Download - Confidence interval
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CONFIDENCE INTERVAL
Dr.RENJU
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OVERVIEW INTRODUCTION
CONFIDENCE INTERVAL
CONFIDENCE LEVEL
CONFIDENCE LIMITS
HOW TO SET?
FACTORS – SET
SIGNIFICANCE
APPLICATIONS
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INTRODUCTION
Statistical parameter
Descriptive statistics :
Describe what is there in our data
Inferential statistics :
Make inferences from our data to more general conditions
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Inferential statistics
Data taken from a sample is
used to estimate a population
parameter
Hypothesis testing (P-values) Point estimation (Confidence intervals)
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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
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ISSUE ???
Variation in sample statistics
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SOLUTION
Estimating a population parameter with a confidence interval
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CONFIDENCE INTERVAL
A range of values so constructed that there is a specified probability of including the true value of a parameter within it
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CONFIDENCE LEVEL
Probability of including the true value of a parameter within a confidence interval Percentage
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CONFIDENCE LIMITS
Two extreme measurements within which an observation lies
End points of the confidence interval
Larger confidence – Wider interval
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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
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HOW TO SET
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CONCEPTS
NORMAL DISTRIBUTION CURVE
MEAN ( µ )
STANDARD DEVIATION (SD)
RELATIVE DEVIATE (Z)
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NORMAL DISTRIBUTION CURVE
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Perfect symmetrySmoothBell shaped
Mean (µ)MedianMode
SD(σ) - 1
Area - 1
0
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RELATIVE DEVIATE (Z)
Distance of a value (X) from mean value (µ) in units of standard deviation (SD)
Standard normal variate
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Z =x – µ SD
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CONFIDENCE LIMITS
From µ - Z(SD)
To µ + Z(SD)
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CONFIDENCE INTERVAL
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FACTORS – TO SET CI
Size of sample
Variability of population
Precision of values
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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
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SKEWED DISTRIBUTION
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VARIABILITY OF POPULATION
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POPULATION STATISTICS
Repeated samples Different means Standard normal curve
Bell shape
Smooth
Symmetrical
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POPULATION STATISTICS
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Population mean (µ) Standard error - Sampling
(SD/√n)
Z = x – µ SD/√n
Confidence limits
From µ - Z(SE)
To µ + Z(SE)
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95%
95% sample means are within 2 SD of population mean
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PRECISION OF VALUES
Greater precision Narrow confidence interval
Larger sample size
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PRECISION OF VALUES
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SIGNIFICANCE
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95% Significance
Observed value within 2 SD of true value
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CONFIDENCE INTERVAL AND Α ERROR
Type I error Two groups
Significant difference is detected Actual – No difference exists False Positive
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Confidence level is usually set at 95%
(1– ) = 0.95
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MARGIN OF ERROR
n
σzME α/ 2 x
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Margin of error
Reduce the SD (σ↓)
Increase the sample size (n↑)
Narrow confidence level (1 – ) ↓
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P VALUE
95% CI corresponds to hypothesis testing with P <0.05
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SIGNIFICANCE
If CI encloses no effect,
difference is non significant
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P value – Statistical significance
Confidence Interval – Clinical significance
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APPLICATIONS
CLINICAL TRIALS
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Margin of error
Increase the sample size
Reduce confidence level
Dynamic relation
Confidence intervals and
sample size
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EXAMPLE
Series of 5 trials
Equal duration
Different sample sizes
To determine whether a novel
hypolipidaemic agent is
better than placebo in
preventing stroke
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Smallest trial 8 patients
Largest trial 2000 patients
½ of the patients in each trial – New
drug
All trials - Relative risk reduction by
50%
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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
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MORE CONFIDENT - LARGER TRIALS
CI - Range within which the true effect of test drug might plausibly lie in the given trial data
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Greater precision
Narrow confidence intervals
Large sample size
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THERAPEUTIC DECISIONS
Recommend for or against therapy ?
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Minimally Important Treatment Effect Smallest amount of benefit that would justify therapy
Points
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Uppermost point of the bell curve
Observed effect
Point estimate
Observed effect
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Tails of the bell curve
Boundaries of the 95% confidence interval
Observed effect
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TRIAL 1
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TRIAL 2
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CI overlaps the smallest treatment benefit Not Definitive Need narrower Confidence interval
Larger sample size
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TRIAL 3
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TRIAL 4
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CI overlaps the smallest treatment benefit Not Definitive Need narrower Confidence interval
Larger sample size
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CONFIDENCE INTERVALS FOR EXTREME PROPORTIONS
Proportions with numerator – 0 Proportions approaching - 1
Proportions with numerators very close to the corresponding denominators
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NUMERATOR - 0
Rule of 3
Proportion – 0/n
Confidence level – 95%
Upper boundary – 3/n
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EXAMPLE 20 people – Surgery None had serious complications
Proportion 0/20 3/n – 3/20 15%
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PROPORTIONS APPROACHING - 1
Translate 100% into its complement
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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%
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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%
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NUMERATORS VERY CLOSE TO THE DENOMINATORS
Rule
Numerator
X
1 52 73 94 10
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95% Confidence level
Upper boundary –
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CONCLUSION
Confidence interval
Confidence level
Confidence limits
95%
Observed value within 2 SD
Population statistics
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THANK YOU