1. Planning, Performing, and Publishing Research with Confidence Limits
A tutorial lecture given at the annual meeting of the American College of Sports Medicine, Seattle, June
© Will G Hopkins
Physiology and Physical Education
University of Otago
Dunedin NZ

2. Definitions and Mis/interpretations Planning Performing Publishing
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

3. 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%)

4. correlation coefficient
Example: in a study of 64 subjects, the correlation between height and weight was 0.68 (likely range 0.52 to 0.79).
observed
value
lower
confidence
limit
upper
confidence
limit
0.00
0.50 1
correlation coefficient

5. 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.

6. 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 distribution of function (e.g. 2)
Rearranging, for 2.5% of the time, true value > function'(observed value, data, critical value) = upper confidence limit

7. 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 between 0.12 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.    

8. correlation coefficient
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
correlation coefficient
area = p value
= 0.29
no effect
observed effect
(r = 0.25)
distribution of
correlations
for no effect
and n = 20
0.5
-0.5

9. correlation coefficient
"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
correlation coefficient
-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

10. Incredibly interesting information about statistical significance and confidence intervals
p < 0.05
p = 0.05
p > 0.05
Two independent estimates of a normally distributed statistic with equal confidence intervals are significantly different at the 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

11. 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.

12. 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.     
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.    

13. Sample Size via Statistical Significance
Planning Research
Sample Size via Statistical Significance
Sample size must be big enough to be sure you will detect 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%.

14. But 95% likely range doesn't work properly with traditional sample-size estimation (maybe). Example: Correlation of 0.06, sample size of
47.5% % (=95%) likely range:
0.1
correlation coefficient
-0.1
Not significant, but could be substantial.
Huh?
0.1
correlation coefficient
-0.1
47.5% + 30% likely range:
Not significant, and can't be substantial.
OK!

15. correlation coefficient
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.
0.10
correlation coefficient
-0.10
smallest worthwhile
effects
confidence
interval for
N = 380

16. correlation coefficient
But sample size needed to detect or delimit smallest effect is overkill for larger effects.
Example: confidence limits for correlations of 0.10 and 0.80 with a sample size of
0.1
0.3
0.5
0.7
0.9 1
correlation coefficient
-0.1
So why not start with a smaller sample and do more subjects only if necessary? Yes, I call it...

17. correlation coefficient
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
0.7
0.9 1
trivial
small
moderate
large
270
350
380
correlation coefficient
-0.1
nearly
perfect 46
very large

18. Problems with sampling on the fly
Do not sample until you get statistical significance: the resulting 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 get statistical 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.

19. In the Methods Publishing Research
"We show the precision of our estimates of outcome statistics 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
Simple differences 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.

20. coefficient of variation (%)
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. 2 1
coefficient of variation (%) 3
Example: coefficient of variation for 10 subjects in 2 tests
usual
adjusted
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-central F distribution or bootstrapping.
Really awful statistics: bootstrapping.

21. 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.

22. In the Results
In TEXT
Change or difference in means First mention: (95% confidence/likely limits/range to 0.93) or (95% confidence/likely limits/range ± 0.51). Thereafter: (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)...

23. In TABLES Confidence intervals r likely range 0.70 0.37 to 0.87 0.44
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
0.29
0.00
1.00
Variable A
Variable B
Variable C
Variable D
Asterisks r
0.70**
0.44*
0.25
0.00
Variable A
Variable B
Variable C
Variable D

24. In FIGURES
-10 -5 5 10
Change in power (%)
Bars are 95% likely ranges
Told placebo
Not told
Told carbohydrate

25. 4 sea level altitude sea level 3 live low train low 2 1 change in
5000-m
time (%)
live high train high
likely range of
true change -1 -2
live high train low -3 2 4 6 8 10 12 14
training time (weeks)

26. correlation coefficient
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
0.7
0.9 1
trivial
small
moderate
large
correlation coefficient
-0.1
nearly
perfect
very large

27. 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 2
Study 1+2
Equal Confidence Intervals
Study 1
Study 2
Study 1+2
Unequal Confidence Intervals

28. Publishing non-significant outcomes
Publishing only significant effects from small-scale studies 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.

29. Disadvantages of Statistical Significance
Conclusions
Disadvantages of Statistical Significance
Emphasizes testing of hypotheses.
Aim is to detect an effect--effects are zero until proven otherwise.
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.

30. 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 delimit 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.