STAT METH SOC RES 2
STAT METH SOC RES 2 STA 6127
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This 52 page Class Notes was uploaded by Golden Bernhard on Friday September 18, 2015. The Class Notes belongs to STA 6127 at University of Florida taught by Lawrence Winner in Fall. Since its upload, it has received 26 views. For similar materials see /class/206560/sta-6127-university-of-florida in Statistics at University of Florida.
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Date Created: 09/18/15
Model Diagnostics Agricultural Intensity Data ANOVR Sum of Model Squares df Mean Square F Si 1 Regression 4000 800 21680 000a Residual 849 23 037 Total 4848 28 a Predictors Constant POPDRY ALLUV POP POPLVSTK DRYSSN b Dependent Variable LOGAGINT Coel cientsquot Unstandardized Standardized Coef cients Coef cients Model B Std Error Beta Sig 1 Constant 825 072 11388 000 004 001 652 4552 000 DRYSSN 109 028 620 3951 001 ALLUV 193 076 225 2549 018 POPLVSTK 002 001 364 2333 029 POPDRY 001 000 419 2130 044 a Dependent Variable LOGAGINT Histogram Dependent Variable LOGAGINT a lt l Std Dev aw Mean nun N zann L Frequency rznn 45m rt ran nun an mu wan 2cm Regressmn standarmzed Reswduai 2 Studenthed Res dual 39 25b Histogram 0f Studentized Residuals Plot of Studentized Residuals vs POP Scatterplot Dependent Variable LOGAGINT 3 Studentized Residuals versus Standardized Predicted Values Regress on Studermzed Res dua Regress on Standardwzed Pred cted Value Critical Values for OutliersIn uence Measures Studentized Residuals Leverage Values DFFITS DFBETAS Variance In ation Factors Coef cientsa Collinearit Statistics Model Tolerance VIF 1 POP 371 2695 DRYSSN 309 3239 ALLUV 981 1019 POPLVSTK 312 3206 POPDRY 197 5080 a De pendent Variable LOGAGI NT STA 6127 Fertility Example Problem 1317 0 3 Groups Urban Native Urban Migrant Rural Migrant 0 Response Fertility of live births by woman 0 Covariate Education Level 1 1 Way ANOVA to Compare 3 Groups Ignores Education Level De scriptives FERT 95 Confidence Interval for M N Mean Std Deviation Std Error Lower Bound Upper Bound Minimum Maximum 100 15 413 1457 3 494 2 7 200 17 429 1611 391 347 512 2 7 300 20 600 1777 397 517 683 3 10 Total 52 490 1829 254 439 541 2 10 Unadjusted means are 413 UN 429 UM 600 RM ANOVA FERT Sum of Squares df Mean Square F Sig Between Groups 39256 2 19628 7327 002 Within Groups 131263 49 2679 TOtal 170519 51 Reject H0 H1 H2 M F2497327 P002 SSE WSS 131623 R2 BSS TSS 39526 170519 0232 Multiple Comparisons Dependent Variable FERT Mean Difference 300 100 16 300 100 187 200 171 The mean difference is signi cant atthe 05 level Conclude that ill HS lt0H1ltH3 Hz Hslt0H2ltH3H1 H2 0H1 7 H3 2 Simple Linear Regression Relating Fertility to Education Ignores Group Model Summary Adjusted Std Error of Model R R Square R Square the Estimate 1 592al 351 338 1488 a Predictors Constant EDUC R2 351 ANOVAP Sum of Model Squares df Mean Square F Sig 1 Regression 59806 1 59806 27009 000a Residual 110713 50 2214 Total 170519 51 a Predictors Constant EDUC b Dependent Variable FERT Reject H0 10 F15027009 P000 Coef cientsa Unstandardized Standardized n ef in nt 0 ef r ie nt Model B Std Error Beta t Sig 1 Constant 6357 347 18294 000 EDUC 265 051 592 5197 000 a De pendent Variable FERT Fitted Equation Fertility 6357 7 0265EDUC 10 D Linear Regression 8 El El El III I III III El III S39 6 2 III III III III fen 636 O27 educ RSquar 035 I I 00 25 50 75 10 0 3 Analysis of Covariance No Interaction Terms Z1 l for UN 0 ow Z2 l for UM 0 ow Z3 l for RM 0 ow Can only t models with 2 of these variables at a time when you include intercept term Variables Entere dIRemoved Variables Variables Model Entered Removed Method 1 EDUCa Enter 2 Z1 Z2 Enter a All requested variables entered b Dependent Variable FERT Model Summary 3h ange Statistics Adjusted Std Error of R Square Model R R Square R Square the Estimate Change F Change df1 df2 Sig F Change 1 592a 351 338 1488 351 27009 