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# CAT ANALYSIS EPIDEM BIOST 536

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This 43 page Class Notes was uploaded by Ramona Leannon on Wednesday September 9, 2015. The Class Notes belongs to BIOST 536 at University of Washington taught by William Barlow in Fall. Since its upload, it has received 12 views. For similar materials see /class/192283/biost-536-university-of-washington in Biostatistics at University of Washington.

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Date Created: 09/09/15

Lecture 3 Overview of study designs Prospectiveretrospective u Prospective cohort study Subjects followed data collection in real time following a planned study a Retrospective prospective cohort study Data collected prospectively and stored administrative data study analytic plan developed later a Retrospective study Exposure amp other covariates may be collected by interview or record review after disease identification Outcome u Binary a Time to event BIOST 536 Lecture 3 Randomized trial with a binary outcome Treatment Success Failure Total Placebo 0 mg r1 n1 r1 n1 Low dose 20 mg r2 n2 r2 n2 High dose 50 mg r3 n3 r3 n3 What statistical hypotheses could we test NN 3 a Does treatment dose affect probability of success a Does success increase with amount of treatment a Does success increase with actual dose level Risk difference risk ratio odds ratios appropriate Can use asymptotic methods or small sample methods BIOST 536 Lecture 3 Example 1 11 1 1 0 1 49 1 1 2 1 0 1 51 0 1 3 1 20 2 62 1 1 Treatment Y1 YO N 4 1 20 2 38 01 5 1 50 3 66 1 1 Placebo0mg 49 51 100 61 50 3 34 01 77777777777777777777 77 Lowdose2olng 62 38 100 tabulate trt y fwcnt row ch12 1r exact Enumerating sampleespace combinations F gh doseltsolng 66 34 100 stage 3 enumerat1ons 1 stage 2 enumerations 21 stage 1 enumerations 0 1 Y trt 1 0 1 1 Total 1 1 51 49 1 100 1 5100 4900 1 10000 2 1 38 62 1 100 1 3800 6200 1 10000 3 1 34 66 1 100 1 3400 6600 1 10000 Total 1 123 177 1 300 1 4100 5900 1 10000 Pearson chi221 65316 Pr 0038 likelihooderatio chi221 65058 Pr 0039 Fisher39s exact 0040 BIOST 536 Lecture 3 3 Overall test 2 df with treatment categorical gen trt2trt2 gen trt3trt3 logistic y trt2 trt3 fwcnt Logistic regression Number of obs 300 LR chi22 651 Prob gt chi2 00387 Log likelihood 719980468 Pseudo R2 00160 y 1 Odds Ratio Std Err Z Pgtlzl 95 Conf Interval trt2 1 1698174 4876475 184 0065 9672778 2981351 trt3 2020408 5875848 242 0016 1142585 3572643 xi logistic y itrt fwcnt itrt Altrti1e3 naturally coded iltrti1 omitted Logistic regression Number of obs 300 LR chi22 651 Prob gt chi2 00387 Log likelihood 719980468 Pseudo R2 00160 y 1 Odds Ratio Std Err Z Pgtlzl 95 Conf Interval Itrti2 1 1698174 4876475 184 0065 9672778 2981351 Itrti3 1 2020408 5875848 242 0016 1142585 3572643 BIOST 536 Lecture 3 Prospective models with long term follow up Longer followup means a greater likelihood of the outcome not being observed Example of a cancer clinical trial Bone marrow transplant vs chemotherapy alone a Death solid circle is the outcome observations may be censored square due to end of study Person ChemoBMT 6 ChemoBMT 5 Chemo A 3 Chemo I 2 ChemoBMT Chemo llllllllllllllllllllllllllllllllll 125456789101112123456789101112123456789101112 2 Month of Randomization BIOST 536 Lecture 3 Interested in 18 month survival All outcomes could be classified at 18 months some information loss in not using known failure times Person Chemo BMT Chemo BMT Chemo Chemo Chemo BMT Chemo o l l l l l l l l l l l 0 3 6 9 12 15 18 21 24 27 30 33 36 Survival PostRandomization BIOST 536 Lecture 3 Suppose one patient is censored early Not all outcomes at 18 months are known Need to use survival analysis to account for censoring estimate hazard ratios