MAS361. MAS361 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS. Medical Statistics
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1 t r r t r r t r t s s MAS361 SCHOOL OF MATHEMATICS AND STATISTICS Medical Statistics Autumn Semester hours t s 2 r t t 1 t t r t t r s t rs t2 r t s q st s r t r t r 2 t st s rs q st s rr2 q 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 MAS361 1 Turn Over
2 Blank MAS361 2 Continued
3 t r s s r s tr s s rst s r r t ttr t t s r r r t r t r r t r s t t t tr t r s 2s s t rt t r tr r r t P ss ss t r P ss r t ss ss t r t t r s t s2 t r r2 t s q t t t 2s s 2 s r r r t 2 t s 2 r r s t t t r t tr r t r t t s r 2 t r t rs s s s 3 t s 2 r r ts r t2 t2 rr r r t s ss r t P s 2 t t t t 2 t r s t tr s 3 t t t r s r 3 r t tr r s s t t st t st r s s t t t s t t st 2 t t r t t s t r r r r t r r t r X X X X X X X X X X X X r X t s t t t q t t t r s s s s r s r ts t s t t r s s r ts r t t r t r t t r MAS361 3 Question 2 continued on next page
4 t s r2 st t st s r t r r s s s r t r r t r s r r s r n s r t r r t r t r r t r r r r t r t r st 2 t t st t r r t t r s t s t s t s r2 st t st s r s t t t str t ss t s r 2 r s st rt ss t ss t s r s r r t s r t s r2 st t st s t r r t t st 2 s st r s 1 2 s st 2 t r r t t s r tr t s s 3 r r s t t s t r r X X X r X X X t ts r r q r t t r 2 t t t r s 1 t ts t r s r t s t ts t r s r t t t t s 1 s st t s s r r t t ts 1 t s t t 2s s 2 s r t tr 1 t s t t r 3 t 2 s r MAS361 4 Question 2 continued on next page
5 t t s 1 s st t s t t ts r s s t r ts t t t t ts t t t r s 1 2 r r r r 2 r s P t t r 1 st t s 1 r 2 r 3 r 4 s r 5 r 6 s r 7 r 8 s r 9 s r 10 s r 11 r 12 s r 13 s r 14 s r 15 r 16 r t ts t t 2 t2 r r s r s s r t t t r s tr t t t ts r r t ts r r r2 t t r st s rt t2 t s s t t t r t st t s r r r s t r t t t s s r t st t s r t r t 2 r st t r st t s MAS361 5 Question 3 continued on next page
6 r r t s t t s t s t t s t t 2s s s s t t t t t st t s t2 r s r t st t s t2 r t MAS361 6 Question 3 continued on next page
7 t s r2 s r t s 7 t2 s r t r s 7 t2 t2 t r s t s r st rr r r t2 t r s t s r st rr r r t s r t s 7 t2 t2 1 t s Drug A Drug B Time (mths) t t 2 ss t s st t t rt t2 t s r t t r s r s MAS361 7 Question 3 continued on next page
8 t t s s st t t t s r t s r 1 t 2 str t t r t s λ A λ B r s t 2 r t s ss t s st t λ A λ B t rt t2 t s t r 1 t t r s r s P r r r t t st t ss ss t r t r s r t rt t2 t str t s t t r s r s s t rt 2 t t t t s t t t s ss s r s 7 s t r t2 s r r s 7 s t r t2 s r 1 t s t r t2 s t r t2 t r r r tr t st ss ss t r t r s r rt t2 t t t r s r s t ss t s 1 t s r str t s s 1 2 r s r s ss t t 2 r 2s s t ts r t r r t st 2 t r t s s t t r r t r t 2 s s ts t r t tr t t t ts t r r t st 2 s t 2 r tt r r t tr t t tr s t t r t tr t ts r r st r t r r t t r t ts r r 2 t t t r tr t t r tr t t r 2 rs r r r rr t r st r r st r t r t r s s s r t s t t s tr t t tr t t tr t t t r t t s s s s t t tr 2 rs 2 r s t t r r rr 2 rs t r r s r s r MAS361 8 Question 4 continued on next page
9 t 2s s s r r t t t t t s r st s r t t s r t 1 r st s 7 s r t s r2 r t 1 r r st s 7 s r t r ts 1 s 3 Pr 535 s r t 2 t r t s r t 2s s t r s t s 3 r t h 0 (t) t r t s r s s r t t ts t r t s t t t r r r r s s t t t t t st t t 3 r r t r s 2 r tr t t s 2 r tr t t r s t r t 2s s s s r t s r r r st s 7 s r t st 1 t s r2 r t s r r r r st s 7 s r t st 1 t t rr r 3 t r t s r t 1 t 1 t str t t r t 2 sq r s r r t s t r t s MAS361 9 Question 4 continued on next page
10 t 2 t t2 2s s s r r r r t t tt r T t t t r rr r s t t ts t r t r t s t t t r rr t t s r 2 r r t tr t t r r s st t t 1 t t t r rr r 2 r s s t tr t t r s End of Question Paper MAS361 10
11 STANDARD FORMULAE FOR MEDICAL STATISTICS (INCLUDING TABLES OF CRTICAL VALUES) 1 Clinical Trials Formulae Two Sample t-test Separate variances form r = min(n 1,n 2 ) X 1 t r = X 2 S 2 1 n 1 + S2 2 n 2 Two Sample t-test Pooled variance form r = n 1 +n 2 2 X 1 t r = X 2 (n1 1)S1 2+(n2 1)S2 2 n 1+n 2 2 ( 1 n n 2 ) Sample Size Calculations Two sample test for proportions NB number in each group n θ 2(1 θ 2 )+θ 1 (1 θ 1 ) (θ 2 θ 1 ) 2 [Φ 1 (β)+φ 1 (α/2)] 2 Sample Size Calculations Two sample test for means NB number in each group 2σ 2 n (µ 2 µ 1 ) 2[Φ 1 (β)+φ 1 (α/2)] 2 Standard Error for Natural Logarithm of Relative Risk 1 s.e[(log e (RR)] = a 1 a+b + 1 c 1 c+d Standard Error for Natural Logarithm of Odds Ratio 1 s.e[(log e (OR)] = a + 1 b + 1 c + 1 d
12 2 Survival Analysis Formulae Exponential Distributions MLE of rate λ with censoring The mle ˆλ = n i=1 δ i n i=1 t i = T var(ˆλ) ˆλ 2 n i=1 δ. i For any (differentiable, monotonic) function g( ), var(g(ˆλ)) [ {g (λ)} 2 var(λ) ] λ=ˆλ. so e.g. var( 1ˆλ) = var(ˆµ) ˆµ 2 n i=1 δ i Exponential Distributions MLE test W = ˆλ 1 ˆλ 2 N(0,1). ˆλ ˆλ Exponential Distributions LRT test { 2 1 log log 2 ( )log } χ 2 1 T 1 T 2 T 1 +T 2 Log-rank Statistic LR = (O 1 E 1 ) 2 E 1 + (O 2 E 2 ) 2 E 2 χ 2 1
13 3 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(0,1) distribution, the tabulated function is also known as the Φ 1 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 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%
14 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%
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