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 (xbar fore = xbar hind ) 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 = s d 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 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; inputtank $ 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 or retrospective See sections 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 --s 2, the sample variance n= s2s2 22 (t  (1or2),df + t  (1)df ) 2 But you don’t know these Because 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 work SAS or many on-line calculators demo -- need good estimate of s 2 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 --s 2 = 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= s2s2 22 (t  (1or2),df + t  (1)df ) 2 n= (1) 2 (t critical 0.05 for df=19 + t critical 0.1 for df=19 ) 2 n= (1) 2 ( ) 2 2 tailed here, but could be one tailed always one tailed n= * (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)

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 s2s2 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= N=12 What is the probability of finding a true difference of at lease 1g in this example?

=  - t  (1or2),df t  (1)df s2s2 n = t  (1) = t  (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!