Statistik Tidak Berparameter

Objektif Pembelajaran Untuk digunakan dalam pengujian hipotesis apabila tidak boleh membuat sebarang anggapan terhadap taburan yang kita ambil Untuk mengetahui ujian untuk taburan bebas yang digunakan dalam keadaan tertentu Untuk menggunakan dan menjelaskan enam jenis pengujian hipotesis tak berparameter Ujian mengetahui kelemahan dan kelebihan ujian tak berparameter

Statistik Berparameter vs Tidak Berparameter Statistik Berparameter adalah teknik statistik berdasarkan kepada andaian berkaitan populasi dimana sampel data adalah dipungut. –Andaian dimana data yang dianalisis adalah dipilih secara rawak dari populasi yang bertaburan normal. –Memerlukan ukuran kuantitatif yang menghasilkan data bertaraf interval atau perkadaran.

Statistik Berparameter vs Tidak Berparameter Statistik Tidak Berparameter adalah berdasarkan andaian yang kurang populasi dan parameter. –Kadangkala dipanggil sebagai statistik “tidak mempunyai taburan”. –Berbagai-bagai jenis statistik tidak berparameter yang ada untuk digunakan dengan data bertaraf nominal atau ordinal.

Kebaikan Teknik Tidak Berparameter Kadangkala tidak terdapat teknik berparameter alternatif untuk digunakan berbanding teknik tidak berparameter. Beberapa ujian tidak berparameter boleh digunakan untuk menganalisis data nominal. Beberapa ujian tidak berparameter boleh digunakan untuk menganalisis data ordinal. Pengiraan statistik tidak berparameter kurang rumit berbanding kaedah berparameter, terutama untuk sampel yang kecil. Pernyataan kebarangkalian yang diperolehi dari kebanyakan ujian tidak berparameter adalah kebarangkalian yang tepat.

Kelemahan Statistik Tidak Berparameter Ujian tidak berparameter boleh membazirkan data jika ujian berparaeter boleh digunakan untuk data tersebut. Ujian tidak berparameter biasanya tidak digunakan dengan meluas dan kurang dikenali berbanding ujian berparameter. Untuk sampel yang besar, pengiraan bagi kebanyakan ujian tidak berparameter boleh mengelirukan.

7 Ujian Larian

Runs Test Test for randomness - is the order or sequence of observations in a sample random or not Each sample item possesses one of two possible characteristics Run - a succession of observations which possess the same characteristic Example with two runs: F, F, F, F, F, F, F, F, M, M, M, M, M, M, M Example with fifteen runs: F, M, F, M, F, M, F, M, F, M, F, M, F, M, F

Runs Test: Sample Size Consideration Sample size: n Number of sample member possessing the first characteristic: n 1 Number of sample members possessing the second characteristic: n 2 n = n 1 + n 2 If both n 1 and n 2 are  20, the small sample runs test is appropriate.

Runs Test: Small Sample Example H 0 : The observations in the sample are randomly generated. H a : The observations in the sample are not randomly generated.  =.05 n 1 = 18 n 2 = 8 If 7  R  17, do not reject H 0 Otherwise, reject H D CCCCC D CC D CCCC D C D CCC DDD CCC R = 12 Since 7  R = 12  17, do not reject H 0

Runs Test: Large Sample If either n 1 or n 2 is > 20, the sampling distribution of R is approximately normal.

Runs Test: Large Sample Example H 0 : The observations in the sample are randomly generated. H a : The observations in the sample are not randomly generated.  =.05 n 1 = 40 n 2 = 10 If  Z  1.96, do not reject H 0 Otherwise, reject H NNN F NNNNNNN F NN FF NNNNNN F NNNN F NNNNN FFFF NNNNNNNNNNNN R = 13 H 0 : The observations in the sample are randomly generated. H a : The observations in the sample are not randomly generated.  =.05 n 1 = 40 n 2 = 10 If  Z  1.96, do not reject H 0 Otherwise, reject H NNN F NNNNNNN F NN FF NNNNNN F NNNN F NNNNN FFFF NNNNNNNNNNNN R = 13

Runs Test: Large Sample Example  Z =  1.96, do not reject H 0

14 Ujian Mann-Whitney U

Mann-Whitney U Test Nonparametric counterpart of the t test for independent samples Does not require normally distributed populations May be applied to ordinal data Assumptions –Independent Samples –At Least Ordinal Data

Mann-Whitney U Test: Sample Size Consideration Size of sample one: n 1 Size of sample two: n 2 If both n 1 and n 2 are  10, the small sample procedure is appropriate. If either n 1 or n 2 is greater than 10, the large sample procedure is appropriate.

Mann-Whitney U Test: Small Sample Example Service HealthEducational Service H 0 : The health service population is identical to the educational service population on employee compensation H a : The health service population is not identical to the educational service population on employee compensation

Mann-Whitney U Test: Small Sample Example  =.05 If the final p-value <.05, reject H 0. W 1 = = 31 W 2 = = 89 CompensationRankGroup H H H H E H H H E E E E E E E

Mann-Whitney U Test: Small Sample Example Since U 2 < U 1, U = 3. p-value =.0011 <.05, reject H 0.

