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Lecture 4

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Lecture 4

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  1. Lecture 4

  2. For assignment 2, recode the smoking variable • replace smoke=smoke-1

  3. Recap from probability distributions • The binomial distribution describes the probability of x successes in n independent trials, each with probability p of success • The normal distribution may be used to describe cutoffs for some continuous random variables with mean µ and standard deviation  • How do you know if your data are normally distributed? • Histograms (stata: histvarname, normal) • QQ plots – next Biostat class • Other statistical tests • What to do if my data are not normal? • Transformations – like taking the log, or the inverse 1/x … • We will discuss an important property of the normal distribution today that applies to non-normally distributed data

  4. Sampling • When we cannot measure the entire population we take a sample • We estimate the population characteristics, i.e. the mean and variance, using the sample • We use statistical inference to draw conclusions about the how the estimates from the sample relate to the population values Pagano and Gavreau, Chapter 8

  5. To do make inference from our sample to the population, our sample must be representative of the population • Random sample – each individual in the population has equal chance of being selected for the sample • The larger the sample, the more reliable our estimates about the population parameters will be • Because we do not have the entire population, there is some level of uncertainty about our data • Confidence intervals afford us a way to quantify this uncertainty

  6. Sampling distributions • Imagine you conducted a sample of size n from a population and measured random variable X, say systolic blood pressure • Calculate the sample mean, X1 from the xi • Take another sample of size n, calculate X2 • If you repeat for a long time you will have a large collection Xs generated from the samples of size n • The Xs and standard deviations will differ from sample to sample due to sampling variability – each sample will most likley be different

  7. The collection of all of the possible Xs that can be obtained can be thought of as random variables that themselves follow a distribution • This distribution is called the sampling distribution • Imagine having a data set just of the means, the Xs, and plotting the histogram to see the shape of their distribution

  8. Why bother? • Sampling distributions of sample means have special properties that allow us to make inference about the mean of a single sample • We need this theory to be able to calculate confidence intervals or calculate p-values

  9. Central limit theorem • If you have a random variable that comes from a distribution with mean=µ and standard deviation σ, the following is true for the sampling distribution of the sample means from samples of size n • The mean of the sampling distribution (the distribution of all of the possible sample means) is µ • The standard deviation of the sampling distribution is σ/√n • If n is large enough, the shape of the sampling distribution is approximately normal Pagano and Gavreau, Chapter 8

  10. Central limit theorem • Central limit theorem (CLT) • If the underlying population is independently distributed with mean=µ and standard deviation=σ, then if we take a random sample of size n and n is large enough, the distribution of the sample mean is normally distributed. • The mean of the sampling distribution will be µ and standard deviation will be σ/√n. Pagano and Gavreau, Chapter 8

  11. Central limit theorem • If we take a sample from any distribution (could be skewed, or discrete, or whatever) of size n, and take the mean, and we do this over and over, the distribution of the means will be normally distributed with mean=the original distribution mean µ and standard deviation= σ/√n, if n is large enough • The more symmetric the distribution of the raw data (not the means), the smaller the n needed to become normal-like Pagano and Gavreau, Chapter 8

  12. Why does this make sense? • It makes sense that the distribution of means would cluster around the population mean • It makes sense that the variability in the means is smaller than in the raw data because the extreme values are already averaged out • The part about the distribution being normal if n is large enough? Mathematical proof that I can’t do… But we can demonstrate it!

  13. Note • σ is the standard deviation of the original distribution • σ/√n is called the standard error, or more precisely, the standard error of the mean, and it is the standard deviation of the distribution of the sample mean. Pagano and Gavreau, Chapter 8

  14. Central limit theorem example • http://www.surveymonkey.com/s/F5VVHQZ • Download to excel • Change variable names • Import into stata • Histogram using dayofbirth1 • Create average using self and first relative day of birth (n=2) , dayofbirth_avg2 • Make a histogram of dayofbirth_avg2 • Repeat for using first 4, then 8 days of birth

