Methods in Epidemiology. Medical statistics 02/11/2014

Transcription

1 Methods n Epdemology At the end of the course students should be able to use statstcal methods to nfer conclusons from study fndngs Medcal statstcs At the end of the lecture students should be able to dfferentate the related concepts of parameter and estmate to explan the meanng of dstrbuton to dstngush between standard devaton and standard error to nterpret standard normal dstrbuton Methods n epdemology Samplng dstrbuton 1

2 Structure of clncal research Plannng Implementaton queston object Truth Varables to conclusons <mthods n epdemology Medcal statstcs- Samplng dstrbuton fndngs Besde descrbng data the fnal purpose s to draw general conclusons (relatve to the ) Structure of clncal research Plannng Implementaton queston object Varables to θ T The true effect n the relevant we want to know <mthods n epdemology Medcal statstcs- Samplng dstrbuton 2

3 Structure of clncal research Plannng Implementaton queston object Varables to Estmate The effect observed n the -th study, that s the best estmate of the parameter θˆ <mthods n epdemology Medcal statstcs- Samplng dstrbuton Structure of clncal research (example) Plannng Implementaton queston object Varables to Estmate δ µ u µ d ˆ δ = ˆ µ u ˆ µ = Is heght dfferent between genders? d <mthods n epdemology Medcal statstcs- Samplng dstrbuton 3

4 Structure of clncal research (example) Plannng Implementaton queston object Varables to Estmate π π f nf Is the rsk of lung cancer ncreased n smokers compared to non smokers? <mthods n epdemology Medcal statstcs- Samplng dstrbuton ˆ π ˆ π f nf Structure of clncal research Plannng Implementaton queston Error Error object θ T Varables to conclusons θ = ( ˆ ) S Eθ <mthods n epdemology Medcal statstcs- Samplng dstrbuton Estmate θˆ 4

5 Structure of clncal research Plannng Implementaton queston Error Error object Varables to conclusons <mthods n epdemology Medcal statstcs- Samplng dstrbuton Estmate How can we make statements on parameters based on estmates? Structure of clncal research Plannng Implementaton queston Error Error object θ T Varables to conclusons θ = ( ˆ ) S Eθ <mthods n epdemology Medcal statstcs- Samplng dstrbuton Estmate θˆ 5

6 Varablty of estmates Estmates of the study effect change from one study to another s only one of the many s that can be studed The observed estmate θˆ of the study effect s only one of the many estmates that can be observed θˆ Is heght dfferent between genders? Women µ d = cm Men µ u = cm Frequenza Frequenza Altezza (cm) Altezza (cm) The true mean dfference (δ ) of heght between men and women n the whole µ ) s equal to 12.7 cm ( u µ d Students of medcne frst year

7 Is heght dfferent between genders? Women d = cm µˆ µˆ u Men = cm Frequenza Frequenza Altezza (cm) Altezza (cm) The mean dfference of heght ( ˆ δ ) between men and women n the st study ˆ µ ˆ ) s equal to 11.8 cm ( u µ d Whch s the mean gender dfference n heght? Stem and Leaf ˆ µ ˆ ( u µ d ) Stem and leaf dsplay of mean dfferences of heght between men and women ( ˆ µ u ˆ µ d ) n 46 s of 20 students Leafs denote the frst decmal dgt All s are drawn at random and are representatve. 7

8 Varablty of estmates Estmates of the study effect change from one study to another s only one of the many s that can be studed The observed estmate of the study effect s only one of the many estmates that can be observed Samplng dstrbuton s the probablty dstrbuton of all hypothetcal estmates that may be observed as a result of random varablty around the parameter E( ) θˆ We use nformaton from samplng dstrbuton to nfer conclusons on the true effect based on the sngle estmate actually observed θˆ Structure of clncal research Plannng Implementaton queston Error Error object θ T Varables to conclusons θ = ( ˆ ) S Eθ <mthods n epdemology Medcal statstcs- Samplng dstrbuton Samplng dstrbuton Estmate θˆ 8

9 Samplng dstrbuton In most cases the dstrbuton of the means s nearly normal as long as the s are large enough ( ) ES E( θˆ ) E s the mean of the means and s equal to the θˆ θˆ mean Standard error (SE) s a measure of the varablty of estmates θˆ around the mean E( ). SE s θˆ proportonal to SD and n nverse proporton to sze ( ) E θˆ θ S e SE are the parameters of the normal samplng dstrbuton θˆ Standard devaton (SD) and standard error (SE) Please dstngush between the standard error (SE) of the means from the standard devaton (SD) of the observatons x j θˆ θˆ DS x j SE SD Sample means θˆ are less varable around the mean than the ndvdual observatons x j ES E( θˆ ) θˆ 9

10 NO! 10

11 Some examples ( θˆì ) E( θ ˆ ) ES ( θˆì ) µˆ µ ˆ µ ˆ µ = ˆ δ A B µ µ = δ A B σˆ 2 n 1 1 σˆ 2 p + na nb Independent data ˆ µ ˆ µ = ˆ δ A B µ µ = δ A B 2 ˆ σ dff n Pared data πˆ ˆ π ˆ A π B π ˆ π (1 ˆ) π n 1 1 π A π B πˆ (1 ˆ) π + n A n B What happens f sze changes? Dstrbuton of means n = 40 n = 20 n = Varablty of means s less among the means of large s than small s µ µˆ 11

12 Standard normal dstrbuton Mean = 0 ; varance = 1 SE=1 All normal dstrbutons may be transformed to standard normal dstrbuton 99,7% 95,4% 68,3% 0 z ( ˆ θ ) ˆ θ E z = SE( ˆ θ ) ˆ µ E( ˆ µ ) 2 σ n e.g. for the arthmetc mean Standard normal dstrbuton (Tavola B Jekel et al.) 12

13 Standard normal dstrbuton 99,7% 95,4% 68,3% 0 ES=1 One taled z P(z > z) Due code P(z > z) o P(-z < -z) 0 0, ,159 0,683 1,282 0,10 0,20 1,645 0,05 0,10 1,96 0,025 0,05 2,326 0,01 0,02 2,576 0,005 0,01 3,09 0,001 0,002 3,291 0,0005 0,001 Standard normal dstrbuton P=0.025 P=0.025 One taled z P(z > z) Two taled P(z > z) o P(-z < -z) 0 0, ,159 0,317 1,282 0,10 0,20 1,645 0,05 0,10 1,96 0,025 0,05 2,326 0,01 0,02 2,576 0,005 0,01 3,09 0,001 0,002 3,291 0,0005 0,001 13

14 Normal approxmaton to bnomal (x successes n n experments) Approxmaton s better as the number of subjects ncreases [Np>5 ; N(1-p)>5] Samplng dstrbutons Dscrete Probabltes refer to ndvdual possble x values Bnomal, Posson, Total sum s equal to 1 Contnual Probabltes refer to ntervals of x values Normal, Student s t, ch-square, Total area under the curva s equal to 1 14

15 Other contnual samplng dstrbutons 2 χ (Table D Jekel et al.) Student s t (Tavola C Jekel et al.) Both depend on degrees of freedom 15