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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
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Comparing Two Groups of Data
Are the MEANS or MEDIANS the same or different?
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Matched Pairs Tests Is there a direct relation between each observation in the first group and its equivalent in the second group?
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Pairing in Environmental Studies Typically by Time or Location
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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)
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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
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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
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Signed Rank Test Null Hypothesis: Median Difference = 0
Alt. Hypothesis: Median Difference ≠ (two-sided) Alt. Hypothesis: Median Difference > 0 Median Difference < (one-sided)
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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
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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
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Frequency Distribution Signed Rank Statistic, W
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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
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Computing Paired t Test
Compute differences: di = xi – yi Compute mean difference: Dm Test Statistic Compute p value from t table or software
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Dm = 0 Computing Paired t Test Dm = 8.3
If the null hypothesis is true, what value of Dm would you expect? Dm = 0
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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
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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
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