### Ch 6: Making Sense of Statistical Significance: Decision Errors

```Ch 6: Making Sense of Statistical
Significance: Decision Errors,
Effect Size, and Power
Pt 2: Sept. 26, 2013
Statistical Power
• Probability that the study will produce a statistically
significant result when the research hypothesis is in
fact true
– That is, what is the power to correctly reject the null?
– Upper right quadrant in decision table
– Want to maximize our chances that our study has the
power to find a true/real result
• Can calculate power before the study using
predictions of means
– or after study using actual means
Statistical Power
•
Steps for figuring power:
1. Gather the needed information: (N=16)
* Mean & SD of comparison distribution (the distrib of means
from Ch 5 – now known as Pop 2)
* Predicted mean of experimental group (now known as Pop 1)
* “Crashed” example:
Pop 1 “crashed group” mean = 5.9
Pop 2 “neutral group/comparison pop”
μ = 5.5,  = .8, m = sqrt (2)/N
m = sqrt[(.8 2) / 16] = .2
Statistical Power
2. Figure the raw-score cutoff point on the
comparison distribution to reject the null
hypothesis (using Pop 2 info)
• For alpha = .05, 1-tailed test (remember we predicted the
‘crashed’ group would have higher fault ratings), z score cutoff
= 1.64.
• Convert z to a raw score (x) = z(m) + μ
x = 1.64 (.2) + 5.5 = 5.83
• Draw the distribution and cutoff point at 5.83, shade area to
right of cutoff point  “critical/rejection region”
Statistical Power
3. Figure the Z score for this same point, but on the
distribution of means for Population 1 (see ex on board)
• That is, convert the raw score of 5.83 to a z score using
info from pop 1.
– Z = (x from step 2 -  from step 1exp group)
m (from step 1)
– (5.83 – 5.9) / .2 = -.35
– Draw another distribution & shade in everything to
the right of -.35
Statistical Power
4. Use the normal curve table to figure the probability
of getting a score higher than Z score from Step 3
•
•
•
Find % betw mean and z of -.35 (look up .35)…
= 13.68%
Add another 50% because we’re interested in area
to right of mean too.
13.68 + 50 = 63.68%…that’s the power of the
experiment.
Power Interpretation
• Our study (with N=16) has around 64% power to find a
difference between the ‘crashed’ and ‘neutral’ groups if it
truly exists.
– Based on our estimate of what the ‘crashed’ mean will
be (=5.9), so if this is incorrect, power will change.
– In decision error table 1-power = beta (aka…type 2
error), so here:
– Alpha?
– Power?
– Beta?
Influences on Power
• Main influences – effect size & N
• 1) Effect size – bigger d  more power
– Remember formula:
1   2
d 

– Bigger difference between the 2 group means, more
power to find the difference (that difference is the
numerator of d)
– Also, the smaller the population standard deviation,
the bigger the effect size (sd is the denominator)
(cont.)
• Figuring power from predicted effect sizes
– Sometimes, don’t know 1 for formula, can
estimate effect size instead (use Cohen’s guidelines:
.2, .5, .8 or -.2, -.5, -.8)
Predicted1  2  (d )()
Example:
Practical Ways of Increasing the Power of a
Planned Study
• Rule of thumb: try for at least 80% power
– Interpretation of 80% power – we have a .8 probability of
finding an effect if one actually exists
• See Table
• 1) Try to increase effect size before the experiment
(increase diffs betw 2 groups)
– Training/no training group – how could you do this?
• 2) Try to decrease pop SD – use standardization so
subjects in 1 group receive same instructions
• 3) Increase N
• 4) Use less stringent signif level (alpha) – but trade-off
in reducing Type 1 error, so usually choose .05 or .01.
• 5) Use a 1-tailed test when possible
```