Institute for the Advancement of University Learning & Department of Statistics

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1 Istitute for the Advacemet of Uiversity Learig & Departmet of Statistics Descriptive Statistics for Research (Hilary Term, 00) Lecture 5: Cofidece Itervals (I.) Itroductio Cofidece itervals (or regios) are a importat ad ofte uderused area of statistical iferece, previously defied as the sciece of usig a data sample to deduce various populatio attributes. As was true for our presetatio of estimators i Lecture 5, the iterpretatio of cofidece itervals ad the methods of costructig them that we itroduce i this lecture all come from the (classical) Frequetist paradigm. However, oe should be aware that each of the other statistical paradigms (e.g., Bayesia) has its ow techiques for costructig ad its ow iterpretatio of cofidece itervals for populatio attributes. (II.) Defiitios ad Iterpretatio We ca defie a cofidece iterval (CI) as a regio, costructed uder the assumptios of our model, that cotais the true value (the parameter of iterest) with a specified probability. This regio is costructed usig particular properties of a estimator ad, as we will see, is a statemet regardig both the accuracy ad precisio of this estimate. There are two quatities associated with cofidece itervals that we eed to defie: Coverage probability: This term refers to the probability that a procedure for costructig radom regios will produce a iterval cotaiig, or coverig, the true value. It is a property of the iterval producig procedure, ad is idepedet of the particular sample to which such a procedure is applied. We ca thik of this quatity as the chace that the iterval costructed by such a procedure will cotai the parameter of iterest. Cofidece level: The iterval produced for ay particular sample, usig a procedure with coverage probability p, is said to have a cofidece level of p; hece the term cofidece iterval. Note that, by this defiitio, the cofidece level ad coverage probability are equivalet before we have obtaied our IAUL DEPARTMENT OF STATISTICS PAGE 1

2 sample. After, of course, the parameter is either i or ot i the iterval, ad hece the chace that the iterval cotais the parameter is either 0 or 1. Thus, for example, 95% of 95% CIs we costruct will cover the parameter uder repeated samplig. Of course, for ay particular sample we will ot kow if the CI produced cotais the true value. It is a commo (ad atural) mistake, whe cofroted with, for example, a 95% cofidece iterval, to say somethig like: The probability that the average height of 10 year old girls lies betwee x ad y is Ufortuately, whe we do this we are usig a set of rules which we have t bee playig by! The correct iterpretatio, at least for Frequetist cofidece itervals, requires us to ivert our thikig: istead of focusig o the probability of the parameter beig i the iterval, we eed to focus o the probability of the iterval cotaiig the parameter. The differece is subtle but importat, ad arises because, uder Frequetist rules, parameters are regarded as fixed, ukow costats (i.e. they are ot radom quatities). A Simulatio Example To illustrate the above poits, we will coduct a simulatio. Suppose we actually kow the populatio distributio of mouth widths of the Argetiea Wide Mouth Frog (ot a real creature!) to be ormal with mea 55mm ad stadard deviatio 10mm, deoted N(55, 100). The probability desity fuctio of this distributio is show i figure 1. We will cosider makig iferece o the mea mouth width ( µ = 55 mm). To simulate the iferetial process, we simply draw a radom sample of size from the populatio usig a radom umber geerator o a computer (most statistical software packages have this capability). We preted this is our data: each of the poits represets the carefully measured mouth width of a Wide Mouth Frog, obtaied after paistakigly idetifyig, ad the samplig from, the populatio of iterest. From this sample we ca costruct (100 p)% cofidece itervals for the mea mouth width of the Wide Mouth Frog (we will see how to do this later i the lecture). We ca the check to see if the iterval cotais µ, which we kow to be 55mm. By repeatig this process N times, we are able to calculate the proportio of the N itervals cotaiig µ, ad hece check that it is (roughly) equivalet to the cofidece level, p. IAUL DEPARTMENT OF STATISTICS PAGE