1 50 000 2 718b 516 486 1311 165 8201 2 48 001 a Predictors Constant EDUC b Predictors Constant EDUC Z1 Z2 Coefficientsa Unstandardized Standardized 7160 242 Z1 1641 22 a De pendent Variable FERT Coef cientsquot Unstandardized Standardized Cnef39finients Coefficients 95 Confidence Interval for B Model B Std Error Beta t Sig Lower Bound Upper Bound 1 Constant 6357 347 18294 000 5659 7055 EDUC 265 051 592 5197 000 368 163 2 Constant 5758 420 13698 000 4913 6603 EDUC 242 045 540 5325 000 333 150 Z1 239 465 060 515 609 1174 695 Z3 1402 436 377 3213 002 525 2 279 a Dependent Variable FERT 4 Analysis of Covariance Model With Interaction Variables EnteredRemoved Variables Variables Model Entered Re moved Method 1 EDUCa Enter 2 Z1 Z2 Enter 3 EDUC Z1 EDUCZ2a Enter a All requested variables entered b Dependent Variable FERT Model Summary Shange Statistics Adjusted Std Error of R Square Model R R Square R Square the Estimate Change F Change df1 df2 Sig F Change 1 5928 1 338 1488 351 27009 50 000 2 718b 516 486 1311 165 8201 2 48 001 3 719c 517 465 1337 001 065 2 46 937 a Predictors Constant EDUC b Predictors Constant EDUC Z1 Z2 c Predictors Constant EDUC Z1 Z2 EDUCZ1 EDUCZ2 Coefficients Unstandardized Standardized Coefficients Coefficients 95 Confiden e Interval for B Model B Std Error Beta t Sig Lower Bound Upper Bound 1 Constant 6357 347 18294 000 5659 7055 EDUC 265 051 592 5197 000 368 163 2 Constant 7160 365 19604 000 6425 7894 EDUC 242 045 540 5 325 000 333 150 Z1 1641 450 411 3648 001 2546 737 Z2 1402 436 363 3213 002 2 279 525 3 Constant 7268 520 13965 000 6220 8315 EDUC 264 089 590 2 976 005 443 085 Z1 1719 796 430 2159 036 3322 116 Z2 1615 745 418 2167 035 3114 115 EDUCZ1 017 124 032 140 890 232 266 EDUCZ2 040 113 086 353 726 187 267 a Dependent Variable FERT fert fert ru39 mig urb mig fert727 RSquare a 4126 educ 026 fert 5 Rs qu 65 4122 educ are 044 n I u39b nat R nA fert 555 4125 educ q 1 mun I mu 2Sample ttest Independent Samples Dataset Cloud Seeding Experiment Dependent Variable Rain gage Depth mm of rainfall in cloudseeded test area Independent Variable SeededNot seeded on that day Seed1 if Yes 0 if No SPSS Instructions Enter data in two columns One for Seed and other for Depth Using Variable View NAME the variables eg Seed Depth Using Variable View assign VALUES to the Seed variable eg 15 Seeded OE Unseeded 0 Return to Data View and Select 0 ANALYZE 0 COMPARE MEANS 0 INDEPENDENT SAMPLES t TEST 0 Select a Test Variable e g Depth 0 Select a Grouping Variable e g Seed 0 De ne Groups eg 10 Source JMeit139n WWoodley J Flueck 1984 Exploration of ExtendedArea Treatment Effects in FACE2 Using Satellite Imagery Journal of Climate and Applied Meteorology pp63 Paired ttest Dependent Samples Dataset Botox for Migraine Headaches Dependent Variable Headache Frequency per Month Independent Variable Botox Condition Pre Post Treatment SPSS Instructions 0 Enter data in 3 columns one for subject id one for pretreatment headache frequency and one for posttreatment headache frequency Using Variable View Name the variables e g id Pre Post Return to Data View and Select 0 ANALYZE 0 COMPARE MEANS 0 PAIRED SAMPLES t TEST 0 Select 2 Paired Variables eg Pre and Post Source R Behmand T Tucker and B Guyuron 2003 SingleSite Botulinum Toxin Type A Injection for Elimination of Migraine Trigger Points Headache Vol 43 1085 1089 Fisher s Exact Test Dataset Antiseptic as Treatment for Amputation In Contingency Table Form Dependent Variable Occurrence of Death among upper limb amputees 1Death 0Survive Independent Variable Period of Surgery 1PostDiscovery of Antiseptic 0Pre SPSS Instructions 0 Enter data in 3 columns Death status antiseptic status number of cases Using Variable View NAME the variables eg Death Antiseptic Cases 0 Using Variable View assign VALUES to the Death and Antiseptic Variables variable eg IE Yes OE No 0 Return to Data View and Select DATA WEIGHT CASES 0 Click on Weight Cases by 0 Select