BIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov

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1 Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio

2 OUTLINE Iterval estimatios for meas ad proportios iterval estimatio populatio mea: kow populatio proportio populatio mea: ukow Studet s distributio estimatio the size of a sample Iterval estimatios for mea of radom fuctios Simulatio-based cofidece itervals Lecture 5. Iterval estimatio for mea ad proportio

3 POPULATION AND SAMPLE Parameters Populatio parameter A umerical value used as a summary measure for a populatio (e.g., the mea, variace, stadard deviatio, proportio ) POPULATION µ mea variace N umber of elemets (usually N= ) SAMPLE x m, mea s variace umber of elemets Sample statistic A umerical value used as a summary measure for a sample (e.g., the sample mea m, the sample variace s, ad the sample stadard deviatio s) mice.txt 790 mice from differet strais All existig laboratory Mus musculus ID Strai Sex Startig age Edig age Startig weight Edig weight Weight chage Bleedig time Ioized Ca i blood Blood ph Boe mieral desity Lea tissues weight Fat weight 1 19S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ f S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m S1/SvImJ m Lecture 5. Iterval estimatio for mea ad proportio 3

4 Desity INTERVAL ESTIMATION Defiitios Iterval estimate A estimate of a populatio parameter that provides a iterval believed to cotai the value of the parameter. For the iterval estimates i this chapter, it has the form: poit estimate ± margi of error. Distributio of m Margi of error The ± value added to ad subtracted from a poit estimate i order to develop a iterval estimate of a populatio parameter N = m Badwidth merror = kow The coditio existig whe historical data or other iformatio provides a good value for the populatio stadard deviatio prior to takig a sample. The iterval estimatio procedure uses this kow value of о i computig the margi of error. ukow The coditio existig whe o good basis exists for estimatig the populatio stadard deviatio prior to takig the sample. The iterval estimatio procedure uses the sample stadard deviatio s i computig the margi of error. Lecture 5. Iterval estimatio for mea ad proportio 4

5 INTERVAL ESTIMATION: σ KNOWN Iterval Estimatio for the Mea Lecture 5. Iterval estimatio for mea ad proportio 5

6 INTERVAL ESTIMATION: σ KNOWN Iterval Estimatio for the Mea Cofidece level The cofidece associated with a iterval estimate. For example, if a iterval estimatio procedure provides itervals such that 95% of the itervals formed usig the procedure will iclude the populatio parameter, the iterval estimate is said to be costructed at the 95% cofidece level. Cofidece iterval Aother ame for a iterval estimate. m z / For 95 % cofidece = 0.05, which meas that i each tail we have Correspodig z / = 1.96 I Excel use oe of the followig fuctios: = CONFIDENCE.NORM(alpha,, ) = -NORM.S.INV(alpha/)*/SQRT() 0.95 / = / = I Excel before 010: = CONFIDENCE (alpha,, ) = -NORMSINV(alpha/)*/SQRT() Lecture 5. Iterval estimatio for mea ad proportio 6

7 INTERVAL ESTIMATION: σ KNOWN Example: Iterval Estimatio for the Mea A egieer is testig a ew measurig device. He tries to put 500 µl of water ito tubes ad the measure the resultig quatity. Based o 36 measuremets he estimated the average volume of 498 µl. From techical documetatio for the device she leart that the stadard deviatio of the volume is aroud 5 µl. Calculate the 95% ad 99% cofidece iterval for the volume the researcher takes o average. Is the desired volume of 500 µl i the cofidece itervals? m merror m z / =CONFIDENCE.NORM(0.05,5,36) Margial error (merror) =CONFIDENCE.NORM(0.01,5,36) 95% CI: 498 +/ = [ ] 99% CI: 498 +/-.14 = [ ] Lecture 5. Iterval estimatio for mea ad proportio 7

