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Biostat 200 Lecture 5

Biostat 200 Lecture 5. Normal distribution. If X ~ N( µ , σ ) then Z=(X- µ )/ σ To find the proportion of values in the population that exceed some threshold, i.e. P(X>x) Transform x to z=(x - µ )/ σ and look up P(Z>z) using display 1-normal(z) To find the percentiles of the distribution

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Biostat 200 Lecture 5

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  1. Biostat 200 Lecture 5

  2. Normal distribution • If X ~ N(µ,σ) then Z=(X- µ)/σ • To find the proportion of values in the population that exceed some threshold, i.e. P(X>x) • Transform x to z=(x - µ)/σ and look up P(Z>z) using display 1-normal(z) • To find the percentiles of the distribution • E.g. to find the 97.5 percentile, the value for which 97.5% of the values are smaller, find z by using display invnormal(.975) and solve z=(x - µ)/σ for x

  3. For normal distribution • Stata calculates P(Z<z) • If you have z and you want p, use display normal(z) • If you have p and you want z, use display invnormal(p) • For t distribution • Stata calculates P(T>t) • If you have t and you want p, use display ttail(df,t) • If you have p and you want t, use display invttail(df,p)

  4. But usually you want to summarize the values, and make inference about the mean • The Central Limit Theorem says that the distribution of a sample mean X has a normal distribution with mean µ and standard deviation σ/ √n

  5. Imagine that we did the following • Draw a sample of size n from a population • Calculate the sample mean • Save the sample mean as an observation in the data set • Repeat steps 1-3 many times, enough to draw a reasonable histogram The histogram will show you the distribution of the sample means If the samples themselves were large enough (n), that distribution of the sample means will appear normal

  6. Confidence intervals • Using the central limit theorem, we can make probability statements about values of X̅ • These probability statements can be used to construct confidence intervals • Confidence intervals for means are statements about the probability that a given interval contains the true population mean µ • Confidence intervals can be constructed for any point estimate, e.g., odds ratios, hazards ratios, correlations, regression coefficients, standard deviations, … • The methods for constructing these will vary

  7. The width of the confidence interval depends on the confidence level (1-) and n • If σ is unknown, we use the t distribution and the sample standard deviation s in our computation

  8. Confidence intervals for means We know from the normal distribution that P(-1.96 ≤ Z ≤ 1.96) = 0.95 Substituting the formula for Z into the above we get After rearranging, the left and right are the confidence limits And is σ is unknown, Pagano and Gavreau, Chapter 9

  9. Confidence intervals for proportions • As n increases, the binomial distribution approaches the normal distribution • The normal approximation: as n increases • X ~ N( np, √(np(1-p)) ) X is the total count of the successes np is the mean of the binomial √ np(1-p) is the standard deviation of the binomial • X/n = p̂ ~ N( p, √(p(1-p)/n) ) X/n is the proportion of successes p is the population proportion √ (p(1-p)/n) is the standard deviation of X/n if X is binomial • So we can use the normal approximation to calculate confidence intervals for p

  10. Confidence intervals for proportions • But if n is small or p is small or both then the normal approximation is not good and the intervals using the normal approximation are not good • We need to use the binomial distribution • Confidence intervals using the binomial distribution are called exact confidence intervals • Exact confidence intervals are not symmetric (because neither is the binomial for small n or small p)

  11. For p=0.05, even for large n, the distribution of X doesn’t exactly look normal For p=0.35, the distribution does look fairly normal at small sample sizes

  12. E.g. Proportion of smokers in our class data set who smoke 10 or more cigarettes/day . gen cigs10=1 if cigs>=10 & cigs != . (541 missing values generated) . replace cigs10=0 if cigs<10 (23 real changes made) . label values cigs10 yesnol . tab cigs10 cigs10 | Freq. Percent Cum. ------------+----------------------------------- no | 23 82.14 82.14 yes | 5 17.86 100.00 ------------+----------------------------------- Total | 28 100.00 . ci cigs10 Variable | Obs Mean Std. Err. [95% Conf. Interval] -------------+--------------------------------------------------------------- cigs10 | 28 .1785714 .073707 .0273371 .3298058 . ci cigs10, binomial -- Binomial Exact -- Variable | Obs Mean Std. Err. [95% Conf. Interval] -------------+--------------------------------------------------------------- cigs10 | 28 .1785714 .0723789 .0606429 .3689333 How do these confidence intervals differ?