similar to risk ratios Chemo BMT 6 4 Chemo BMT Chemo Chemo Chemo BMT Chemo 1 o o l l l l l l l l l l l 0 3 6 9 12 15 18 21 24 27 30 33 36 Survival PostRandomization BIOST 536 Lecture 3 Casecontrol study Exposure Cases Controls None 0 rads n11 n01 Low 20 rads n12 n02 High 50 rads n13 n03 Total n1 n0 What statistical hypotheses could we test ls exposure associated with casecontrol status ls casecontrol status associated with a trend in exposure level a Is casecontrol status associated with the actual amount of exposure Odds ratios appropriate Can use asymptotic methods or small sample methods BIOST 536 Lecture 3 Example Exposure Cases Controls None 0 rads 49 51 Low 20 rads 62 38 High 50 rads 66 34 Total 177 123 Same data as in the previous example Perform the same logistic regression analyses obtain odds ratios for the associations of disease and exposure category ordinal level of exposure or actual estimated exposure Observational data potential bias in exposure assessment soften the scientific conclusions associations BIOST 536 Lecture 3 10 Casecontrol Decide on criteria for cases and identify all cases possible Decide on criteria for controls and identify a number proportional to the number of cases 1 case to 1 control 12 or 14 May be frequency matched on agegender or other characteristics of the cases logistic regression May be tightly matched to a case in 1m matching on several factors simultaneously conditional logistic regression lnterpret the odds ratios BIOST 536 Lecture 3 11 Cohortembedded casecontrol studies Overall longitudinal cohort available from which possible cases can be identified Identify actual cases among the possible cases intake diagnosis suggests possible myocardial infarction identify actual Ml cases Identify controls anyone without an actual Ml occurring at the same age as the case Decide on number of controls per case and randomly sample from all potential controls the same age as the case Assemble information retrospectively that was collected prior to the case s diagnosis date Perform a matched casecontrol analysis conditional logistic regression Interpret the odds ratios BIOST 536 Lecture 3 12 Estimation MOP 20tate Mtate 1 RRt 0 El Cl Cl 1 2 Measures for prospective designs Incidence risk ratio Risk difference Attributable fraction Typically compare exposed incidence rate to unexposed incidence rate using the ratio Incidence risk ratio may not depend on age time even though the incidence rates do Atate 0 l A M ame o e Mathematically convenient so extensively used May prefer a risk difference instead RD I lm x100 20rar 21tatdose e9 RD t11t 10tatdose e at dose e9 Cannot be used for casecontrol studies without more information BIOST 536 Lecture 3 13 Attributable fraction Exposed attributable risk a Cole amp McMahon 1971 a Used descriptiver primarily Population attributable risk 1 Levin 1953 2 Need to knowthe proportion of the population in the exposed group at time t p 1 RR t ARltIgtMgtltgtI Mt Population Attributable Risk an an Fmvailnu Cunsunr W m p AW100 p11lttgt17p10lttgt pRRlttgt71 pRRt 1712 PAR z ass as 97 1 Expusure Praviienc a u 02 n 4 Study designs Exposed Unexposed Disease a b n 1 No Disease c d n 0 m 1 m 0 1 1 2 x 2 Tables can have either the table total n column totals m1 m0 or row totals n1 no fixed by design Crosssectional design Only n fixed Choose n subjects at random and ascertain both exposure and disease status Very inefficient for studying association since disease andor exposure may be rare Can estimate disease prevalence in unexposed and exposed b a I70 I p1t m0 m1 Estimates may not be stable for small a b m1 BIOST 536 Lecture 3 15 Exposed UneXposed Prospective cohort Disease a b n 1 No Disease c d n 0 m 1 m 0 1 1 Choose subjects from a population that does not have disease at