8 Desity INTERVAL ESTIMATION Populatio Proportio p ( 1 ) p p( 1 p) Samplig distributio for Distributio proportio of p p z / p(1 p) if p5 ad (1-p) N = Badwidth = 0.03 Excel variat 1: calculate ad p =CONFIDENCE.NORM(alpha, SQRT(p*(1-p)), ) Excel variat : calculate ad p calculate st.dev σ p =SQRT(p*(1-p)/) calculate z α/ statistics =-NORM.S.INV(alpha/) = z α/ * σ p for 95% cofidece z 0.05 = 1.96 Lecture 5. Iterval estimatio for mea ad proportio 8

9 Margial Error p(1-p) INTERVAL ESTIMATION Populatio Proportio: Some Practical Aspects p z / p(1 p) The ormal distributio is applicable oly whe eough data poits are observed. The rule of thumb is: p5 ad (1-p) p. The maximal margial error is observed whe p= The estimatio of the sample size ca be obtaied: z / p(1 p) E p5 ad (1-p) p where p is a best guess for or the result of a prelimiary study Lecture 5. Iterval estimatio for mea ad proportio 9

10 INTERVAL ESTIMATION Example: Populatio Proportio pacreatitis.txt 1. Defie a 95% cofidece iterval for ever-smokig proportio of people comig to a hospital.. How may patiets you would eed to have error less tha 1% Thik whether you would like to use pooled groups (other, pacreatitis) or make idepedet aalysis for each? Why? = 17 Never 56 p= st.dev= Error= z= Error= = p error = % Lecture 5. Iterval estimatio for mea ad proportio 1 p z z / / p(1 p) E p(1 p) for 95% cofidece z 0.05 =

11 INTERVAL ESTIMATION: σ UNKNOWN Populatio Mea: Ukow Assume that we have a sample of 0 mice ad would like to estimate a average size of a mice i populatio. Weight m =.73 s = 8.84 m s As we replace s, we itroduce a additioal error ad this chage the distributio from z to t (Studet) Note: ot a realistic scale here for illustratio oly Lecture 5. Iterval estimatio for mea ad proportio 11

12 INTERVAL ESTIMATION: σ UNKNOWN Studet distributio t-distributio A family of probability distributios that ca be used to develop a iterval estimate of a populatio mea wheever the populatio stadard deviatio is ukow ad is estimated by the sample stadard deviatio s. Degrees of freedom A parameter of the t-distributio. Whe the t distributio is used i the computatio of a iterval estimate of a populatio mea, the appropriate t distributio has 1 degrees of freedom, where is the size of the simple radom sample. Lecture 5. Iterval estimatio for mea ad proportio 1

13 INTERVAL ESTIMATION: σ UNKNOWN Iterval Estimatio for the Mea i Case of Ukow Weight m =.73 s = 8.84 s(m) = 1.98 t =.09 m.e. = 4.14 Variat 1 (Excel 010) use: =CONFIDENCE.T(alpha,s,) Variat (Excel 010) use: =-T.INV(alpha/,-1)*s/SQRT() m t ( 1) / s 0.95 / = / = I old Excel use: = TINV(alpha,-1) *s/sqrt() Lecture 5. Iterval estimatio for mea ad proportio 13

14 INTERVAL ESTIMATION Populatio Mea: Practical Advices Advice 1 Populatio m t ( 1) / s ot ormal ormal symmetric skewed highly skewed ay ~ 10 ~ Advice if >100 you ca, i priciple, use z-statistics istead of t-statistics (error will be <1.5%) Lecture 5. Iterval estimatio for mea ad proportio 14

15 Lecture 5. Iterval estimatio for mea ad proportio 15 INTERVAL ESTIMATION Determiig the Sample Size Let s focus o aother aspect: how to select a proper umber of experimets.? ), ( ), ( E E E m / / E z z E / E z / ) (1 E p p z

