Epidemiology Principle of Biostatistics Chapter 5 Probability Distributions (continued) John Koval

Transcription

1 Epidemiology 9509 Principle of Biostatistics Chapter 5 Probability Distributions (continued) John Koval Department of Epidemiology and Biostatistics University of Western Ontario

2 What was covered previously 1. probability P(A) sets P(A and B); P(A or B) 2. probability distributions 2.1 discrete equiprobable bernoulli binomial poisson 2.2 continuous uniform normal 3. calculating probabilities 3.1 discrete Pr(X = x) 3.2 continuous intervals: Pr(X < a), Pr(a < X < b)

3 What is being covered now Using SAS to 1. calculate probabilities 2. calculate and plot probability distributions

4 Calculating probabilities SAS function PDF title calculate binomial probability ; data binom1; prob = pdf( binomial, 4, 0.4, 10); output ; proc print data=binom1;

5 binomial probability calculate binomial probability Obs prob Does this agree with previous calculations?

6 binomial probability calculate binomial probability Obs prob Does this agree with previous calculations? 0.251, Lecture Chapter 5, page 8

7 Calculating probability distribution title "calculate binomial probability distribution ; data binom2; do x = 0 to 10 by 1; prob = pdf( binomial, x, 0.4, 10); output; end; proc print data=binom2; proc gplot; plot prob*x; run;

8 binomial probability distribution calculate binomial probability distribution Obs x prob

9 GPLOT of pdf

10 Calculating cumulative probabilities values up to and including SAS function CDF title calculate cumulative binomial probability ; data binom3; prob = cdf( binomial, 7, 0.4, 20); output ; proc print data=binom3; run;

11 binomial cumulative distribution calculation calculate cumulative binomial probability Obs prob Does this agree with previous calculations?

12 binomial cumulative distribution calculation calculate cumulative binomial probability Obs prob Does this agree with previous calculations? , using R, Lecture Chapter 5, page 30

13 cumulative continuous probabilities Pr(X ( < b) ) = Pr Z N < b µ σ ) = Φ ( b µ σ Φ() given by SAS function PROBNORM

14 example Recall normal approximation to binomial want Pr(X norm < 7.5) = Pr(Z N < ( ) = Φ(.228) title calculate Normal probability ; data norm1; prob =probnorm(-0.228); output; proc print data=norm1; run; ;

15 binomial cumulative distribution calculation calculate Normal probability Obs prob Does this agree with previous calculations?

16 binomial cumulative distribution calculation calculate Normal probability Obs prob Does this agree with previous calculations? by linear interpolation, see lecture Chapter 5, page 30

17 Probability of interval Pr(17 < X < 22) = Pr ( < Z N < ) 5 = Pr( 0.6 < Z N < 0.4) = Φ(0.4) Φ( 0.6) title calculate Normal probability for interval ; data norm2; a=-0.6; b=0.4; proba =probnorm(a); probb = probnorm(b); probint = probb - proba; output; proc print data=norm2; run;

18 binomial cumulative distribution calculation calculate Normal probability for interval Obs a b proba probb probint Does this agree with previous calculations?

19 binomial cumulative distribution calculation calculate Normal probability for interval Obs a b proba probb probint Does this agree with previous calculations? , see lecture Chapter 5, page 26

20 Plotting normal density function not usually done in practice data norm3; do x = 0 to 10 by 0.05; density = pdf( normal, x, 4, 1.55); output ; end; proc gplot data = norm3; plot density*x; symbol interpol=join;

21 GPLOT of pdf of Normal N(4,2.4)

22 normal approximation to binomial title Normal approximation to binomial ; data normbinom; n=20; pi=0.4; mu = n*pi; var = n*pi*(1-pi); sd = sqrt(var); do i = 0 to by 0.025; binompdf = pdf( binomial, floor(i), pi, n); x = i-0.5; normpdf = pdf( normal, x, mu, sd); output normbinom; end;

23 normal approximation to binomial(continued) proc gplot data=normbinom; plot binompdf * x normpdf * x/ haxis=-1 to 21 by 1 vaxis=0 to 0.2 by 0.05 overlay; symbol interpol=join;

24 GPLOT of normal approximation to Bin(20,0.4)

25 another normal approximation to binomial non-symmetric distribution Bin(10,.2) data normbinom2; n=10; pi=0.2; mu = n*pi; var = n*pi*(1-pi); sd = sqrt(var); do i = 0 to by 0.025; binompdf = pdf( binomial, floor(i), pi, n); x = i-0.5; normpdf = pdf( normal, x, mu, sd); output normbinom2; end;

26 non-symmetric distribution (continued) proc gplot data=normbinom2; plot binompdf * x normpdf * x / haxis=-1 to 11 by 1 vaxis=0 to 0.5 by 0.05 overlay; symbol interpol=join;

27 Normal approximation to Bin(10,0.2) original distribution is asymmetric not a good fit to the normal