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# 68 Class Note for STAT 30100 with Professor Gundlach at Purdue

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Date Created: 02/06/15
Chapter 13 TwoWay Analysis 0fVariance Learning goals for this chapter 0 Know how twoway ANOVA is related to oneway ANOVA and 2sample comparison of means techniques 0 Test the standard deviations to see if it is OK to pool the variances Understand why it is important to be able to pool the variances for twoway ANOVA Explain and check the assumptions that must be met for doing twoway ANOVA Calculate R2 and the estimate for 039 0 Write the 3 sets of hypotheses for twoway ANOVA I Use the F test statistics and Pvalues from SPSS to do twoway ANOVA hypothesis tests 0 Write conclusions to twoway ANOVA tests in terms of the story including using the words population mean Interpret means plots in terms of the two main effects and potential interaction 0 Understand that summary statistics and graphs refer to the sample data and hypothesis tests give us information about the population parameter I Recognize the response variable factors number of levels for each factor and the total number of observations 0 Identify whether the best statistical technique for a story is lsample mean matched pairs 2sample comparison of means oneway ANOVA twoway ANOVA or summary statistics Chapter 7 Twosample comparison of means t tests 1 categorical variable for sorting and 1 quantitative variable for measurement Example Are the mean taste ratings of chewy granola bars the same as those for crunchy granola bars if you conduct a taste test scale of 110 Chapter 12 F tests compare the means of several populations 1 categorical variable for sorting and 1 quantitative variable for measurement Example Are the mean taste ratings of Quaker Kellogg s and Nature Valley granola bars the same if you conduct a taste test scale of 110 Chapter 13 F tests compare the means of populations that are classified in 2 ways 2 categorical variables for sorting and 1 quantitative variable for measurement Example Do brand texture chewy vs crunchy andor their interaction make a difference to the mean taste ratings scale of 110 for granola bars What s similar for Two Way ANOVA Just as in One way ANOVA we still assume the data are approximately normal the groups have the same standard deviation even if the means may be different pool to estimate the standard deviation use F statistics for signi cance tests What s different for Two Way ANOVA We can look at each categorical variable separately and we can look at their interaction With oneway ANOVA it was impossible to look at interaction Example from 4th edition of M ampM 39 Each of the following situations is a 2way study design For each case identify the response variable and both factors and state the number of levels for each factor I and J and the total number of observations N a b A study of smoking classifies subjects as nonsmokers moderate smokers or heavy smokers Samples of 80 men and 80 women are drawn from each group Each person reports the number of hours of sleep he or she gets on a typical night The strength of concrete depends upon the formula used to prepare it An experiment compares 6 different mixtures Nine specimens of concrete are poured from each mixture Three of these specimens are subjected to 0 cycles offreezing and thawing 3 are subjected to 100 cycles and 3 specimens are subjected to 500 cycles The strength of each specimen is then measured Four methods for teaching sign language are to be compared Sixteen students in special education and sixteen students majoring in other areas are the subjects for the study Within each group they are randomly assigned to the methods Scores on a final exam are compared Example from 4 11 edition of M ampM In the course of a clinical trial of measures to prevent coronary heart disease blood pressure measurements were taken on 12866 men Individuals were classi ed by age group and race The means for systolic blood pressure are given in the following table 3539 4044 4549 5054 5559 White 131 1323 1352 1394 142 NonWhite 1323 1342 1372 1413 1441 Note that we are not given raw data on these 12 866 men The table above is the mean for each raceage combination This means we can t use ANOVA We 11 just use graphing anal marginal means to describe the situation a Plot the means with age on the xaXis and blood pressure on the yaXis For each racial group connect the points for the different ages b Describe the patterns you see Does there appear to be a difference between the 2 racial groups Does diastolic blood pressure appear to vary with age If so how does it vary Is there an interaction between race and age c Compute the marginal means Find the differences between the white and nonwhite mean blood pressures for each age group Use this information to summarize numerically the patterns in the plot category mean white 13598 Nonwhite 13782 3539 13165 4044 13325 4549 1362 5054 14035 5559 14305 How do you know from looking at a means plot whether there is interaction 0 If any lines cross each other then you have an interaction 0 If the lines are all fairly parallel to each other then you do not have an interaction 0 If the lines aren t parallel but don t cross each other either then you might have an interaction If you have 2 factors why not just do 2 separate One Way ANOVAs I It is more ef cient to study 2 factors simultaneously rather than separately 0 We can reduce the residual variation in a model by including a second factor