time t0 and follow until t1 Ascertain exposure prior to t0 and disease status in to t1 Total n or the column totals m1 m0 may be fixed by design a Greater efficiency usually if m1 m0 Estimate the risk of developing disease in the interval to t1 by exposure status p 03 61 O m 1710 0 m1 If only n fixed by design no sampling on exposure then risk of developing the disease in t0 t1 among the population is Clb n1 pltrgt mlmO n a Otherwise need to weight by exposure probabilities BIOST 536 Lecture 3 16 Prospective cohort facts Cl If the risk ratio does not depend on t then M RR RR om gt 1 p10 1 1900 Means the probability of not having disease in the interval if exposed is the probability of not having disease in the interval for the unexposed raised to the power of the risk ratio Prob of no disease Prob of no disease RR in interval if exposed in interval if unexposed If the disease probability pi t is small then 710 3 RR 170 I So the risk ratio is estimated by WWW BIOST 536 Lecture 3 Exposed UneXposed Casecontrol de31gn Disease a b n 1 No Disease c d n 0 m 1 m 0 1 1 Choose subjects from a population that do or do not have disease in the interval to t1 Typically no n1 are fixed by design The proportions below do not estimate disease risk among exposed and unexposed a b 1710 1700 1 O Instead we estimate probability of exposure conditional on case Status i gt Pr exposed disease in t0t1 PrED quot1 i gt Pr exposed no disease in t0t1 PrE5 no Withoutmore information cannotget PrDEandPrDE BIOST 536 Lecture 3 18 Casecontrol design estimation Let Y disease and Xexposure and adopt a logistic regression model e 0 1X PrYX 6 W Then the odds ratio is estimated as e 0 1 1e o l PDE 1 PDE 1e 1 e 1 e 1 PDIE e o e quot 1 PDE Hem 1 i 1em We cannot estimate PDE sin e we need Bo to do that We do get an estimated Bo from our logistic regression but it is not the right Bo more on that later BIOST 536 Lecture 3 19 Casecontrol design estimation Cannot estimate the risk difference Cannot estimate the population attributable risk without estimates of exposure rates in the population Casecontrol analysis also assumes that sampling rates for exposed and unexposed individuals are the same otherwise bias can result Later will control for other sources of bias in casecontrol analysis BIOST 536 Lecture 3 20 Lecture 8 Model fit and diagnostics Grouped versus individual data a Individual outcome data binary yi Binomial lpi Each individual has specific covariate values and a binary outcome 01 Data may be grouped it there are individuals with exactly the same values may be less likely it we have true continuous variables Can be analyzed by logistic regression Example ii OKDOOQO XUlprJNH l logistic y age Logistic regression Number of obs 10 LR chi2l 009 Prob gt chi2 07628 Log likelihood 768859106 Pseudo R2 00066 y l Odds Ratio Std Err z Pgtlzl 95 Conf Interval age l 9742461 0844814 7030 0764 821972 115473 BIOST 536 Lecture 8 1 Example continued Get counts of the crossclassification of y and covariates and use frequency weights ill 24 ll le 4o 2i 3io 26 2i 4io 37 ll 5li 26 ii i 7777777777777 Wl 6l0 40 ll 7ll 37 ll 8l0 45 ll 7777777777777 77 logistic y age fwcnt Logistic regression Number of obs 10 LR chi2l 009 Prob gt chi2 07628 Log likelihood 768859106 Pseudo R2 00066 y l Odds Ratio Std Err z Pgtlzl 95 Conf Interval age l 9742461 0844814 7030 0764 821972 115473 Same results as before and we still use logistic regression If we had recorded age in months or days then would not have tied data As the number of observations goes to infinity then theoretically could still have a single value at each level of x BIOST 536 Lecture 8 2 Grouped data binomial Model the number of cases yi based on the denominator ni when all individuals have covariate combination xi There may be a fixed