Two sample test (def 8.1) vs one sample test : Hypotesis testing: Two samples (Chapter 8) Example 8.2. Matched pairs (Example 8.6)

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1 Hypotesis testig: Two samples (Chapter 8) Medical statistics 00 Two sample test (def 8.) vs oe sample test : Two sample test: Compare the uderlyig parameters of two differet groups, where the values i both groups are ukow. Oe sample test: Compare the uderlyig parameter i a group with a kow value, for example 0 or a kow populatio mea. Example 8. Is there a relatioship betwee use of oral cotraceptives (OC) ad blood pressure (BP)? Several study desig are possible. Logitudial study (follow-up study) - eq 8. Idetify a group opregat premeopausal wome i childbearig age (6-49) who are ot curretly OC users ad measure their BP (baselie) After year: Idetify a study group who have remaied opregat ad have become OC users. Measure the BP i the study group. Compare baselie ad year values 3 4 Cross-sectioal study - eq 8. Idetify both a group of OC users ad a group of o-oc users amog opregat premeopausal wome i childbearig age (6-49), ad measure their BP Compare the BP i the two groups Matched pairs (Example 8.6) Does fertility differ betwee OC users ad diaphragm (IUD) users? Group cosists of 0 OC users. For each woma i Group, idetify a IUD user with same age (withi 5 years), race, parity, socio-ecoomic status. Registrer time to become pregat after stoppig cotraceptio. 5 6

2 Matched versus idepedet samples: differet methods Two samples are matched if every observatio i the first sample is related to a specific observatio i the secod sample (for example logitudial study or matched pairs) Two samples are idepedet if the observatios i the first sample are ot related to the observatios i the secod sample (for example cross-sectioal study) Matched pairs. Eample from Box, Huter & Huter: Statistics for Experimeters d ed. (005) 7 8 Paired t-test or cofidece iterval: For each pair of observatios, compute the differece d = x -x Expected differece is Δ =E(D) H 0 : Δ =0 agaist H : Δ 0 (alt >0 or <0) Perform a oe sample t-test or compute a cofidece iterval for Δ based o the differeces d, d,, d Repetitio: If X, X,..., X are idepedet N( μσ, ): X μ The: Z = ~ N(0,). σ / If σ is ukow, use S = ( Xi X) = Xi X i= i= X μ The T = ~ t S / Z or T is used to set up a hypotesis test or cofidece iterval for μ. If is large, the T is approximately N ( μσ, ) 9 0 Example 8.5 (Table 8.) =0, d = 4.80, s =0.85=4.566 Two sided test, t=3.3 Fid 0.00 < p < 0.0 from Table 5 i Appedix EXCEL: =TDIST(3,3;9;) gives the value p=

3 95% cofidece iterval for Δ: d t s, α / / / 0 = t-test ad cofidece iterval for two idepedet samples observatios, assumed idepedet N(μ, σ ) observatios, assumed idepedet N(μ, σ ) H 0 : μ = μ agaist H : μ μ Equivalet: H 0 : μ -μ =0 agaist H : μ - μ 0 Assume for the preset equal variace, σ = σ = σ That is,.53 to 8.07 (mmhg) 3 4 Estimator for μ μ: X X ( μ μ ) ~ 0, σ σ hece: N ( ) If σ σ σ σ σ X X ~ N μ μ, X X ( μ μ ) ~ 0, σ = = the N ( ) But σ is ukow ad is estimated by pooled estimate of the variace : S = ( X i X) ( Xi X ) i= i= = S S We use that X X ( μ μ) ~ t S 5 6 Example 8.9 Cardiovascular Disease, Hypertesio Suppose a sample of eight 35- to 39-year-old opregat, premeopausal OC users who work i a compay are idetified who have mea systolic blood pressure of 3.86 mm Hg ad sample stadard deviatio of 5.34 mm Hg. A sample of twety-oe 35- to 39 year-old opregat, premeopausal o-oc users are similarly idetified who have mea systolic blood pressure of 7.44 mm Hg ad sample stadard deviatio of 8.3 mm Hg. What ca be said about the uderlyig mea differece i blood pressure betwee the two groups? 7 Example 8.0 equal variace =8, x =3.86, s =5.34 =, x =7.44, s =8.3 H 0: μ -μ =0 x x = 5.4, s 7 0 = = = ( 0) t = = Degrees of freedom: 8-=7, reject H 0 at 5% level if 0.74 >.05 P-value f.ex. EXCEL TFORDELING(0,74;7;)=

