This presentation provides an overview of statistical inference procedures in biostatistics, including: hypothesis test for population average; hypothesis test for comparing means; and, power and sample size

Baek, Jonggyu

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University of Massachusetts Medical School eScholarship@UMMS PEER Liberia Project UMass Medical School Collaborations in Liberia 2019-2 Introduction to Biostatistics - Lecture 2: Statistical Inference Procedures Jonggyu Baek University of Massachusetts Medical School Let us know how access to this document benefits you. Follow this and additional works at: https://escholarship.umassmed.edu/liberia_peer  Part of the Biostatistics Commons, Family Medicine Commons, Infectious Disease Commons, Medical Education Commons, and the Public Health Commons Repository Citation Baek J. (2019). Introduction to Biostatistics - Lecture 2: Statistical Inference Procedures. PEER Liberia Project. https://doi.org/10.13028/xjxb-cf87. Retrieved from https://escholarship.umassmed.edu/liberia_peer/9 This material is brought to you by eScholarship@UMMS. It has been accepted for inclusion in PEER Liberia Project by an authorized administrator of eScholarship@UMMS. For more information, please contact Lisa.Palmer@umassmed.edu. Introduction to Biostatistics2/28/2019Jonggyu Baek, PhDLecture 2: CTS605A - Lecture Notes,     Jonggyu Baek, PhD2• Statistical Inference Procedures–Hypothesis test for population average–Hypothesis test for comparing means–Power and sample sizeStatistical InferenceTwo broad areas of statistical inference:• Estimation: Use sample statistics to estimate the unknown population parameter.– Point Estimate: the best singlevalue to describe the unknownparameter.– Standard Error (SE): standard deviation of the sample statistic. Indicates how precise is the point estimate.– Confidence Interval (CI): the range with the most probable values for the unknown parameter with a (1-α)% level of confidence.• Hypothesis Testing: Test a specific statement (assumption) about the unknown parameter.CTS605A - Lecture Notes,     Jonggyu Baek, PhD3Statistical Inference for population average μEstimation:   Point Estimate   &   Standard Error• Suppose X a variable (e.g., systolic BP, hypertension, # of prior complications) from a population of size N with average μ and standard deviation σ.• We select a random sample x1, x2, …, xn of size n• Point Estimate of μ :  തx• Standard error of തx : Standard Deviation of all possible  തx ‘s• From the central limit theorem (CLT), for n large (n  30):തx  N(μ,σn)• If σ also unknown we can estimate from the sample standarddeviation s.CTS605A - Lecture Notes,     Jonggyu Baek, PhD4The Central Limit Theorem (CLT)Suppose X from a population (N) with μ and σ.• If we take random samples (n) with replacement from the population, for large “n” the distribution of the sample mean തx is approximately normally distributed with μതx = μand σതx = σn, i.e.:തx  N(μ,σn)Importance:• The distribution of the sample mean (ത𝐱) is approximately normal even if X does not follow N(μ, σ).