Epidemiology Principle of Biostatistics Chapter 11 - Inference about probability in a single population. John Koval

Transcription

1 Epidemiology 9509 Principle of Biostatistics Chapter 11 - Inference about probability in a single population John Koval Department of Epidemiology and Biostatistics University of Western Ontario

2 What is being covered inference about single population with probability/proportion using single sample with sample proportion 1. hypothesis test 2. confidence interval 3. power and sample size 4. using SAS

3 Single sample π can be thought of as 1. proportion of items in a population eg, proportion of smokers in Canada 2. probability of selecting a item in a population eg, probability of selecting a smoker at random We use the latter for our probability models Probability of Heads in coin toss of success with a new drug of selecting a smoker mathematically recorded as a number X = 1 as opposed to X = 0 sample of size n sample mean x = xi n also called sample proportion p ( or ˆπ)

4 probability of event x(= x i ) successes in n trials is binomial pdf is ( ) n π x (1 π) n x x eg, π = 0.2, n = 10 ( ) x x x can get these from SAS function PDF (see previous notes) or tables at the back of some textbooks

5 B(10,0.2) Probability asymmetric distribution

6 inference - hypothesis test For testing hypothesis about probability in a single population H o : π = π o against H A : π > π o with α = 0.05

7 example For 36 students, 9 are smokers n = 36,p = 0.25 at α = 0.05, test H o : π = 0.20 against H A : π > 0.20

8 exact computation for binomial need SAS program p-value = Pr(X B 9 π = 0.20) = 1 Pr(X B 8 π = 0.20) 8 ( ) 36 = x x x x=0

9 exact computation (more) title "cumulative binomial for n=36 x=8 pi=0.2"; options ps=24 ls=64; data cumbinom368; n=36; x=9; xm1=x-1; pi=0.2; prob = cdf( binomial, xm1, pi, n); pvalue = 1 - prob; output cumbinom368; proc print data=cumbinom368; quit;

10 exact computation (continued) cumulative binomial for n=36 x=8 pi=0.2 Obs n x xm1 pi prob pvalue try normal approximation

11 Normal approximation for proportion Note: p is an average (of a binomial) for large n, CLT holds Mean: E( X) = E(X B) n = π Var(P) = Var ( ) X n = 1 n 2Var(X) = 1 n 2nπ(1 π) = π(1 π) n ) (= σ2 n

12 example (continued) for hypotheses test use π o need nπ o 5 n(1 π o ) 5 check nπ o = 36(0.2) = 7.2, n(1 π o ) = 36(0.8) = 28.8

13 example (more) This is smaller than exact p-value = Pr(p > 0.25 π = 0.20) = Pr Z N > (0.8) 36 ( ) 0.05 = Pr Z N > ( = Pr Z N > 0.05 ) = Pr(Z N > 0.75) = z 0.75 = too liberal (optimistic) - leads to more Type I error rejects H o more often than 0.05

14 better approximation continuity correction p-value = Pr(p > 0.25 π = 0.20) ( = Pr Z N > /72 ) ( = Pr Z N > ) = Pr(Z N > ) = Pr(Z N > 0.542) = z = (by linear interpolation) This is conservative (larger p-value than exact) we reject less often than 0.05 make fewer type I errors

15 SAS program title "pvalue 1-sided test binom "; title2 "exact and two approximations"; data binom; input n x pi; p = x/n; num = p - pi;numcc = p - pi - 1/(2*n); den = (pi*(1-pi))/n; sqden = sqrt(den); zobs = num/sqden; zobscc=numcc/sqden; po = probnorm(-zobs); pocc = probnorm(-zobscc); xb= x-1; prob = cdf( binomial,xb, pi, n); pvalue = 1 - prob; datalines; ;

16 print results proc print data=binom; var n x pi; proc print data=binom; po pocc pvalue; quit;

17 output of pvalue program pvalue 1-sided test binom exact and two approximations Obs n x pi Obs po pocc pvalue

18 smaller sample sample of size 10, with 3 smokers same null and alternative hypotheses Exact computation prob = cdf("binomial", 2, 0.2, 10); pvalue = 1 - prob; calculate binomial probability Obs prob pvalue

19 small sample (normal approximation) continuity correction p value = Pr(p > 3/10 π = 0.20) = Pr(Z N > ( ) 1/2n 0.2(0.8) 10 ( = Pr Z N > /20 ) ( = Pr Z N > ) ( = Pr Z N > 0.05 ) = Pr(Z N > 0.395) = z.395 = by linear interpolation This is again conservative not bad approximation for such a small sample.

20 Output of SAS program for small sample pvalue 1-sided test binom exact and two approximations Obs n x pi Obs po pocc pvalue