number of covariate combinations Usually do not have a true continuous covariate but usually a categorical variable that may be modeled as a grouped linear variable The number of covariate combinations is fixed so as the sample size goes to then the number of observations in each cell goes to Analyzed by blocked logit regression blogit in Stata BIOST 536 Lecture 8 3 Example consider age as categorical in years blogit cases 11 age Logistic regression for grouped data Number of obs 10 LR chi2l 009 Prob gt chi2 07628 Log likelihood 768859106 Pseudo R2 00066 foutcome Odds Ratio Std Err z Pgtz 95 Conf Interval age 9742461 0844814 7030 0764 821972 115473 BIOST 536 Lecture 8 Grouped versus ungrouped data Grouped data a View the number of covariate combinations as finite a Often do diagnostics at the level of the covariate combination eg How much does this particular covariate combination affect the model fit a May compare fitted values to observed values across the different covariate combinations a Stata has more limited choices for the blogit command BIOST 536 Lecture 8 5 Grouped versus ungrouped data Individual data a Number of covariate combinations can be finite or theoretically infinite a Often do diagnostics at the level of the individual e ng much does this particular person affect the model fit u Stata has more choices for diagnostics for logistic command a Can always turn grouped data into individual level data but can take longer to run or may have memory problems in Stata BIOST 536 Lecture 8 6 Grouped versus ungrouped data Start with grouped data and fit a blocked logit model level Cases Total 0 23 400 1 30 370 Xi blogit cases total iage X Grouped linear variable in the model Categorical variable in the model Assumptions about the association of exposure and outcome BIOST 536 Lecture 8 Grouped data analysis Xi blogit cases total iage X iage Iage7173 OI Logistic regression for grouped data naturally coded Number of obs Iage41 omitted 2810 LR chi23 66311 Prob gt chi2 00000 Log likelihood 713233737 Pseudo R2 02003 ioutcome Odds Ratio Std Err z Pgtlzl 95 Conf Interval Iage72 2241811 3112071 582 0000 1707795 2942811 Iage 3 1175013 1519547 1905 0000 9119353 1513985 X 1 2123945 1306102 1225 0000 1882779 2396002 blogit Logistic regression for grouped data Number of obs 2810 LR chi23 66311 Prob gt chi2 00000 Log likelihood 713233737 Pseudo R2 02003 outcome Coef Std Err z Pgtlzl 95 Conf Interval Iage72 8072842 1388195 582 0000 535203 1079365 Iage73 2463865 1293217 1905 0000 2210399 2717331 X 1 7532753 0614942 1225 0000 632749 8738016 cons 73025445 1315639 72300 0000 73283305 72767584 estat ic Model Obs llnull llmodel df AIC BIC 2810 71654931 71323374 4 2654747 2678511 Note NObs used in calculating BIC see R BIC note BIOST 536 Lecture 8 Grouped versus ungrouped data Change to individual data with counts and fit a logistic model Age group Exposure level y cnt x 0 1 1 Xi logistic y iage X fwzcnt BIOST 536 Lecture 8 Individual data analysis with counts Xi logistic y iage X fwcnt iage flage7173 naturally coded 41age1 omitted Logistic regression Number of obs 2810 LR chi23 7 66311 Prob gt chi2 00000 Log likelihood 713233737 Pseudo R2 02003 y 1 Odds Ratio Std Err z Pgtlzl 95 Conf Interval Iage72 2241811 3112071 582 0000 1707795 2942811 Iage73 1175013 1519547 1905 0000 9119353 1513985 X 1 2123945 1306102 1225 0000 1882779 2396002 logistic coef Logistic regression Number of obs 2810 LR chi23 66311 Prob gt chi2 00000 Log likelihood 713233737 Pseudo R2 02003 y Coef Std Err z Pgtlzl 95 Conf Interval flage72 8072842 1388195 582 0000 535203 1079365 1age73 2463865 1293217 1905 0000 2210399 2717331 X 1 7532753 0614942 1225 0000 632749 8738016 cons 73025445 1315639 72300 0000 73283305 72767584 BIOST 536 Lecture 8 Grouped versus ungrouped data Change