Llsk_confidence Interval Printout

8/11/2019 Llsk_confidence Interval Printout

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Confidence Intervals

Prof. Benjamin HK YipDivision Family MedicineSchool of Public Health and Primary Care

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Overview

Outline! Background! Confidence intervals (CIs)! Examples

Learning Objectives! to understand CI construction! to be able to name 3 factors that affect CIs! to be able to interpret CIs found in literature

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Motivation

! Research/Clinical Questions: Do HK youngadults have a low BMD (L2-L4)?

! Ex: In this particular class, the sample mean ofBMD (L2-L4) is 0.96 g/cm 2.

! Questions:

" How meaningful is this sample mean?" Will you trust this estimates?

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Statistical inference

! Methods for drawing conclusions about a population from a sample data," Parameter estimation and Confidence interval" Hypothesis testing (p-values)

! Question: What allows us to make valid inferencesabout a population based only on a sample ?" Probability (see previous lecture)" Random process, i.e., randomization is the key

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Why should I sample instead of usingthe entire population?

The reasons to avoid using entire population arefollowing:

! Cost ( ! , $) and time! Impractical! Inaccurate

" There is a lot of error to control and monitor

" Lists are rarely up to date.! Random sampling

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Terminology

Population parameter! a quantity that describes a population.

Sample statistics! an estimate of the population parameter

Statistical inference! process of drawing conclusions about a population

based on observations in a sample

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Framework for statistical inference

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Sampling

Inference

Population Sample

Random sampling, study design

Statistical estimation, hypothesis testing

! 2

"

#

x

s2

r

p


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Example of Population, sample and

parameters

Arbitrary population:! Objective: Smoking = > cancer! Population: ???! The underlying truth process, which is

universally true for the Population

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Definitions

Confidence interval, CI! a range of values that probably contain the

population value

Confidence limits! the values that state the boundaries of the

confidence interval

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Construction

Most CIs have the following form:

Sample +/- (critical value)x(SE of sample statistics)statistics

margin of error

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Construction! The sample statistics is point estimate based on sampled

data (eg, sample mean, sample proportion)

! The critical value represents the desired confidence levelbased on distribution theory (normal, t, Poisson).

! The SE of sample statistics is a measure of the precisionof the sample estimation. In case the estimate is about a

mean (central tendency) then it can also be called asStandard error of the mean (SEM). SE differ to SD, butthey are related (see later slides).

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Critical value

! Decide distribution which the desired CI is based on." Continuous: Normal or Student- t" Count: Poisson

" Binary: Binomial! In general, normal (z-table) is the default distribution,

given the sample size is large enough (Central limittheorem).

! Decide type I error rate (

"): Incorrect to claim asignificant results (False positive). In general:

" " = 0.05

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Recall

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Pr ! 1.96 < z < 1.96( )= 0.95


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Population Sample

Mean

UnbiasedEstimator

m or

# Standard deviation

UnbiasedEstimator

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x

SD =1

n ! 1 x

i ! x( )2"

SE and SD

SE =!

N =

SD

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SE vs SD! Standard Deviation tells you the variability of your

data.

! Standard Error of the mean, SEM, tells you howgood is your estimate of the mean (accuracy). Its ingeneral smaller than SD, but dont let this be areason for you to choose to use it!

! Which one to use is depending on the content, youwant to describe the variability of the data or theaccuracy of your mean estimation?

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Probability and Confidence Interval

! From the CLT we know that

! From a N(0,1)-table we have

! Rearranging gives

! Thus, the interval is a 95% CI for !

November 08, 2012Benjamin Yip 19

! SE

~ z ~ N (0, 1)

Pr ! 1.96 < !

SE < 1.96

"#$

%&'

= 0.95

Pr( ! 1.96 " SE < < + 1.96 " SE ) = 0.95

..96.1 e s!


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*Theory behind CI

! Constructing a CI is simple, only need 3components: statistics (e.g., mean), SE of thestatistics, and desired % CI.

! However, the logic behind is more complicated. Itinvolves three type of standard deviation (SD): SD ofthe population parameter, SD of the sample, and SD

of the sampling distribution.

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Only 1 CI missed the true mean.

Indicates the true mean (75mmHg)

*95% CI for the mean diastolic BP for 20simulated studies, 50 subjects in simulation


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An example! Suppose that you would like to know the effect of a

newly developed drug (drug A) and a current drug(drug B) on systolic blood pressure (SBP).

! Let say 35 patients were randomly assigned toreceive drug A and another 35 assigned to drug B.The average (mean) SBP among drug A and drug Bpatients was 107 mmHg (SD=19) and 125 (20)mmHg, respectively.

! Construct 95% CI for each group, do the CIs overlapand what is the interpretation?

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Mean =

z1-" /2 =

SE = SD/sqrt(N) =

95% CI =

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95% CI = mean z1! ! /2 SE = mean z1! ! /2SD

N

95% CI for Drug A


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Graph the CIs

100 110 120 130 140

100 110 120 130 140

Drug A

Drug B

Non-overlapping CIs indicating a true (i.e., signicant) mean difference: Drug A is more effective to lower SBP than drug B.


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In general:

27Sourse: http://www.measuringusability.com/blog/ci-10things.php


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Factors that affect the width of a CI are:

! Targeted confidence level, 1- " (higher % wider CI)

! Sample size, N(larger sample size, shorter CI)

! Variability or standard deviation, # (or SD)(higher SD, wider CI)

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mean z1! ! /2SD

N


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Factors that affect the width of a CI are:Targeted CI, 1- !

! Intuition: a higher confidence interval level withoutimproving data quality means a larger margin of

error.

! As the targeted confidence interval increases, the CIwidth increases, given all other quantities remainunchanged.

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Factors that affect the width of a CI are:Sample size, N

! Intuition: a larger sample size means moreinformation, which implies better inference

! As the sample size increases, the CI widthdecreases, given all other quantities remainunchanged.

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Factors that affect the width of a CI are:Variability, SD

! Intuition: more variability or larger spread meansmore difficult to estimate population value withoutlarge amounts of data

! As the variability increases, the CI width increases,given all other quantities remain unchanged

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5 things to know about CI

1. CI tells you the most likely range of the unknownpopulation statistics (e.g., mean, proportion).

2. CI provides both the location and precision of a

measure3. Three things influence the width of a CI ( " , N, SD)4. Our CI estimated from sample data may or may not

contain the population average.

5.

Overlap in CIs is a quick way to check for statisticalsignificance. However, the term significance ismore related to hypothesis testing.

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