# Principles of Hypothesis Testing for Public Health

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1 Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine Fall 2011 Answers to Questions I Usually Get Around Now ITT is like generalizing to real life I am not a fan of stratification Except by clinic/site Not everyone agrees with me OK to adjust for (some) variables Baseline covariates Cannot stratify a continuous variable At least rarely can you do it well Some variables are not ok, or you just upgraded to a fancy model! Objectives Formulate questions for statisticians and epidemiologists using P-value P Power Type I and Type II errors Identity a few commonly used statistical tests for comparing two groups 1

2 Outline Estimation and Hypotheses How to Test Hypotheses Confidence Intervals Regression Error Diagnostic Testing Misconceptions Appendix Estimation and Hypotheses Inference How we use Hypothesis Testing Estimation Distributions Hypothesis testing Sides and Tails Statistical Inference Inferences about a population are made on the basis of results obtained from a sample drawn from that t population Want to talk about the larger population from which the subjects are drawn, not the particular subjects! 2

3 Use Hypothesis Testing Designing a study Reviewing the design of other studies Grant or application review (e.g. NIH study section, IRB) Interpreting study results Interpreting other s study results Reviewing a manuscript or journal Interpreting the news I Use Hypothesis Testing Do everything on previous slide Analyze the data to find the results Program formulas not presented here in detail Anyone can analyze the data, too, but be careful Analysis Follows Design Questions Hypotheses Experimental Design Samples Data Analyses Conclusions 3

4 What Do We Test Effect or Difference we are interested in Difference in Means or Proportions Odds Ratio (OR) Relative Risk (RR) Correlation Coefficient i Clinically important difference Smallest difference considered biologically or clinically relevant Medicine: usually 2 group comparison of population means Estimation and Hypotheses Inference How we use Hypothesis Testing Estimation Distributions Hypothesis testing Sides and Tails Estimation: From the Sample Point estimation Mean Median Change in mean/median Interval estimation Variation (e.g. range, σ 2, σ, σ/ n) 95% Confidence interval 4

5 Pictures, Not Numbers Scatter plots Bar plots (use a table) Histograms Box plots Not Estimation See the data and check assumptions Graphs and Tables A picture is worth a thousand t-tests Vertical (Y) axis can be misleading Like the Washington Post Weather, Though 5

6 Estimation and Hypotheses Inference How we use Hypothesis Testing Estimation Distributions Hypothesis testing Sides and Tails Distributions Parametric tests are based on distributions Normal Distribution (standard normal, bell curve, Z distribution) Non-parametric tests still have assumptions, but not based on distributions 2 of the Continuous Distributions Normal distribution: N( μ, σ 2 ) μ = mean, σ 2 = variance Z or standard normal = N(0,1) t distribution: t ω ω = degrees of freedom (df) Usually a function of sample size Mean = X (sample mean) Variance = s 2 (sample variance) 6

7 Binary Distribution Binomial distribution: B (n, p) Sample size = n Proportion yes = p Mean = np Variance = np(1-p) Can do exact or use Normal Many More Distributions Not going to cover Poisson Log normal Gamma Beta Weibull Many more Estimation and Hypotheses Inference How we use Hypothesis Testing Estimation Distributions Hypothesis Testing Sides and Tails 7

8 Hypothesis Testing Null hypothesis (H 0 ) Alternative hypothesis (H 1 or H a ) Null Hypothesis For superiority studies we think for example Average systolic blood pressure (SBP) on Drug A is different than average SBP on Drug B Null of that? Usually that there is no effect Mean = 0 OR = 1 RR = 1 Correlation Coefficient = 0 Sometimes compare to a fixed value so Null Mean = 120 If an equivalence trial, look at NEJM paper or other specific resources Alternative Hypothesis Contradicts the null There is an effect What you want to prove If equivalence trial, special way to do this 8

9 Example Hypotheses H 0 : μ 1 = μ 2 H A : μ 1 μ 2 Two-sided test H A : μ 1 > μ 2 One-sided test 1 vs. 2 Sided Tests Two-sided test No a priori reason 1 group should have stronger effect Used for most tests One-sided test Specific interest in only one direction Not scientifically relevant/interesting if reverse situation true Use a 2-Sided Test Almost always If you use a one-sided test Explain yourself Penalize yourself on the alpha sided test becomes a sided test 9