to individual data with one observation per record Can use expand command in Stata a expand cnt will take the grouped data and replicate each observation n times where n cnt in this case n Then we have individual level data expand cnt 2792 observations created Xi logistic y iage X iage flage7173 naturally coded flageil omitted Logistic regression Number of obs 2810 LR chi23 66311 Prob gt chi2 00000 Log likelihood 713233737 Pseudo R2 02003 y l Odds Ratio Std Err z Pgtlzl 95 Conf Interval flage72 l 2241811 3112071 582 0000 1707795 2942811 flage73 l 1175013 1519547 1905 0000 9119353 1513985 X l 2123945 1306102 1225 0000 1882779 2396002 BIOST 536 Lecture 8 11 Model t and diagnostics Types of diagnostics u Comparison of predicted values to observed a Specific measures of goodnessoffit HosmerLemeshow test deviance Pearson x2 Akaike u Cstatistic Pearson and deviance residuals Influence diagnostics BIOST 536 Lecture 8 12 Comparison of predicted values to Observed Not very interesting for individual level data since observed values are 0 or 1 and predicted values are estimated probabilities Can be more useful for grouped data Com arison of observed versus ex ected redicted values Model yi Binomial n1 pi 63031Xi1wgk Xik logit Pi 0 1 X11 k Xik 3 I31 163031Xnm kXik Predicted value is n1 P1 The maximum likelihood equations guarantee that z yi ni i 0 and z xijyi nifi0 for a11j1k For categorical variables the sum of the observed values in that category equals the sum of the expected values I gt Tabling observed versus expected by covariates may not be that useful BIOST 536 Lecture 8 13 Example Esophageal data Data grouped by alcohol tobacco and age group 88 observed covariate combinations out of 96 possible 4 alcohol groups x 4 tobacco groups x 6 age groups Fit a model using age as a categorical covariate and tobacco as a grouped linear covariate Grouped data analysis blogit will give predicted values for each of the 88 cells A expected 61 2 mi pl for denominator ml of covariate combination 1 1 88 Overall sum ofgpbservgd cases sum of expected cases 2 yr Zei i1 i1 Sum of observed cases in categoryj of a categorically modeled covariate sum of expected cases in categoryj BIOST 536 Lecture 8 14 Example summ Variable 1 Obs Mean Std Dev Min Max age 1 88 3386364 1650021 1 6 alc 1 88 2454545 1123511 1 4 tob 1 88 2409091 1120718 1 4 cases 1 88 2272727 2753169 0 17 cents 1 88 8806818 1213512 0 60 tot 1 88 1107955 127227 1 60 list in l5 1 age alc tob cases conts tot 1 1 1 1 1 2534 039 09 0 40 40 1 2 1 2534 039 1019 0 10 10 1 3 1 2534 039 2029 0 6 6 1 4 1 2534 039 30 0 5 5 1 5 1 2534 4079 09 0 27 27 1 list in 8488 1 age alc tob cases conts tot 1 1 1 84 1 75 4079 30 1 0 1 1 85 1 75 80119 0 1 0 1 1 86 1 75 80119 1019 1 0 1 1 87 1 75 120 09 2 0 2 1 88 1 75 120 1019 1 0 1 1 BIOST 536 Lecture 8 Example xi blogit cases tot iage tob or iage Iage71 6 naturally coded Iageil omitted Logistic regression for grouped data Number of obs 975 LR chi26 15390 Prob gt chi2 00000 Log likelihood 41779231 Pseudo R2 01555 ioutcome Odds Ratio Std Err z Pgtz 95 Conf Interval Iage72 5732821 6096968 164 0101 7130152 4609332 Iage73 3372291 3443578 345 0001 4557532 2495286 Iage74 5727469 582569 398 0000 7801243 4204958 Iage75 7652069 7828554 424 0000 1030257 5683453 Iage76 5786546 6154725 382 0000 719546 4653505 tob 163521 1407873 571 0000 1381298 1935797 predict exp option n assumed predicted no of cases list age alc tob cases exp tot in 8488 age alc tob cases exp tot 84 75 40 79 30 1 5610738 1 85 75 80 119 0 9 1 2262176 1 86 75 80 119 10 19 1 3234369 1 87 75 120 0 9 2 4524353 2 88 75 120 10 19 1 3234369 1 umm cases exp Variable Obs Mean Std Dev Min Max cases 88 2272727 2753169 0 17 exp 88 2272727 2684869 0050269 1338307 BIOST 536 Lecture 8 Example bysort age summ cases exp gt age 