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…….

In SAS open solutions analysis analyst

Statistics one-sample t-test (or whichever you want)

to use other “analyst” functions must have read in data set

Difference you want to detect

Calc from variance

Increase minimum difference you care about, n goes down. Easier to detect big difference

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?

=

- t (1or2),df t (1)df s2 n

=

1

- 2.201t (1)11 1.5682

12

= 2.766- 2.201t (1)11

= 0.57t (1)11

df = 11

Find the closest value, is approximate because table not “fine grained”

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!