  15. Random draws from the chi-square distribution (µ=2)

  16. Means of 5 random draws from the chi-square distribution (µ=2)

  17. Means of 10 random draws from the chi-square distribution (µ=2)

  18. Means of 20 random draws from the chi-square distribution (µ=2)

  19. Means of 30 random draws from the chi-square distribution (µ=2)

  20. Distributions of the means of chi-square random draws

  21. Distributions of the means of binomial random draws

  22. Using the CLT • Suppose you sampled from a HIV-infected population with mean µ CD4 count = 250 cells/mm3 and standard deviation σ = 200 cells/mm3. • If we select repeated samples of size 50, what proportion of the samples will have a mean value of less than 100 cells/mm3 ? • Using the CLT, we know that the mean of all the samples, X, will follow a normal distribution with mean µ=250 and standard error σ/ √n=200/ √50 • Then we know that (X-250)/(200/ √50) ~ N(0,1) • If the mean cutoff value is 100, we want P(Z<100) then z=(100-250)/(200/ √50) = -150 / (200/ 7.07) = -150/28.3 = -5.3 P(Z<-5.3) = ?

  23. Using the CLT • What level of CD4 count is the lower 10th percentile of the mean values? • P(Z<=z)=.10 for what value of z? • Table A.3 give the value P(Z>=z) • P(Z<=z) = - P(Z>=z) for the same value of z • The value of z for which P(Z>=z) = 0.10 is ____ • The lower 10th percentile cutoff for z is ____ • Now we need to transform back to get X • Using

  24. Using the CLT • What level of CD4 count is the lower 2.5th percentile of the mean values? • P(Z<=z)=.025 for what value of z? • Remember the tails of the standard normal distribution or look up in Table A.3 • The value of z for which P(Z<=z) = 0.025 is ____ • The value of z for which P(Z>z) = 0.025 is ____ • Now we need to transform back to get X • Using

  25. Using the CLT • What level of CD4 count is the upper 2.5th percentile of the mean values? • P(Z<=z)=.025 for what value of z? • The value of z for which P(Z>=z) = 0.025 is ____ • The lower 2.5th percentile cutoff for z is ____ • Now we need to transform back to get X • Using

  26. Now we have the lower and upper 2.5% percentiles of the distribution of the sample means. • The interior area contains 95% of the sample means. • 95% of the means from sample size 50 lie within the 95% confidence bounds (194.6, 305.4) • If we selected a sample of size 50 and the sample mean was outside these percentiles, we might suspect it came from an underlying population with a different population mean and standard deviation, or that a rare (5% probability) event had occurred.

  27. The confidence interval for the mean depends on the sample size, n. If the sample size was 300, what would be the interval? • -1.96 <= (X – 250 )/(200/ √ 300) <= 1.96 • The lower and upper limits would be: 227.4 <= X <= 272.6 Which are narrower than the limits for n=50 (194.6, 305.4) • As n increases, the width of the interval decreases

  28. Confidence intervals for means • X, the sample mean, is a point estimate of , the population mean • Different samples will yield different Xs, so we cannot be certain how our estimate differs from  • Interval estimation provides a range of reasonable values that contain the population parameter (in this case ) with a certain degree of confidence • This interval is called a confidence interval Pagano and Gavreau, Chapter 9

  29. Confidence intervals for means • We put together what we learned about the normal distribution and the central limit theorem in order to construct confidence intervals • By the CLT, Xfollows a normal distribution if n is sufficiently large X ~ N(,/√n) • So, follows a standard normal distribution Z ~ N(0,1) Pagano and Gavreau, Chapter 9

  30. Confidence intervals for means • We know from examining the standard normal distribution that P(-1.96 ≤ Z ≤ 1.96) = 0.95 95% 2.5% 2.5% Pagano and Gavreau, Chapter 9

  31. Confidence intervals for means • P(-1.96 ≤ Z ≤ 1.96) = 0.95 Substituting the formula for Z into the above we get Multiplying by σ/√n , adding X , and multiplying by -1, we get Pagano and Gavreau, Chapter 9

  32. Confidence intervals for means Thus the lower 95% confidence limit for µ is And the upper 95% confidence limit for µ is So we are 95% confident that the interval we calculate using the above includes  Pagano and Gavreau, Chapter 9

  33. Confidence intervals for means • An important subtlety: • X is a random variable •  is a population parameter that is fixed in perpetuity; it has the same value irrespective of the sample • is either in the interval you calculate or it is not • What is random is the interval because it is based on the sample (X - 1.96/√n , X + 1.96/√n ) Pagano and Gavreau, Chapter 9