10 Never Accept Anything Reject the null hypothesis Fail to reject the null hypothesis Failing to reject the null hypothesis does NOT mean the null (H 0 ) is true Failing to reject the null means Not enough evidence in your sample to reject the null hypothesis In one sample saw what you saw Outline Estimation and Hypotheses How to Test Hypotheses Confidence Intervals Regression Error Diagnostic Testing Misconceptions Experiment Develop hypotheses Collect sample/conduct experiment Calculate l test t statistic ti ti Compare test statistic with what is expected when H 0 is true 10

11 Information at Hand 1 or 2 sample test? Outcome variable Binary, Categorical, Ordered, Continuous, Survival Population Numbers (e.g. mean, standard deviation) Example: Hypertension/Cholesterol Mean cholesterol hypertensive men Mean cholesterol in male general (normotensive) population (20-74 years old) In the year old male population the mean serum cholesterol is 211 mg/ml with a standard deviation of 46 mg/ml One Sample: Cholesterol Sample Data Have data on 25 hypertensive men Mean serum cholesterol level is 220mg/ml ( X= 220 mg/ml) Point estimate of the mean Sample standard deviation: s = 38.6 mg/ml Point estimate of the variance = s 2 11

12 Compare Sample to Population Is 25 enough? Next lecture we will discuss What difference in cholesterol is clinically i ll or biologically i ll meaningful? Have an available sample and want to know if hypertensives are different than general population Situation May be you are reading another person s work May be already collected data If you were designing up front you would calculate the sample size But for now, we have 25 people Cholesterol Hypotheses H 0 : μ 1 = μ 2 H 0 : μ = 211 mg/ml μ = POPULATION mean serum cholesterol for male hypertensives Mean cholesterol for hypertensive men = mean for general male population H A : μ 1 μ 2 H A : μ 211 mg/ml 12

13 Cholesterol Sample Data Population information (general) μ = 211 mg/ml σ = 46 mg/ml (σ 2 = 2116) Sample information (hypertensives) X = 220 mg/ml s = 38.6 mg/ml (s 2 = ) N = 25 Experiment Develop hypotheses Collect sample/conduct experiment Calculate test statistic Compare test statistic with what is expected when H 0 is true Test Statistic Basic test statistic for a mean point estimate of μ - target value of μ test statistic = σ point estimate of μ σ = standard deviation (sometimes use σ/ n) For 2-sided test: Reject H 0 when the test statistic is in the upper or lower 100*α/2% of the reference distribution What is α? 13

14 Vocabulary Types of errors Type I (α) (false positives) Type II (β) (false negatives) Related words Significance Level: α level Power: 1- β Unknown Truth and the Data Truth H 0 Correct H A Correct Data Decide H 0 1- α β fail to reject True Negative False Negative H 0 Decide H A reject H 0 α False Positive 1- β True Positive α = significance level 1- β = power Type I Error α = P( reject H 0 H 0 true) Probability reject the null hypothesis given the null is true False positive Probability reject that hypertensives µ=211mg/ml when in truth the mean cholesterol for hypertensives is

15 Type II Error (or, 1- Power) β = P( do not reject H 0 H 1 true ) False Negative Probability we NOT reject that male hypertensives cholesterol l is that t of the general population when in truth the mean cholesterol for hypertensives is different than the general male population Power Power = 1-β = P( reject H 0 H 1 true ) Everyone wants high power, and therefore low Type II error Cholesterol Sample Data N = 25 X = 220 mg/ml μ = 211 mg/ml s = 38.6 mg/ml (s 2 = ) σ = 46 mg/ml (σ 2 = 2116) α = 0.05 Power? Next lecture! 15

