MAS6012. MAS Turn Over SCHOOL OF MATHEMATICS AND STATISTICS. Sampling, Design, Medical Statistics

Transcription

1 t r r r t s t SCHOOL OF MATHEMATICS AND STATISTICS Sampling, Design, Medical Statistics Spring Semester hours t s 2 r t t t t r t t r s t rs t2 r t s s rs r t r t 2 r t st s rs q st s r rt r s t r s Please leave this exam paper on your desk Do not remove it from the hall str t r r r ts t t 2 st t Turn Over

2 Blank 2 Continued

3 t r ss r r tr t t 2 r t s t s t r t r t r t 2st r ss r s t ts r t r s tr t t t t r r P r t r P s t r s t r t s t 2 t t s t 2 t rs t r st 2 r r r t t sts t t r 2 s t r ss r t t r r t t t t P r t 2 t t t t r r t r r r s ts r r tr t 2 rt s t s t s r t t r 2s s r s st tt r 2s s 2 r r s P r r t t st 2 s st t s s r t t ts t r s t ts r s t tr t r s 3 s r t r P s r t r t t t s t s 2 t t t ss t tr r s 2 s 2 s r t t t ts t s t r s 3 Turn Over

4 t rts t s q st r st 2 rs r t s r ts rr tr s ts r tr t t t r t r t r t t st 2 r tr s ts r ts t t rs r t t P ss t r2 t rs r s r r r t2 r r t r s 2 rs t r t r r r t s t s r t t t t t tr s t s t t t ss ss t r t2 s ss t t rr rs r t 2 t r t st r r ss 2s s t t s t s t st s t s t ss ss t r t r t r t t t r t r s s s rs r t rr rs r t rr t2 s r 2 s 2 r s r r s 2 ts r s r r t 3 Pr 2 2 st r r ss 2s s t r r t 2s s t t2 rt r s s t t t t t t r t2 rs r t r t t t r t r r r t 4 Question 2 continued on next page

5 t r r s t st t r ts t tt t s r tr r ts r t st t str t s s r r r t r t ts t s s t t s t r s st r s t s s r s r t s t ss t t t s r s r r s s r t t r t r st t t t s r r t t s s r st t t r t r q r t r t t r st t s 2 t s t t s s st t t t s r t s r t 2 str t r t s ss t st t t t s r r t t s r s t r t str t t r t λ t S(t) = e λt 2 t t str t ss t rt s r r t r s r t s st 2 s t t r t r t r r t r t r 2 s ss t r r t ss s t r t s t ss ss t s ss t t r t q r s 5 Turn Over

6 r st s tt r rt tr 2 s s t ts s rt 2s s q t t s 2s s t t t t t r t 2 rr t 2 s t r t t r t s s t t r s s rt r st s tt s t t t r rr t t t r t t s rt t s t t r t t rs r r r r s s t r t t s t s r t s r t s r Pr r2 t r st s t r t s s t2 t r rr t t r st r 2s s r t r t r s s s t t t r rr s st t s s r t r r rr s r t t 2rs s s s t2 r ss t2 s P t r r 2 s r t r r t ts t t t t t t r s t s 2 tt 2s s 2s s s r r t st t s 2s s t r s r t 2s s s r 7 s s t 2s s t r t2 2 st t r t2 s r t r t s s s t2 Estimated probability of survival Other GN AN PKD Time 6 Question 3 continued on next page

7 t r t 2s s s r r 2s s t s r r 2s s s r 7 s s s st t s r2 2s s t s r r r 2s s s r 7 s s s st t t rr r 3 t r t s s s s s s s P t t str t t r t 2 sq r s r r t s t r t s 2s s t s Pr s s s s r t s 2s s s r 2 t2 tt t st t st r r s t t t t t r s r s t t ts t r t r s r r rr s s t2 s s t r r s s t t t rt r t t t r rr t r 2r t s s t2 t r r t t t t t t s r s t t r rt 3 r s t r 2 r t t r rt 3 r s r s 7 Turn Over