  13. E.g. proportion of smokers in the sample ci smoke Variable | Obs Mean Std. Err. [95% Conf. Interval] -------------+--------------------------------------------------------------- smoke | 546 .0531136 .0096062 .0342438 .0719833 . ci smoke, binomial -- Binomial Exact -- Variable | Obs Mean Std. Err. [95% Conf. Interval] -------------+--------------------------------------------------------------- smoke | 546 .0531136 .0095974 .0358559 .0753919 . • How do these confidence intervals differ?

  14. Hypothesis testing • Confidence intervals tell us about the uncertainty in a point estimate of the population mean or proportion (or some other entity) • Hypothesis testing allows us to draw a conclusion about a population parameter that we have estimated in a sample • Remember, we are taking samples from the population and we never know the truth Pagano and Gavreau, Chapter 10

  15. Hypothesis testing • To conduct a hypothesis test about the mean of a population, we postulate a value for it, and call that value µ0. • We write this H0 : µ= µ0 • We call this the null hypothesis • We set an alternative hypothesis for all other values of µ • We write this: HA : µ≠ µ0 • We call this the alternative hypothesis Pagano and Gavreau, Chapter 10

  16. Hypothesis testing for a mean • After formulating the hypotheses, draw a random sample • Compare the mean of the random sample, X , to the hypothesized mean µ0 • Is the difference between the sample mean X and the hypothesized population mean µ0 likely due to chance (remember you have taken a sample), or too large to be attributed to chance? • If too large to be due to chance, we say we have statistical significance

  17. Just like a 95% confidence interval has a 5% probability of not including the population parameter (e.g. the mean or the proportion), there is some probability that we will incorrectly reject the null hypothesis • Like confidence intervals, this is because we are taking a sample and do not know the true distribution • We set that probability in advance, before collecting the data and doing the analysis • That probability is called the significance level • The significance level is denoted by  , and is frequently set to 0.05.

  18. Criminal trials in the US – innocent until proven guilty • In hypothesis testing, we assume the null hypothesis is true, and only reject the null if there is enough evidence that the sample did not come from the hypothesized population (e.g. that the mean is not µ0) • In jury trials there is a chance that someone who is innocent is found guilty, but our method of innocent until proven guilty is in place to minimize this • In hypothesis testing, a true null hypothesis might be mistakenly rejected • This mistake is called a Type I error • The probability of this mistake is chosen in advance, and is the significance level, which is referred to as 

  19. The probability of obtaining a mean as or more extreme as the observed sample mean X bar, given that the null hypothesis is true, is called the p-value of the test, or p.

  20. Terminology • Before the test, set the significance level,  • This is the probability of rejecting the null hypothesis when it is true (also called a Type I error) • When you run the test, you get a probability of observing a mean as extreme as you did, given that the null hypothesis is true • This probability is the p-value • If the p-value is less than , (e.g. =0.05 and p=0.013), then the test is called statistically significant • If the p-value is ≥ , the test is called “not statistically significant”

  21. Hypothesis testing • 3 steps • Specify the null and alternative hypothesis • E.g. null: Is the level of CD4 count in a sample of new HIV positives in Uganda below 200 cells/mm3? • Determine the compatibility of the data with the null hypothesis • We do this by using the data to calculate a test statistic that will be compared to a theoretical statistical distribution, e.g., the standard normal distribution or the t-distribution • If the test statistic is very large, then our data are very unlikely under the null hypothesis • Either: • Reject the null OR • Fail to reject the null Pagano and Gavreau, Chapter 10

  22. 1. State the hypotheses • In epidemiology the null hypothesis is often that there is no association between the exposure and the outcome E.g., • The difference in disease risk=0 • Rexposed -Rnot exposed=0 • The ratio of risk/prevalence/etc=1 • Rexposed /Rnot exposed=1 • Clinical trials: The mean value of something does not differ between two groups • The alternative hypothesis is usually an association or risk difference • E.g., • Rexposed -Rnot exposed>0 • Rexposed /Rnot exposed>1 • The mean response in the treated group is greater than in the untreated group • H0 (the null) and HA (the alternative) must be mutually exclusive (no overlap) and include all the possibilities Pagano and Gavreau, Chapter 10

  23. 2. Determine the compatibility with the null. • We determine if there is enough evidence to reject the null hypothesis (i.e. calculate the test statistic and look up the p-value) • If the null is true, what is the probability of obtaining the sample data as extreme or more extreme? • This probability is call the p-value Pagano and Gavreau, Chapter 10

  24. 3. Reject or fail to reject • If the p-value (the probability of obtaining the sample data if the null is true) is sufficiently small (we often use 5%) then we reject the null and say the test was statistically significant. • If we fail to reject the null, it does not mean that we accept it. • We might fail to reject the null if the sample size is too small or the variability in the sample was too large Pagano and Gavreau, Chapter 10

