# © aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   1.

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© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   1

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   2 TOOLS WILL NEED  Ordinal scales  Probability (the unit normal table)  Introduction to hypothesis testing  Correlation

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   3 Preview  People have a passion for rankings things  Nile is the longest river in the world  The Labrador retriever is the number one registered dog in the US  English is the fourth common native language in the world (after Chinese, Hindi, and Spanish)  Universitas Indonesia is number 201 top university in the world (THES, 2009)

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   4 Preview  Part of fascination with rank is that they are easy to obtain and they are easy to understand  What is your favorite ice-cream flavor?  Ordinal scales are less demanding and less sophisticated than the interval or ratio scales  easier to use  ordinal scales can cause some problems for statistical analysis

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   5 Preview  Because ordinal data (ranks) provide limited information they must be used and interpreted carefully  Standard statistical methods such as means, t test, or analysis F variance should not be used when data are measured on an ordinal scale

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   6 DATA FROM AN ORDINAL SCALE  Ordinal values (ranks) only tell the direction from one score to another, but provide no information about the distance between scores  In a horse race, for example, we know that the second-place horse is somewhere between the first- and the third-place horses  a rank of second is not necessarily halfway between first and third

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   7 OBTAINING ORDINAL MEASUREMENT 1.Ranks are simpler. “He is little taller than I am” 2.The original score may violate some of the basic assumption that underlie certain statistical procedures.  the homogeneity of variance assumption 3.The original score may have unusually high variance 4.Occasionally, an experiment produce undetermined, or infinite, score

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   8  Rank the following scores 3 44799912 14 34035143 Boy’s score8, 17, 14, 21 Girl’s score18, 25, 23, 21, 34, 28, 32, 30, 13 LEARNING CHECK

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   9 THE MANN-WHITNEY U TEST An Alternative to The Independent-Measures t Test

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   10 THE MANN-WHITNEY U TEST … is designed to use the data from two separate samples to evaluate the difference between two treatment (or two population)  The calculations for this test require that the individual scores in the two samples

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   11 THE MANN-WHITNEY U TEST  A real difference between the two treatments should cause the scores in one sample to be generally larger than the score in the other sample  If the two sample are combined and all the scores placed in rank order on a line, the scores from one sample should be concentrated at one end of the line, and the scores from the other sample should be concentrated at the other end

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   12 The scores from the two samples are clustered at opposite ends of the rank ordering 121734561416789101112151318 In this case, the data suggest a systematic difference between the two treatment (or two samples) Sample from treatment A Sample from treatment B

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   13 THE MANN-WHITNEY U TEST  On the other hand, if there is no treatment difference, the large and small scores will be mixed evenly in the two samples because there is no reason for one set of scores to be systematically larger or smaller than the other

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   14 The scores from the two samples are intermixed evenly along the scale 121734561416789101112151318 In this case, the data indicating no consistent difference between the two treatment (or two samples) Sample from treatment A Sample from treatment B

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   15 THE NULL HYPOTHESIS FOR THE MANN-WHITNEY U TEST  Because the Mann-Whitney test compares two distributions (rather than two means), the hypotheses tend to be somewhat vague H 0 : There is no difference between treatments  therefore, there is no tendency for the ranks in one treatment condition to be systematically higher (or lower) than the ranks in the other treatment condition

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   16 CALCULATION OF THE MANN-WHITNEY U Sample A: 27, 2, 9, 48, 6, 15 Sample B: 71, 63, 18, 68, 94, 8  Combine the two samples and all 12 scores are placed in rank order 2, 6, 8, 9, 15, 18, 27, 48, 63, 68, 71, 94  Each individual in sample A is assigned 1 point every score in sample B that has a higher rank

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   17 CALCULATION OF THE MANN-WHITNEY U RANK ORDERED SCORES POINTS FOR SAMPLE A POINTS FOR SAMPLE B SCORESAMPLE 12A6 points 26A 38B4 points 49A5 points 515A5 points 618B2 points 727A4 points 848A4 points 963B0 points 1068B0 points 1171B0 points 1294B0 points U A + U B = n A n B 30 + 6 = 6(6)

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   18 RANKING WITHOUT SCORES  We assumed we had obtained a score for each individual. However, it is not necessary to have a set previously obtained scores  For example, a researcher could observe a group of 12 preschool children (6 boys and 6 girls) and rank them in terms aggressive behavior

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   19 COMPUTING U FOR LARGE SAMPLES RANK ORDERED SCORES SCORESAMPLE 12A 26A 38B 49A 515A 618B 727A 848A 963B 1068B 1171B 1294B U A = n A n B + n A (n A +1) 2 - Σ R A U B = n A n B + n B (n B +1) 2 - Σ R B

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   20 THE MANN-WHITNEY U … is the smaller U See Table B.9A To be significant for any given n A and n B, the obtained U must be equal to or less than the critical value in the table.