8 tr s s t r t t ss t r r s r,2 3 r r rt ts t t t tr r rt ts r r t r r r t r r s r s EY ij = µ+α i, r Y ij s t r s s t jt rt t r i r i,j 3 r t t s tr X r t s s t r t rs µ,α,α 2,α 3 t s r t t s r t r s t r s 2 2 t str t α +α 2 +α 3 = 0 r t t s tr t r t rs µ,α,α 2 r s s st t t t r r m rt ts t t t r t r t r r m 2t r r r t EY ij = µ + α i r i,j =,2,3 t str t α + α 2 + α 3 = 0 X T X t r s m t r X s t s tr t t X T X = 9t 2 (m 2t) t t r t t t s D t s r m = 000 r s s r r t t r s t t rs x,x 2,x 3,x 4 ) rs t t s t + s t t r s ts r Pr t s r t rs t t t t r t t ts r x x 2 x 4 t t r t t t s str t r r t s t r t rs r s str t t r t t r s s 2 r s rt r s 8 Continued

9 st t r s st 2 t r Y t s t r2 r x s s t t s r t s t 2 s t t s r t rr r t r σ 2 s r s r t it s r t EY i = β 0 +β x 3 i. st t r r s s t s t r s ts {x,x 2,x 3,x 4 } = {,0,,} t t r s s r t s s ts Y,Y 2,Y 3,Y 4 r s t 2 X s t s tr s t t (X T X) = ( 3 4 ) t r s t st sq r s st t rs ˆβ 0 ˆβ t r s σ 2 r t s s r s st t r t s t r r (ˆβ 0, ˆβ ) T s r t r t r t r r s r t t r t s st 2 t r t 2 r rr t r s r t rt s r t s t tr t r r (ˆβ 0, ˆβ ) T t r s Y,Y 2,Y 3,Y 4 t t r t EY i = β 0 + β x 3 i t s {x,x 2,x 3,x 4 } = {, 0,, } s t r D t r G t 2 s t r q r r s s s t x 0 s t r r s ξ t r t tr G t G t r t tr t s ξ t x 0 r f(x 0 ) T t r t s tr r ξ rr s t t x 0 t t G = G ( f(x 0 ) T G f(x 0 ) ) 2 t r s t s A s s r p p tr B s p n tr C s n p tr I s t n n t t2 tr t A BC = A I CA B r s st t r s t r t r t rr t s {x,x 2,x 3,x 4 } = {,0,,} r t EY i = β 0 + β x 3 i s t 2 t D t t t r r t s s r s 9 Turn Over

10 s r 2 t t st t t s r 2 st r t st ts t rs t2 r t t tr t P t s 3 s 3 st st t t s r 2 st r t st ts t rs t2 s t st r s st t r x st r s st t t r r t t s x st t t 2 ss t s 2 r s t r s r 2 s t t t t s str t t s sts r s t t t t t s s 3 s s 2 t s s 3 s str t s t t s 2 t r s s t r x st s t t t t st t q str t s sts st t r str t s t s s t s s st t r str t s t s r 2 r s t str t s s 3 s 2 s t r x st s t t t t st t t t st t r str t s k t s s s t s s st t r str t t s r 2 r t q s s 3 s t str t t k t 2 ss r s s st t t s r s s t s t st 2 s s r s s r 2 st ts t t r x i s t it st r t st t s r 0 i= x i = i= x 2 i =,0,250 s t t r t t st 2 t s s 3 s r t t t t r r t s r t r t End of Question Paper 0

11 FORMULA SHEET & CRTICAL VALUES Clinical Trials Formulae Two Sample t-test Separate variances form r = min(n,n 2 ) X t r = X 2 S 2 n + S2 2 n 2 Two Sample t-test Pooled variance form r = n +n 2 2 X t r = X 2 (n )S 2+(n2 )S2 2 n +n 2 2 ( n + n 2 ) Sample Size Calculations Two sample test for proportions NB number in each group n θ 2( θ 2 )+θ ( θ ) (θ 2 θ ) 2 [Φ (β)+φ (α/2)] 2 Sample Size Calculations Two sample test for means NB number in each group n 2σ 2 (µ 2 µ ) 2[Φ (β)+φ (α/2)] 2 Standard Error for Natural Logarithm of Relative Risk s.e[(log e (RR)] = a a+b + c c+d Standard Error for Natural Logarithm of Odds Ratio s.e[(log e (OR)] = a + b + c + d 2 Survival Analysis Formulae Exponential Distributions MLE of rate λ with censoring The mle ˆλ = n i= δ i n i= t i = T For any (differentiable, monotonic) function g( ), var(ˆλ) ˆλ 2 n i= δ. i so e.g. var(g(ˆλ)) [ {g (λ)} 2 var(λ) ] λ=ˆλ var( ˆλ) = var(ˆµ) ˆµ 2 n i= δ i Exponential Distributions MLE test W = ˆλ ˆλ 2 N(0,) ˆλ2 + ˆλ Exponential Distributions LRT test { 2 log + 2 log 2 ( + 2 )log } + 2 χ 2 T T 2 T +T 2 Log-rank Statistic LR = (O E ) 2 E + (O 2 E 2 ) 2 E 2 χ 2

