Populations and Samples Bios 662

Transcription

1 Populations and Samples Bios 662 Michael G. Hudgens, Ph.D. mhudgens :29 BIOS Populations and Samples

2 Random Variables Random sample: result of independently selecting elements at random from a population Def 4.8 A random variable is a variable associated with a random sample PV Rao (p 786, 1998) A rv is a variable whose value is determined by the observed characteristics of an item randomly selected from a popluation BIOS Populations and Samples

3 Probability Functions Def 4.9 The probability mass function is a function that for each possible value of a discrete rv takes on the probability of that value occurring Def 4.10 The probability density function is a curve that specifices, by means of the area under the curve over an interval, the probability that a contiunous rv falls within the interval BIOS Populations and Samples

4 Probability Functions Probability mass function Probability density function Number of Boys in Families with Eight Children x BIOS Populations and Samples

5 Cumulative Distribution Function Def 4.9 The cumulative distribution function for a rv X is F (x) = Pr[X x] If X is discrete, where p X is the pmf of X F (x) = y x p X (y) If X is continuous, F (x) = where f is the pdf of X x f(y)dy BIOS Populations and Samples

6 Population quantile Intuitive defintion: The p th quantile of X, say ζ p, should be such that F (ζ p ) = Pr[X ζ p ] = p Formally: ζ p = inf{x : F (x) p} If F is continuous F (ζ p ) = p BIOS Populations and Samples

7 Quantiles: Example F(x) F(x) x x BIOS Populations and Samples

8 Mean and Variance Mean or expected value of X If X is discrete, µ = E(X) = y yp X (y) If X is continuous, Variance µ = E(X) = yf(y)dy σ 2 = V ar(x) = E{(X µ) 2 } BIOS Populations and Samples

9 Skewness and Kurtosis Skewness α 3 = E{(X µ)3 } σ 3 Kurtosis α 4 = E{(X µ)4 } σ 4 BIOS Populations and Samples

10 Parameters and Statistics Definition: A parameter is a numerical characteristic of a population Definition: A statistic is a numerical characteristic of a sample Notation: Greek letters typically denote parameters; English letters denote statistics Example: µ = population mean; σ 2 population variance Ȳ sample mean; s 2 sample variance BIOS Populations and Samples

11 Parameters and Statistics Parameters are fixed constants Statistics are random variables Statistics have probability distributions We will use statistics and probability theory to draw conclusions (inference) about parameters BIOS Populations and Samples

12 Sampling Distributions Defintion 4.15 The probability function of a statistic is called the sampling distribution of the statistic. Eg, when sampling from a population, the sample mean Ȳ is a rv becuase its value depends on chance, namely, on which sample is obtained. The probability distribution of the random variable Ȳ is called the sampling distribution of the mean. BIOS Populations and Samples

13 Sampling Distributions Result 4.1 If a rv Y has a population mean µ and a population variance σ 2, the sampling distribution of the mean (Ȳ ) has mean µ and variance σ2 /n Definition 4.16 The standard deviation of the sampling distribution is called the standard error Eg the standard error of Ȳ is σ/ n BIOS Populations and Samples

14 Normal or Gaussian Distribution PDF: { f(x; µ, σ) = 1 σ 2π exp 1 2 ( ) } x µ 2 σ CDF: F (x; µ, σ) = µ mean, σ 2 variance x f(y; µ, σ)dy X N(µ, σ 2 ) [beware X N(µ, σ)] BIOS Populations and Samples

15 Normal Distribution density N(25, 900) N(25, 225) N(100, 81) N(100, 225) x BIOS Populations and Samples

16 Z N(0, 1) Standard Normal Distribution PDF: φ(z) = 1 2π exp{ 1 2 z2 } CDF: Φ(z) = z φ(y)dy N(0, 1) is a standard normal distribution BIOS Populations and Samples

17 Standard Normal Distribution density Z BIOS Populations and Samples

18 Properties of Standard Normal Distribution A rv w/ pdf f is symmetric about µ if f(µ + x) = f(µ x) for all x Z N(0, 1) is symmetric about 0 φ(z) = φ( z) for all < z < Thus i.e. Pr[Z z] = Pr[Z z] Φ( z) = 1 Φ(z) BIOS Populations and Samples

19 Standard Normal Distribution density Pr[Z <.1] = Pr[.1 < Z] = Z BIOS Populations and Samples

20 Standard Normal Distribution R > pnorm(-.1,0,1) [1] > 1-pnorm(.1,0,1) [1] > qnorm( ,0,1) [1] BIOS Populations and Samples

21 Standard Normal Distribution SAS data; x=probnorm(-.1); y=cdf( NORMAL,-.1,0,1); z=quantile( NORMAL, ); run; proc print; run; Obs x y z BIOS Populations and Samples

22 Properties of a Random Variable Let X be a random variable Suppose Y = ax + b where a and b are constants Then E(Y ) = ae(x) + b V ar(y ) = a 2 V ar(x) If X N(µ, σ 2 ) and Y = ax + b, then Y N(aµ + b, (aσ) 2 ) BIOS Populations and Samples

23 Conversion to Standard Normal Suppose Y N(µ, σ 2 ) Let Then Z = Y µ σ Z N(0, 1) In words: a normal random variable can be standardized by subtracting its mean and dividing by its standard deviation BIOS Populations and Samples