the variable Cases ANALYZE DESCRIPTIVE STATISTICS CROSSTABS 0 Select Antiseptic as Rows 0 Select Death as Columns 0 Click on Statistics and click on Chi Square Source J Lister 1870 Effects of the Antiseptic System of Treatment on the Salubrity ofa Surgical Hospital The Lancet 14 64042 McNemar s Test Dataset Silicone Breast Implant Ruptures In Contingency Table Form Dependent Variable RuptureLeak Reporting Status in surgery on silicone gel breast implants 1Yes 0No Independent Variable Reporter 1Se1f Report 2Surgical Record SPSS Instructions Enter data in three columns Self Report Surgical Record Number of cases 0 Using Variable View Name the Variables eg Self Surgical Cases 0 Using Variable View Asssign Values to the levels ofthe variables eg IE Yes OENo 0 Return to Data View and Select DATA WEIGHT CASES 0 Click on Weight Cases by 0 Select the variable Cases ANALYZE DESCRIPTIVE STATISTICS CROSSTABS 0 Select Self as Rows 0 Select Surgical as Columns 0 Click on Statistics and click on McNemar Source Brown and Pennello 2002 Replacement Surgery and Slicone Breast Implant Rupture Journal of Women s Health amp Gender Based Medicine Vol 11 pp255264 ChiSquared Test for Independence Dataset Union Army Deaths by Rank and Duty In Contingency Table Form Dependent Variable Mortality status for Union troops during Civil War 1Died during war 0survived war Independent Variable RanldDuty staus 1PrivateInfantry 2PrivateNoninfantry 30f cer Infantry 40f cerNoninfantry SPSS Instructions Enter data in 3 columns ranldduty status mortality status number of cases Using Variable View Name the Variables eg rankduty Death Cases Using Variable View Asssign Values to the levels of the variables Return to Data View and Select 0 DATA 0 WEIGHT CASES 0 Click on Weight Cases by 0 Select the variable Cases ANALYZE DESCRIPTIVE STATISTICS CROSSTABS 0 Select rankduty as Rows 0 Select Death as columns Click on Statistics and click on Chi Square Click on Cells and click on Expected Column Percentages ConditionalMarginal Distributions of Death Survive for each ranldduty group Adj Standardized Residuals adjusted residuals for each cell Source C Lee 1999 Selective Assignment of Military Positions in the Union Army Implications for the Impact of the Civil War Social Science History Vol 23 pp 6797 Measures of Association for Ordinal Variables Dataset Price and Quality Ratings In Contingency Table Form Dependent Variable Beer drinker s assessment of beer taste 0undrinkable 1Poor 2Fair 3Good 4Very Pleasant Independent Variable Price condition assigned to beer pretasting 1Low 2Medium 3High SPSS Instructions Enter data in 3 columns Price Condition Taste Quality number of cases Using Variable View Name the Variables eg Price Taste Cases Using Variable View Assign Values to the levels of the variables Return to Data View and Select 0 o WEIGHT CASES 0 Click on Weight Cases by 0 Select the variable Cases ANALYZE DESCRIPTIVE STATISTICS CROSSTABS 0 Select price as Rows 0 Select taste as columns Click on Statistics and click on Ordinal Measures Gamma and Kendall s Tau b Source McConnell 1968 An Experimental Evaluation of the PriceQuality Relationship The Journal of Business Vol 41 pp439444 Simple Linear Regression Dataset Tombstone Weathering Dependent Variable Tombstone Surface Recession Rate Independent Variable 100Year Mean S02 concentration SPSS Instructions Enter data into 2 columns S02 Concentration Tombstone Recession Rate 0 Using Variable View Name the variables e g s02 recrate Return to Data View and Select GRAPHS To obtain Plot with Fitted Equation INTERACTIVE SCATTERPLOT 0 Move recrate to vertical up down axis 0 Move 02 to horizontal right left axis 0 Click on Fit tab 0 Select Regression as Method ANALYZE To t Model REGRESSION LINEAR 0 Identify recrate as Dependent Variable 0 Identify 02 as Independent Variable TRANSFORM To make log transformation on recrate COMPUTE 0 Select a name for Target Variable eg logrrate 0 Give