12 3 Design Formulae Linear Model formulae ˆβ = (X T X) X T Y and ˆβ N{β,σ 2 (X T X) } Prediction Variance var ŷ(x 0 ) = σ 2 f(x 0 ) T (X T X) f(x 0 ) Standardized Prediction Variance Confidence Regions, σ 2 unknown d(x) = nf(x) T (X T X) f(x) = f(x) T M f(x) p ˆσ 2 (ˆβ β) T X T X(ˆβ β) has an F p,n p distribution, provided n > p Balanced Incomplete Block Design Notation t = number of treatments k = number of units in a block b = number of blocks r = number of applications of each treatment λ = number of times each pair of treatments appears together in a block Balanced Incomplete Block Design Relationships t > k bk = rt r(k ) = λ(t ) Balanced Incomplete Block Design - Unreduced Design ( ) ( ) ( ) t t t 2 b = r = λ = k k k 2 Efficiency of Balanced Incomplete Block Design compared to a Randomized Block design t k Adding an extra point x Deleting an existing point x G = G ( +f(x) T G f(x) ) G = G ( f(x) T G f(x) ) Adding a new point y and deleting an existing point x { ( f(x) G 2 = G T G f(x) )( +f(y) T G f(y) ) + ( f(x) T G f(y) ) } 2 4 Sample Surveys and Computer Experiments Formulae Population variance S 2 = N N (X i X) 2 = N and for a binary characteristic (X i = or 0 for each i), S 2 = NP( P) N ( N ) Xi 2 N X 2 i=

13 Variance of sample mean for simple random sampling ( var x = n ) S 2 N n. Sample size to achieve given confidence interval width for simple random sampling n N +N(d/(2Sz α/2 )) 2 Stratified estimate of population mean and its variance x st = N l N i x i and var x st = l ( ) 2 Ni f i S N n i. 2 i Allocation methods Optimal allocation: n i NiSi ci Neyman allocation: n i = nnisi l NiSi Cluster estimate of population mean and its variance x cl = lk l K x ij and var (x cl ) = f l L L (X i X) 2 Regression estimator of population mean and its variance x lr = x ˆβ(y Y) and var x lr f n S2 X( ρ 2 ) Approximate variance of the Peterson estimator, Chapman estimator and approximate variance n: size of st sample, m: size of 2nd sample. Var( ˆN p ) = mn2 (m r) r 3, ˆN c = (n+)(m+), r + Var( ˆN c ) = (n+)(m+)(n r)(m r) (r +) 2. (r +2) Variance identity Var(Y) = Var X {E(Y X)}+E X {Var(Y X)}. 5 Tables of Percentage Points (also known as Quantiles or Critical Values) for Three Standard Distributions The tables contain values of quantiles q such that P[X q] = p for various probabilities p when X has the specified distribution (which may depend on particular degrees of freedom ν). In these tables, p has been expressed as a percentage rather than a decimal. The relevant R commands for generating the q are also shown. For the N(0,) distribution, the tabulated function is also known as the Φ function. STANDARD NORMAL DISTRIBUTION PERCENTAGE POINTS qnorm(p) where p is percentage, e.g. for 95%, p = % 66.7% 75.0% 80.0% 87.5% 90.0% 95.0% 97.5% 99.0% 99.5% 99.9% qnorm

14 CHI-SQUARED PERCENTAGE POINTS qchisq(p, ν) where p is percentage, e.g. for 95%, p = 0.95 ν 60.0% 66.7% 75.0% 80.0% 87.5% 90.0% 95.0% 97.5% 99.0% 99.5% 99.9% STUDENT S t PERCENTAGE POINTS qt(p, ν) where p is percentage, e.g. for 95%, p = 0.95 ν 60.0% 66.7% 75.0% 80.0% 87.5% 90.0% 95.0% 97.5% 99.0% 99.5% 99.9%