Biostatistics Second Term, Final Examination December 17, 2015 Answer Key

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1 Botattc 4.6 Secod Term, 5-6 Fal Eamato December 7, 5 Awer Key Itructo: You wll have two hour for th eamato. Pleae wrte o thee age ad crcle oe reoe for each queto. There Stata outut ad formula age whch cota elected Z ad value at the ed for your ue. Queto -5 are queto of geeral kowledge: ) Whe calculatg the 95% cofdece terval for the dfferece roorto ug two deedet amle, the tadard error of the dfferece roorto baed o: (Crcle oly oe reoe.) a) Ug the ooled roorto calculatg the SE of the dfferece roorto. b) Ug the earate etmated roorto calculatg the SE of the dfferece roorto. The comutato of the tadard error ot baed o the aumto that the ull hyothe true becaue cofdece terval cotructo doe ot requre tatemet of a hyothe. Accordgly, the SE ue the ot etmate for the roorto baed o grou ad the ot etmate for the roorto baed o grou : q q c) Ug the MSE a the bet etmate of the amle varace. d) Ug the otmal roorto of.5 calculatg the SE of the dfferece roorto. e) Ug the earate true oulato roorto calculatg the SE of the dfferece roorto. ) Suoe that a Aaly of Varace (ANOVA) wa erformed to vetgate mea BMI acro four age-geder grou. Whch of the followg tatemet mot accurately decrbe what the MSB? (Crcle oly oe reoe). a) The amle varace of the four BMI mea (oe for each age-geder grou), calculated by ummg the weghted quared dfferece betwee each grou mea ad the overall mea. The MSB = y y ( ) where the umber of grou.

2 Botattc 4.6 Fal Eamato- Awer Key b) The amle varace of BMI value a artcular age-geder grou, calculated by oolg the amle varace for each of the four age-geder grou. c) The amle varace of dvdual BMI value about the overall mea, gorg the age-geder grou. d) The roorto of varablty BMI that ca be elaed by dfferece mea BMI betwee age-geder grou. e) The roorto of varablty BMI that caot be elaed by dfferece mea BMI betwee age-geder grou. 3) The lk fucto ued a geeralzed lear model allow the outcome varable Y to be ereed a a lear combato of the X varable, X, X,., X. Whch of the followg the terretato of the regreo coeffcet for the varable X whch the model ecfed wth famly a bomal ad the lk a logt? (Crcle oly oe reoe). a) The dfferece log rk of the outcome aocated wth each ut creae X, adutg for the other X. b) The dfferece log odd of the outcome aocated wth each ut creae X, adutg for the other X. Th geeralzed lear model ecfed a: logt(y) = log(odd of Y)=log{P(Y=)/P(Y=)} = β +β X +.+ β X c) The dfferece rk of the outcome aocated wth each ut creae X, adutg for the other X. d) The dfferece mea outcome aocated wth each ut creae the X. e) The mea outcome whe each of the X equal zero. 4) Suoe that a lear regreo Model A, BMI regreed oto age uch that E[BMI] = β+βage. Suoe that a dfferet lear regreo Model B, age regreed oto BMI uch that E[age] = α+αbmi. What ca be ad about ther reult? (Crcle oly oe reoe). a) The loe are the ame (β = α) but the tercet tycally are dfferet (β α). b) The loe tycally are dfferet (β α) but the tercet are the ame (β = α). c) Both the loe are the ame (β = α) ad the tercet are the ame (β = α). d) The loe tycally are dfferet (β α ) ad the tercet tycally are dfferet (β α ). I Model A, β the eected BMI whe age=; β the chage eected BMI wth each oe ut creae age. I Model B, α the eected age whe BMI=; α the chage eected age wth each oe ut creae BMI. Thu, the ut eed ot be the ame, ad the etmated tercet are tycally ot the ame ad the etmated loe are tycally ot the ame. 5 Joh Hok Uverty Deartmet of Botattc /8/6

