1. Chapter 4 Comparing Two Groups of Data
BAE 6520 Applied Environmental Statistics
Biosystems and Agricultural Engineering Department
Division of Agricultural Sciences and Natural Resources
Oklahoma State University
Source
Dr. Dennis R. Helsel & Dr. Edward J. Gilroy
2006 Applied Environmental Statistics Workshop
and
Statistical Methods in Water Resources

2. Comparing Two Groups of Data
Are the
MEANS or MEDIANS
the same or different?

3. Matched Pairs Tests
Is there a direct relation between each observation in the first group and its equivalent in the second group?

4. Pairing in Environmental Studies Typically by Time or Location

5. Sign Test Null Hypothesis: Prob [x>y] = 1/2
Alt. Hypothesis: Prob [x>y] ≠ 1/ (two-sided)
Alt. Hypothesis: Prob [x>y] > 1/2
Prob [x>y] < 1/2
(one-sided)

6. Computing Sign Test Compute differences, di = xi – yi
Record sign of di, + or –
Test Statistic: S = # of positive differences
Compute p value of S from table or software

7. If the null hypothesis is true, how many S’s would you expect?
Computing Sign Test
S = 5
If the null hypothesis is true, how many S’s would you expect?
S = 3

8. Signed Rank Test Null Hypothesis: Median Difference = 0
Alt. Hypothesis: Median Difference ≠ (two-sided)
Alt. Hypothesis: Median Difference > 0
Median Difference < (one-sided)

9. Computing Signed Rank Test
Compute differences, di = xi – yi
Rank differences in order of absolute value
Signed Rank – put algebraic sign on rank based on difference
Test Statistic W = sum of positive signed ranks
Compute p value of W from table or software

10. S = (1+2+3+4+5+6)/2 = 10.5 Computing Signed Rank Test
W = = 16
If the null hypothesis is true, how value for W would you expect?
S = ( )/2 = 10.5

11. Frequency Distribution Signed Rank Statistic, W

12. Paired t Test Null Hypothesis: Mean Difference = 0
Alt. Hypothesis: Mean Difference ≠ (two-sided)
Alt. Hypothesis: Mean Difference > 0
Mean Difference < (one-sided)
Assumption
Both groups follow a normal distribution

13. Computing Paired t Test
Compute differences: di = xi – yi
Compute mean difference: Dm
Test Statistic
Compute p value from t table or software

14. Dm = 0 Computing Paired t Test Dm = 8.3
If the null hypothesis is true, what value of Dm would you expect?
Dm = 0

15. Two Sample t Test Null Hypothesis: Mean X = Mean Y
Alt. Hypothesis: Mean Difference ≠ (two-sided)
Alt. Hypothesis: Mean Difference > Mean Difference < (one-sided)
Assumptions
Both groups follow a normal distribution
Both groups have the same variance

16. Computing Two Sample t Test
Test Statistic
If null hypothesis is true, t will be close to zero
If null hypothesis is false, | t | will be large