• Sample mean is very useful for statistical inference.CTS605A - Lecture Notes,       S. Chrysanthopoulou, PhD5Normal Distribution Examples:CTS605A - Lecture Notes,       S. Chrysanthopoulou, PhD60.1.2.3.4y-10 -5 0 5 10xN(0,1)0.1.2.3.4y-10 -5 0 5 10xN(2,1)0.05.1.15.2y-10 -5 0 5 10xN(2,2)0.05.1.15.2y-10 -5 0 5 10xN(0,2)The Standard Normal Distribution തx  N(μ, Τ𝜎 𝑛) can be transformed to a Z  N(0, 1):Z = ഥx−μΤ𝜎 𝑛• N(0, 1) is called the standard normal distribution• Z is the standardized value of തx• Standardized values make comparable variables that are measured in different units, or have different variabilityCTS605A - Lecture Notes,       S. Chrysanthopoulou, PhD7020406080100Frequency0 5 10 15 20yy~N(10,3)020406080100Frequency-4 -2 0 2 4zz~N(0,1)Statistical Inference for population average μEstimation: Confidence Interval • Confidence Interval (CI): a range of values that are likely to cover the true parameter value with a level of confidence (1-α)% assigned to it. The most common choice for 𝛼 is 5%.• Usually CIs are symmetric around the point estimate.• From the central limit theorem (CLT), for n large (n  30):തx  N(μ,σn)• Hence,Z = തx−μൗσ n Ν(0, 1)CTS605A - Lecture Notes,     Jonggyu Baek, PhD8• E.g., (1-α)=95% CI for μ 95% CI for average μ : [ തx - 1.96( ൗσn) , തx + 1.96( ൗσn)]How we derived its 95% CI?• 95% of Z around 0 is between -1.96 and 1.96[or 𝑍0.025 = −1.96 and 𝑍0.975 = 1.96] • Remember that Z does not have any scalebecause it is standardized. We need the scale back to calculate 95% CI. CTS605A - Lecture Notes,     Jonggyu Baek, PhD9Statistical Inference for population average μEstimation: Confidence Interval ത𝑋 ത𝑋 ത𝑋• Based on the percentiles of the N(0,1) there are some commonly reported CIs:CTS605A - Lecture Notes,     Jonggyu Baek, PhD10(1-α)% CI α α/2 1-α/2 Zα/2 Z1-α/280% 20 10 90 -1.28 1.2890% 10 5 95 -1.64 1.6495% 5 2.5 97.5 -1.96 1.9699% 1 0.5 99.5 -2.58 2.58Statistical Inference for population average μEstimation: Confidence Interval Example of CIs: The Framingham Heart Study• Can you calculate 95% CIs based only on descriptive statistics for the systolic blood pressure?95% CI : [ തx - 1.96( ൗσn) , തx + 1.96( ൗσn)]  = [136.32- 1.960.21 , 136.32+ 1.960.21]  = [135.91, 136.73] CTS605A - Lecture Notes,     Jonggyu Baek, PhD11Example of CIs: The Framingham Heart Study• Is there any way to calculate 95% CI directly?CTS605A - Lecture Notes,     Jonggyu Baek, PhD12• Suppose X continuous from a population with mean μ and standard deviation σ.• What is the value of μ?• We select a random sample from that population and try to make inference about μ.CTS605A - Lecture Notes,     Jonggyu Baek, PhD13Hypothesis Testing for the mean μ• Null hypothesis (H0):– An explicit statement about an unknown parameter the validity of which you wish to test, e.g., μ=μ0• Alternative hypothesis (H1):– An alternative statement about the unknown parameter used to compare your null with, e.g., • μ  μ0 (two-sided test)• μ < μ0 (one-sided test)• μ > μ0 (one-sided test)• Errors:– Type I :   reject H0 | H0 is true (crucial)– Type II:   do not reject H0 | H1 is true (moderate)CTS605A - Lecture Notes,     Jonggyu Baek, PhD14Statistical Inference for population average μKey Concepts in Hypothesis TestingThink of Type I error as the “presumption of innocence” according to which “everyone is presumed innocent until proven guilty”:“It is better that ten guilty persons escape than that one innocent suffer” from the principle of Blackstone formula:– H0 : a person is innocent– H1 : a person is guilty• Without enough evidences, a person is innocentWhat about this?– H0 : a person is guilty– H1 : a person is innocent• Without enough evidences, a person is guiltyCTS605A - Lecture Notes,     Jonggyu Baek, PhD15Statistical Inference for population average μKey Concepts in Hypothesis Testing• What is the value of μ? (e.g., the population mean of systolic BP is 136. • Hypothesis Test:H0: μ = μ0(=136)CTS605A - Lecture Notes,     Jonggyu Baek, PhD16Hypothesis Testing for the mean μ• What is the value of μ?