3 Botattc 4.6 Fal Eamato- Awer Key e) The correlato coeffcet for the lear aocato betwee BMI ad age tycally dfferet from the correlato coeffcet for the lear aocato betwee age ad BMI. 5) Suoe that vetgator are tereted whether a reoe dffer by treatmet a certa eermet. They defe the model a log [(P(Y=)/P(Y=)] = β+βx where Y=reoe where =o, =ye; X= for lacebo ad for drug. From th model, what rereet the odd of reoe the drug grou? (Crcle oly oe reoe). a) e(β) b) e(β+β) c) e(β +β ) Whe X= (drug), we would wrte: log [(P(Y=)/P(Y=)] = β +β () d) β+β e) β+β Queto 6 through cocer data from a logtudal tudy of heart deae Lo Agele Couty emloyee. Baele meauremet of everal varable were take ad the emloyee were the followed for year. Follow-u meauremet ad dago of chemc heart deae were of teret. Varable of teret are defed a: chol = erum choleterol at follow-u (mg%), cholce = baele erum choleterol cetered at 5 (meda value) = (chol-5) (mg%) agece = baele age cetered at 47 year (meda age) = (age - 47) bce= baele ytolc blood reure cetered at (meda value) = (b- ) age_chol= agece*cholce= (age - 47) * (chol-5) hd = f dago of chemc heart deae durg follow-u; f o dago. Model A D are gve at the back of the eam. 6) Lookg at the regreo reult Model A ad B, whch model rovde a more rece redcto of the outcome? (Crcle oly oe reoe). a) Model A becaue baele erum choleterol ela % of the total varato follow-u erum choleterol wherea Model B oly ela %. b) Model B becaue the etmated redual tadard devato larger, 46. mg (%), veru 4.3 mg (%) Model A. c) Both rovde rece redcto ce the regreo coeffcet are tattcally gfcatly dfferet from zero. d) Model B becaue the F-tattc maller. e) Model B becaue the tadard error of the regreo coeffcet maller. 5 Joh Hok Uverty Deartmet of Botattc /8/6 3

4 Botattc 4.6 Fal Eamato- Awer Key 7) Lookg at the regreo reult Model A D, would you coclude that baele age a cofouder of the relatoh betwee aocato betwee follow-u erum choleterol ad baele erum choleterol? If o, why? (Check oly oe reoe that correctly addree both the cocluo ad the reao). a) No, becaue the etmated baele erum choleterol coeffcet Model A ad C are both tattcally gfcatly dfferet from zero. b) No, becaue the etmated baele erum choleterol coeffcet Model A ad C are very mlar. Baele age a cofouder of the aocato betwee baele ad follow-u erum choleterol oly f t aocated wth both varable but doe ot fall the caual athway betwee them. If baele age wa a cofouder, the we would eected the etmated baele choleterol coeffcet would chage ubtatally magtude whe baele age cluded the model (Model C) a comared to the model that oly clude baele choleterol (Model A). The gudele emloyed whether the etmated baele choleterol coeffcet chage by 5%-% or more betwee model. I th cae, the etmated baele choleterol coeffcet aromatg.39 mg%/mg% both Model A ad C. c) Ye, becaue the etmated teracto coeffcet for baele erum choleterol ad baele age Model D tattcally gfcatly dfferet from zero. d) Ye, becaue the etmated age coeffcet Model C ubtatally le tha the etmated age coeffcet Model B. e) Ye, becaue the etmated baele erum choleterol coeffcet the teracto Model D tattcally gfcatly dfferet from zero. 8) I Model D, the cotat term rereet: (Crcle oly oe reoe). a) The eected follow-u erum choleterol level for emloyee whoe baele value of age, erum choleterol ad ytolc blood reure equal. b) The eected follow-u erum choleterol level for emloyee havg baele meda value of age, erum choleterol ad ytolc blood reure. Model D ca be wrtte a: E[chol] = β +β (chol-5) +β (age- 47) +β 3 (age-47)(chol-5) +β 4 (b-) Thu, β the eected outcome for emloyee wth cetered value of the covarate. All of the covarate are cetered at ther meda value. c) The chage eected follow-u erum choleterol level for each oe year creae baele age, holdg both baele erum choleterol ad ytolc blood reure cotat. d) The chage eected follow-u erum choleterol level for each oe mg% creae baele erum choleterol, holdg both baele age ad ytolc blood reure cotat. 5 Joh Hok Uverty Deartmet of Botattc /8/6 4

