confidence limits in statistics

7/28/2019 Confidence Limits in Statistics

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Planning, Performing, and

Publishing Research withConfidence Limits

A tutorial lecture given at the annual meeting of theAmerican College of Sports Medicine, Seattle, June 4 1999.

Will G Hopkins

Physiology and Physical Education

University of Otago

Dunedin NZ

[email protected]


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Outline

Definitions and Mis/interpretations

Planning Sample size

Performing Sample size "on the fly"

Publishing Methods, Results, Discussion

Meta-analysis

Publishing non-significant outcomes

Conclusions

Dis/advantages


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Definitions and Mis/interpretations

Confidence limits: Definitions

"Margin of error" Example: Survey of 1000 voters

Democrats 43%, Republicans 33%

Margin of error is 3% (for a result of 50%...)

Likely range of true value "Likely" is usually 95%.

"True value" = population value

= value if you studied the entire population.

Example: Survey of 1000 voters

Democrats 43% (likely range 40 to 46%)

Democrats - Republicans 10% (likely range 5 to 15%)


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Example: in a study of 64 subjects, the correlation between

height and weight was 0.68 (likely range 0.52 to 0.79).

correlation coefficient

observed

value

0.00

0.50 1

upper

confidence

limit

lower

confidence

limit


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Confidence interval: difference between the upper

and lower confidence limits.

Amazing facts about confidence intervals(for normally distributed statistics)

To halve the interval, you have to quadruple sample size.

A 99% interval is 1.3 times wider than a 95% interval.

You need 1.7 times the sample size for the same width.

A 90% interval is 0.8 of the width of a 95% interval.

You need 0.7 times the sample size for the same width.


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How to Derive Confidence Limits Find a function(true value, observed value, data) with a

known probability distribution.

Calculate a critical value, such that for 2.5% of the time,function(true value, observed value, data) < critical value.

probability

function (e.g. (n-1)s2/2)

area =0.025

critical value

probability

distributionof function(e.g. 2)

Rearranging, for 2.5% of the time,true value > function'(observed value, data, critical value)

= upper confidence limit


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Mis/interpretation of confidence limits

Hard to misinterpret confidence limits for simple

proportions and correlation coefficients. Easier to misinterpret changes in means.

Example: The change in blood volume in a study

was 0.52 L (likely range 0.12 to 0.92 L).

For 95% of subjects, the change was/would be between0.12 and 0.92 L.

The average change in the population would be between

0.12 and 0.92 L.

The change for the average subject would be between

0.12 and 0.92 L. There may be individual differences in the change.


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P value: Definition

The probability of a more extreme absolute value

than the observed value if the true value was zero

or null.

Example: 20 subjects, correlation = 0.25, p = 0.29.

probability


area =p value= 0.29

no effect

observed effect(r = 0.25)

distribution ofcorrelations

for no effectand n = 200 0.5-0.5


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"Statistically Significant": Definitions

P < 0.05

Zero lies outside the confidence interval. Examples: four correlations for samples of size 20.

0.00 0.50 1


-0.50

r likely range P

0.70 0.37 to 0.87 0.007

0.44 0.00 to 0.74 0.05

0.25 -0.22 to 0.62 0.29

0.00 -0.44 to 0.44 1.00


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Incredibly interesting information about statistical

significance and confidence intervals

Two independent estimates of a normally distributed statistic

with equal confidence intervals are significantly different atthe 5% level if the overlap of their intervals is less than 0.29

(1 - 2/2) of the length of the interval. If the intervals are very unequal...

p < 0.05 p = 0.05 p > 0.05

p < 0.05 p = 0.05 p > 0.05


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Type I and II Errors

You could be wrong about significance or lack of it.

Type I error = false alarm. Rate = 5% for zero real effect.

Type II error = failed alarm. Traditional acceptable rate = 20% for smallest worthwhile

effect. Lots of tests for significance implies more chance of

at least one false alarm: "inflated type I error". Ditto type II error?

Deal with inflated type I error by reducing the p value.

Should we adjust confidence intervals? No.


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Mis/interpretation of P > 0.05(for an observed positive effect)

The effect is not publishable.

There is no effect.

The effect is probably zero or trivial.

There's a reasonable chance the effect is < zero.

Mis/interpretation of P < 0.05(for an observed positive effect)

The effect is probably big.

There's a < 5% chance the effect is zero.

There's a < 2.5% chance the effect is < zero.

There's a high chance the effect is > zero.

The effect is publishable.


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Planning Research

Sample Size via Statistical Significance

Sample size must be big enough to be sure you willdetect the smallest worthwhile effect. To be sure: 80% of the time.

Detect: P < 0.05.

Smallest worthwhile effect: what impacts your subjects

correlation = 0.10

relative risk = 1.2 (or frequency difference = 10%)

difference in means = 0.2 of a between-subject standard deviation

change in means = 0.5 of a within-subject standard deviation

Example: 760 subjects to detect a correlation of 0.10. Example: 68 subjects to detect a 0.5% change in a

crossover study when the within-subject variation is 1%.


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But 95% likely range doesn't work properly with

traditional sample-size estimation (maybe).Example: Correlation of 0.06, sample size of 760...

47.5% + 47.5% (=95%) likely range:

0.10


-0.1

Not significant, but

could be substantial.

Huh?

0.10


-0.1

47.5% + 30% likely range:

Not significant, and

can't be substantial.OK!


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Sample Size via Confidence Limits

Sample size must be big enough for acceptable

precision of the effect. Precision means 95% confidence limits.

Acceptable means any value of the effect within these

limits will not impact your subjects.

Example: need 380 subjects to delimit a correlation of

zero.