4 Pr( t T t) = α, where t = t =.05, α / Two idepedet samples, uequal variace X X ( μ μ) t t S solve with respect to μ μ (eq 8.3) : observatios, assumed idepedet N(μ, σ ) observatios, assumed idepedet N(μ, σ ) H 0 : μ = μ agaist H : μ μ Uequal variace, σ σ 9.5 μ μ Two samples, σ σ : We use Satterthwaite s method : X X ( μ μ) ~ t d ' S S approximately, Where the degrees of freedom d is computed from,s,,s. ( S / S / ) d ' = ( S / ) /( ) ( S / ) /( ) Example 8. (exteded) Uequal variace t = x x = 0.8 s s d ' = 5.04 ( d'' = 5) p value = 0.43 Two idepedet samples, test for uequal variace observatios, assumed idepedet N(μ, σ ) observatios, assumed idepedet N(μ, σ ) H 0 : σ = σ agaist H : σ σ Equivalet: H 0 : σ /σ = agaist H : σ /σ Reject H 0 if S /S deviates much from Uder H 0 : S /S F -, - (Fisher distributed with - ad - degrees of freedom) SPSS uses Levee s test istead of Fisher s test 3 Example 8.6 F = S /S = 8.3 /5.34 =.4 Reject H 0 : σ /σ = at level α=0.05 if F > F 0,7,0.975 = 4.47 (FINV(0,05;0;7) i EXCEL) or F< F 0,7, = 0.33 (FINV(0,975;0;7) i EXCEL) Alteratively: p-value = * = 0.67 (FDIST(,4;0;7)) Coclusio: Do ot reject H 0 4 4

5 Equatio 8.4 Lower p-percetile is a F-distributio with d ad d degrees of freedom is the iverse of the upper p-percetile i a F-distributio with d ad d degrees of freedom: F = / F d, d, p d, d, p Roser, Figure 8.0 Strategy for testig the equality of meas i two idepedet, ormally distributed samples Sigificat Perform F test for the equality of two variaces i Equatio 8.5 Not sigificat (Useful if the table cotais oly upper percetiles) Perform t test assumig uequal variaces i Equatio 8. Perform t test assumig equal variaces i Equatio BUT: Navidi: Statistics for Egieers ad Scietists, 006, page : Do t Assume the Populatio Variaces are Equal Just Because the Sample Variaces are Close 7 the expressio assumig equal variaces requires that the populatio variaces be equal, or early so. I situatios where the sample variaces are early equal, it is temptig to assume that the populatio variaces are early equal as well. However, whe the sample sizes are small, the sample variaces are ot ecessarily good approximatios to the populatio variaces. Thus it is possible that the sample variaces be close eve whe the populatio variaces are fairly far apart. I geeral, populatio variaces should be assumed equal oly whe there is kowledge about the processes that produced the data that justifies this assumptio. 8 Hece: t-test or cofidece iterval for differece betwee the meas of two idepedet, ormally distributed samples: the expressio ot assumig equal variaces produces good results i almost all cases, whether the populatio variaces are equal or ot. (Exceptios ca occur whe the sample sizes are very differet.) Therefore, whe i doubt, use the expressio ot assumig equal variaces. Assume Equal variace (eq 8.) uequal variace (eq 8.) The truth Equal variace uequal variace correct Approximately same aswer as above Gives wrog aswer correct 9 Use t-test for uequal variace, or a o-parametric method, if i doubt! 30 5

6 Percet Percet If data are ot ormally distributed: 30% Ma Kvie t-tests give approximately correct results whe there is limited variatio i data t-tester are useless if extreme observatios or outliers. No-parametric methods ca always be used, ad are almost as powerful as the t-test (uless small sample sizes). F-test for comparig variaces is ot robust agaist departures from the ormal distributio. 0% 0% 0% 5,00 0,00 5,00 30,00 35,00 5,00 0,00 5,00 30,00 35,00 bmi bmi Approximately ormally distributed - t-test is OK Kvier 0-5 år Kvier år 5% 40 0% 30 5% 0 0% 0 geder 5% Percet 0 - ot at all - a little 3 - partly 4 - very much female male -0,00-5,00 0,00 5,00 gsfer -0,00-5,00 0,00 5,00 gsfer do you feel depressed? Limited variatio i data. T-test is OK or use oparametric methods T-test is useless use oparametric methods

7 Cocetratio of serum IgM (g/l) i 98 healthy childre, 6 moths - 6 years old (Altma, 99)