5 Botattc 4.6 Fal Eamato- Awer Key e) The chage eected follow-u erum choleterol level for a oe mg% creae baele erum choleterol ad oe year creae age take together, holdg ytolc blood reure cotat. 9) I Model D, oe would coclude that there a tattcally gfcat teracto betwee baele age ad baele erum choleterol, after adutg for baele age, baele erum choleterol ad baele ytolc blood reure, baed o: (Crcle oly oe reoe). a) F(4,95)=.5, =. b) t= 7.68, =. c) t= -.95, =.53 d) t= -.7, =.8 I Model D, we tet th teracto by tetg Ho: β 3 = ug t = (b 3 -)/SEb 3 e) t=85.4, =. ) From Model D, what the terretato of b3, the age_chol teracto coeffcet? (Crcle oly oe reoe). a) The dfferece the average follow-u erum choleterol level betwee dvdual havg baele value of age ad erum choleterol above the meda value a comared to below the meda value, holdg baele ytolc blood reure cotat. b) The dfferece average follow-u erum choleterol level er ut creae baele ytolc blood reure, betwee dvdual wth hgh value of baele erum choleterol ad thoe wth low value of baele erum choleterol, adutg for baele age. c) The dfferece average follow-u erum choleterol level er ut creae baele ytolc blood reure, holdg baele age ad baele erum choleterol cotat. d) The dfferece average follow-u erum choleterol level er ut creae baele erum choleterol, holdg baele age ad baele ytolc blood reure cotat. e) The dfferece, betwee dvdual varyg by oe year of baele age, the chage average follow-u erum choleterol level er ut creae baele erum choleterol, holdg baele ytolc blood reure cotat. The model : E[chol] = β +β (chol-5) +β (age- 47) +β 3 (age-47)(chol-5) +β 4 (b-) A a eamle: For age=48: E[chol] = β +β (chol-5) +β (48-47) +β 3 (48-47)(chol-5) +β 4 (b-) = (β +β ) + (β +β 3 )(chol-5) +β 4 (b-) For age=47: E[chol] = β +β (chol-5) +β (47-47) +β 3 (47-47)(chol-5) +β 4 (b-) = β + β (chol-5) +β 4 (b-) 5 Joh Hok Uverty Deartmet of Botattc /8/6 5

6 Botattc 4.6 Fal Eamato- Awer Key By ubtracto, β 3 the dfferece loe (chage eected follow-u choleterol er ut creae baele choleterol) betwee emloyee dfferg by oe year of age, holdg baele ytolc blood reure cotat. ) Ug the reult of Model D: For dvdual wth the ame baele age, erum choleterol level ad ytolc blood reure, we could etmate that aromately 68% of the oulato would have follow-u erum choleterol value the followg rage of the true mea follow-u erum choleterol level: (Crcle oly oe reoe.) a) ± 39.5 mg (%) Aumg a ormal dtrbuto, we eect that aromately 68% of the oulato value for dvdual wth the ame baele age would fall wth ± σ of the mea, where σ the redual tadard devato. We etmate σ by the quare root of the MSE = 39.5 % b) ± mg (%) c) ± mg (%) d) ± 56.9 mg (%) e) ±.97 mg (%) ) The lot below how the redual veru redctor lot (rvlot) for Model D ug the redctor of cetered baele age. Whch of the lear regreo aumto(), f ay, aear to be ubtatally volated, baed o th lot? (Crcle oly oe reoe.) a) Ideedece of obervato b) Equal varace c) Normal dtrbuto d) Lear relatoh e) Noe of the aumto aear to be ubtatally volated. The lot how radom catter aroud zero wth o aaret atter. 5 Joh Hok Uverty Deartmet of Botattc /8/6 6