3 pdf µ (55mm) Mouth Width (mm) Figure 1: The populatio distributio of the mouth width of the Argetiea Wide Mouth Frog While we are makig use of the computer, we will also repeat the procedure described above for differet sample sizes ( = 10, 50, 100, ad 500) ad cofidece levels (p = 0.95 ad 0.99). This will allow us to empirically determie: 1. How, for a give cofidece level, the sample size effects the width of a iterval; ad. How, for a give sample size, the cofidece level effects the width of a iterval. The results of our simulatio are give i table 1. The umber of simulatios was N = Graphical represetatios of the coverage are give i figures ad 3. Note that we use N = 100 for these graphs, so that we ca distiguish idividual CIs. Cofidece Level (%) Observed Coverage (%) = 10 = 50 = 100 = Table 1: Observed Coverage for 95- ad 99- % CIs for a ormal mea, with varyig sample size. Figures based o N = 1000 simulatios IAUL DEPARTMENT OF STATISTICS PAGE 3

4 As see i table 1, the observed coverage appears to be approximately equivalet to the cofidece level, as we would expect from the defiitios above. Note that the observed coverage does ot deped o the sample size. Simulatio Simulatio Mouth width (mm) Sample size = Mouth width (mm) Sample size = 50 Simulatio Simulatio Mouth width (mm) Sample size = Mouth width (mm) Sample size = 500 Figure : Oe hudred 95% CIs for mea mouth width of the Argetiea Wide Mouth Frog, based o samples of size 10, 50, 100, ad 500. Pik (dark) lies represet CIs that do ot iclude the true mea Figures ad 3 show ad 99 -% CIs, for each of the four sample sizes = 10, 50, 100, 500, respectively. Each cofidece iterval is represeted by a horizotal lie. The blue (light) lies are those CIs that cover the true mea mouth width (the dotted vertical lie); the pik (dark) lies are those that do ot. We calculate the observed coverage from these graphs i the obvious way: 100 # pik( dark) lies. 100 Note that this represets a measure of the accuracy of our estimatio procedure. We are also i a positio to aswer the two questios posed above: 1. For ay give cofidece level, a icrease i sample size will yield arrower, or more precise, cofidece itervals. We ca see this pheomeo clearly i each figure: the legths of the horizotal lies decrease dramatically as the sample size icreases from 10 to 500 (the x axis for each graph has bee held fixed at the =10 rage so we ca see this more clearly). The reaso for this will become apparet i a later sectio, but for ow we will simply ote that the legth of a IAUL DEPARTMENT OF STATISTICS PAGE 4

5 CI depeds o (amog other thigs) the stadard error of a appropriate estimator. As you kow, the stadard error has a iverse relatioship with the sample size ad so will decrease as gets larger. At a ituitive level, a larger sample will cotai more iformatio about the populatio parameter of iterest, ad we ca therefore be more precise i our estimatio of it. Simulatio Simulatio Mouth width (mm) Sample size = Mouth width (mm) Sample size = 50 Simulatio Simulatio Mouth width (mm) Sample size = Mouth width (mm) Sample size = 500 Figure 3: Oe hudred 99% CIs for the mea mouth width of the Argetiea Wide Mouth Frog, based o samples of size 10, 50, 100, ad 500. Pik (dark) lies represet CIs that do ot iclude the true mea. For ay give sample size, a icrease i cofidece level will yield wider itervals. This is harder to see from the graphs, but if we look carefully at, for istace, the = 50 case, we otice that the itervals appear to lie roughly withi the mm rage i figure, whereas they appear to lie slightly outside this rage i figure 3. We will see the techical reaso for this icrease i iterval width whe we describe the procedure for costructig cofidece itervals for a ormal mea. However, we ca see ituitively why this should be so: if we wat to be more cofidet that a iterval will cotai the true value, we should make that iterval wider. Ideed, i the extreme, a 100% CI for the mea mouth width of the Argetiea Wide Mouth Frog is (, )! I ay evet, the importat poit to ote is that there is a trade off betwee precisio (iterval legth) ad accuracy (coverage). Oe - Sided Cofidece Itervals I the above simulatio, we costructed what are kow as two sided cofidece itervals. Two sided cofidece itervals are used whe we are iterested i iferrig IAUL DEPARTMENT OF STATISTICS PAGE 5