instructions e g logrrateloglogrrate 0 Repeat Process for Graph and Model t Source Meierding 1993 Marble Tombstone Weathering and Air Pollution in North America Annals of the Association of Geographers Vol83 4 pp 568588 Multiple Linear Regression Dataset Japanese Emigration to Paci c Northwest 18801915 Dependent Variable Emigrants per 1 million residents Independent Variables land tenant farmers change in ratio of tenant farmlands average farm area government laborers in Hawaii existence of pioneer immigrants SPSS Instructions 0 Enter data into 6 columns emigrants tenant farmers change tenant farmlands average farm area government work in Hawaii pioneer immigrants 0 Using variable view Name the variables e g emigrant pctfarm chgfarm farmarea govhaw pioneer 0 Return to Data view and select ANALYZE To obtain descriptive statistics DESCRIPTIVE STATISTICS DESCRIPTIVES 0 Enter variable names of interest eg emigrant pctfarm chgfarm farmarea govhaw pioneer ANALYZE To obtain pairwise simple correlations CORRELATE BIVARIATE 0 Enter variable names of interest eg emigrant pctfarm chgfarm farmarea govhaw pioneer ANALYZE To t regression model REGRESSION LINEAR 0 Enter dependent variable e g emigrant 0 Enter independent variables e g pctfarm chgfarm farmarea govhaw pioneer o STATISTICS To obtain Partial Correlations 0 Click on Part and Partial Correlations ANALYZE To obtain Partial correlations directly CORRELATE PARTIAL 0 Enter correlation variables eg pctfarm chgfarm farmarea govhaw pioneer 0 Enter variables to control for e g emigrant Source Murayama 1991 Information and Emigrants Interprefectural Differences of Japanese Emigration to the Paci c Northwest 18801915 The Journal ofEconomz39c History Vol51 1 pp 125147 1Way Analysis of Variance Dataset Mollusc Nervous Impulse Rates Dependent Variable Mollusc Impulse Rate Independent Variable Species 5 levels SPSS Instructions 0 Enter data into two columns species number impulse rate Using variable view Name the variables eg species impulse Using variable view give Values to the levels of species Return to data view and select 0 ANALYZE O COMPARE MEANS 0 One Way ANOVA 0 Enter Dependent variable eg Impulse Rate 0 Enter Factor eg Species number 0 Under Post hoc select method of multiple comparisons eg Bonferroni Tukey Scheffe Source Jenkins and Carlson 1903 The Rate of the Nervous Impulse in Certain Molluscs American Journal ofPhysiology Vol 8 pp 2517268 2Way Analysis of Variance Dataset Thalidomide for Weight Gain in HIV Patients with and without TB Dependent Variable 21day weight gain in HIV patients Factor A TB Status lPositive 0Negative Factor B Treatment lThalidomide 0Placebo SPSS Instructions 0 Enter data into three columns e g weight gain tb treatment 0 Using variable view Name the variables e g wtgain tb tX 0 Using variable view give Values to the levels of tb and tX 0 Return to data view and select 0 ANALYZE 0 GENERAL LINEAR MODEL 0 UNIVARIATE 0 Enter Dependent Variable e g thain 0 Enter Fixed Factors e g tb tX 0 Under post hoc select factors whose levels are to be compared eg tb tX Note that the default is to t the full factorial interaction model To t the additive effects model Select MODEL CUSTOM Highlight the model factors e g tb tX Under Build terms choose MAIN EFFECTS Enter them into model with arrow Source Klausner Makonkawkeyoon Akarasewi et al 1996 The Effect of Thalidomide on the Pathogenesis of Human Immunodeficiency Virus Type 1 and M tuberculosis Infection Journal of Acquired Immune De ciency Syndromes and Human Retrovirology 11247257 Randomized Block Design Dataset Caffeine and Endurance Dependent Variable Endurance time on Bicycle Factor A Caffeine Dose Fixed Factor Factor B Athlete Random Factor SPSS Instructions 0 Enter data into three columns eg time dose athlete 0 Using variable view Name the variables e g timedoseath1ete 0 Return to data view and select 0 ANALYZE 0 GENERAL LINEAR MODEL 0 UNIVARIATE Enter Dependent Variable e