• Hypothesis Test:H0: μ = μ0(=136)• Random sample:ത𝐱CTS605A - Lecture Notes,     Jonggyu Baek, PhD17Hypothesis Testing for the mean μ• What is the value of μ?• Hypothesis Test:H0: μ = μ0 (?)• Random sample:ത𝐱• If ത𝐱 close to μ0 → H0 probable• If ത𝐱 far from μ0 → H0 not probableCTS605A - Lecture Notes,     Jonggyu Baek, PhD18Hypothesis Testing for the mean μ• What is the value of μ?• Hypothesis Test:H0: μ = μ0 (?)• Random sample:ത𝐱• If ത𝐱 close to μ0 → H0 probable• If ത𝐱 far from μ0 → H0 not probableCTS605A - Lecture Notes,     Jonggyu Baek, PhD19Hypothesis Testing for the mean μ• Test Statistic:– A summary measure of your sample, with known distribution under H0, used for testing the null hypothesis (H0), e.g., Z0 = തx−μ0ൗσ nH0~ N(0, 1)• Critical points:– Values (percentiles) of the known distribution of the test statistic above or below which the probability of Type I Error is α%, e.g.,Zα, Ζα/2, Ζ1-α/2, t1-α/2, d.f. ,     etc.CTS605A - Lecture Notes,     Jonggyu Baek, PhD20Statistical Inference for population average μKey Concepts in Hypothesis TestingTest Statistic• Example: Hypothesis testing about the population mean μ, at α% level of significance • H0: μ=μ0• H1: μ  μ0  μ = μ1  μ0• CLT → തx  N(μ,σn)  Z = തx−μൗσ n N(0, 1)• If H0 is true: Z0 = തx−μ0ൗσ nH0~ N(0, 1)– Z0 close to 0   → H0 probably true• Z0 “much” different from 0    → H0 probably NOT trueCTS605A - Lecture Notes,     Jonggyu Baek, PhD21Statistical Inference for population average μHypothesis Test• Example: Hypothesis testing about the population mean μ, at α% level of significance • H0: μ=μ0• H1: μ  μ0  μ = μ1  μ0• CLT → തx  N(μ,σn)  Z = തx−μൗσ n N(0, 1)• If H0 is true: Z0 = തx−μ0ൗσ nH0~ N(0, 1)– Z0 close to 0   → H0 probably true• Z0 “much” different from 0    → H0 probably NOT trueCTS605A - Lecture Notes,     Jonggyu Baek, PhD22Statistical Inference for population average μHypothesis TestHow “much”?• Example: Hypothesis testing about the population mean μ, at α% level of significance • H0: μ=μ0• H1: μ  μ0  μ = μ1  μ0• CLT → തx  N(μ,σn)  Z = തx−μൗσ n N(0, 1)• If H0 is true: Z0 = തx−μ0ൗσ nH0~ N(0, 1)– Z0 close to 0   → H0 probably true• Z0 “much” different from 0    → H0 probably NOT trueCTS605A - Lecture Notes,     Jonggyu Baek, PhD23Statistical Inference for population average μHypothesis TestHow “much”? Critical Z point (Zc) CTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ  μ024Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ  μ025Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nCritical PointsCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ  μ026Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nRejection RegionCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ  μ027Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nDecision Rule:Reject H0 if Z0<Zα/2 or   Z0>Z1 - α /2CTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ < μ028Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nCritical PointCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ < μ029Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nRejection RegionCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ < μ030Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nDecision Rule:Reject H0 if Z0<ZαCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ > μ031Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nCritical PointCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ > μ032Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nRejection RegionCTS605A - Lecture Notes,     Jonggyu Baek, PhD• Example: Hypothesis testing about the population mean μ, at α%H0: μ=μ0H1: μ > μ033Statistical Inference for population average μHypothesis TestTest statistic:        Z0 = തx−μ0ൗσ nDecision Rule:Reject H0 if Z0>Z1-α• Decision Rule:– What values of the test statistic would indicate the H0 is probably not supported by the observed data, hence it should be rejected.