7 Botattc 4.6 Fal Eamato- Awer Key Redual agece Queto 3 through 9 volve the ame data et of Lo Agele health deartmet emloyee. The followg Table how the cro-tabulato of umber of emloyee wth a dago of chemc heart deae by baele erum choleterol. We defe: Y=hd where hd = f ye; f o durg follow-u X= hchol (dchotomzed a low ad hgh) where hchol= f hgh; f low. TABLE. tab hd hchol hchol hd Total Total ) I the comutato of the ch-quare tattc for a tet of o aocato betwee chemc heart deae ad hgh baele erum choleterol level: What are the eected umber wth chemc heart deae the low ad hgh baele erum choleterol grou, reectvely? (Crcle oly oe reoe.) a) 67, 65 b) 7, 4 c) 3,36 The overall roorto wth hd = 68/=.34. Aumg o aocato betwee dh ad hgh baele erum choleterol, we would eect (.34)(94)=3 ad (.34)(6)=36 dvdual wth hd the low ad hgh baele erum choleterol grou, reectvely. d) 7, 36 e) 6, 6 5 Joh Hok Uverty Deartmet of Botattc /8/6 7

8 Botattc 4.6 Fal Eamato- Awer Key 4) Suoe a logtc regreo of chemc heart deae (hd) o hgh baele erum choleterol (hchol) mg (%) ( year) erformed: log(odd of hd) = β +β hchol Baed o the data Table, what b, the etmated regreo coeffcet for hchol? (Crcle oly oe reoe). a) log(7/67) b) 4/65 c).56 d) log(.56) ce β the log OR for hd comarg thoe wth hgh veru low baele erum choleterol. From the Table, the OR = (4/65)/(7/67) =.56. e) (67 4)/(7 65) Model through 3 vetgate the aocato betwee chemc heart deae ad hgh baele erum choleterol: MODEL. glm hd hchol, famly(bomal) lk(logt) OIM hd Coef. Std. Err. z P> z [95% Cof. Iterval] hchol _co MODEL. glm hd hchol, famly(bomal) lk(log) OIM hd Coef. Std. Err. z P> z [95% Cof. Iterval] hchol _co MODEL 3. glm hd hchol, famly(bomal) lk(detty) OIM hd Coef. Std. Err. z P> z [95% Cof. Iterval] hchol _co Coeffcet are the rk dfferece. 5) Whch of the Model -3 how that there o tattcally gfcat aocato betwee chemc heart deae ad hgh baele erum choleterol level a gve by the etmated relatve rk?: (Crcle oly oe reoe.) 5 Joh Hok Uverty Deartmet of Botattc /8/6 8

9 Botattc 4.6 Fal Eamato- Awer Key a) Model : RR=.45; (95% CI: -.5,.4) b) Model : RR=.56; (95% CI:.86,.76) c) Model : RR=.35; (95% CI:.9,.) The lk(log) gve u a model ereed a: log(rk of hd) = β +β hchol where b the etmated log(rr) of hd by hgh veru low baele erum choleterol. We eoetate the etmated coeffcet ad aocated 95% cofdece lmt how above Model order to etmate the RR. d) Model 3: RR=.; (95% CI: -.3,.3) e) Model 3: RR=.9; (95% CI:.,.38) 6) We would wrte Model 3 a: (Crcle oly oe reoe.) a) log [P(Y=)] = β+βhchol b) log [P(Y=)/P(Y=)] = β+βhchol c) P(Y=)/P(Y=) = β+βhchol d) P(Y=) = β +β hchol whch gve by the lk(detty). e) E[log Y] = β+βhchol Model X-Z how the reult of a logtc regreo of chemc heart deae o dfferet covarate: MODEL X. logt hd chol hd Coef. Std. Err. z P> z [95% Cof. Iterval] chol _co MODEL Y. logt hd chol agece hd Coef. Std. Err. z P> z [95% Cof. Iterval] chol agece _co Joh Hok Uverty Deartmet of Botattc /8/6 9

