Nonparametric Statistics Previously we have done inferences relative to a population parameter Quantitative data - Population Mean - µ Qualitative data - Population Proportion - p Now we infer location in a general sense without a parameter We have Ordinal data or Quantitative data not normally distributed. Rank-Sum Tests - Use the observations only to assign ranks to the data Ex: "Dance Fever" Couple Judge 1 2 3 A 82 84 86 B 88 82 85 C 82 92 72 Total 252 258 243

Two-Sample Rank-Sum Test Wilcoxon - Rank Observations from Low to High - Assign 1 to Min Obs Mann-Whitney - Rank Observations from High to Low - Assign 1 to Max Obs (Note: The author ranks from Low to High which is incorrect) WSU St Mary 95 91 78 76 81 80 Wwsu = 93 75 86 88 77 71 WStMary = 92 83 79 90 89 72 94 82

Mann-Whitney U Test: If n1 ≥ 10 and n2 ≥ 10, the Rank Sums are Normally Distributed H0: Loc(WSU) = Loc(StMary) HA: Loc(WSU) > Loc(StMary) R: Z > 1.645

Ordinal Data: Likert Scale - 1-Strongly Favor 2-Favor 3-Neutral 4-Oppose 5-Strongly Oppose Likert Male M-Rank F-Rank Female Rank Ave Rank 1 SF 10 5 2 F 35 25 3 N 40 35 4 O 10 25 5 SO 5 10 100 100 H0: Loc(M) = Loc (F) HA: Loc(M) ≠ Loc (F) R: Z > 1.96 Z < -1.96

Correction for Ties: (More then 25% Obs Involved in Ties) For this type data we should also do a second test: Homogenity of Data Male Female SA 10 30 A 20 20 N 40 0 D 20 20 SD 10 30 H0: Data Homogeneous HA: Data Not Homogeneous R: χ2 > χ2α,df=k-1 χ2 > 9.49

Kruskal-Wallis Test - k groups of Observations 1 2 3 … k Use Observations to X11 X12 X13 X1k Assign Ranks to Data X21 X22 X23 X2k … … … … W1 W2 W3 Wk Rank Sums H0: All Group Locations Equal HA: Not All Locations Equal R: K > χ2α,df=k-1

Example 3: 1 2 3 50 67 50 48 72 44 53 71 43 48 74 45 51 66 43 H0: Loc(1)=Loc(2)=Loc(3) HA: Not All Loc Equal R: K > χ2.05,df=2 K > 5.99

WSU-StMary Example Wwsu = 83 WStMary = 127 H0: Loc(WSU)=Loc(StMary) HA: Loc(WSU)≠Loc(StMary) R: K > χ2.10,df=1 K > 2.706

Cookie Mix Example: Income Pre-Mix Mix Own Ranks Ave Rank <30K 24 16 161-200 180.5 30-60K 100 20 41-160 100.5 >60K 36 4 1-40 20.5 H0: Both Locations Equal HA: Not Equal R: K > χ2.05,df=1 K > 3.84