g time Enter Fixed Factor eg dose Enter Random Factor e g athlete Under post hoc select factors whose levels are to be compared eg dose Select MODEL Select CUSTOM Highlight the model factors eg dose athlete Under Build terms choose MAIN EFFECTS Enter them into model with arrow Source WJPasman MAvan Baak AEJeukendrup and Ade Haan 1995 The Effect of Different Dosages of Caffeine on Endurance Performance Time International Journal of Sports Medicine Vol16 pp225230 Repeated Measures Design Multivariate Approach Dataset Rogaine Clinical Trial in Women Multivariate Responses on Time Dependent Variable Hair weight at target site Factor A Within subjects Period of evaluation weeks 8 16 24 32 Factor B Between subjects Treatment lMinoxodil 0Placebo SPSS Instructions 0 Enter data into 6 columns eg treatment subject periodl period4 0 Using variable View Name the variables e g tXsubjectthvW2vW3vW4 0 Return to data View and select 0 ANALYZE 0 GENERAL LINEAR MODEL 0 REPEATED MEASURES Give a name to the Within Subject Factor e g hairwt Specify the number oflevels eg 4 click on De ne Highlight the variables wtl wt4 and select them as the levels of the Within subject factor Select the Between Subject Factor e g tX Under Post hoc you can request comparisons among levels of the between subjects factor however they will only be computed if there are 3 or more groups e g tX Source VH Price and E Menefee 1990 Quantitative Estimation of Hair Growth 1 Androgenetic Alopecia in Women Effect of Minoxidil The Journal of Investigative Dermatology 95683687 Analysis of Covariance Dataset Head Size and Brain Weight Dependent Variable Brain weight grams Independent Variables Gender Fixed Factor Head sizecovariate in cm3 SPSS Instructions Enter data into 3 columns e g gender head size brain weight Using variable View Name the variables e g gender headsz brainwt Using variable View give Values to any categorical variables Return to data View and select ANALYZE GENERAL LINEAR MODEL UNIVARIATE 0 Select the Dependent Variable e g brainwt 0 Select the Fixed Factor eg gender 0 Select the Covariate e g headsz o If the xed factor has more than 2 levels can select Post hoc tests 0 Note this ts the model without interaction between grouping variable and covariate To t model with interaction Select Model Custom Highlight the model factors eg gender headsz Under Build terms choose MAIN EFFECTS Enter them into model with arrow Under Build terms choose INTERACTION Enter them into model with arrow To obtain Adjusted Means Select Options Click on grouping variable e g gender and click arrow key Click on Compare main effects and select a con dence interval adjustment if more than 2 levels e g Bonferroni Source RJ Gladstone 1905 A Study of the Relations of the Brain to the Size of the Head Biometrika Vol4 pp105123 Logistic Regression Quantitative andor Dummy Predictors Data NFL Field Goal Attempts 2003 Dependent Variable Field Goal Attempt Outcome 1Success 0Failure Independent Variable Distance Yards SPSS Instructions 0 Enter data into 2 columns eg eld goal outcome yards Using variable View Name the variables e g outcome yards Using variable View give Values to any categorical variables Return to data View and select 0 ANALYZE o REGRESSION o BINARY LOGISTIC o Assign the Dependent variable e g outcome 0 Assign the Independent variables aka Covariates e g yards Probit models can be t in a similar manner use the PROBIT option under REGRESSION Sources WWWjtswcom and ESPNCOM Logistic Regression Qualitative Predictors Data Union Army Deaths by Rank and Duty from contingency table Dependent Variable Survival outcome of civil war soldiers lDie 0Survive Independent Variables Private lPrivate 0Nonprivate Infantry lInfantry 0Noninfantry SPSS Instructions Enter the data in 4 columns e g Death status private infantry number of soldiers Using Variable View assign Names to variables and Values to levels of categorical variables Return to Data View