6 two poits betwee which the populatio quatity lies. This is almost always the case i practice. If, however, we have very strog prior kowledge/beliefs regardig the process uder ivestigatio, we might cosider costructig a oe sided cofidece iterval. These CIs are appropriate whe we are iterested i either a upper or lower boud for µ, but ot both. The followig example illustrates the distictio. Example: (Pagao ad Gauvreau, 1993). Cosider the distributio of haemoglobi levels for the populatio of childre uder the age of 6 years who have bee exposed to high levels of lead. Suppose this populatio is ormally distributed with mea µ (ad stadard deviatio σ ), o which we wish to make iferece. Uder usual circumstaces, we simply wat to locate µ, with the focus of the iferece beig betwee which two poits does µ lie? I this circumstace, a sided CI is appropriate. Suppose, however, that we have some extra iformatio regardig the process uder ivestigatio. Specifically, suppose it is kow that childre with lead poisoig ted to have lower levels of haemoglobi tha childre who do ot. We might therefore be iterested i a upper boud o µ. Hece, we would costruct a oe sided cofidece iterval for of the form (-, ], where µ deotes this µ µ u u upper boud. Note that we would trucate this iterval below at 0, givig [0, a], as haemoglobi levels caot be egative. (III.) Costructig Cofidece Itervals Because this is a itroductory course, we will focus most of our attetio o the costructio of cofidece itervals for meas of ormally distributed populatios. Our approach will be to make the assumptios required by such costructios explicit. Typically, the most importat assumptio is that our data are idepedet ad (approximately) ormally distributed. The reaso for this will be made clear presetly. Before cotiuig, we give a word of warig. If the assumptios of a procedure are ot satisfied i a particular istace, the procedure is ot valid ad should ot be used. However, we are free to explore ways i which to satisfy such assumptios. For example, a trasformatio may iduce ormality i o - ormal data. As aother example, takig differeces of paired (depedet) data will yield idepedet observatios. For otatioal coveiece, we itroduce the quatity α = 1 cofidece level. Rearragig, we ca express this relatioship, i its more usual form, as IAUL DEPARTMENT OF STATISTICS PAGE 6

7 cofidece level = 1 - α. We will see α agai whe we discuss hypothesis tests. I the preset cotext, ote that it defies the probability that the cofidece iterval does ot cotai the true parameter, ad is thus a measure of iaccuracy, or error. I fact, it is what we estimated by the proportio of pik lies i figures & 3. We would obviously like α to be quite small, although we should bear i mid the accuracy/precisio trade off discussed above. A Motivatig Example Recall that, by the cetral limit theorem, the sample mea, X, ca be regarded as beig σ ormally distributed with mea µ ad stadard deviatio whe is large eough. We ca thus calculate the probability that X lies withi a certai distace of µ (provided σ, the populatio variace, is kow) by usig the stadard ormal, or Z, trasform. Although ulikely i most applicatios, for the purposes of the followig illustratio, we will assume we kow σ. Our example cocers the aual icomes of UK statisticias. Suppose that the distributio of aual icomes (i pouds) withi this professio has a stadard deviatio of σ = 350. For a radom sample of = 400 statisticias, we would like to kow the probability that the sample mea, X, lies withi 100 of the true, or populatio, mea; that is: ( µ < 100)? Pr X =. We ca thik of the quatity X µ as a measure of the error i estimatig µ by X. The use of the absolute value reflects that error ca arise because we either uderestimate or overestimate the populatio mea. The probability statemet above the becomes: What is the probability that the (absolute) estimatio error is less tha 100? We kow that X ~ N( µ, σ ). Recall from lecture that we ca aswer probability questios regardig ormally distributed radom variables (i this case X ) by usig the Z trasform. The Z trasform for the sample mea is Z = X µ X µ =. se..( X) σ The radom variable Z has a stadard ormal distributio (i.e. has mea 0 ad variace 1). We thus calculate the above probability to be IAUL DEPARTMENT OF STATISTICS PAGE 7

8 Pr ( X µ < 100) = Pr X µ < σ = Pr ( Z < 0.615) ( < 0.615) ( < 0.615) Pr( < = Pr Z < = Pr Z Z ) = 0.46, usig stadard ormal tables. Figure 4 shows a stadard ormal pdf with the area correspodig to this probability. desity area = Z Figure 4: Area uder Z correspodig to a estimatio error of 100 Note three importat poits: 1. The area uder the stadard ormal curve betwee ad (0.46) is the same as the area betwee µ-100 ad µ+100 uder the samplig distributio of X (a ormal distributio with mea µ ad stadard deviatio ).. The populatio mea, µ, is ukow. 3. If the sample size (400) or the estimatio error ( 100) is made larger, the calculated probability will become larger. IAUL DEPARTMENT OF STATISTICS PAGE 8

