confidence interval

CONFIDENCE INTERVAL

Dr.RENJU

OVERVIEW INTRODUCTION

CONFIDENCE INTERVAL

CONFIDENCE LEVEL

CONFIDENCE LIMITS

HOW TO SET?

FACTORS – SET

SIGNIFICANCE

APPLICATIONS

INTRODUCTION

Statistical parameter

Descriptive statistics :

Describe what is there in our data

Inferential statistics :

Make inferences from our data to more general conditions

Inferential statistics

Data taken from a sample is

used to estimate a population

parameter

Hypothesis testing (P-values) Point estimation (Confidence intervals)

POINT ESTIMATE Estimate obtained from a sample

Inference about the population

Point estimate is only as good as the sample it represents

Random samples from the population - Point estimates likely to vary

ISSUE ???

Variation in sample statistics

SOLUTION

Estimating a population parameter with a confidence interval

CONFIDENCE INTERVAL

A range of values so constructed that there is a specified probability of including the true value of a parameter within it

CONFIDENCE LEVEL

Probability of including the true value of a parameter within a confidence interval Percentage

CONFIDENCE LIMITS

Two extreme measurements within which an observation lies

End points of the confidence interval

Larger confidence – Wider interval

A point estimate is a single number A confidence interval contains a certain set of possible values of the parameter

Point EstimateLower Confidence Limit

UpperConfidence Limit

Width of confidence interval

HOW TO SET

CONCEPTS

NORMAL DISTRIBUTION CURVE

MEAN ( µ )

STANDARD DEVIATION (SD)

RELATIVE DEVIATE (Z)

NORMAL DISTRIBUTION CURVE

Perfect symmetrySmoothBell shaped

Mean (µ)MedianMode

SD(σ) - 1

Area - 1

0

RELATIVE DEVIATE (Z)

Distance of a value (X) from mean value (µ) in units of standard deviation (SD)

Standard normal variate

Z =x – µ SD

CONFIDENCE LIMITS

From µ - Z(SD)

To µ + Z(SD)

CONFIDENCE INTERVAL

FACTORS – TO SET CI

Size of sample

Variability of population

Precision of values

SAMPLE SIZE

Central Limit Theorem

“Irrespective of the shape of the underlying distribution, sample mean & proportions will approximate normal distributions if the sample size is sufficiently large”

Large sample – Narrow CI

SKEWED DISTRIBUTION

VARIABILITY OF POPULATION

POPULATION STATISTICS

Repeated samples Different means Standard normal curve

Bell shape

Smooth

Symmetrical

POPULATION STATISTICS

Population mean (µ) Standard error - Sampling

(SD/√n)

Z = x – µ SD/√n

Confidence limits

From µ - Z(SE)

To µ + Z(SE)

95%

95% sample means are within 2 SD of population mean

PRECISION OF VALUES

Greater precision Narrow confidence interval

Larger sample size

PRECISION OF VALUES

SIGNIFICANCE

95% Significance

Observed value within 2 SD of true value

CONFIDENCE INTERVAL AND Α ERROR

Type I error Two groups

Significant difference is detected Actual – No difference exists False Positive

Confidence level is usually set at 95%

(1– ) = 0.95

MARGIN OF ERROR

n

σzME α/ 2 x

Margin of error

Reduce the SD (σ↓)

Increase the sample size (n↑)

Narrow confidence level (1 – ) ↓

P VALUE

95% CI corresponds to hypothesis testing with P <0.05

SIGNIFICANCE

If CI encloses no effect,

difference is non significant

P value – Statistical significance

Confidence Interval – Clinical significance

APPLICATIONS

CLINICAL TRIALS

Margin of error

Increase the sample size

Reduce confidence level

Dynamic relation

Confidence intervals and

sample size

EXAMPLE

Series of 5 trials

Equal duration

Different sample sizes

To determine whether a novel

hypolipidaemic agent is

better than placebo in

preventing stroke

Smallest trial 8 patients

Largest trial 2000 patients

½ of the patients in each trial – New

drug

All trials - Relative risk reduction by

50%

QUESTIONS In each individual trial, how

confident can we be regarding

the relative risk reduction

Which trials would lead you to

recommend the treatment

unequivocally to your patients

MORE CONFIDENT - LARGER TRIALS

CI - Range within which the true effect of test drug might plausibly lie in the given trial data

Greater precision

Narrow confidence intervals

Large sample size

THERAPEUTIC DECISIONS

Recommend for or against therapy ?

Minimally Important Treatment Effect Smallest amount of benefit that would justify therapy

Points

Uppermost point of the bell curve

Observed effect

Point estimate

Observed effect

Tails of the bell curve

Boundaries of the 95% confidence interval

Observed effect

TRIAL 1

TRIAL 2

CI overlaps the smallest treatment benefit Not Definitive Need narrower Confidence interval

Larger sample size

TRIAL 3

TRIAL 4

CI overlaps the smallest treatment benefit Not Definitive Need narrower Confidence interval

Larger sample size

CONFIDENCE INTERVALS FOR EXTREME PROPORTIONS

Proportions with numerator – 0 Proportions approaching - 1

Proportions with numerators very close to the corresponding denominators

NUMERATOR - 0

Rule of 3

Proportion – 0/n

Confidence level – 95%

Upper boundary – 3/n

EXAMPLE 20 people – Surgery None had serious complications

Proportion 0/20 3/n – 3/20 15%

PROPORTIONS APPROACHING - 1

Translate 100% into its complement

EXAMPLE Study on a diagnostic test 100% sensitivity when the test is performed for 20 patients who have the disease.

Test identified all 20 with the disease as positive – 100%

No falsely negatives – 0%

95% Confidence level

Proportion of false negatives - 0 /20

3/n rule

Upper boundary - 15% (3 /20 )

Sensitivity

Lower boundary

Subtract this from 100%

100 – 15 = 85%

NUMERATORS VERY CLOSE TO THE DENOMINATORS

Rule

Numerator

X

1 52 73 94 10

95% Confidence level

Upper boundary –

CONCLUSION

Confidence interval

Confidence level

Confidence limits

95%

Observed value within 2 SD

Population statistics

THANK YOU