Create a variable that is the crossproduct of Private and Infantry o TRANSFORM To create a new independent variable 0 COMPUTE 0 Select name for Target variable eg privinf 0 Give instructions eg o Privinfwrivate kinfantry Weight the cases by selecting 0 DATA 0 WEIGHT CASES 0 Click on Weight cases by and select the variable representing the number of cases To t the model select 0 ANALYZE o REGRESSION o BINARY LOGISTIC 0 Select the Dependent variable e g Death 0 Select the Covariates e g Private Infantry Privinf Source C Lee 1999 Selective Assignment of Military Positions in the Union Army Implications for the Impact of the Civil War Social Science History Vol 23 1 pp 67 97 Loglinear Model Dataset Poverty and Migration In form of a 2X2X2 Contingency Table Variable 1 Poverty 1Yes 0No Variable 2 Female 1Yes 0No Variable 3 White 1Yes 0No SPSS Instuctions 0 Enter the data in 4 columns e g poverty female white number of cases 0 Using Variable View assign Names to variables and Values to levels of categorical variables 0 Weight the cases by selecting 0 DATA 0 WEIGHT CASES 0 Click on Weight cases by and select the variable representing the number of cases 0 To fit the model select 0 ANALYZE o LOGLINEAR 0 GENERAL 0 Select all variables in model for Factors 0 Select MODEL Click Custom Highlight all 3 variables e g poverty female white Under Build Terms select Main Effects and hit arrow button Under Build Terms select All 2 way Interaction and hit arrow button Click Continue 0 Select OPTIONS 0 Click Estimates Source D Wenk and C Hardesty 1993 The Effects of RuraltoUrban Migration on the Poverty Status of Youth in the 1980s Rural Sociology 581 7692 Cumulative Logit Model Ordinal Regression Dataset Price and Quality Ratings In Contingency Table Form Dependent Variable Quality Rating of Beer 0Undrinkable 1Poor 2Fair 3Good 4VeryPleasant Independent Variable Perceived Price Low Medium High Operationalized by 2 Dummy Variables Medium and High SPSS Instructions 0 Enter the data in 4 columns e g quality medium high number of cases 0 Using Variable View assign Names to variables and Values to levels of categorical variables 0 Weight the cases by selecting 0 DATA 0 WEIGHT CASES 0 Click on Weight cases by and select the variable representing the number of cases 0 To t the model select 0 ANALYZE o REGRESSION o ORDINAL 0 Select the Dependent Variable e g quality 0 Select the Factors e g medium and high PY Sijrice Note that SPSS ts the model log 1PY S j Pr1ce a MM HH Source McConnell 1968 An Experimental Examination of the PriceQuality Relationship The Journal of Business 414 439444 Halo Effect SPSS Output BetweenSubjects Factors N Essyqual 1 30 2 30 Attract 1 20 2 20 3 20 Descriptive Statistics Dependent VariableScore Essyqu al Attract Mean Std Deviation N 1 1 179000 481938 10 2 179000 359991 10 3 154990 469881 10 Total 170997 440538 30 2 1 148990 331088 10 2 134000 599073 10 3 87010 367999 10 Total 123333 509456 30 Total 1 163995 430865 20 2 156500 533550 20 3 121000 538836 20 Total 147165 529833 60 Tests of BetweenSu bjects Effects Dependent VariableScore 7va H1 Sum 61 Samoa 3 33133 31 Mean 333313 5 S13 CDYYEE E MD E 533 339 5 117663 5353 333 MEYDEM 12333 522 1 12333 522 657376 333 373331 333 763 1 333 763 17231 333 211333 2 135 533 5 335 333 33733315733133 36575 2 13233 325 333 Enm 1367313 53 13776 73131 NEED 735 63 031136133 73131 3 33333133 355 73313313333333133 23E Estimated Marginal Means of Score 1333 1533 1033 Estimated Marginal Means E1 13333 3 33 Attract Model Summary Adjusted R Std Error of the Change StatiStiCS Model R R Square Square Estimate R Square Change F Change df1 df2 1 5778 333 297 444107 333 9325 3 2 596b 355 296 444705 022 925 2 a Predictors Constant Acnt Egd Ahi b Predictors Constant Acnt Egd Ahi EgdAcnt EgdAhi ANOVAc Model Sum of Squares df Mean Square F Sig 1 Regression 551769 3 183923 9325 0008 Residual 1104494 56 19723 Total 1656263 59 2 Regression 588344 5 117669 5950 000b