Transcript paired ttests and power

Paired-Sample Hypotheses
-Two sample t-test assumes samples are independent
-Means that no datum in sample 1 in any way associated
with any specific datum in sample 2
-Not always true
Ex:
Are the left fore and hind limbs of deer equal?
1) The null (xbarfore = xbarhind) might not be true,
meaning a real difference between fore and hind
2) Short / tall deer likely to have similarly short /tall fore
and hind legs
Examples of paired means
NPP on sand and rock from a group of mesocosms
Sand
NPP
Rock
NPP
*******Will give code later, you can try if you want
Examples of paired means
Do the scores from the first and second exams in a
class differ? Paired by student.
More……..
Don’t use original mean, but the difference within
each pair of measurements and the SE of those
differences
d
t=
sd
mean difference
t=
SE of differences
- Essentially a one sample t-test
-  = n-1
Paired-Sample t-tests
-Can be one or two sided
-Requires that each datum in one sample correlated
with only one datum in the other sample
-Assumes that the differences come from a normally
distributed population of differences
-If there is pariwise correlation of data, the pairedsample t-test will be more powerful than the “regular”
t- test
-If there is no correlation then the unpaired test will be
more powerful
-Example code for paired test
-make sure they line up by appropriate pairing unit
data start;
infile ‘your path and filename.csv' dlm=',' DSD;
input tank $ light $ ZM $ P $ Invert $ rockNPP sandNPP;
options ls=100;
proc print;
data one; set start;
proc ttest;
paired rockNPP*sandNPP;
run;
Power and sample sizes of t-tests
To calculate needed sample size you must know:
significance level (alpha)
power
surmised effect (difference)
variability
a priori
To calculate the power of a test you must know:
significance level (alpha)
surmised effect (difference)
variability
sample size
a priori
or retrospective
See sections 7.5-7.6 in Zar, Biostatistical Analysis for references
Power and sample sizes of t-tests
To estimate n required to find a difference, you need:
-- , frequency of type I error
-- , frequency of type II error; power = 1- 
-- , the minimum difference you want to find
--s2, the sample variance
Only one variable can be missing
n=
s2
2
(t(1or2),df + t (1)df)2
But you don’t know these
Because you don’t know n!
--Iterative process. Start with a guess and continue
with additional guesses, when doing by hand
Or
--tricky let computer do the work
SAS or many on-line calculators
demo
-- need good estimate of s2
Where should this come from?
Example: weight change (g) in rats that were forced to
exercise
Data:
1.7, 0.7, -0.4, -1.8, 0.2, 0.9, -1.2, -0.9, -1.8, -1.4,-1.8,-2.0
Mean= -0.65g
--s2=1.5682
--Find diff of 1g
--90% chance of detecting difference (power)
power=1- 
= 0.1 (always 1 sided)
--=0.05, two sided
Start with guess that N must =20, df=19
n=
s2
2
(t(1or2),df + t (1)df)2
2 tailed here, but could be one tailed
n=
1.5682
(tcritical 0.05 for df=19 + tcritical 0.1 for df=19)2
always one tailed
(1)2
1.5682
n=
(2.09 + 1.328)2
(1)2
n= 1.5682 * (3.418)2
n= 18.3
Can repeat with df= 18 etc…….
In SAS open solutions  analysis  analyst
Statistics  one-sample t-test (or whichever you want)
Difference you want to detect
Calc from variance
to use other “analyst”
functions must have
read in data set
Increase minimum difference you care
about, n goes down. Easier to detect big
difference
Very useful in planning experimentseven if you don’t have exact values for
variance….. Can give ballpark estimates
(or at least make you think about it)
Calculate power (probability of correctly rejecting
false null) for t-test

t  (1)df = s2 - t (1or2),df
n
--Take this value from t table
Back to the exercising rats…….
Data:
1.7, 0.7, -0.4, -1.8, 0.2, 0.9, -1.2, -0.9, -1.8, -1.4,-1.8,-2.0
Mean= -0.65g
--s2=1.5682
N=12
What is the probability of finding a true difference of at
lease 1g in this example?

t  (1)df
=
s2
n
- t (1or2),df
1
t  (1)11
=
t  (1)11
= 2.766- 2.201
t  (1)11
= 0.57
1.5682
12
- 2.201
Find the closest
value,  is
approximate
because table not
“fine grained”
df = 11
If  > 0.25, then
power < 0.75
--Can use SAS Analyst and many other packages
(e.g. JMP,………) to calculate more exact power
values
--For more complicated designs….. Seek professional
advise!