• P-value:– The exact level of significance, i.e., the probability of observing a value as extreme or more extreme than the calculated test statistic under the null hypothesis H0, e.g.,p-value = P(Z > Z0)CTS605A - Lecture Notes,     Jonggyu Baek, PhD34Statistical Inference for population average μKey Concepts in Hypothesis Testing1. Set the null hypothesis H0 and alternative hypothesis H12. Set a level of significance α%.3. Calculate a test statistic4. decision rule or5. P-value of the test statistic (preferred)6. conclusionCTS605A - Lecture Notes,     Jonggyu Baek, PhD35Statistical Inference for population average μSteps in Hypothesis Testing• We will cover examples for three cases– 1) Single population: one sample t-test• Interested in the population mean– 2) Two independent population: two sample t-test• Interested in comparing two population means– 3) Two dependent population: Paired t test• Interested in comparing mean changes within subjects (before vs. after)CTS605A - Lecture Notes,     Jonggyu Baek, PhD36Statistical Inference for population average μ• Example: We want to test the following hypothesis about the population mean μ of the systolic blood pressure of the Framingham Heart Study population, at α=5% level of significance:H0: μ=130 vs H1: μ  130• Test statistic: Z0 =തx−μ0ൗσ n=തx−μ0ൗs n=136.32−130ൗ22.8 11627= 29.91• Conclusion:       Z0 = 29.91      reject H0 if |Z0| > 1.96• p-value = P(Z>|Z0|) = P(Z<-Z0) + P(Z > Z0) = 2*P(Z>29.91) < 0.0001CTS605A - Lecture Notes,     Jonggyu Baek, PhD37Statistical Inference for population average μCase 1: One-Sample: two-sided hypothesis Test• Example: We want to test the following hypothesis about the population mean μ of the systolic blood pressure of the Framingham Heart Study population, at α=5% level of significance:H0: μ=130 vs H1: μ > 130• Test statistic: Z0 =തx−μ0ൗσ n=തx−μ0ൗs n=136.32−130ൗ22.8 11627= 29.91• Conclusion:       Z0 = 29.91  reject H0: if Z0 > 1.68• p-value = P(Z > Z0) = P(Z>29.91) < 0.0001CTS605A - Lecture Notes,     Jonggyu Baek, PhD38Statistical Inference for population average μOne-sided hypothesis Test• Example: We want to test the following hypothesis about the population mean μ of the systolic blood pressure of the Framingham Heart Study population, at α=5% level of significance:H0: μ=130 vs H1: μ < 130• Test statistic: Z0 =തx−μ0ൗσ n=തx−μ0ൗs n=136.32−130ൗ22.8 11627= 29.91• Conclusion:       Z0 = 29.91      reject H0: if Z0 < -1.68• p-value = P(Z < Z0) = 1CTS605A - Lecture Notes,     Jonggyu Baek, PhD39Statistical Inference for population average μOne-sided hypothesis TestTwo Independent Samples• Case 2: two-independent populations (two-samples)• X1 ‘sysbp’ of people without previous CHD, with μ1 and unknown σ1• X2 ‘sysbp’ of people with previous CHD, with μ2 and unknown σ2Hypothesis Testing for μ1-μ2• Null hypothesis (H0): μ1-μ2=0   ⟹   μ1=μ2• Alternative hypothesis (H1): – μ1-μ2  0  ⟹   μ1  μ2 (two-sided test), or– μ1-μ2 < 0 ⟹   μ1 < μ2 (one-sided test), or– μ1-μ2 > 0  ⟹   μ1 > μ2 (one-sided test)CTS605A - Lecture Notes,    Jonggyu Baek, PhD40Two Independent Samples• Case 2: two-independent populations (two-samples)– Case 2.A: Known variances• X1 ‘sysbp’ of people without previous CHD, with μ1 and known σ1• X2 ‘sysbp’ of people with previous