Residual 1067919 54 19776 Total 1656263 59 a Predictors Constant Acnt Egd Ahi b Predictors Constant Acnt Egd Ahi EgdAcnt EgdAhi c Dependent Variable Score Coefficientsa Standardized i 39 39 Coefficient Coefficient 95 Confidence Interval Model B Std Error Beta t Sig Lower Bound UpperE 1 Constant 9717 1147 8474 000 7420 Egd 4766 1147 454 4157 000 2469 Ahi 4300 1404 386 3061 003 1486 Acnt 3550 1404 319 2528 014 737 2 Constant 8701 1406 6187 000 5882 Egd 6798 1989 647 3418 001 2811 003 2211 Ahi 6198 1989 556 3116 Acnt 4699 1989 422 2363 022 712 EgdAhi 3797 2813 269 1350 183 9436 EgdAcnt 2298 2813 163 817 417 7937 a Dependent Variable Score Analysis of Covariance 0 Combines linear regression and AN OVA Can be used to compare treatments after controlling for quantitative factor believed to be related to response eg pretreatment score 39 Can be used to compare regression equations among 3 groups e g common slopes andor intercepts Model X quantitative 21Zg1 dummy variables EY a X AZ 4234 Tests for Additive Model Relation for group i i1g 1 EYaw Relation for group g EYa i Hoe l g10 Controlling for covariate no differences among treatmmts Interaction Model Regression slopes between Y and X are allowed among groups 507 a 13X 16121 lag12341 3le v ygXZg1 Grou p i i1vgl a rxm 79X 39 Groupg EYa No interaction means common slopes rlyg l0 Inference in C OVA Model EU a M Z 421 71le ywxz Consuuct 3 Se a of independent variables a ZIA39azIZDquot39VquotZgls WWXZga d Fit Complete model containing all 3 Obtain or equivalently and cng Fit Reduced model X 2122Zg 1 Obtain or equivalently 123 and HO3731O N0 interaction Test Statistic 39 39 R R Inference in ANCOVA Test for Differences controlling for covariat c 50 a X IZI gZx Fit Conlplete model containing X Z1 Zzr Zg1 Obtain SSEC or equivalently RC2 and 6 Fit Reduced model X SSE or equivalently R32 and 62 H0 i6g0 No group differences Test Statistic Inference in COVA Test for Effect of Covariate controlling for qualitative variable EY a X 321 3le 11701550 No covariate effect Test Statistic b tabs 39 A O b Adjusted Means O Goal Compare the g group means after controlling for the eovariate Unadjusted Means 171wqu Adjusted Means 171 Obtained by evaluating regression equation at X E 39 Comparing adjusted based on regression equation bi bi bj Multiple Comparisons of Adjusted Means Comparisons of each group with group g A bi i1gl Comparisons among the other g l groups 4 2 152 bji tac 2Ng J05 63 2C0Vbi bi Variances and covariances are obtained i39om computer packages SPSS SAS Regression Model Building 0 Setting Possibly a large set of predictor variables including interactions Goal Fit 3 parsimonious model that explains variation in Ywith a set of predictors Automated Procedures and all possible regressions Backward Elimination Top down approach Forward Selection Bottom up approach Stepwise Regression Combines FomardBackward CI Statistic Summarizes each possible model where best model be has on statistic Backward Elimination Select a signi cance level to stay in the model eggs SLS020 generally 05 is too low too many variables to be removed Fit the full model with all possible predictors Consider ae predictor lowest tstatistie highest Pvalue If P gt SL8 remove the and t model without variable must re t model here because regression cae icients change If P s SLS step and keep current model Continue until all predictors have Pvalues below SL S Forward Selection Choose a signi cance level to enter the model eg SLE O20 generally 05 is too low causing too few variables to be entered Fit all simple regression models Consider the predictor with the highest t statistie IQWest Pvalue IfP S SLE variable t all two variable models that include predictor If gt SLE stop and keep previous model Continue until no predictors have P s SLE Stepwise Reession 0 Select SLS and SLE SLEltSLS 0t like Forward Selection Bottom up process variables must have P s SLB to enter t Retests all old