10 Botattc 4.6 Fal Eamato- Awer Key 7) Baed o Model Y aromately what the odd rato for chemc heart deae comarg 45 year old ad 5 year old, after adutg for baele erum choleterol? (Crcle oly oe reoe.) a). b).87 We ca wrte: log(odd of hd) = β +β chol+β (age-47) Whe age=45: log(odd of hd 45 year) = β +β chol+β (45-47) Whe age=5: log(odd of hd 5 year) = β +β chol+β (5-47) By ubtractg, the dfferece log odd = log(or) = β Thu, we ca etmate the logor = b = (-.7) = -.4; The etmate OR = e(-.4) =.87 c).99 d).6 e) ) Coder the reult from Model X ad Y: I the aocato betwee IHD ad baele erum choleterol modfed by age? (Crcle oly oe reoe.) a) No, becaue the etmated choleterol coeffcet mlar both the mle lear regreo Model X ad the multle lear regreo Model Y. b) No, becaue the age coeffcet Model Y ot tattcally gfcatly dfferet from zero. c) Ye, becaue the etmated choleterol coeffcet tattcally gfcat dfferet from zero both Model X ad Y. d) Ye, becaue the chage log odd of IHD creae wth creag baele erum choleterol level, after adutg for baele age. e) The comaro of Model X ad Y doe ot ae effect modfcato. A comaro of Model X ad Y allow a aemet of cofoudg of the IHD ad baele erum choleterol by age. Model Z how the reult of a logtc regreo of chemc heart deae o baele erum choleterol, age, ad dcator for categore,, 3, 4, 5 of ocoecoomc tatu (SES): MODEL Z. logt hd chol agece.e hd Coef. Std. Err. z P> z [95% Cof. Iterval] chol.7899 b agece b e b b b b _co b Joh Hok Uverty Deartmet of Botattc /8/6

11 Botattc 4.6 Fal Eamato- Awer Key 9) I Model Z, the rght had de of the equato equal β+βchol+ βagece+ β3e + β4e3+β5e4 + β6e5 What the terretato of the eoetated (b5-b4) from thee reult? (Crcle oly oe reoe.) a) The odd of IHD are reduced by 6% for emloyee SES category 4 a comared to SES category 3, after cotrollg for baele erum choleterol ad age. b) The odd of IHD are.5 tme greater for emloyee SES category 4 a comared to SES category 3, after cotrollg for baele erum choleterol ad age. From the model, b 5 = dfferece aduted log(odd) betwee e 4 ad e ad b 4 = dfferece aduted log(odd) betwee e 3 ad e. By ubtractg b 5 -b 4 = dfferece aduted log(odd) betwee e 4 ad e 3. Thu, the aduted OR of IHD for e 4 a comared to e 3 = e(b 5 -b 4 )= e(.99 ( -.)) = e(.399) =.49 ~.5. c) The uaduted log odd rato of IHD for emloyee SES category 4 a comared to SES category 3.4, after cotrollg for baele erum choleterol ad age. d) The odd of IHD decreae by % wth each year of age, for emloyee SES category 4 a comared SES category 3, after cotrollg for baele erum choleterol. e) The log odd of IHD creae by.4 wth each category creae SES, after cotrollg for baele erum choleterol ad age. For queto, the vetgator further vetgated the relatoh betwee chemc heart deae (IHD) ad baele erum choleterol (dchotomzed a low veru hgh) by detfyg each emloyee who develoed IHD ad matchg that ero, baed o age ad e, to a emloyee who had ot develoed IHD The cout data o the reultg matche are tabulated below. TABLE 3 Emloyee develog IHD Hgh baele erum choleterol Low baele erum choleterol Emloyee ot develog IHD Hgh baele erum Low baele erum choleterol choleterol 3 6 ) Baed o thee data whch of the followg the bet etmate of the odd rato for the aocato betwee IHD (IHD veru o IHD) ad erum choleterol level (hgh veru low) amog dvdual of the ame age ad e? (Crcle oly oe reoe.) a).47 b).6 5 Joh Hok Uverty Deartmet of Botattc /8/6

