PairedSample Hypotheses Two sample ttest assumes samples are

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 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 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 $ rock. NPP sand. NPP; options ls=100; proc print; data one; set start; proc ttest; paired rock. NPP*sand. NPP; 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 --s 2, the sample variance Only one variable can be missing n= s 2 2 (t (1 or 2), 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 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. 65 g --s 2=1. 5682 --Find diff of 1 g --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= s 2 2 (t (1 or 2), 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 = s 2 - t (1 or 2), 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. 65 g --s 2=1. 5682 N=12 What is the probability of finding a true difference of at lease 1 g in this example?

t (1)df = s 2 n - t (1 or 2), df 1 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!