16 INTERVAL ESTIMATION Summary Iterval Estimatio Proportio -? Mea µ -? Esure that p 5 (1-p) 5 Stadard deviatio is kow Stadard deviatio is ukow (use s) Normal statistics, z Studet s statistics, t p z / z α/ = -NORM.S.INV(α/) p(1 p) m z / t α/ (-1) = T.INV(α/, -1) m t ( 1) / s =CONFIDENCE.NORM(α, SQRT(p*(1-p)),) =CONFIDENCE.NORM(α,σ,) =CONFIDENCE.T(α,s,) Lecture 5. Iterval estimatio for mea ad proportio 16

17 Part II SIMULATION-BASED CONFIDENCE INTERVALS FOR RANDOM FUNCTIONS Lecture 5. Iterval estimatio for mea ad proportio 17

18 INTERVAL ESTIMATIONS FOR RANDOM FUNCTIONS Sum ad Square of Normal Variables Distributio of sum or differece of ormal radom variables The sum/differece of (or more) ormal radom variables is a ormal radom variable with mea equal to sum/differece of the meas ad variace equal to SUM of the variaces of the compouds. x y Normal distributio E x y Ex Ey x y x y Distributio of sum of squares o k stadard ormal radom variables The sum of squares of k stadard ormal radom variables is a with k degree of freedom. if k i1 x x,..., x i 1 k Normal distributio with d. f. k What to do i more complex situatios? x y? x? log x? Lecture 5. Iterval estimatio for mea ad proportio

19 INTERVAL ESTIMATIONS FOR RANDOM FUNCTIONS Terrifyig Theory Try to solve aalytically? Simplest case. E[x] = E[y] = 0 Lecture 5. Iterval estimatio for mea ad proportio

20 INTERVAL ESTIMATIONS FOR RANDOM FUNCTIONS Practical Approach Two rates where measured for a PCR experimet: experimetal value (X) ad cotrol (Y). 5 replicates where performed for each. From previous experiece we kow that the error betwee replicates is ormally distributed. Q1: provide a iterval estimatio for the fold chage X/Y (=0.05) Q: provide a iterval estimatio for the log fold chage log (X/Y) # Experimet Cotrol Mea StDev Let us use a umerical simulatio Lecture 5. Iterval estimatio for mea ad proportio

21 INTERVAL ESTIMATIONS FOR RANDOM FUNCTIONS Practical Approach 1. Geerate sets of ormal radom variable with meas ad stadard deviatios correspodig to oes of experimetal ad cotrol set. Mea StDev I Excel go: Tools Data Aalysis: Radom Number Geeratio If you do ot have Data Aalysis tool approximate ormal distributio by sum of uiform: N( x, m x, ) x m x x 1 i1 U( x i ) 6 = RAND() U(x) Lecture 5. Iterval estimatio for mea ad proportio

22 INTERVAL ESTIMATIONS FOR RANDOM FUNCTIONS Practical Approach s m 1. Recalculate stadard deviatio s s m. Geerate sets of ormal radom variable with meas ad stadard deviatios s m correspodig to oes of experimetal ad cotrol set (assume you perform series by 5 experimets i each). s Mea StDev sim.m sim.s Mea StDev Build the target fuctio. For Q1 build X/Y X/Y.m X/Y.s mi max Study the target fuctio. Calculate summary, build histogram if ecessary Lecture 5. Iterval estimatio for mea ad proportio 5. If you would like to have 95% iterval, calculate.5% ad 97.5% percetiles. I Excel use fuctio =PERCENTILE(data,0.05) E[X/Y] [.55, 3.48 ]

23 INTERVAL ESTIMATIONS FOR RANDOM FUNCTIONS Practical Approach Q: provide a iterval estimatio for the log fold chage log(x/y) Mea Stadard Devi E[log(X/Y)] [ 1.35, 1.80 ] Simulatio Normal.50% % Lecture 5. Iterval estimatio for mea ad proportio

24 QUESTIONS? Thak you for your attetio to be cotiued Lecture 5. Iterval estimatio for mea ad proportio 4