Mann-Whitney U Test: Formulas for Large Sample Case

Incomes of PBS and Non-PBS Viewers PBSNon-PBS 24,50041,000 39,40032,500 36,80033,000 44,30021,000 57,96040,500 32,00032,400 61,00016,000 34,00021,500 43,50039,500 55,00027,600 39,00043,500 62,50051,900 61,40027,800 53,000 n 1 = 14 n 2 = 13 H o : The incomes for PBS viewers and non-PBS viewers are identical H a : The incomes for PBS viewers and non-PBS viewers are not identical

Ranks of Income from Combined Groups of PBS and Non-PBS Viewers IncomeRankGroupIncomeRankGroup 16,0001Non-PBS39,50015Non-PBS 21,0002Non-PBS40,50016Non-PBS 21,5003Non-PBS41,00017Non-PBS 24,5004PBS43,00018PBS 27,6005Non-PBS43, PBS 27,8006Non-PBS43, Non-PBS 32,0007PBS51,90021Non-PBS 32,4008Non-PBS53,00022PBS 32,5009Non-PBS55,00023PBS 33,00010Non-PBS57,96024PBS 34,00011PBS61,00025PBS 36,80012PBS61,40026PBS 39,00013PBS62,50027PBS 39,40014PBS

PBS and Non-PBS Viewers: Calculation of U

PBS and Non-PBS Viewers: Conclusion

25 Ujian Pemeringkatan Tanda Padanan-Pasangan Wilcoxon

Wilcoxon Matched-Pairs Signed Rank Test A nonparametric alternative to the t test for related samples Before and After studies Studies in which measures are taken on the same person or object under different conditions Studies or twins or other relatives

Wilcoxon Matched-Pairs Signed Rank Test Differences of the scores of the two matched samples Differences are ranked, ignoring the sign Ranks are given the sign of the difference Positive ranks are summed Negative ranks are summed T is the smaller sum of ranks

Wilcoxon Matched-Pairs Signed Rank Test: Sample Size Consideration n is the number of matched pairs If n > 15, T is approximately normally distributed, and a Z test is used. If n  15, a special “small sample” procedure is followed. –The paired data are randomly selected. –The underlying distributions are symmetrical.

Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example Family PairPittsburghOakland 11,950 1,760 21,840 1,870 32,015 1,810 41,580 1,660 51,790 1,340 61,925 1,765 H 0 : M d = 0 H a : M d  0 n = 6  =0.05 If T observed  1, reject H 0.

Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example Family PairPittsburghOaklanddRank 11,950 1, ,840 1, ,015 1, ,580 1, ,790 1, ,925 1, T = minimum( T +, T - ) T + = = 18 T - = = 3 T = 3 T = 3 > T crit = 1, do not reject H 0.

Wilcoxon Matched-Pairs Signed Rank Test: Large Sample Formulas

Airline Cost Data for 17 Cities, 1997 and 1999 City dRankCity dRank H 0 : M d = 0 H a : M d  0

Airline Cost: T Calculation

Airline Cost: Conclusion

35 Ujian Kruskal-Wallis

Kruskal-Wallis Test A nonparametric alternative to one-way analysis of variance May used to analyze ordinal data No assumed population shape Assumes that the C groups are independent Assumes random selection of individual items

Kruskal-Wallis K Statistic

Number of Patients per Day per Physician in Three Organizational Categories Two Partners Three or More PartnersHMO H o : The three populations are identical H a : At least one of the three populations is different

Patients per Day Data: Kruskal-Wallis Preliminary Calculations n = n 1 + n 2 + n 3 = = 18 Two Partners Three or More PartnersHMO PatientsRankPatientsRankPatientsRank T 1 = 29T 2 = 52.5T 3 = 89.5 n 1 = 5n 2 = 7n 3 = 6

Patients per Day Data: Kruskal- Wallis Calculations and Conclusion

41 Ujian Friedman

Friedman Test A nonparametric alternative to the randomized block design Assumptions –The blocks are independent. –There is no interaction between blocks and treatments. –Observations within each block can be ranked. Hypotheses – H o : The treatment populations are equal – H a :At least one treatment population yields larger values than at least one other treatment population

Friedman Test

Friedman Test: Tensile Strength of Plastic Housings Supplier 1Supplier 2Supplier 3Supplier 4 Monday Tuesday Wednesday Thursday Friday H o :The supplier populations are equal H a :At least one supplier population yields larger values than at least one other supplier population

Friedman Test: Tensile Strength of Plastic Housings

Supplier 1Supplier 2Supplier 3Supplier 4 Monday3412 Tuesday3214 Wednesday2314 Thursday3214 Friday j R 2 j R

Friedman Test: Tensile Strength of Plastic Housings

48 Korelasi Pemeringkatan Spearman

Spearman’s Rank Correlation Analyze the degree of association of two variables Applicable to ordinal level data (ranks)