thought to in uence the response lurking variable 0 We can investigate interactions between the factors 0 If the twoway ANOVA test nds that a main effect is signi cant then you can go back to doing oneway ANOVA for that factor to nd which levels are signi cantly different from each other Two Way ANOVA Table Source Sum of Degrees of Mean square F df df Pvalue Squares Freedom A SSA DFAI1 M 39 ff t MSA SSA FA MSA fin e ec DFA MSE 0 B SSB DFBJ1 M 39 ff t MSB SSB FB MSB feign e ec DFB MSE 0 DFB DFE AB SSAB DFAB SSAB MSAB Interaction MSAB F llXJl DFAB AB MSE OfA and B DFAB DFE E SSE DFE rror MSE SSE N IJ DFE T t 1 SST N l o a MST SST DF T The hypothesis tests are H0 main effect A 0 H0 main effect B 0 H0 interaction of A and B0 Use Pvalue from F A Use Pvalue from F 3 Use Pvalue from interaction of A and B Example One way to repair serious wounds is to insert some material as a scaffold for the body s repair cells to use as a template for new tissue Scaffolds made from extracellular material ECM are particularly promising for this purpose Because they are made from biological material they serve as an effective scaffold and are then reabsorbed One study compared 6 types of scaffold material Three of these were ECMs and the other three were made of inert materials There were 3 mice used per scaffold type The response measure was the of glucose phosphated isomerase Gpi cells in the region of the wound A large value is good indicating that there are many bone marrow cells sent by the body to repair the tissue Here are the data for 2 weeks 4 weeks and 8 weeks after the repair G i Material quot 2 weeks 4 weeks 8 weeks me an Mew Qata Ivanstmm Analvze Qvaphs mm 70 5539 60 DEI E as email be E ECM1 75 70 65 ll 6 quot l7 V GPl l Mammal Tlme l l i 65 70 65 1 l 7n Ecw 2 weeks 60 60 60 2 75 Ecw 2 weeks 3 e5 Ecw 2 weeks ECM2 65 65 70 A 55 Ecw 4 weeks 70 65 60 5 7n Ecw 4 weeks w A 5 7n Ecw Aweeks 89 75 7 ED Ecw 8 weeks ECM3 60 701 80 E E5 Ecw 8 weeks V a B5 Ecw Eweeks 7 70 1D ED ECMZ 2 weeks 50 15 M E5 ECMZ 2weeks 12 7n ECMZ 2 weeks MAT1 45 25 13 en ECMZ Aweeks 50 14 E5 ECMZ 4 weeks 5 15 e5 ECMZ 4 weeks 15 en ECMZ a weeks MAT2 10 17 7D ECMZ 8 weeks a an ECMZ a weeks 15 19 an ECM3 2 weeks 30 2D ED ECM3 2 weeks MAT 25 21 75 ECM3 2 weeks 22 75 ECM3 4 weeks 25 23 7D ECM3 4 weeks 24 75 ECM3 4 weeks 25 7n ECM3 a weeks 25 an ECM3 a weeks l nt Vanaaewew Using SPSS Analyze 9 General LinearModel 9 Univariate Move Gpi into Dependent Variable box Move Time and Material into Fixed Factors box To get a means plat click Plots box move Material into the Horizontal Axis box Move Time into the Separate Lines box Click Add and Continue To get summary statistics click Options box Move Materia and Time into the Display Means box Click Descriptive Statistics box and click Continue Click OK a Make a table giving the sample size mean and standard deviation for each of the materialbytime combinations Is it reasonable to pool the variances Descriptive Statistics Dependent Variable GPI Material Time Mea n Std Deviation N ECM1 2 7000 5000 3 4 6500 8660 3 8 6333 2887 3 Total 661 1 6009 9 ECM2 2 6667 7638 3 4 6333 2887 3 8 6333 5774 3 Total 6444 5270 9 ECM3 2 7167 10408 3 4 7333 2887 3 8 7333 5774 3 Total 7278 6180 9 MAT1 2 4833 2887 3 4 2333 2887 3 8 2167 5774 3 Total 3111 13411 9 MAT2 2 1000 5000 3 4 667 2887 3 8 667 2887 3 Total 778 3 632 9 MAT3 2 2667 2887 3 4 1167 2887 3 8 1000 5000 3 Total 1611 8580 9 Total 2 4889 24648 18 4 4056 28330 18 8 3972 28619 18 Total 4306 27064 54 Because the sample sizes in this experiment are very small we expect a large amount of variability in the sample standard deviations Although they vary more than we would prefer we will proceed with the ANOVA b Make a plot of the means of the combinations Describe the main features of the plot Estimated Marginal Means of GPI Meleiial ECM3 4ECNH iECMZ 4MAT l 4MAT3 MATZ ED i Estimated Marginal Means T o o o o u l i i 2 4 a Time c Make a lable ofthe sample size and mean for each lype ofmalenal Make a plol othe means oflnemalenals Give a shortsummary omeni Crpl depends on the lype ofmaterlal 1 Maierial Denende lVallable Fl 95 Cun dence lnlewal Maleiial Mean Sid Enm aneianund Unnei Enund 5ch BE 1 M 1742 E2 578 E9 544 ECM2 E4 444 1742 ED all E7 978 ECM3 72 77a 1742 E9 245 7B all MAN 31 ill 1 742 27 57a 34 E44 MATZ 7 77a 1742 4 245 ll all MATS law 1742 12578 19544 01 Make a lable ofthe sample size mean and slandaid em for each lime palod Make a plol othe means othe limes Give a snail summary ofthe Crpl depends on the lime period 211m 03337133711 vaname 6P1 95 0313337133 1n137va1 T1733 Mean Sm Enur L3w37333n3 113337333713 2 43 339 1232 46 391 51337 4 43 556 1232 33 357 43 354 E 39 722 1232 37 224 42 221 Run the analysis ofvan39ance Report the F statistics and PValues for the main effects and the interaction What are the hypotheses you are testing What do you conclude Write a short paragraph summarizing the results ofyour analysi s 3515 a 331wean 333335 Weds 5313a1muiuime 331 773311133quot Samba squuaus 31 Mean 3333 r 31 33113312133131 27327533 17 2225725 31435 333 3123231 133134137 1 133134137 3334331 333 1113 25373322 7174137 232347 333 rm 325333 432533 13322 333 Mateual39 71m 1341337 13 134137 2314 331 Euvr 333333 33 27313 r3231 123325 333 54 331133131 r3121 23323 322 52 2 3 313131 375331135131 3 3132133 33 HD39 main effect time 0 H main effect time 7 0 HD main effect maten39al 0 H main effect maten39al r 0 F PvaluP conclusion HD39 interaction 0 H interaction 70 F PvaluP conclusion

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