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
dt =
s d
mean differencet =
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 paired-sample t-test will be more powerful than the “regular” t- test
-If there is no correlation then the unpaired test will be more powerful
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;
-Example code for paired test
-make sure they line up by appropriate pairing unit
significance level (alpha) surmised effect (difference)
variability sample size
To calculate the power of a test you must know:
To calculate needed sample size you must know:
significance level (alpha)power
surmised effect (difference)variability
Power and sample sizes of t-tests
a priori
a priorior 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
n=s2
2(t(1or2),df + t (1)df)2
But you don’t know theseBecause you don’t know n!
Only one variable can be missing
--Iterative process. Start with a guess and continue with additional guesses, when doing by hand Or
--tricky let computer do the workSAS or many on-line calculatorsdemo
-- need good estimate of s2
Where should this come from?
Example: weight change (g) in rats that were forced to exerciseData: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
n=
1.5682
(1)2
(tcritical 0.05 for df=19 + tcritical 0.1 for df=19)2
n=
1.5682
(1)2
(2.09 + 1.328)2
2 tailed here, but could be one tailed
always one tailed
n= 1.5682 * (3.418)2
n= 18.3Can repeat with df= 18 etc…….
to use other “analyst” functions must have read in data set
Difference you want to detect
Calc from variance
Very useful in planning experiments- even 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 (1or2),df t (1)df s2 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=1.5682N=12
What is the probability of finding a true difference of at lease 1g in this example?
df = 11
Find the closest value, is approximate because table not “fine grained”
If > 0.25, then power < 0.75