12 Botattc 4.6 Fal Eamato- Awer Key 5 Joh Hok Uverty Deartmet of Botattc /8/6 c).6 d).3 e).8 Th a matched ar cae-cotrol tudy ad the etmated odd rato baed o the dcordat ar uch that: OR =b/c= 3/ Where b= (umber of ar whch em develog IHD have hgh baele erum choleterol ad ther matched em ot develog IHD have low baele erum choleterol) ad c= (umber of ar whch em develog IHD have low baele erum choleterol ad ther matched em ot develog IHD have hgh baele erum choleterol). Fal Eam Formula Sheet - Imortat Formula for Stattcal Iferece Poulato z RR / Oe Samle q z z : H t : H SE (log ) e q q RR OR q q SE (log OR) e q q Two Samle where : H t z t / ( ) z z / a a z q z q

13 Botattc 4.6 Fal Eamato- Awer Key H H : d : d a b r c d r c c N χ χ d d t d z q q (O E ) E Nad bc rrcc where OR b SE (log OR) c e b c b c χ b c z q z z z q q z z / Oe-Way ANOVA H : 3 Source SS df Betwee Grou T. T y y y y N Wth Grou ( T. y y y ) N Total y y N y N( y..) y T N Boferro * Boferro t y y MSW MSW 5 Joh Hok Uverty Deartmet of Botattc /8/6 3

14 Botattc 4.6 Fal Eamato- Awer Key Aocato r y y y y ( ( ) y ) ( y ) y y Smle Lear Regreo y b b where b y b y y ( y) y y b r or r b ( ) y z. ( ded).58 z.(-ded).33 z5. ded. 96 z5. ded. 645 z. ded. 8 z. ded. 84 ANOVA for Regreo Source SS df y Regreo y y Redual y Total y y y y SS r Regreo SST Tal Probablty Z -ded -ded (ame a =3.84) Joh Hok Uverty Deartmet of Botattc /8/6 4

15 Botattc 4.6 Fal Eamato- Awer Key Model A-D correod to queto 6. Varable of teret are defed a: chol = erum choleterol at follow-u (mg %), cholce = baele erum choleterol cetered at 5 (meda value) = (chol-5) (mg %) agece = baele age cetered at 47 year (meda value) = (age - 47) bce= baele ytolc blood reure cetered at (meda value) = (b- ) age_chol= agece*cholce= (age - 47) * (chol-5) hd = f dago of chemc heart deae durg follow-u; f o dago. MODEL A. ge cholce= chol-5. regre chol cholce Source SS df MS Number of ob = F(, 98) = Model Prob > F =. Redual R-quared = Ad R-quared =.96 Total Root MSE = chol Coef. Std. Err. t P> t [95% Cof. Iterval] cholce _co MODEL B. ge agece= age -47. regre chol agece Source SS df MS Number of ob = F(, 98) = 4.5 Model Prob > F =.348 Redual R-quared = Ad R-quared =.74 Total Root MSE = 46.4 chol Coef. Std. Err. t P> t [95% Cof. Iterval] 5 Joh Hok Uverty Deartmet of Botattc /8/6 5

16 Botattc 4.6 Fal Eamato- Awer Key agece _co MODEL C. regre chol cholce agece Source SS df MS Number of ob = F(, 97) = 3.7 Model Prob > F =. Redual R-quared = Ad R-quared =.39 Total Root MSE = 4.57 chol Coef. Std. Err. t P> t [95% Cof. Iterval] cholce agece _co MODEL D. ge age_chol= agece*cholce. ge bce= b-.. regre chol cholce agece age_chol bce Source SS df MS Number of ob = F(4, 95) =.5 Model Prob > F =. Redual R-quared = Ad R-quared =.779 Total Root MSE = 39.5 chol Coef. Std. Err. t P> t [95% Cof. Iterval] cholce agece age_chol bce _co Joh Hok Uverty Deartmet of Botattc /8/6 6