9 We coclude that, prior to observig ay data (but give that we pla to sample 400 statisticias), there is a 46% chace that the value of the estimate X will lie at a distace o greater tha 100 from the ukow quatity µ. Note that we have fixed the (absolute) distace of X from µ, ad the calculated the probability of this evet occurrig usig the properties of the samplig distributio of X. A quick review of its defiitio makes it clear that to costruct a cofidece iterval, we eed to do the opposite: fix the probability (the cofidece level, p) ad the determie the (absolute) distace δ of X from µ. The followig sectio examies this procedure. The Geeral Costructio Techique Cotiuig the example, suppose we wat to costruct a 95% CI for the mea aual icome of UK statisticias. As stated at the ed of the previous sectio, this is equivalet to fidig a value δ such that ( µ < δ ) 95 Pr X = 0.. That is, we wat there to be a 95% chace that X lies withi δ uits of µ, ad hece a 5% chace that X lies at least δ uits from µ. Recall that we deote the latter probability α, ad treat it as a measure of error. I this case, it is measurig the probability that X does ot lie withi δ uits of µ. The procedure for determiig δ is similar to that set out i the above example. We agai use the Z trasform to give [ ] δ Pr X µ < δ = Pr Z < σ δ = Pr σ < Z δ < σ = Pr Z = < δ σ Pr Z δ < σ IAUL DEPARTMENT OF STATISTICS PAGE 9

10 δ It is clear that we eed to fid a poit with the property that 95% of the area σ uder a stadard ormal curve lies betwee it ad its egative value. This is equivalet to fidig the poit above which.5% of the probability lies: as stated i lecture 1, this value is the th quatile of Z ad is deoted. Because Z is symmetric about 0, we immediately kow that.5% of the area uder the stadard ormal curve lies to the left of - (that is, - = ), ad thus 95% of the area lies betwee these two z0.975 z z z poits. A graphical represetatio of this argumet is give i figure 5. Desity Area = 0.95 Area = 0.05 Area = 0.05 z = z Z z Figure 5: Poits defiig the cetral 95% of area uder a stadard ormal curve We ca fid z usig stadard ormal tables: it is approximately Thus or δ σ = 1.96 δ = σ. Oce X is kow (i.e. after we obtai a radom sample of size = 400), a 95% CI for the mea aual icome of UK statisticias would be give by X ± IAUL DEPARTMENT OF STATISTICS PAGE 10

11 This idea geeralizes to accommodate ay level of cofidece we desire. Deotig the cofidece level as 1 α, ad the ( 1 α ) th quatile of Z as z, we ca write 1 α δ = z σ α. 1 Hece, X lies withi δ uits of µ with probability 1 α. That is, whe is large ad σ is kow, a 100 ( 1 α )% CI for µ is X ± z σ α. 1 Critical Values of the Stadard Normal It is coveiet at this poit to make a small diversio i order to discuss some importat quatiles of the stadard ormal distributio. Such quatiles are also referred to as critical values of the radom variable Z. Figure 6 shows the (symmetric) values betwee which the stadard Normal distributio accumulates 99%, 95% ad 90% of probability. The correspodig itervals ca be foud from stadard Normal tables, ad are [-.58,.58], [-1.96, 1.96] ad [-1.645, 1.645] respectively. Note that if, for istace, 1 - α = 0.99, the area below -.58 is equal to α =0.005, which is the same area above.58 by symmetry. desity %; z=.58 desity %; z= 1.96 desity %; z= Z Z Z Figure 6: Some importat values for the stadard ormal distributio IAUL DEPARTMENT OF STATISTICS PAGE 11

