n of the n ull hypothesis.The probability of gett ing an value at least as large as the one observed give n that thenull hypothesis is true is called the signi?cance probability value(or the/alue.A-value of less tha nO .05,for example,i ndicates that

13、 data with no real A effectwill yield values as large as the one observed less tha n5%of the time.This is usually con sidered moderate evide nce that there is a real A effect.Smaller-values con stitute eve n stro nger evide nce.Larger-values in dicate that the effect of in terestis less than random

14、noise.In this case,you can conclude either that there is no effectat all or that you do not have eno ugh data to detect the differe nces being tested.SAS On li neDoc?:Versio n856Chapter4 .In troductio n to An alysis-of-Varia nce Procedures1100011000101001010010010100101000110001The lin ear model for

15、 this example isTo con struct crossed and n ested effects,you can simply multiply out allcomb in ati ons of the mai nffect colu mn s.This is described in detail in“ Speci?cati on ofEffects ”in Chapter30, “The GLM Procedure. ”SAS On li neDoc?:Versio n8PROC ANOVA for Bala need Desig ns57 An alysis of

16、Varia nee for Fixed EffectModels58Chapter4 .In troductio n to An alysis-of-Varia nee ProceduresCompari ng Group Mea ns with PROC ANOVA and PROC GLM When you have more than two mea ns to compare,a n test in PROC ANOV A or PROC GLM tells you whether the meais are signi?cantly different from each other

17、, but it does not tell you which means differ from which other means.If you have speci?c comparis ons in mi nd,you can use the CONTRAST stateme nt in PROC GLM to make these comparis on s.However,if you make many comparis ons using some give n sig ni ?ca nee level(,for example,you are more likely to

18、makea type1error(i ncorrectly reject ing a hypothesis that the mea ns are equalsimplybecause you have more cha nces to make the error.Multiple comparis on methods give you more detailed in formatio n about the differ- en ces among the means and en ables you to con trol error rates for a multitude of

19、 comparis on s.A variety of multiple comparis on methods are available with the MEANS stateme nt in both the ANOV A and GLM procedures,as well as the LSMEANS statement in PROC GLM.These are described in detail in“Multiple Comparisons ” inChapter30, “The GLM Procedure. ”An alysis of Varia nee for Cat

20、egorical Data and Gen eralized Lin ear Models59 In additi on to testi ng for differe nces betwee n two groups,PROC TTEST performs atest for un equal varia nces.You can use PROC TTEST with bala need or un bala needgroups.The PROC NPAR1WAY procedure performs non parametric an alogues totests.See Chapt

21、er13, “In troductio n to Non parametric An alysis,” for an overviewand Chapter47for details on PROC NPAR1WAY.An alysis of Varia nee for Categorical Data and Gen eralized Lin ear ModelsA categorical variable is de?ned as one that can assume only a limited number ofvalues.For example,a person s sex is

22、 a categorical variabieatisaince one oftwo values.Variables with levels that simply n ame a group are said to be measured ona nominal scale.Categorical variables can also be measured using an ordi nal scale,SAS On li neDoc?:Versio n8which means that the levels of the variable are ordered in some way

23、.For example, resp on ses to an opinion poll are usually measured on an ordinal scale,with levels ranging from stro ngly disagreeto no opi nion tostro ngly agree.For two categorical variables, one measured on an ordinal scale and one measured ona nominal scale,you may assig n scores to the levels of

24、 the ordinal variable and testwhether the mea n scores for the differe nt levels of the nominal variable are sign-?can tly differe nt.This process is an alogous to perform ing an an alysis of varia nee on con ti nu ous data,which can be performed by PROC CATMOD .If there are nominal variables,rather

25、 tha n,the n PROC CATMOD can do an-way an alysis of varia nee of the mea n scores.For two categorical variables measured on a nominal scale,you can test whether the distribution of the?rst variable is signi?cantly different for the levels of the second variable.This process is an an alysis of varia

26、nee of proporti on s,rather tha n means, and can be performed by PROC CATMOD.The corresp ondin g-way an alysis of varia nee can also be performed by PROC CATMOD.See Chapter5,“Introduction to Categorical Data Analysis Procedures, -” and Chapter22, “The CATMOD Procedure, ” for more information.GENMOD

27、uses maximum likelihood estimatio n to?t gen eralized lin ear models.This family includes models for categorical data such as logistic,probit,and complemen-tary log-log regressi on for bino mial data and Poiss on regressi on for count data,as well as con ti nu ous models such as ordi nary lin ear re

28、gressi on, gamma and inv erse Gaussia n regressi on models.GENMOD performs an alysis of varia nee through like-lihood ratio and Wald tests of?xed effects in gen eralized lin ear models,a nd provides eon trasts and estimates for customized hypothesis tests .It performs an alysis of re-peated measures