24 Suppose Y N(µ, σ 2 ) Computation of Probabilities Let Then Z = Y µ σ Pr[a < Y < b] = Pr[ a µ σ < Z < b µ σ ] = Φ( b µ σ ) Φ(a µ σ ) BIOS Populations and Samples

25 Table 1 (p 818 text): Standard normal distribution. Let Z be a normal random variable with mean zero and variance one. For selected values of z, three values are tabled: (1) the two-sided p-value, or Pr[ Z z]; (2) the one-sided p-value, or Pr[Z z]; and (3) the cumulative distribution function at z, or Pr[Z z]. Two One Cumu. z sided sided dist BIOS Populations and Samples

26 Example Intraocular pressure (IP) is used to diagnose glaucoma Assume IP is normally distributed with mean µ = 16 mmhg and variance σ 2 = 9 mmhg If pressure greater than 20 mmhg is considered abnormal, what proportion of the population is abnormal? Pr[X > 20] = Pr[ X 16 3 > ] = Pr[Z > 1.33] = 1 Φ(1.33) = = BIOS Populations and Samples

27 Example (continued) What proportion of the population has IP between 4 and 18? Pr[4 < X < 18] = Pr[ < X 16 3 < ] = Pr[ 4 < Z < 2/3] = Φ(2/3) Φ( 4) = Φ(2/3) 1 + Φ(4) = = BIOS Populations and Samples

28 Assessing Normality How do we assess whether the normal distribution model fits a particular set of data? One graphical approach: quantile-quantile (QQ) plot Plot quantiles of the observed data distribution versus the quantiles of the normal distribution Straight line indicates normality assumption reasonable BIOS Populations and Samples

29 QQ Plot Example Table 4.3 from text (p. 81) Cumul Endpoint Freq Pct BIOS Populations and Samples

30 QQ Plot Example: Table 4.3 from text Normal Q Q Plot Sample Quantiles Theoretical Quantiles > qqnorm(galton,type="l") BIOS Populations and Samples

31 QQ Plot Normal Q Q Plot Theoretical Quantiles Sample Quantiles Normal Q Q Plot Theoretical Quantiles Sample Quantiles > par(mfcol=c(1,2)); qqnorm(rnorm(1000,0,1)); qqnorm(rexp(1000,3)) BIOS Populations and Samples

32 QQ Plot x x rexp(1000, 3) rnorm(1000, 0, 1) > x <- rexp(1000,3) > par(mfcol=c(1,2)); qqplot(rexp(1000,3),x); qqplot(rnorm(1000,0,1),x) BIOS Populations and Samples

33 QQ Plot Sample quantiles Sample quantiles Theoretical quantiles of Exp(1/3) Theoretical quantiles of N(0,1) > x <- rexp(1000,3); probs <- seq(.01,.99,length=100) > qx <- quantile(x,probs); tqexp <- qexp(probs,3); tqnorm <- qnorm(probs,0,1) > par(mfcol=c(1,2)); plot(tqexp,qx,xlab="theoretical quantiles of Exp(1/3)",ylab="Sample quantiles") > plot(tqnorm,qx,xlab="theoretical quantiles of N(0,1)",ylab="Sample quantiles") BIOS Populations and Samples

34 Some approximations The interval x ± s will contain approx 68% of the observations The interval x ± 2s will contain approx 95% of the observations Assuming Y N(µ, σ 2 ) Pr[µ σ < Y < µ + σ] = Pr[ 1 < Z < 1] = Pr[µ 2σ < Y < µ + 2σ] = Pr[ 2 < Z < 2] = BIOS Populations and Samples

35 Some approximations Do these approximations hold for non-normal data? Not in general. Consider X Exp(λ) such that EX = λ and V (X) = λ 2. For λ = 1/3, Pr[0 X 2/3] = 0.86 BIOS Populations and Samples

36 Some approximations Consider X = W Y +(1 W )Z where W Bern(1/2), Y N(10, 1), and Z N(0, 1). Can show Pr[µ X σ X X µ X + σ X ] 0.54 Histogram of x Density x BIOS Populations and Samples

37 Some approximations The following holds for any rv Y with mean µ and variance σ 2 Pr [µ Kσ Y µ + Kσ] 1 1 K 2 K 1. This is Chebyshev s inequality (note typo in text page 100) Eg, if K = 2, Pr [µ 2σ Y µ + 2σ] 0.75 i.e. we would expect at least 75% of observations to be within two standard deviations of the mean for any underlying distribution BIOS Populations and Samples

38 Central Limit Theorem (CLT) Let Y 1, Y 2,..., Y n be independent and identically distributed (iid) random variables with Define E(Y i ) = µ and, V ar(y i ) = σ 2 > 0 Z n = Ȳ µ σ/ n Then the distribution function of Z n converges to a standard normal distribution function as n. BIOS Populations and Samples

39 Central Limit Theorem (CLT) In words - see Result 4.3 page 84 of van Belle et al If a random variable Y has population mean µ and population variance σ 2, then the sample mean Ȳ, based on n observations, is approx normally distributed with mean µ and variance σ 2 /n for sufficiently large n BIOS Populations and Samples

40 Notes on CLT The CLT applies to any distribution of the Y s The approximation improves as n gets large Check out Rice Virtual Lab in Statistics BIOS Populations and Samples

41 Result 4.2 If Y is normally distributed with mean µ and variance σ 2, then Ȳ, based on a random sample of n observations, is normally distributed with mean µ and variance σ 2 /n. This is true regardless of sample size. BIOS Populations and Samples