16 Z Test Statistic and N(0,1) Want to test continuous outcome Known variance Under H 0 X μ 0 ~ N(0,1) σ / n Therefore, X μ0 Reject H 0 if > 1.96 (gives a 2-sided α =0.05 test) σ / n σ σ Reject H 0 if X > μ or X < μ n n Experiment Develop hypotheses Collect sample/conduct experiment Calculate test statistic Compare test statistic with what is expected when H 0 is true Reference distribution Assumptions about distribution of outcome variable Z or Standard Normal Distribution 16

17 Z or Standard Normal Distribution Z or Standard Normal Distribution How to test? Rejection interval Like a confidence interval but centered on the null mean Z test or Critical Value N(0,1) distribution and alpha t test or Critical Value t distribution and alpha P-value Confidence interval 17

18 General Formula (1-α)% Rejection Region for Mean Point Estimate Z σ Z μ, μ + n 1 α/2 1 α/2 σ n Note that +Z (α/2) = - Z (1-α/2) 90% CI : Z = % CI : Z = % CI : Z = 2.58 Cholesterol Rejection Interval Using H 0 Population Information N(211, 46 2 ) Reject H o if 220 is outside of (193,229) Normal Distribution Cholesterol Rejection Interval Using H 0 Sample Information t (df=24, 211, ) Reject H o if 220 is outside of (195,227) t Distribution (df = 24) 18

19 Side Note on t vs. Z If s = σ then the t value will be larger than the Z value BUT, here our sample standard deviation (38.6) was quite a bit smaller than the population sd (46) HERE intervals using t look smaller than Z intervals BUT Because of sd, not distribution How to test? Rejection interval Like a confidence interval but centered on the null mean Z test or Critical Value N(0,1) distribution and alpha t test or Critical Value t distribution and alpha P-value Confidence interval Z-test: Do Not Reject H 0 X μ Z = = = < σ / n 46/ 25 19

20 Z or Standard Normal Distribution Determining Statistical Significance: Critical Value Method Compute the test statistic Z (0.98) Compare to the critical value Standard Normal value at α-level (1.96) If test statistic > critical value Reject H 0 Results are statistically significant If test statistic < critical value Do not reject H 0 Results are not statistically significant T-Test Statistic Want to test continuous outcome Unknown variance (s, not σ) Under H 0 X μ 0 ~ t ( n 1) s/ n Critical values: statistics books or computer t-distribution approximately normal for degrees of freedom (df) >30 20

21 Cholesterol: t-statistic X μ Using data T = = = 1.17 s/ n 38.6 / 25 For α = 0.05, two-sided test from t(24) distribution ib ti the critical value = T = 1.17 < The difference is not statistically significant at the α = 0.05 level Fail to reject H 0 Almost all Critical Value Tests: Exact Same Idea Paired tests 2-sample tests Continuous data Binary data See appendix at end of slides How to test? Rejection interval Like a confidence interval but centered on the null mean Z test or Critical Value N(0,1) distribution and alpha t test or Critical Value t distribution and alpha P-value Confidence interval 21

22 P-value Smallest α the observed sample would reject H 0 Given H 0 is true, probability of obtaining a result as extreme or more extreme than the actual sample MUST be based on a model Normal, t, binomial, etc. Cholesterol Example P-value for two sided test X = 220 mg/ml, σ = 46 mg/ml n = 25 H 0 : μ = 211 mg/ml H A : μ 211 mg/ml 2* PX> [ 220] = 0.33 Determining Statistical Significance: P-Value Method Compute the exact p-value (0.33) Compare to the predetermined α-level (0.05) If p-value < predetermined α-level Reject H 0 Results are statistically significant If p-value > predetermined α-level Do not reject H 0 Results are not statistically significant 22

23 P-value Interpretation Reminders Measure of the strength of evidence in the data that the null is not true A random variable whose value lies between 0 and 1 NOT the probability that the null hypothesis is true. How to test? Rejection interval Like a confidence interval but centered on the null mean Z test or Critical Value N(0,1) distribution and alpha t test or Critical Value t distribution and alpha P-value Confidence interval Outline Estimation and Hypotheses How to Test Hypotheses Confidence Intervals Regression Error Diagnostic Testing Misconceptions 23