  25. Hypothesis testing p-value, the result of your statistical test • If p<, reject the null  test is statistically significant • If p≥, do not reject the null  test is not statistically significant 1 0 Significance level, set a priori Pagano and Gavreau, Chapter 10

  26. Tests of one mean • We want to test whether a mean is equal to some hypothesized value • If we believe there might be deviations in either direction, we use a two-sided test • Two sided test: • Null hypothesis:H0: μ=μ0 • Alternative hypothesis: HA: μ≠μ0 • If we only care about values above or below a certain value, we use a one-sided test • One sided test: • Null hypothesis:H0: μ≥μ0 • Alternative hypothesis: HA: μ<μ0 or • Null hypothesis:H0: μ≤μ0 • Alternative hypothesis: HA: μ>μ0 Pagano and Gavreau, Chapter 10

  27. Lexicon • For one sided tests, people often say they are testing the hypothesis that the mean is less than or more than xxx (μ0). When they say this they are usually stating the alternative hypothesis. • For two sided tests, people often say they are testing the hypothesis that the mean is μ0 . When they say this they are stating the null hypothesis. • Since you know that the null is the complement of the alternative, you don’t usually state both in practicality (but we will for this class)

  28. Tests of one mean • The distribution of a sample mean if n is large enough is normal. For a normally distributed random variable, calculate the Z statistic • If the standard deviation (σ) is not known, calculate the t-statistic. • Compare your test statistic to the appropriate distribution to get your p-value • Find P(Z>zstat) or P(T>tstat) for a one-sided hypothesis test • Find 2*P(Z>zstat) or *2P(T>tstat) for a two-sided hypothesis test Pagano and Gavreau, Chapter 10

  29. Test of one mean, one sided Non-pneumatic anti-shock garment for the treatment of obstetric hemorrhage in Nigeria • Mean initial blood loss of 83 women 1413.1 ml, SD=491.3 • Our question: Are these women hemorrhaging (>750 ml blood loss)? • H0: μ≤750 HA: μ>750 =0.05 • T-statistic: • p-value: P(T>12.3) Data adapted from Miller, S et al. Int J Gynecol Obstet (2009) Pagano and Gavreau, Chapter 10

  30. Test of one mean • 12.3 is off the graph • So the probability of observing a sample mean of 1413 with n=83 and SD=431 if the true mean is <=750 is very very low Pagano and Gavreau, Chapter 10

  31. Test of one mean, one sided hypothesis, Stata P(T>test stat) . display ttail(82, 12.3) 1.308e-20 We reject the null P(T>t) Use ttail(df,tstat) Pagano and Gavreau, Chapter 10

  32. Test of one mean, one sided hypothesis, Stata Another way: Stata immediate code for ttests: ttestisamplesizesamplemeansamplesdhypothesizedmean ttesti 83 1413.1 491.3 750 One-sample t test ------------------------------------------------------------------------------ | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- x | 83 1413.1 53.92718 491.3 1305.822 1520.378 ------------------------------------------------------------------------------ mean = mean(x) t = 12.2962 Ho: mean = 750 degrees of freedom = 82 Ha: mean < 750 Ha: mean != 750 Ha: mean > 750 Pr(T < t) = 1.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 0.0000  We reject the null Pagano and Gavreau, Chapter 10

  33. Test of one mean, two sided hypothesis Example: Do children with congenital heart disease walk at a different age than healthy children? Healthy children start walking at mean age of 11.4 months • Null hypothesis: H0: μ=11.4 months (μ0) • Alternative hypothesis: HA: μ≠ 11.4 months • Significance level=0.05 • Data: Sample of children with congenital heart defects, n=9, sample mean=12.8, sample SD=2.4 Pagano and Gavreau, Chapter 10

  34. Calculate the test statistic and its associated p value • P value is P(T<-1.75) + P(T>1.75) = 2*P(T>1.75) display ttail(8,1.75)*2 .11823278 ----- Or run ttesti in Stata ttesti 9 12.8 2.4 11.4 One-sample t test ------------------------------------------------------------------------------ | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- x | 9 12.8 .8 2.4 10.9552 14.6448 ------------------------------------------------------------------------------ mean = mean(x) t = 1.7500 Ho: mean = 11.4 degrees of freedom = 8 Ha: mean < 11.4 Ha: mean != 11.4 Ha: mean > 11.4 Pr(T < t) = 0.9409 Pr(|T| > |t|) = 0.1182 Pr(T > t) = 0.0591  Fail to reject the null