12 (IV.) Specific Cofidece Itervals: Cofidece Itervals for Meas I the followig sectios we assume data have bee procured via a radom sample. That is, we assume observatios are idepedet ad idetically distributed. (IV.a.) Kow Variace There are two possible situatios we will cosider i this sectio. Both yield idetical expressios for costructig cofidece itervals o a populatio mea. The first is precisely the situatio we cosidered i the previous example. Whe is large ad the populatio variace, σ, is kow, the samplig distributio of X is N( µ, σ ), regardless of the distributio of the populatio from which the sample was take. This is a direct cosequece of the Cetral Limit Theorem. The secod situatio arises whe we sample from a ormally distributed populatio. Agai, give that we kow σ, the samplig distributio of X is N( µ, σ ), regardless of the sample size. Cofidece Iterval Formula Whe or σ, the populatio variace, is kow ad either: i. The sample size,, is large ii. The sample is ormally distributed, a 100 ( 1 α )% CI for µ is X ± z σ α, 1 where r.v. z deotes the ( 1 α ) 1 α th quatile of a stadard ormal Example: Cosider the beard legths (i cubits) of the populatio of Biblical Prophets. Suppose we kow that the radom variable which measures beard IAUL DEPARTMENT OF STATISTICS PAGE 1

13 legth, deoted X, is distributed ormally with ukow mea µ cubits ad stadard deviatio ½ cubits (i.e. X~ N( µ, ¼)). A sample of size =30 was obtaied from this populatio, ad the sample mea calculated to be X = 1.3 cubits. What is a 95% CI for the mea beard legth µ of this populatio, based o the iformatio obtaied from the sample? A: First we eed to check that either the assumptio of ormality is satisfied (usig, for example, a true histogram or a goodess of fit test) or that the sample size is large. Imagie we costructed a true histogram of the data, ad it looked ormal. Sice we have satisfied oe of the its assumptios, we may use the procedure defied above. (Note that, by a commoly asserted rule of thumb, = 30 classifies as beig large eough, so we have satisfied the other assumptio as well). Sice the level of cofidece required is 0.95, z ad α must be Therefore, z = = 1.96 (approximately) from stadard ormal tables. We kow σ =¼, = 30, 1 α X = 1.3. Therefore, a 95% CI for mea beard legths of Biblical Prophets is X ± z σ = ± α 30 =[1.1, 1.48]. Sample Size Calculatio Prior to the collectio of data, it would be useful to determie the sample size required to achieve a certai level of precisio i our iterval estimate for µ. We are i a positio to be able to write a explicit formula relatig iterval legth (precisio) to sample size. Recall that δ z σ α = 1. Solvig this equatio for, we obtai z = Note that will icrease if 1 α δ σ. δ decreases (i.e. we require greater precisio); or σ icreases (i.e. there is more dispersio i the populatio); or α decreases (i.e. we require greater accuracy). Figure 7 illustrates these ideas. IAUL DEPARTMENT OF STATISTICS PAGE 13

14 sample size vs delta; alpha= delta sigma= 0.1 sigma= 0.5 sigma= 0.5 sigma= 1 Figure 7: Sample size as a fuctio of δ ad σ Example (cotiued): Suppose that before we collected data from our populatio of hirsute Prophets, we decided that we wated a 95% CI for mea beard legth to be o greater tha 0.1 cubits i legth. Note that our previous 95% CI for µ, based o a sample of size 30, was approximately 0.36 cubits i legth. We are thus askig for a more precise (iterval) estimate of µ. How large should our sample size be to achieve this iterval legth (precisio)? A: We require δ to be 0.05, because we add ad subtract δ from X whe we costruct the CI: thus the width of a (two sided) CI is δ. We kow that α is 0.05, so z = 1.96 as before. Agai, σ =¼. Therefore 1 α z = 1 α δ σ = = We eed a sample of at least 385 prophets i order to costruct a 95% CI for mea beard legth which is at most 0.1 cubits wide. Note that we did ot require a data sample to make this calculatio. IAUL DEPARTMENT OF STATISTICS PAGE 14