variables that have already entered must 39haVe P s SLS to stay in model 39 Continues until no new variables be entered and no old variables need to be removed All Possible Regressions Cp Fits every possible model If potential predictor variables were ZKl models Label the Square Error far the model containing all predictors as MSEK For each model compute and C1 where p is the number of including intercept in model n2p IE Select the model with the fewest predictors that has Cy as p Reession Diagnostics Model Assumptions Regression mctien correctly speci ed eg linear Conditional distribution of Y is manna distibution Conditional distribution of Y has deviation Observations on Y statistiwa independent Residual plots can be to check the assumptions Histogram stem andleafplet should be moundshaped mum Plot of Residuals versus each predictor should be random cloud 39 115th or inverted U gt Nonlinear relation 39 Funnel shaped z Nonmm Variance Plot of Residuals versus Time order Time series data should he rande eloud If appears not independent Detecting In uential Observations 9 Studentized Residuals Residuals divided by their estimated tastatisties Observatinns with values larger 3 in value considered outliers o Leverage Values Hat Measure nf hnw far an nbsmation isfrom the others in terms ofthe levels of the independent variables not dependent variable Observations with values larger than 2klln are considered to be potentially in uential where k is the number of and n sample 0 Measure ef how much an observation has e eeted its value regression Values lmger than 2sqrtkln in absolute value considered highly in uential Use standardized DFFI TS in SPSS Detecting In uential Observations o DFBETAS Measm of how much an Observation has effected the ofa regressinn coef cient there is one DFBETA for regression coefficient including the intercept Values larger than leqr n in absolute value are considered in uential 6 Cook s I Measure of of each nbservatinn the group of regression ene cicnts as well as the goup of values Values than 411 onnsidered in uential 6 COVRATIO Measure of the of each nbservation on variances and errors of the regression coef cients and their outside the l l 3kln are commer in uential Obtaining In uence Statistics and Studentized Residuals in SPSS Choose REGRESSION and input the Dependent variable and 61 ofIndependent variables from your model of interest39mo ssibly having been chosen via an automated made selection methad 0 Undar smusncs meet Dhg39nos cs Diagnostics and All and 0 Under PLOTS select wsmsmm Alsochm These give a plot of warms values andahis togmm mammmMSIz Select com 0 Undar SAVE select Zack s van Ratio sandman Drums scande BFFITS Salem moremus win beaddodto youro nal am wmksheet Variance In ation Factors Variance Inflation Factor VIF Measure of how higrly correlated each independent variable is with the other predictors the model Used to identify Multicollinearity Values larger than 10 for a predictor imply large in ation of errors of regression coef cients due to this variable being in model 1 In ated standard errors lead to small tstatisties for partial regression coef cients and con dence intervals Nonlinearity Polynomial Regression When relation between Y andX is not linear polynomial models can be t that approximate the relationship Within a particular range of X 139 General form of model EY a 1Xm ka Second order model most Widely 6386 allows one bend EY ozgt 1XBZX2 Must be very not to extrapolate beyond observed X levels Generalized Linear Models GLM class of linear models that are made up of 3 components Random Systematic and Link Function Random component Identi es dependent variable Y and its probability distribution Systematic Component Identi es the set of explanatwa variables X1AXk Link Function Identi es a function of the mean that is a linear function of the explanatory variables gm a txl kxk
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