  35. Hypothesis testing • One sided hypotheses : HA: μ<μ0 HA: μ>μ0 • For a one sided hypothesis, you calculate either P(Z<z) or P(Z>z) for the above alternative hypotheses respectively. • You are only looking at one tail of the distribution • For P(Z<z) to be <0.05, z must be < -1.645 . diinvnormal(.05) -1.6448536 • For P(Z>z) to be <0.05, your test statistic z must be >1.645 . diinvnormal(.95) 1.6448536

  36. Hypothesis testing • Two sided hypotheses HA: μ≠μ0 • For a two sided hypothesis, you are considering the probability that μ>μ0 or μ<μ0 , so you calculate P(Z>z) or P(Z<-z) which is both tails of the distribution • This is then 2*P(Z>z) • For 2*P(Z>z) to be <0.05, your test statistic z must be >1.96 . di invnormal(.975) 1.959964

  37. Hypothesis testing • So at the same significance level, here =0.05, less evidence is needed to reject the null for a one sided test as compared to a two sided test • So a two sided test is more conservative • What if you ran a test and got z=1.83? If your hypothesis was one-sided, you’d reject. If your hypothesis was two-sided, you’d fail to reject. • Therefore it is very important to specify your test a priori! • For clinical trials, most people use two-sided hypotheses • That way no one will suspect you of changing your hypothesis test a posteriori • On the other hand, does it really make sense to have an alternative hypothesis that a new treatment is either better or worse than the old one?

  38. Inference on proportions • Variables that are measured as successes or failures, disease or no disease, etc., can be considered binomial random variables • x=number of successes • n=number of “trials” • x/n = proportion diseased Pagano and Gavreau, Chapter 14

  39. Hypothesis test of a proportion • Under the CLT, the sampling distribution of an estimated proportion p̂, if n is large enough, is approximately a normal distribution with mean p and standard deviation=√(p(1-p)/n) Pagano and Gavreau, Chapter 14

  40. Hypothesis test of a proportion • Therefore we can test that a sample proportion is equal to, greater than, or less than some hypothesized p0using the Z statistic Pagano and Gavreau, Chapter 14

  41. Hypothesis test of a proportion • Example: Micronutrient intake of black women in South Africa • Study: Pre-menopausal women randomly selected based on geographic location • Micronutrient intake was determined using a Quantitative Food Frequency Questionnaire developed for South Africans • Question: Do more than 10% of the women have the micronutrient intake of less than the RDA? Pagano and Gavreau, Chapter 14 Data adapted from Hattingh Z. et al, West Indian Med J 2008: 57 (5):431

  42. Hypothesis test of a proportion • Null hypothesis: • Fewer than 10% of the women aged 25-34 have dietary folate levels of less than 268 µg (a cutoff based on RDA) • H0: p0 < 0.10 • Alternative hypothesis: • 10% or more of the women aged 25-34 have dietary folate levels of less than 268 µg • HA: p0≥ 0.10 • Significance level set at = 0.05 Pagano and Gavreau, Chapter 10

  43. Hypothesis test of a proportion • The data: 158/279=56.6% had folate levels of less than the cutoff, 268 µg • Using the normal approximation to the binomial distribution (formula on slide 40): • Z= (.566-0.10)/√(0.10*0.90/279) =25.9 • The chance of observing a z statistic of this large (25.9) is very very small (<0.05) • So we reject the null hypothesis Pagano and Gavreau, Chapter 14

  44. Hypothesis testing of a proportion # prtestisamplesizeobservedphypothp . prtesti 279 . 566 .1 One-sample test of proportion x: Number of obs = 279 ------------------------------------------------------------------------------ Variable | Mean Std. Err. [95% Conf. Interval] -------------+---------------------------------------------------------------- x | .566 .0296723 .5078434 .6241566 ------------------------------------------------------------------------------ p = proportion(x) z = 25.9458 Ho: p = 0.1 Ha: p < 0.1 Ha: p != 0.1 Ha: p > 0.1 Pr(Z < z) = 1.0000 Pr(|Z| > |z|) = 0.0000 Pr(Z > z) = 0.0000 ------- display 1-normal(25.9) 0 Pagano and Gavreau, Chapter 14

  45. Hypothesis testing of a proportion • Another example • Null hypothesis: • Fewer than 10% of the women aged 25-34 have dietary zinc levels of less than 5 mg • H0: p < 0.10 • Alternative hypothesis: • 10% or more of the women aged 25-34 have dietary zinc levels of less than 5 mg • HA: p ≥ 0.10 • Significance level = 0.05 Pagano and Gavreau, Chapter 14

