Lecture Outline. BIOST 514/517 Biostatistics I / Applied Biostatistics I. Paradigm of Statistics. Inferential Statistic.

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1 BIOST 514/517 Biostatistics I / Applied Biostatistics I Kathlee Kerr, Ph.D. Associate Professor of Biostatistics iversity of Washigto Lecture 11: Properties of Estimates; Cofidece Itervals; Stadard Errors; Iferece for proportios November 8 ad 13, 2013 Lecture Outlie Properties of Estimates (Iferetial Statistics) variability bias mea squared error cosistecy efficiecy Cofidece Itervals for Populatio Parameters Cofidece Itervals for Proportio Asymptotic vs. Exact methods Estimatig Stadard Errors Comparig proportios: risk differece, odds ratio; 2 Iferetial Statistic A iferetial statistic or a estimate is computed o a sample ad used to estimate a populatio parameter sample mea used to estimate populatio mea sample media used to estimate populatio media proportio of a sample that is hypertesive used to estimate the proportio of the populatio that is hypertesive etc Paradigm of Statistics Populatio parameters are real but ukow umbers Iferetial statistics computed o samples are used to estimate populatio parameters We do t expect to exactly estimate a populatio parameter. We use statistical theory to uderstad the error i our estimate

2 Error of Estimates There are two kids of error that estimates ca have: 1. Variability 2. Bias variability: measures precisio Aother way to say this is: there are two desirable properties of estimates: 1. Precisio 2. Accuracy bias: measures accuracy 5 target the true value of the populatio parameter 6 Estimates ad Error We like estimates that are precise ad accurate. Said differetly, we like estimates that have low variability ad little or o bias. Who is the best player? 7 8 2

3 This statistic shows Statistic A low (actually o) bias ad low variability, i.e. high precisio sample statistic values true parameter value Statistic C This statistic shows high bias ad low variability, i.e. high precisio sample statistic values true parameter value This statistic shows low (actually o) bias ad high variability, Statistic B i.e. low precisio sample statistic values true parameter value This statistic shows high bias Statistic D ad high variability, i.e. low precisio sample statistic values true parameter value 9 Estimates ad Error I the precedig slide, the distributios represet the samplig distributio of the statistic Most of the statistics we use are ubiased the expected value of the samplig distributio is the true value of the populatio parameter biasedess is desirable but ot a absolutely ecessary property of good estimates 10 Estimator of populatio mea µ: Sample mea: Expected value: 1 j1 Example: j E[ ] biased estimator for the populatio mea. Mea Squared Error For a estimator T of a populatio parameter ϴ, the mea squared error of T is E[(T- ϴ ) 2 ] MSE is related to the bias ad variability of T. Specifically, MSE(T) = var(t) + Bias 2 (T) Variace: 2 Var [ ] / Precisio icreases with the sample size. Stadard Error: Var[ ]

4 Cosistecy Good estimators are cosistet. Roughly, this meas that as the sample icreases, the samplig distributio of the estimators becomes more cocetrated aroud the true value of the populatio parameter. There are differet precise mathematical defiitios of becomes more cocetrated correspodig to otios of weak ad strog cosistecy. For example, a ubiased estimator whose variace decreases as icreases is cosistet. E.g., the mea Cosistecy A estimator ca be biased ad still be cosistet Cosistecy A estimator ca be biased ad still be cosistet. Some estimates of the variace are biased yet cosistet (depeds whether we divide by or -1) Efficiecy A desirable property of a estimator is that it is efficiet. Roughly, this meas that it uses as much of the iformatio i the data as possible. We wo t get ito the exact techical defiitios of efficiecy For example, for a sample 1, 2,, suppose we wat to estimate the mea. is ubiased ad cosistet. is also ubiased ad cosistet. But it is less odd efficiet tha

5 Efficiecy Suppose we kow our variable is Normally distributed i the populatio. The the mea ad the media are the same parameter μ. The sample mea ad the sample media are both ubiased estimates of μ. However, the media is about 64% as efficiet i estimatig μ as the mea. Estimatig μ usig the media is like throwig out a radom 1/3 of your data ad the usig the mea. Cofidece Itervals Cofidece Itervals Cofidece Itervals aswer questios of the followig sort: For what values of the populatio parameter are the data fairly typical? For what values of the populatio parameter are the data cosistet? Cofidece Itervals Fairly typical is defied with respect to the samplig distributio: Not i the upper extreme of the samplig distributio or, Not i the lower extreme of the samplig distributio or, I either of the tails of the samplig distributio