24 General Formula (1-α)% CI for μ X Z σ Z, X + n 1 α/2 1 α/2 σ n Construct an interval around the point estimate Look to see if the population/null mean is inside Cholesterol Confidence Interval Using Population Variance ( Z ) N(220, 46 2 ) CI for the Mean, Unknown Variance Pretty common Uses the t distribution Degrees of freedom tn 1,1 α/ 2s tn 1,1 α/ 2s X, X + n n 2.064* *38.6 = 220, = (204.06,235.93) 24

25 Cholesterol Confidence Interval Using Sample Data ( t ) t (df=24, 220, ) But I Have All Zeros! Calculate 95% upper bound Known # of trials without an event (2.11 van Belle 2002, Louis 1981) Given no observed events in n trials, 95% upper bound on rate of occurrence is 3 / (n +1) No fatal outcomes in 20 operations 95% upper bound on rate of occurrence = 3 / (20 + 1) = 0.143, so the rate of occurrence of fatalities could be as high as 14.3% Hypothesis Testing and Confidence Intervals Hypothesis testing focuses on where the sample mean is located Confidence intervals focus on plausible values for the population mean In general, the best way to estimate a confidence interval is to bootstrap (details: see a statistician) 25

26 CI Interpretation Cannot determine if a particular interval does/does not contain true mean Can say in the long run Take many samples Same sample size From the same population 95% of similarly constructed confidence intervals will contain true mean Think about meta analyses Interpret a 95% Confidence Interval (CI) for the population mean, μ If we were to find many such intervals, each from a different random sample but in exactly the same fashion, then, in the long run, about 95% of our intervals would include the population mean, μ, and 5% would not. Do NOT interpret a 95% CI There is a 95% probability that the true mean lies between the two confidence values we obtained from a particular sample We can say that we are 95% confident that the true mean does lie between these two values. Overlapping CIs do NOT imply nonsignificance 26

27 Take Home: Hypothesis Testing Many ways to test Rejection interval Z test, t test, or Critical Value P-value Confidence interval For this, all ways will agree If not: math wrong, rounding errors Make sure interpret correctly Take Home Hypothesis Testing How to turn questions into hypotheses Failing to reject the null hypothesis DOES NOT mean that the null is true Every test t has assumptions A statistician can check all the assumptions If the data does not meet the assumptions there are non-parametric versions of tests (see text) Take Home: CI Meaning/interpretation of the CI How to compute a CI for the true mean when variance is known (normal model) How to compute a CI for the true mean when the variance is NOT known (t distribution) In practice use Bootstrap 27

28 Take Home: Vocabulary Null Hypothesis: H 0 Alternative Hypothesis: H 1 or H a or H A Significance Level: α level Acceptance/Rejection Region Statistically Significant Test Statistic Critical Value P-value, Confidence Interval Outline Estimation and Hypotheses How to Test Hypotheses Confidence Intervals Regression Error Diagnostic Testing Misconceptions Regression Continuous outcome Linear Binary outcome y Logistic Many other types 28

29 Linear regression Model for simple linear regression Y i = β 0 + β 1 x 1i + ε i β 0 = intercept β 1 = slope Assumptions Observations are independent Normally distributed with constant variance Hypothesis testing H 0 : β 1 = 0 vs. H A : β 1 0 In Order of Importance 1. Independence 2. Equal variance 3. Normality (for ANOVA and linear regression) More Than One Covariate Y i = β 0 + β 1 x 1i + β 2 x 2i + β 3 x 3i + ε i SBP = β 0 + β 1 Drug + β 2 Male + β 3 Age β 1 Association between Drug and SBP Average difference in SBP between the Drug and Control groups, given sex and age 29

30 Testing? Each β has a p-value associated with it Each model will have an F-test Other methods to determine fit Residuals See a statistician and/or take a biostatistics class. Or 3. Repeated Measures (3 or more time points) Do NOT use repeated measures AN(C)OVA Assumptions quite stringent Talk to a statistician Mixed model Generalized estimating equations Other An Aside: Correlation Range: -1 to 1 Test is correlation is 0 With N=1000, easy to have highly significant ifi (p<0.001) 001) correlation = Statistically significant that is No where CLOSE to meaningfully different from 0 Partial Correlation Coefficient 30