  46. Hypothesis testing of a proportion • The data: 31/279=11.1% had zinc levels of less than 5 mg prtesti 279 .111 .1 One-sample test of proportion x: Number of obs = 279 ------------------------------------------------------------------------------ Variable | Mean Std. Err. [95% Conf. Interval] -------------+---------------------------------------------------------------- x | .111 .0188066 .0741397 .1478603 ------------------------------------------------------------------------------ p = proportion(x) z = 0.6125 Ho: p = 0.1 Ha: p < 0.1 Ha: p != 0.1 Ha: p > 0.1 Pr(Z < z) = 0.7299 Pr(|Z| > |z|) = 0.5402 Pr(Z > z) = 0.2701 • The data are NOT inconsistent with the null hypothesis, therefore we fail to reject the null Pagano and Gavreau, Chapter 14

  47. Hypothesis testing of a proportion • Remember that if n or p are small, you may need to use an exact test • The commands in Stata to do this are display binomialtail(n,k,p) or bitesti samplesize observedp hypothp

  48. Hypothesis tests versus confidence intervals • You can reach the same conclusion with confidence intervals as with two-sided hypothesis tests • Reject the null if the 95% confidence interval does not include the hypothesized value μ0 • Confidence intervals give us more information: the range of reasonable values for μ and the uncertainty in our estimate X . • Many statisticians and others prefer using confidence intervals to using hypothesis tests.

  49. Hypothesis test controversy (e.g. in psychology) • Lyken, D. L. (1991). What’s wrong with psychology? In D. Cicchetti & W.M. Grove (eds.), Thinking Clearly about Psychology, vol. 1: Matters of Public Interest, Essays in honor of Paul E. Meehl(pp. 3 – 39). Minneapolis, MN: University of Minnesota Press. • Meehl, P. E. (1967). Theory-testing in psychology and physics: A methodological paradox. Philosophy of Science, 34, 103-115. • Meehl, P. E. (1978). Theoretical risks and tabular asterisks: Sir Karl, Sir Ronald, and the slow progress of soft psychology. Journal of Consulting and Clinical Psychology, 46, 806-834. • Rozeboom, W. W. (1960). The fallacy of the null hypothesis significance test. Psychological Bulletin, 57, 416-428. • Bakan, D. (1966). The test of significance in psychological research. Psychological Bulletin, 66, 423-437. [optional] • Cohen, J. (1994). The earth is round (p < .05). American Psychologist, 49, 997-1003. • Gigerenzer, G. (1993). The superego, the ego, and the id in statistical reasoning. In G. Keren & C. Lewis (Eds.), A handbook for data analysis in the behavioral sciences: Methodological issues (pp. 311-339). Hillsdale, NJ: Lawrence Erlbaum Associates. • Schmidt, F. L. & Hunter, J. E. (1997). Eight common but false objections to the discontinuation of significance testing in the analysis of research data. In Lisa A. Harlow, Stanley A. Mulaik, and James H. Steiger (Eds.) What if there were no significance tests? (pp. 37-64). Mahwah, NJ: Lawrence Erlbaum Associates. • Oakes, M. (1986). Statistical inference: A commentary for the social and behavioral sciences. New York: Wiley. (Chapter 2 [A Critique of Significance Tests]) [optional]Frick, R. W. (1996). The appropriate use of null hypothesis testing. Psychological Methods, 1, 379-390. • Hagen, R. L. (1997). In praise of the null hypothesis statistical test. American Psychologist,52, 15-24. • Wilkinson, L., & the Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594-604. • Wainer, H. (1999). One cheer for null hypothesis significance testing. Psychological Methods, 6, 212-213. • Mulaik, S. A., Raju, N. S., & Harshman, R. A. (1997). There is a time and place for significance testing. In Lisa A. Harlow, Stanley A. Mulaik, and James H. Steiger , Eds. What if there were no significance tests? (pp. 65-116). Mahwah, NJ: Lawrence Erlbaum Associates. [optional] • Abelson, R. P. (1997). On the surprising longevity of flogged horses: Why there is a case for the significance test. Psychological Science, 8, 12-15. • Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method. American Psychologist, 56, 16-26. • Scarr, S. (1997). Rules of evidence: A larger context for the statistical debate. Psychological Science, 8, 16-17. • Greenwald, A. G., Gonzalez, R., Harris, R. J., & Guthrie, D. (1996). Effect sizes and p values: What should be reported and what should be replicated? Psychophysiology, 33, 175-183. • Nickerson, R. S. (2000). Null hypothesis significance testing: A review of an old and continuing controversy. Psychological Methods, 5, 241-301.[optional] • Harris, R. J. (1997). Significance tests have their place. Psychological Science, 8, 8-11. [optional] http://www.uic.edu/classes/psych/psych548/fraley/

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