15 (IV.b.) Ukow Variace Because σ is ukow, we eed to estimate it. The procedure for costructig the CI the proceeds as described above, but with σ replaced by its estimate. However, usig a estimate of the populatio variace chages the samplig distributio of the trasformed sample mea. I particular, whe we use the sample variace, S, to estimate σ, the radom variable X µ T = S o loger has a stadard ormal distributio, but rather a Studet s t distributio with (- 1) degrees of freedom, deoted. Degrees of freedom will be discussed i a future lecture. t 1 This distributio was amed after its discoverer. Studet was the pseudoym of the Eglish statisticia W.S. Gossett, who published this classic result i Gossett took a first i Chemistry from Oxford i 1899 ad worked most of his adult life i the Guiess Brewery, i Dubli. It was his employer s request that he use a pseudoym i his statistical publicatios. He died i Studet s t distributio is symmetric with respect to 0 ad has heavier tails tha the stadard ormal pdf. It approaches the stadard ormal distributio as the umber of degrees of freedom (ad hece sample size) icrease. Figure 8 shows several t pdfs together with a Normal pdf for compariso. desity df=1 df= df=4 df=8 df=16 df=3 Std. Normal T Figure 8: Studet s t pdfs compared with a stadard ormal pdf IAUL DEPARTMENT OF STATISTICS PAGE 15

16 Cofidece Iterval Formula Whe σ, the populatio variace, is ukow ad the sample is ormally distributed, a 100 ( 1 α )% CI for µ is X ± t S (1 α 1 ), where (1 1 α ) 1 α th quatile of a Studet s t radom variable with -1 degrees of freedom, ad t deotes the ( ) 1 1 i= 1 ( ) X i X S =. Critical values of Studet s t distributios, for differet degrees of freedom, ca be foud either by usig statistical tables or computer software. Note that we lose some iformatio i ot kowig σ. We thus itroduce a added level of ucertaity by havig to estimate this quatity. By otig that t 1 ( 1 α ) z 1 a, we see that the t distributio attempts to compesate for this ucertaity by makig the cofidece iterval wider, for give ad α, tha it would otherwise have bee. Of course, this is ot always true as the width of this CI also depeds o S, the sample stadard deviatio, which will vary betwee samples. Example: The data give below are from two similar experimets desiged to measure acceleratio due to gravity. The data give are 10 3 (measuremet i cm/sec - 980). Experimet 1: Experimet : For this data, we ca calculate the followig 95% CIs: df X s.e.( X ) t 1( 1 α ) Lower Boud Upper Boud Experimet Experimet IAUL DEPARTMENT OF STATISTICS PAGE 16

17 Sample Size Calculatio Ufortuately, calculatig sample size i this particular istace is ot as straightforward as i the previous sectio. A explicit formula caot be give. However, several iteractive sample size calculators exist o the web. See, for example, Note that these procedures ofte make the simplifyig assumptio that the populatio variace is kow. (V.) Coclusio Cofidece itervals are highly uderrated ad uderused i may areas of scietific research. However, iterval estimatio is a importat area of statistical iferece, because a radom iterval simultaeously provides iformatio regardig both estimate accuracy ad precisio. It is ufortuate that, due to igorace of this area, jourals prefer the ubiquitous p-value (we will deal with p-values i lecture 7). The importat poits to take from this lecture are: The iterval is radom, ot the parameter. Thus, we talk of the probability of the iterval cotaiig the parameter, ot the probability of the parameter lyig i the iterval. The width of a iterval is a measure of precisio. The cofidece level of a iterval is a measure of accuracy. The width of a CI depeds o two thigs: i. the size of the estimator s stadard error (which depeds o the sample size); ad ii. the level of cofidece we require (which depeds o the samplig distributio of the particular statistic we use to costruct the CI). Various formulae for CIs have bee give. However, special care must be take to esure that the assumptios required by these formulas are satisfied. If these assumptios are ot or caot be satisfied, a differet procedure must be used. Note that we have oly covered a very small subset of situatios for which cofidece itervals ca be costructed. I particular, we have oly discussed situatios for which we ca derive mathematical expressios for these CIs, ad eve the oly for meas of ormally distributed uivariate populatios. The material i lecture 6 will ot be preseted i these semiars, but should be read as it cotais ideas o what to do if we caot derive mathematical formulae for, or satisfy the typical assumptios of, CIs for meas ad other populatio quatities (such as the media ad variace). IAUL DEPARTMENT OF STATISTICS PAGE 17

18 (VI.) Referece PAGANO, M., ad GAUVREAU, K. (1993) Priciples of Biostatistics. Wadsworth Belmot Califoria, USA. MCB (I-000), KNJ (III-001), JMcB (III-001) IAUL DEPARTMENT OF STATISTICS PAGE 18

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