31 Do Not Use Correlation. Use Regression Some fields: Correlation still popular Partial regression coefficients High correlation is > 0.8 (in absolute value). Maybe 0.7 Never believe a p-value from a correlation test Regression coefficients are more meaningful Analysis Follows Design Questions Hypotheses Experimental Design Samples Data Analyses Conclusions Outline Estimation and Hypotheses How to Test Hypotheses Confidence Intervals Regression Error Diagnostic Testing Misconceptions 31

32 Is α or β more important? Depends on the question Most will say protect against Type I error Multiple comparisions Need to think about individual and population health implications and costs Omics False negative (Type II error) Miss what could be important Are these samples going g to be looked at again? False positive (Type I error) Waste resources following dead ends HIV Screening False positive Needless worry Stigma g False negative Thinks everything is ok Continues to spread disease For cholesterol example? 32

33 What do you need to think about? Is it worse to treat those who truly are not ill or to not treat those who are ill? That answer will help guide you as to what amount of error you are willing to tolerate in your trial design Outline Estimation and Hypotheses How to Test Hypotheses Confidence Intervals Regression Error Diagnostic Testing Misconceptions Little Diagnostic Testing Lingo False Positive/False Negative (α, β) Positive Predictive Value (PPV) Probability diseased given POSITIVE test result Negative Predictive Value (NPV) Probability NOT diseased given NEGATIVE test result Predictive values depend on disease prevalence 33

34 Sensitivity, Specificity Sensitivity: how good is a test at correctly IDing people who have disease Can be 100% if you say everyone is ill (all have positive result) Useless test with bad Specificity Specificity: how good is the test at correctly IDing people who are well Example: Western vs. ELISA 1 million people ELISA Sensitivity = 99.9% ELISA Specificity = 99.9% 1% prevalence of infection 10,000 positive by Western (gold standard) 9990 true positives (TP) by ELISA 10 false negatives (FN) by ELISA 1% Prevalence 990,000 not infected 989,010 True Negatives (TN) 990 False Positives (FP) Without confirmatory test Tell 990 or ~0.1% of the population they are infected when in reality they are not PPV = 91%, NPV = % 34

35 1% Prevalence total test positive by ELISA 9990 true positive 990 false positive 9990/10980 = probability diseased GIVEN positive by ELISA = PPV = 0.91 = 91% 989,020 total test negatives by ELISA 989,010 true negatives 10 false negatives / = NPV = % 0.1% Prevalence 1,000 infected ELISA picks up FN 999,000 not infected 989,001 True Negatives (TN) 999 False Positives (FP) Positive predictive value = 50% Negative predictive value = % 99% PPV 99.99% NPV 10% Prevalence 35

36 Prevalence Matters (Population You Sample to Estimate Prevalence, too) Numbers look good with high prevalence Testing at STD clinic in high risk populations Low prevalence means even very high h sensitivity and specificity will result in middling PPV Calculate PPV and NPV for 0.01% prevalence found in female blood donors Prevalence Matters PPV and NPV tend to come from good cohort data Can estimate PPV/NPV from case control studies but the formulas are hard and you need to be REALLY sure about the prevalence Triple sure High OR Does Not a Good Test Make Diagnostic tests need separation ROC curves Not logistic regression with high OR Strong association between 2 variables does NOT mean good prediction of separation 36

37 What do you need to think about? How good does the test need to be? 96% sensitivity and 10% specificity? 66% AUC? (What is that?) Guide you as to what amount of differentiation, levels of sensitivity, specificity, PPV and NPV you are willing to tolerate in your trial design Outline Estimation and Hypotheses How to Test Hypotheses Confidence Intervals Regression Error Diagnostic Testing Mistakes & Misconceptions Avoid Common Mistakes: Hypothesis Testing Mistake: Have paired data and do not do a paired test OR do not have paired data and do a paired test If you have paired data, use a paired test If you don t then you can lose power If you do NOT have paired data, do NOT use a paired test You can have the wrong inference 37

