1. 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

2. 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

3. Examples of paired means Do the scores from the first and second exams in a class differ? Paired by student. More……..

4. 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

5. 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

6. 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

7. 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 7.5-7.6 in Zar, Biostatistical Analysis for references

8. 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

9. --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?

10. 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 =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

11. n= s2s2 22 (t  (1or2),df + t  (1)df ) 2 n= 1.5682 (1) 2 (t critical 0.05 for df=19 + t critical 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= 18.3 Can repeat with df= 18 etc…….

13. In SAS open solutions  analysis  analyst

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

15. to use other “analyst” functions must have read in data set Difference you want to detect Calc from variance

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

18. 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)

19. 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

20. 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?

21. =  - t  (1or2),df t  (1)df s2s2 n = 1 - 2.201 t  (1)11 1.5682 12 =2.766- 2.201t  (1)11 =0.57t  (1)11

22. df = 11 Find the closest value,  is approximate because table not “fine grained” If  > 0.25, then power < 0.75

23. --Can use SAS Analyst and many other packages (e.g. JMP,………) to calculate more exact power values --For more complicated designs….. Seek professional advise!