0 0.10


-0.10

smallest worthwhile

effects confidence

interval for

N = 380


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But sample size needed to detect or delimit

smallest effect is overkill for larger effects.

Example: confidence limits for correlations of 0.10 and0.80 with a sample size of 760...

0.1 0.3 0.5 0.70 0.9 1


-0.1

So why not start with a smaller sample and do

more subjects only if necessary?

Yes, I call it...


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Performing Research

Sample Size "On the Fly"

Start with a small sample; add subjects until you

get acceptable precision for the effect.Acceptable precision defined as before.

Need qualitative scale for magnitudes of effects.

Example: sample sizes to delimit correlations...

155

0.1 0.3 0.5 0.70 0.9 1

trivial small moderate large

270350

380

correlation coefficient-0.1

nearlyperfect

46

very large


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Problems with sampling on the fly

Do not sample until you get statistical significance: theresulting outcomes are biased larger than life.

Sampling until the confidence interval is acceptable

produces bias, but it is negligible.

But researchers will rush into print as soon as they getstatistical significance.

And funding agencies prefer to give money once

(but you could give some back!).

And all the big effects have been researched anyway?No, not really.


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Publishing Research

In the Methods

"We show the precision of our estimates of outcomestatistics as 95% confidence limits (which define the

likely range of the true value in the population from

which we drew our sample)."

Amazingly useful tips on calculating confidence limits Simpledifferences between means: stats program.

Other normally distributed statistics: mean and p value.

Relative risks: stats program.

Correlations: Fisher's z transform.

Standard deviations and other root mean square variations:chi-squared distribution.


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Coefficients of variation: standard deviation of 100x natural log

of the variable. Back transform for CV>5%.

Use the adjustment of Tate and Klett to get shorter intervals for

SDs and CVs from small samples.

21

coefficient of variation (%)

0 3

Example:

coefficient of

variation for 10

subjects in 2 tests

usualadjusted

Ratios of independent standard deviations: F distribution.

R2 (variance explained): convert to a correlation.

Use the spreadsheet at sportsci.org/stats for all the above.

Effect-size (mean/standard deviation): non-centralF distribution or bootstrapping.

Really awful statistics: bootstrapping.


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Bootstrapping (Resampling) for confidence limits Use for difficult statistics, e.g. for grossly non-normal

repeated measures with missing values. Here's how... For a large-enough sample, you can recreate (sort of) the

population by duplicating the sample endlessly.

Draw 1000 samples (of same size as your original) from

this population.

Calculate your outcome statistic for each of these

samples, rank them, then find the 25th and 975th place-

getters. These are the confidence limits.

Problems

Painful to generate. No good for infrequent levels of nominal variables.


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In the Results

In TEXT

Change or difference in meansFirst mention:

...0.42 (95% confidence/likely limits/range -0.09 to 0.93) or

...0.42 (95% confidence/likely limits/range 0.51).

Thereafter:

...2.6 (1.4 to 3.8) or 2.6 ( 1.2) etc. Correlations, relative risks, odds ratios, standard deviations

ratios of standard deviations: can't use because the

confidence interval is skewed:

...a correlation of 0.90 (0.67 to 0.97)...

...a coefficient of variation of 1.3% (0.9 to 1.9)...


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In TABLES Confidence intervals

r likely range

0.70 0.37 to 0.87

0.44 0.00 to 0.74

0.25 -0.22 to 0.62

0.00 -0.44 to 0.44

Variable A

Variable B

Variable C

Variable D

P values

r p

0.70 0.007

0.44 0.05

0.25 0.290.00 1.00

Variable A

Variable B

Variable CVariable D

Asterisks

r

0.70**

0.44*

0.250.00

Variable A

Variable B

Variable CVariable D


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In FIGURES

-10 -5 0 5 10

Change in power (%)

Bars are 95% likely ranges

Told placebo

Not told

Told carbohydrate


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-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12 14

live low

train low

live high

train high

live high

train low

sea level altitude sea level

change in

5000-m

time (%)

training time (weeks)

likely range of

true change


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In the Discussion

Interpret the observed effect and its 95%

confidence limits qualitatively. Example: you observed a moderate correlation, but the

true value of the correlation could be anything between

trivial and very strong.

0.1 0.3 0.5 0.70 0.9 1

trivial small moderate large


-0.1

nearlyperfectvery large


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Meta-Analysis

Deriving a single estimate and confidence interval

for an effect from several studies. Here's how it works for two:

Study 1

Study 2Study 1+2

Equal Confidence Intervals

Study 1

Study 2

Study 1+2

Unequal Confidence Intervals


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Publishing non-significant outcomes

Publishing only significant effects from small-scalestudies leads to publication bias.

Publishing effects with confidence limits regardless

of magnitude is free of bias.

Many smaller studies are probably better than a

few larger ones anyway.

So bully the editor into accepting the paper about

your seemingly inconclusive small-scale study.


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Conclusions

Disadvantages of Statistical Significance

Emphasizes testing of hypotheses.Aim is to detect an effect--effects are zero until provenotherwise.

Have to understand Type I and II errors.

Hard to understand; easy to misinterpret.

Have to consider sample size. Focuses on statistically significant effects.

Advantages of Statistical Significance

Familiar.

All stats programs give p values.

Easy to put asterisks in tables and figures.


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Disadvantages of Confidence Limits

Unfamiliar.

Not always available in stats programs. Cluttersome in tables.

Display in time series can be a challenge.

Advantages of Confidence Limits

Emphasizes precision of estimation.Aim is to del imi t an effect--effects are never zero.

Only one kind of "error".

Meaning is reasonably clear, even to lay readers.

No confusion between significance and magnitude.

Journals now require them.

confidence limits in statistics

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