38 Avoid Common Mistakes: Hypothesis Testing Mistake: assume independent measurements Tests have assumptions of independence Taking multiple samples per subject? Statistician MUST know Different statistical analyses MUST be used and they can be difficult! Mistake: ignore distribution of observations Histogram of the observations Highly skewed data - t test and even nonparametric tests can have incorrect results Avoid Common Mistakes: Hypothesis Testing Mistake: Assume equal variances (and the variances are not equal) Did not show variance test Not that t good of a test t ALWAYS graph your data first to assess symmetry and variance Mistake: Not talking to a statistician Estimates and P-Values Study 1: 25±9 Stat sig at the 1% level Study 2: 10±9 y Not statistically significant (ns) 25 vs. 10 wow a big difference between these studies! Um, no. 15±

39 Comparing A to B Appropriate Statistical properties of A-B Statistical properties of A/B NOT Appropriate Statistical properties of A Statistical properties of B Look they are different! Not a big difference? 15?!? Distribution of the difference 15±12.7 Not statistically significant Standard deviations! Important. Study 3 has much larger sample size! 2.5±0.9 3 Studies. 3 Answers, Maybe Study # 3 is statistically significant Difference between study 3 and the other studies Statistical Different magnitudes Does study 3 replicate study 1? Is it all sample size? 39

40 (Mis)conceptions P-value = inferential tool? Yes Helps demonstrate that population means in two groups are not equal Smaller p-value larger effect? No Effect size is determined by the difference in the sample mean or proportion between 2 groups (Mis)conceptions A small p-value means the difference is statistically significant, not that the difference is clinically significant. YES A large sample size can help get a small p-value. YES, so do not be tricked. Failing to reject H 0 means what? There is not enough evidence to reject H 0 YES H 0 is true! NO NO NO NO NO! Analysis Follows Design Questions Hypotheses Experimental Design Samples Data Analyses Conclusions 40

41 Questions? Appendix Formulas for Critical Values Layouts for how to choose a test Do Not Reject H 0 Z X μ = = = 0.98 < 1.96 σ / n 46/ 25 σ =X > μ = * = NO! n 25 σ =X < μ = * = NO! n 25 41

42 Paired Tests: Difference Two Continuous Outcomes Exact same idea Known variance: Z test statistic Unknown variance: t test statistic H 0 : μ d = 0 vs. H A : μ d 0 Paired Z-test or Paired t-test d d Z = or T = σ / n s/ n 2 Samples: Same Variance + Sample Size Calculation Basis Unpaired - Same idea as paired Known variance: Z test statistic Unknown variance: t test statistic H 0 : μ 1 = μ 2 vs. H A : μ 1 μ 2 H 0 : μ 1 - μ 2 = 0 vs. H A : μ 1 - μ 2 0 Assume common variance x y x y Z = or T = σ 1/ n+ 1/ m s 1/ n+ 1/ m 2 Sample Unpaired Tests: 2 Different Variances Same idea Known variance: Z test statistic Unknown variance: t test statistic H 0 : μ 1 = μ 2 vs. H A : μ 1 μ 2 H 0 : μ 1 - μ 2 = 0 vs. H A : μ 1 - μ 2 0 x y x y Z = or T = σ / n+ σ / m s / n+ s / m

43 One Sample Binary Outcomes Exact same idea For large samples Use Z test statistic Set up in terms of proportions, not means pˆ p0 Z = p (1 p ) / n 0 0 Two Population Proportions Exact same idea For large samples use Z test statistic pˆ1 pˆ2 Z = pˆ ˆ ˆ ˆ 1(1 p1) p2(1 p2) + n m Normal/Large Sample Data? No Binomial? Yes No Independent? Yes Expected 5 Yes 2 sample Z test for proportions or contingency table No No Nonparametric test McNemar s test Fisher s Exact test 43

44 Yes Normal/Large Sample Data? Yes Inference on means? Yes Independent? Yes No Variance Paired t known? No Variances equal? No Inference on variance? Yes F test for variances Z test Yes T test w/ pooled variance No T test w/ unequal variance 44

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