6 Cofidece Itervals: Thought Exercise We wat to estimate the mea LDL cholesterol level amog seior citizes. The mea i our sample of 50 seior citizes is 132 mg/dl. 132 Ask: If the true mea i the populatio were # mg/dl, would a sample mea of 132 be surprisig? Ask: If the true mea i the populatio were 130 mg/dl, would a sample mea of 132 be surprisig? 132 samplig distributio of the mea of a sample of size 50 whe the populatio mea is 130 As: No. If the true mea i the populatio were 130 mg/dl, a sample mea of 132 would ot be surprisig. 22 Ask: If the true mea i the populatio were 140 mg/dl, would a sample mea of 132 be surprisig? samplig distributio of the mea of a sample of size 50 whe the populatio mea is Ask: If the true mea i the populatio were 100 mg/dl, would a sample mea of 132 be surprisig? samplig distributio of the mea of a sample of size 50 whe the populatio mea is As: No. If the true mea i the populatio were 140 mg/dl, a sample mea of 132 would ot be very As: Yes. If the true mea i the populatio were 100 mg/dl, a sample mea of 132 would be very surprisig. surprisig

7 Cofidece Itervals: Thought Exercise We wat to estimate the mea LDL cholesterol level amog seior citizes. The mea i our sample of 50 seior citizes is 132 mg/dl. I this example, we might report that with 95% cofidece, the mea cholesterol level amog seior citizes is betwee 122 ad 142 or Now the math: Cofidece Iterval for the Populatio Mea Questio: Whe we do ot kow the populatio mea, how ca we use the sample to estimate the populatio mea, ad use our kowledge of probability to give a rage of values cosistet with the data? Parameter: Estimator: The data are cosistet with a mea cholesterol level amog seior citizes betwee 122 ad Parameter: Estimator: Give adequate sample size, usig the CLT, we ca state: ~ N, P / 27 Rearragig: P P P / 1.96 / 1.96 / / 1.96 / / 1.96 / P The 95% cofidece iterval for is: 1.96 /, 1.96 / 28 7

8 Iterpretatio Simulatio Study: Cofidece itervals Correct: If we repeat the procedure of takig a sample of the same size ad costructig a 95% cofidece iterval o the sample, about 95% of those cofidece itervals will cotai the true value. Icorrect: There is a 95% chace that the 95% cofidece iterval cotais the true value 29 Simulatio of 100 data sets: 95% CIs were computed for each data set. The true parameter value is i purple 95% of these itervals cotai the true value! 30 Cofidece Iterval: Populatio Mea We showed that the 95% cofidece iterval for is: 1.96 /, 1.96 / Wat a 100(1-α) % cofidece iterval (ote: α is i the iterval (0,1)]. Develop: Pz z / Cofidece Iterval: Populatio Mea A 100(1-α) % cofidece iterval for is z 1, z This formula requires kowledge of the populatio variace ( 2 ). I practice, we do ot kow the populatio variace. 100(1-α) % cofidece iterval for is z1, z

9 Cofidece Iterval: Populatio Mea sually, the populatio variace is ukow. We ca estimate it with s 2 (the sample variace)! s j1 2 j Normal ad t distributios The statistic T s / has a t-distributio with -1 degrees of freedom. We ca use this distributio to obtai a cofidece iterval for whe is ot kow Cofidece Iterval: Populatio Mea A 100(1-α) % cofidece iterval for whe the populatio variace is ukow is give by t s t 1,1, 2 1, 1 2 s Critical value (quatile) i the t-distributio with (-1) degrees of freedom. t-distributio ad cofidece Itervals Wheever we make a cofidece iterval for the mea of a cotiuous variable, we must also estimate the populatio variace. So techically, we should make cofidece itervals with the t-distributio istead of the Normal distributio. A t-distributio is cetered at 0 ad has heavier tails tha a Normal distributio. A t-distributio is parameterized by its degrees of freedom. Ofte, we are OK to gloss over this detail, sice a t- distributio is very close to Normal for large degrees of freedom