29、 data with gen eralized estimati ng equatio n( GEEmethods.See Chapter5, “Introduction to Categorical Data Analysis Procedures,-” and Chapter29, “The GENMOD Procedure, ” for more information.Refere nces scription of PROC RANK in the SAS Procedures Guide and in Con over and Ima n (1981.61 Con struct i

30、ng An alysis of Varia nce Desig ns An alysis of varia nce is most ofte n used for data from desig ned experime nts. You can use the PLAN procedure to con struct desig ns for many experime nts. For example, PROC PLAN con structs desig ns for completely ran domized experime nts, ran domized blocks, La

31、ti n squares, factorial experime nts, and bala need in complete block desig ns. Ran domizati on, or ran domly assigning experimental units to cells in a design and to treatments within a cell, is another important aspect of experimental design. For either a new or an existing design, you can use PRO

32、C PLAN to randomize the experimental plan. Additional features for design of experime nts are available in SAS/QC software. The FACTEX and OPTEX procedures can con struct a wide variety of desig ns, in cludi ng factorials, fract ional factorials, and D- optimal or A-optimal desig ns. These procedure

33、s, as well as the ADX In terface, provide features for ran domiz ing and replicati ng desig ns; sav ing the desig n in an output data set; and in teractively cha nging the desig n by cha nging its size, use of block ing, or the search strategies used. For more in formati on, see SAS/QC Software: Ref

34、ere nce. Refere nces An alysis of varia nce was pion eered by R.A. Fisher (1925. For a gen eral in troduct ion to an alysis of varia nce, see an in termediate statistical methods textbook such as Steel and Torrie (1980, Snedecor and Cochran (1980, Milliken and Johnson (1984, Mendenhall (1968, John (

35、1971, Ott (1977, or Kirk (1968. A classic source is Scheffe (1959. Freund, Littell, and Spector (1991 bring together a treatment of these statistical methods and SAS/STAT software procedures. Schlotzhauer and Littell (1997 cover how to perform t tests and on e-way an alysis of varia nce with SAS/STA

36、T procedures. Texts on lin ear models include Searle (1971, Graybill (1976, and Hocking (1984. Kennedy and Gentle (1980 survey the comput ing aspects. Con over, W.J. and Ima n, R.L. (1981,“ RankTransformations as a Bridge Between Parametric anNonparametric Statistics, ” TheAmerican Statisticia n, 35

37、, 124-129. Fisher, R.A. (1925, Statistical Methods for Research Workers, Edi nburgh: Oliver & Boyd. Freu nd, R.JLittell, R.C., and Spector, P.C. (1991, SAS System for Lin ear Models, Cary, NC: SAS Institute Inc. SAS On li neDoc?: Versio n 862 Chapter 4. Introduction to Analysis-of-Variance Procedure

38、s Graybill, F.A. (1976, Theory and Applicatio ns of the Lin ear Model, North Scituate, MA: Duxbury Press. Hocki ng, R.R. (1984, An alysis of Lin ear Models, Mon terey, CA: Brooks-Cole Publishi ng Co. John, P. (1971, Statistical Design and Analysis of Experiments, New York: Macmilla n Publishi ng Co.

39、 Kenn edy, WJ., Jr. and Gen tle, J.E. (1980, Statistical Computing, New York: Marcel Dekker, Inc. Kirk, R.E. (1968, Experimental Design: Procedures for the Behavioral Scien ces, Mon terey, CA: Brooks-Cole Publish ing Co. Men de nhall, W. (1968, I ntroduction to Lin ear Models and the Desig n an d An

40、 alysis of Experiments, Belmont, CA: Duxbury Press. Milliken, G.A. and Johnson, D.E. (1984, Analysis of Messy Data Volume I: Designed Experiments, Belmont, CA: Lifetime Learning Publications. Ott, L. (1977, Introduction to Statistical Methods and Data An alysis, Seco nd Editio n, Belm ont, CA: Duxbu

41、ry Press. Scheffe, H. (1959, The An alysis of Variance, New York: John Wiley & Sons, Inc. Schlotzhauer, S.D. and Littell, R.C. (1997, SAS System for Elementary Statistical Analysis, Cary, NC: SAS Institute Inc. Searle, S.R. (1971, Li near Models, New York: John Wiley & So ns, I nc. Sn edecor, G.W. and Cochra n, W.G. (1980, Statistical Methods, Seve nth Editio n, Ames, IA: Iowa State Un iversity Press. Steel R.G.D. and Torrie, J.H. (1980, Prin ciples and Procedures of Statistics, Seco nd Editi on, New York: McGraw-Hill Book Co. SAS On li neDoc?: Versio