CHD, with μ2 and known σ2Hypothesis Testing for μ1-μ2• Test statistic: Z0 =ഥx1 − ഥx2𝑠12n1+𝑠22n2H0~ N(0, 1)• Decision Rules by H1:    Testing H0: μ1-μ2=0 vs :CTS605A - Lecture Notes,    Jonggyu Baek, PhD41H1 Reject H0 if:μ1-μ2  0 Z0 < Zα/2 or Z0 > Z1-α/2μ1-μ2 < 0 Z0 < Zαμ1-μ2 > 0 Z0 > Z1-αTwo Independent Samples• Case 2: two-independent populations (two-samples)• X1 ‘sysbp’ of people without prevchd, with μ1 and unknown σ1• X2 ‘sysbp’ of people with prevchd, with μ2 and unknown σ2Hypothesis Testing for μ1-μ2H0: μ1=μ2 vs. H1: μ1  μ2CTS605A - Lecture Notes,    Jonggyu Baek, PhD42Two Dependent Samples• Case 3: two-dependent populations (two-samples)• X1 ‘sysbp’ of people at baseline, with μ1 and unknown σ1• X2 ‘sysbp’ of people 6yrs after baseline, with μ2 and unknown σ2• Suppose variable: d=x1-x2 from population with μd and σdHypothesis Testing for μd• Null hypothesis (H0): μd=0• Alternative hypothesis (H1): – μd  0  (two-sided test), or– μd < 0 (one-sided test), or– μd > 0 (one-sided test)CTS605A - Lecture Notes,    Jonggyu Baek, PhD43Looks familiar? This is then same as one-sample t-test!Power = 1- P(Type II error) = 1 - P(do not reject H0 | H1 is true) = 1 – β= P(reject H0 | H1 is true)• E.g., the hypothesis: H0: μ=μ0 vs H1: μ = μ1 > μ0• The power of this test is:Power = P(reject H0 | H1 is true) = P( Z0 > Z1-α | μ = μ1 > μ0 )CTS605A - Lecture Notes,     Jonggyu Baek, PhD44Power and Sample Size DeterminationCTS605A - Lecture Notes,     Jonggyu Baek, PhD45Power and Sample Size DeterminationReference: Krzywinski and Altman, “Power and sample size”, Nature Methods 10, 1139-1140 (2013). Power is a function of 1) standard deviation (𝜎),2) sample size (n), 2) mean difference (or effect size), 3) type I error (𝛼). • The power of the test is:Power = P(reject H0 | H1 is true) = P(Z1 > Z1-α -μ1−μ0ൗσn)        (2)• The power of the test depends on:❖ n (standard deviation)𝜎 ↑  Power ❖ n (sample size)n   Power ❖ α (significance level)α   Power ❖ μ1 − μ0 (Effect Size)ES   Power CTS605A - Lecture Notes,     Jonggyu Baek, PhD46Power and Sample Size DeterminationCase 1: Single population (one-sample):H0: μ=100 vs   H1: μ100• at α=5% level of significance.• We want a powerful test with power 80% power.• The test will reject the null hypothesis if the true mean is 5 unitsdifferent from 100 (either smaller or larger – two-sided test).Namely, |μ-μ0|=5.• Suppose we know that standard deviation of the outcome variableσ=9.5• What is the required sample size?CTS605A - Lecture Notes,     Jonggyu Baek, PhD47Sample Size DeterminationSample Size DeterminationCase 1: single population (one-sample)H0: μ=100 vs   H1: μ100 (two-sided test)CTS605A - Lecture Notes,    Jonggyu Baek, PhD48The total N = 31Sample Size DeterminationCase 2: two dependent populations (two-samples)with unknown variance of the differencesExample: Suppose sd=7. We want to test the hypothesis:H0: μ1=μ2=100 vs   H1: μ1μ2• at α=5% level of significance.• We want to detect |μ1-μ2|=5.• With power=80%What is the required sample size?CTS605A - Lecture Notes,    Jonggyu Baek, PhD49Sample Size DeterminationCase 2: two dependent populations (two-samples)H0: μ1=μ2=100 vs   H1: μ1μ2 (two-sided test)Assume:→ unknown varianceof the differences,i.e., sd=7CTS605A - Lecture Notes,    Jonggyu Baek, PhD50N = 32 per group. The total N = 64. 

Introduction to Biostatistics - Lecture 2: Statistical Inference Procedures

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Introduction to Biostatistics - Lecture 2: Statistical Inference Procedures

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