10 t-based critical values df (-1) Critical value for 95% CI Cofidece Iterval: Geeral form For may parameters that we estimate, we make cofidece itervals via aalogous methods. ˆ 100 (1-) % cofidece 1- critical value (std err of ˆ ) 2 Parameter of iterest iterval for : Normal 1.96 Estimate The stadard error of the estimator is Var[ˆ] To fid the critical value we eed to kow the samplig distributio of the estimator. 38 Stadard errors We have see that the stadard error of the mea is SD()/. It ca be estimated by s/. We eed stadard errors for other estimates of other parameters. Example: Differece i Meas Suppose we have 1 observatios i group 1 with stadard deviatio s 1 ad 2 observatios i group 2 with stadard deviatio s 2. There are two widely used estimates for the stadard error of the differece i the two group meas

11 Example: Differece i Meas SE equal estimates the stadard error uder the assumptio that Groups 1 ad 2 have the same variace. Sometimes called a pooled variace estimate SE uequal estimates the stadard error without assumig that Groups 1 ad 2 have the same variace. This is the oe we have see already. This is the oe you should kow (be able to derive). Sice we rarely kow that the true populatio variaces are the same i the two groups, it makes sese to use this oe. Example: Differece i Meas SE uequal estimates the stadard error without assumig that Groups 1 ad 2 have the same variace. The Cetral Limit Theorem tells us that as log as 1 ad 2 are ot too small, 1 2 will have approximately a Normal distributio cetered at the true differece i populatio meas. The stadard deviatio of this samplig distributio is cosistetly estimated with SE uequal. We ca make cofidece itervals usig critical values from the Normal distributio or, whe s are small, a t distributio Which t-distributio? There are two ways of calculatig the degrees of freedom: Satterwaite ad Welch. Extremely techical ad uiterestig. Let your software do it. Example: Proportios A populatio proportio is ofte a parameter of iterest. Sice a proportio is also a mea, we could use what we already kow. However, there is a mea-variace relatioship for biary variables. For a biary variable with true populatio mea p, the true populatio stadard deviatio is [p(1-p)]. It is covetioal to estimate the stadard error of our estimate of p usig this formula rather tha from s. Example: Proportios FEV data. Estimatig the proportio of kids who smoke.. ge is_smoker= smoke==1. tab is_smoker is_smoker Freq. Percet Cum Total

12 Example: Proportios FEV data. Estimatig the proportio of kids who smoke. Oe optio is to treat is_smoker like a cotiuous variable, ad use s to estimate the stadard error of our estimate. Example: Proportios FEV data. Estimatig the proportio of kids who smoke. A 2 d optio is ackowledge that is_smoker is a biary variable ad use the mea-variace relatioship to estimate the stadard error of our estimate.. ci is_smoker Variable Obs Mea Std. Err. [95% Cof. Iterval] is_smoker ci is_smoker, biomial wald -- Biomial Wald --- Variable Obs Mea Std. Err. [95% Cof. Iterval] is_smoker Exact Distributio Here, we do ot have to rely o asymptotic theory A biary variable must be Beroulli Sums of idepedet Beroulli radom variables must be biomial We ca use the exact biomial distributio to compute our probabilities (Well, computers ca) Biomial Distributio Probability theory provides a formula for the distributio of biomial radom variables Data,, Y i1 1 For k 0,1,, : i ~ B 1, p 1 iid ~ B, p Pr ( Y k)! k p (1 p) k!( k)! k