25 34 Variable Obs Mean Std Dev Min Max cases 15 0666667 2581989 0 1 exp 15 0666667 0598642 0050269 2010757 gt age 35 44 Variable Obs Mean Std Dev Min Max cases 15 6 9856108 0 3 exp 15 6 4705423 0844457 1688915 Etc rest deleted Sum of observed cases in each age category sum of expected cases in age category Not true for grouped linear variable but still constrained by MLE equations where tobacco takes on values 1 to 4 Z tobaccoi yi ni13i 0 elt1y1 e1lt2y2 e2lt3y3 e3lt4y e 0 BIOST 536 Lecture 8 Example bysort tob summ cases exp gt tob 0 9 Variable Obs Mean Std Dev Min Max cases 24 325 4024382 0 17 exp 24 3262584 3923558 0050269 1338307 gt tob 10 19 Variable Obs Mean Std Dev Min Max cases 24 2416667 233902 0 8 exp 24 2242071 2308286 0081939 70663 gt tob 20 29 Variable Obs Mean Std Dev Min Max cases 20 165 1663066 0 5 exp 20 2023726 2096349 0133293 7415762 gt tob 30 Variable Obs Mean Std Dev Min Max cases 20 155 1877148 0 5 exp 20 1370687 1169193 0432265 335127 May actually learn something here about the fit of the model In general few diagnostics available in blogit prefer to work with logistic command BIOST 536 Lecture 8 18 Some model checking tools Residuals Calculated for each covariate combination Let M be the number of observed covariate patterns depends on which model was fit Fori1M1et yinun1ber of cases for covariate patterni n1 2 number of observations for covariate pattern 139 0 Pearson residual 1 yi ni pi i A A lni pi 1 pi 0 Standardized Pearson residual 139 1 S Standardized 1 1 where hi comes from the quothatquot matrix 1 BIOST 536 Lecture 8 19 Some model Checking tools Residuals continued Deviance residuals np mid t l i di 2 yilog y niy10gLyi if yiltnzl3i and quot91 nip quotid Pi di2yi10g y niyi10g j yi 271i i and quot1 gt1 139 d1 2ni10gl ifyi0 di2ni10g i yini Related to overall Pearson goodnessoffit and deviance BIOST 536 Lecture 8 20 Some model checking tools Overall goodness Of t Summary measures when there are k number of predictors for the m covariate combinations 2 Pearson X2 I has a x2 distribution with M k1 df l 2 Deviance i has a x2 distribution with M k1 df l Akaike criterion Deviance 2 k 1 Akaike allows comparison of nonnested models by penalizing models that have many predictors eg Model 1 has deviance 790 with 3 covariates Akaike 870 Model 2 has deviance 820 with 1 covariate Akaike 860 gt Model 2 is preferred If the number of observations is large compared to the number of covariate configurations then we can use these to test the model fit BIOST 536 Lecture 8 21 Some model checking tools Overall goodness Of t Grouped dat grouped by covariate configuration Deviance and Pearson chisquared are measures of goodness offit For grouped data df groups parameters n k1 Both measures should have a x2 distribution with nk1 df Since the mean of a chisquare distribution is equal to its df then in a good model we expect them to be close to the df If either is much larger than its df then we have a poor fit If either is much smaller than its df then we probably have too many sparse cells ie too many covariates in the model This tests general departures from the model omnibus test but lacks power for detecting specific departures BIOST 536 Lecture 8 22 Some model checking tools Overall goodness Of t Ungrouped data Suppose one of the covariates is continuous so the data cannot be grouped Can we still use these as a measure of fit No These tests only work when the sample size is much greater than the number of parameters If n gt oo then the number of parameters in the saturated model goes to infinity for ungrouped data Deviance no longer has a chisquared distribution so cannot be used to assess model fit We can use the HosmerLemeshow test instead BIOST 536 Lecture 8 23

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