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# 252 Class Note for STAT 30100 with Professor Zhao at Purdue

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Date Created: 02/06/15

Chapter 13 TwoWay Analysis of Variance Chapter 7 Twosample comparison of means t tests 1 categorical and 1 quantitative variables 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 and 1 quantitative variables 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 classi ed in 2 ways 2 categorical and 1 quantitative variables 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 classified 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 and marginal means to describe the situation a Plot the means with age on the x aXis 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 It is more efficient to study 2 factors simultaneously rather than separately We can reduce the residual variation in a model by including a second factor thought to in uence the response lurking variable We can investigate interactions between the factors If the twoway ANOVA test finds that a main effect is significant then you can go back to doing oneway ANOVA for that factor to find which levels are significantly different from each other Two Way ANOVA Table Source Sum of Degrees of Mean square F df df Pvalue Squares Freedom A SSA DFA Il SSA MSA Main effect MSA A f A DFA MSE 0 DFA DFE B SSB DFB Jl SSB MSB Main effect MSB B f B DFB MSE 0 DFB DFE AB SSAB DFAB MSAB SSAB 7 MSAB Interaction IlJ l DFAB ABT MSE ofA and B DFAB DFE Error SSE DFE MSE SSE N IJ D FE Total SST Nl MST SST DF T The hypothesis tests are H0 main effect A 0 Use Pvalue from F A H0 main effect B 0 Use Pvalue from F B H0 interaction of A and B0 Use Pvalue from interaction of A and B Example Exercise 1320 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 Gpi Matenal 2 weeks 4 weeks 8 weeks 70 55 60 ECM1 75 70 65 65 70 65 60 60 60 ECM2 65 65 70 70 65 60 80 75 70 ECMS 6 0 70 8039 75 75 7O 50 15 MAT 45 25 50 25 5 MAT2 10 15 30 MATS 25 Using SPSS Analyze 9 General LinearMoalel 9 Univariate Move Gpi into Dependent Variable box Move Time anal Material into FixealFactors box To get a means plot click Plots box move Material into the Horizontal Axis box Move Time into the Separate Lines box Click Aalal anal Continue To get summary statistics click Options box Move Material anal Time into the Display Means box Click Descriptive Statistics box anal 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 Mean Std Deviation N ECM1 2 7000 5000 3 4 6500 8660 3 8 6333 2887 3 Total 6611 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 3632 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 othe means othe combinations Describe the mam features of Estimated Marginal Means of GPI ault Estimated Marginal Means T 139 J o C the plot Make a table othe sample size and mean for each type ofmatenal Make a Time MmeHai ECM3 4ECNH iECMZ 4MAT39i 4MAT3 MATZ plot othe means othemaienals Gwe a shortsummary othe n Gpi depends on the type ofmaienal 1 Material DependeniVanabie em 95 Cunhdence iniervai Maienai Mean Std Enuv Luwev Euund Uppev Emma EOM1 BB 111 1742 E2 578 E9 544 ECMZ E4 444 1 742 ED 911 E7 978 ECM3 72 77a 1 742 E9 245 7B 311 MAT1 31 111 1 742 27 57a 34 E44 MATZ 7 77a 1 742 4 245 11 311 MATS 15111 1742 12578 19544 d Make a table of the sample size mean and standard error for each time period Make a plot ofthemeans ofthe times Give a short summary ofthe Crpi depends on the time period 211m Depenuentvananle GM 95 03771337733 interval Time Mean Stu Ermr Luwer 333773 Upperammu 2 43 339 1232 46 391 51337 4 43 556 l 232 33 357 43 354 E 39 722 l 232 37 224 42 221 e Run the analysis ofvariance 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 analysis rssis 3i astwesn Sume s Weds hopsnigammme cpl Yypam um Samba squuaus 31 Mean 33333 F 33 3313313133131 27327533 17 2225725 31435 333 intercept 133134137 133134137 3334331 333 1113 25373322 7174137 232347 33 runs 325333 432533 13322 333 Mstsusi Time 1341337 13 134137 2314 331 uw 333333 33 27313 mil 123325 333 54 331133131 rum 23323 322 52 i 3 333131 375 4373331 3 3133133 33 H3 main effect maten39al 0 H17 main effect time 0 Ha main effect maten39al 0 Ha main effect time 0 F F PvaluP PvaluP conclusion conclusion H17 interaction 0 Ha interaction 0 F PvaluP conclusion

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