13 Exact Poit Estimate Still use the sample mea Data,, 1 Poit estimate : iid ~ B 1, p E p Var p1 p 1 pˆ i i1 1 i i Pr Pr Exact Cofidece Itervals se the biomial distributio (But let a computer do it for you) A exact 100(1- )% cofidece based o observatio Y k is where a iterative search is used to fid k! i!( i)! i Y k; pˆ ˆ 1 ˆ p p i0! i!( i)! iterval i i Y k; pˆ ˆ 1 ˆ L pl pl / 2 ik pˆ, pˆ L i for p / Example: Proportios FEV data. Estimatig the proportio of kids who smoke. 3 rd optio: biomial exact cofidece iterval. ci is_smoker, biomial -- Biomial Exact -- Variable Obs Mea Std. Err. [95% Cof. Iterval] is_smoker Example: Proportios 1 st optio would be a uusual choice i practice. It is valid, but should be better to use a meavariace relatioship whe we have it. 2 d optio is commoly used 3 rd optio is commoly used. Exact cofidece itervals are better whe they are possible because we avoid makig a distributioal approximatio. Optio 3 is the preferred method, but will be similar to Optio 2 uless p or (1-p) is small. Exact biomial cofidece itervals are the default i STATA whe the biomial optio is used, but ot i other software. 13

14 Proportios: 0 evets i trials 2-sided cofidece itervals fail i case where there are either 0 or evets observed i Beroulli trials However, we ca derive oe-sided cofidece bouds i these cases. Proportios: 0 evets i trials: pper Cofidece Boud Exact upper cofidece boud whe there are 0 successes or evets i trials Suppose Y ~ B(, p) ad Y 0 is observed Exact 100(1- )% upper cofidece boud for p is pˆ Pr Y 0; pˆ 1 pˆ pˆ 1 1/ Large sample approximatio 1 pˆ log 1 pˆ log Large sample approximatio Three over rule log(.05) = So for 0 evets i trials upper cofidece boud is approximately 3/ For small pˆ so for large pˆ log 1 pˆ log pˆ 99% upper cofidece boud log(0.01)= se 4.6/ as 99% upper cofidece boud 14

15 Approximatio vs Exact Whe =0 evets observed i Beroulli trials 95% boud 99% boud Exact 3/ Exact 4.6/ evets i trials We ca also use the three over rule to fid the lower cofidece boud for p whe every trial of trials has a evet Lower 95% cofidece boud is 1-3/ Impress your frieds! Compute cofidece itervals durig a elevator ride! Comparig proportios: risk differece Sometimes we are iterested i comparig rates across groups. For example, i the FEV data we might be iterested i comparig smokig rates for boys ad girls. We estimate the proportio of girls ad boys who smoke with their correspodig sample proportios. We estimate the differece i smokig rates with the differece i sample proportios, which has stadard error: Comparig proportios: risk differece Boys Girls Smoker Not Smoker Total I the sample, 12.3% of girls smoke ad 7.8% of boys. We estimate the differece i smokig rates betwee girls ad boys is 4.5% with 95% cofidece iterval -0.1% to 9.1%. I large samples, CIs ca be computed from Normal approximatio

16 Comparig proportios There are other ways to compare two proportios Relative risk (risk ratio) odds ratio These provide examples where we make CIs usig trasformatios. Trasformatio to improve CLT: OR For some statistical summaries, it is stadard to calculate stadard errors ad cofidece itervals for some trasformatio of the summary. Oe example is the odds ratio. Exposed exposed Disease a b Not Disease c d The odds ratio ca be show to be ad/bc (HW6) Its stadard error is estimated by Trasformatio to improve CLT: OR We usually work with the logarithm of the odds ratio, whose stadard error is estimated by Trasformatio to improve CLT: OR A 95% cofidece iterval for log(or) is Although the samplig distributios for both OR ad log(or) approach Normal distributios as the sample size icreases, it happes faster for log(or) This meas that for a give sample size, a CI for the log(or) is more reliable tha for the OR. 63 Sice 95% of sample log odds ratios are i this iterval, the 95% of sample odds ratios are i the expoetiated iterval This cofidece iterval will ot be symmetric aroud the poit estimate Similar results hold for the relative risk (RR). Best to make cofidece itervals for the log(rr) ad expoetiate

17 . csi STATA. csi , or STATA Exposed exposed Total Cases Nocases Total Risk Poit estimate [95% Cof. Iterval] Risk differece Risk ratio Prev. frac. ex Prev. frac. pop chi2(1) = 3.74 Pr>chi2 = Exposed exposed Total Cases Nocases Total Risk Poit estimate [95% Cof. Iterval] Risk differece Risk ratio Prev. frac. ex Prev. frac. pop Odds ratio (Corfield) chi2(1) = 3.74 Pr>chi2 =