  34. Interpreting confidence intervals for means • The probability that the interval contains the true population mean is 95% • If we were to select 100 random samples from the population and calculate confidence intervals for each, approximately 95 of them would include the true population mean µ (and 5 would not) Pagano and Gavreau, Chapter 9

  35. Confidence intervals for means • 90% confidence interval • Replace 1.96 in the formula with 1.64 • 99% confidence interval • Replace 1.96 in the interval with 2.58 Generic formula: Where 100%*(1-) is the % of the confidence interval E.g. for a 95% confidence interval, =0.05, and we use z0.025 =1.96 Pagano and Gavreau, Chapter 9

  36. Confidence intervals for means • How to get a tighter interval? • Decrease the confidence level • Increase n Pagano and Gavreau, Chapter 9

  37. Uniform distribution demonstration

  38. Confidence intervals for means • What to do when σ is not known? (In practice, always) • By the Central limit theorem, follows a normal distribution, if n is sufficiently large • Can we substitute s, the sample standard deviation for ? • s is not a reliable estimate of  if n is small • If X is normally distributed, and a sample of size n is chosen, then follows a Student’s t distribution with n-1 degrees of freedom This is denoted tn-1 Pagano and Gavreau, Chapter 9

  39. Student’s t distribution • The mean of the t distribution is 0 and the standard deviation is 1 • The t distribution is symmetric and bell-shaped, but has heavier tails than the standard normal – extreme values are more likely to occur • For small n, the tails are fatter • For large n, the t distribution approaches (i.e. becomes indistinguishable from) the standard normal distribution Pagano and Gavreau, Chapter 9

  40. The t-distribution

  41. Student’s t distribution • There are separate curves for each degree of freedom (df) • Table A.4 gives t value for selected P(T>t) and selected df • Better to use Stata: • P(T>t) is calculated using ttail ***** **** note that normal() is P(Z<z)!!! • The code is ttail(df,t) • E.g., P(T>1.95) n=20 display ttail(19,1.95) .03304428 • To find the value for which P(T>t) use invttail(df,p) • The answer for this the t cutoff is denoted t19,.05 display invttail(19,.05) 1.7291328 Pagano and Gavreau, Chapter 9

  42. Confidence intervals for means • So using the t-distribution, the general formula for a 1-% confidence interval for a mean is: • The formula for a 95% confidence interval for a mean is: Where df =n-1 Pagano and Gavreau, Chapter 9

  43. Confidence intervals for means • Remember that when n is large, the t distribution approaches the normal distribution • E.g. z0.025 = 1.96 • While tn-1,0.025 =  Pagano and Gavreau, Chapter 9

  44. Confidence intervals for means • Example • CD4 cell count among HIV positives diagnosed at Mulago Hospital • N=270 • Sample mean = 296.9 • Sample SD = 255.4 • t269,.025=1.969 • 95% CI = ( 296.6 – 1.969*255.4/√270, 296.6 + 1.969*255.4/√270) = (266.0-327.2) Pagano and Gavreau, Chapter 9

  45. Note that some statistical output gives you the SE or the SEM, which stand for standard error or standard error of the mean. • This is s/ √n which is the standard deviation of the distribution of X • Remember, if X is a random variable with mean µ and standard deviation , if n is large enough, X is normally distributed with mean µ and standard deviation / √n

  46. . summ ncigs Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- ncigs | 545 .1963303 1.405081 0 20 . mean ncigs Mean estimation Number of obs = 545 -------------------------------------------------------------- | Mean Std. Err. [95% Conf. Interval] -------------+------------------------------------------------ ncigs | .1963303 .060187 .0781028 .3145577 -------------------------------------------------------------- . ci ncigs Variable | Obs Mean Std. Err. [95% Conf. Interval] -------------+------------------------------------------------------------- ncigs | 545 .1963303 .060187 .0781028 .3145577

  47. Normal approximation to the binomial distribution • Remember that binomial distributions are used to describe the number of success in n trials P(X=x) • The parameters of the binomial distribution are n and p, and the mean=np and standard deviation=square root of (np(1-p)) • As n increases, the binomial distribution more closely resembles the normal distribution

  48. Binomial approximation to normal distribution • Note that the binomial distribution approaches normality at smaller sample sizes when p is closer to 0.5 • Therefore you could use the normal distribution to look up the probability of observing X or more (or less) successes, using np as the mean and square root[ np(1-p)] as the standard deviation • Considered valid when np and np(1-p) ≥5