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Reporting in Literature The original scores measured in questionaire score, were rank-ordered and a Mann-Whitney U-test was used to compared the ranks for the n=6 group A versus n=6 group B. The results indicate there is significant difference between group A and group B, U=6, p<.05 one-tailed with the sum of the ranks equal to 27 for group A and 61 for group B. 21

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   22 HYPOTHESIS TESTS WITH THE MANN-WHITNEY U  A large difference between the two treatments (or samples) causes all the ranks from sample A to cluster at one end of the scale all the ranks from sample B to cluster at the other.  At the extreme, there is no overlap between two sample  the Mann-Whitney U will be zero because one of the sample gets no point at all

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   23 Psychosis such as schizophrenia often expressed in the artistic work produced by patients. To test the reliability of this phenomenon, a psychologist collected 10 painting done by schizophrenic patient and another 10 painting by normal college student. A professor in art department was asked to rank order all 20 paintings in term bizarreness. Schizophrenic patents: 1, 3, 4, 5, 6, 8, 9, 11, 12, 14 Student: 2, 7, 10, 13, 15, 16, 17, 18, 19, 20 Test at the.01 level of significance LEARNING CHECK

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   24 THE WILCOXON SIGNED-RANKS T TEST An Alternative to The Repeated-Measures t Test

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   THE WILCOXON SIGNED-RANKS T TEST  … is designed to evaluate the difference between two treatments, using the data from repeated-measures experiment.  The data for the Wilcoxon test consist of the difference scores from repeated-measures design.  The test requires that the difference be ranked from smallest to largest in term of their absolute values. 25

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   43 14 20 48 21 4 +25 +5 +18 +7 -8 621534621534 TreatmentsDifferenceRank ABCDEFABCDEF 18 9 21 30 14 12 Participants  Siapa saja yang nilainya menurun?  Rankingnya berapa saja?  Total ranking mereka yang nilainya turun?

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Hyperactive children often treated with a stimulant such as Ritalin to improve their attention spans. In one test of this drug treatment, hyperactive children were given a boring task to work on. A psychologist recorded the amount of time (in seconds) each child spent on the task before becoming distracted. Each child’s performance was measured before he or she received the drug and again after the drug was administered. 27

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   135 309 150 224 25 345 43 188 15 +107 +294 -33 +176 -5 +112 +22 +78 +3 694827351694827351 TreatmentsDifferenceRank ABCDEFGHIABCDEFGHI 28 15 183 48 30 233 21 110 12 Participants WITHOUT

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   29 The KRUSKAL-WALLIS Test An Alternative to The Independent-Measures ANOVA

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Kruskal-Wallis Test Berbeda dengan analisis Mann-Whitney U Test yang terbatas untuk membandingkan 2 kelompok (treatment) yang terpisah, analisis Kruskal-Wallis H Test digunakan untuk mengevaluasi perbedaan urutan individu dari 3 kelompok atau lebih yang independen (between subjects) 30

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   31 Ilustrasi Penelitian Seorang peneliti ingin mengetahui pengaruh kelembapan (kadar air) udara terhadap performa kerja karyawan dalam mengetik. 3(tiga) kelompok karyawan dipilih secara acak untuk ditempatkan pada 3(tiga) ruangan secara terpisah. Ketiga ruangan tersebut diatur agar memiliki kelembapan udara rendah (60%), sedang (75%), dan tinggi (90%). Kecepatan mengetik diukur dengan urutan menyelesaikan mengetik ulang suatu tulisan yang diberikan peneliti.

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Kruskal-Wallis Test  Apabila hasil pengukuran (original data) DV memiliki varians yang besar dan terdapat undetermined/infinite score, maka akan lebih tepat menggunakan analisis Kruskal-Wallis dibandingkan dengan One-Way ANOVA  Skala pengukuran DV merupakan skala ordinal (rank-order) atau data numerik (skala interval/rasio) diubah dalam bentuk rank-order (ordinal) 32

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Kruskal-Wallis Test Tujuan penelitiannya adalah untuk mengetahui apakah kelompok treatment yang satu akan memiliki ranking yang secara konsisten lebih tinggi (atau lebih rendah) dibandingkan kelompok lainnya. 33

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Hipotesis Kruskal-Wallis Test  H 0 : tidak ada kecenderungan bahwa ranking pada kelompok treatment tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain. Dengan demikian, tidak ada perbedaan yang signifikan di antara kelompok treatment  H A : sedikitnya ranking pada satu kelompok treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain. Dengan demikian, terdapat perbedaan yang signifikan di antara kelompok treatment. 34

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Hipotesis Kruskal-Wallis Test  H 0 : tidak ada kecenderungan bahwa ranking pada kelompok treatment tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain. Dengan demikian, tidak ada perbedaan yang signifikan di antara kelompok treatment  H A : sedikitnya ranking pada satu kelompok treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain. Dengan demikian, terdapat perbedaan yang signifikan di antara kelompok treatment. 35

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 36 Kelembapan Udara RendahSedangTinggi Rank?

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 37 Kelembapan Udara RendahSedangTinggi 1

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 38 Kelembapan Udara RendahSedangTinggi 1 2

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 39 Kelembapan Udara RendahSedangTinggi 1 2 3,5

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 40 Kelembapan Udara RendahSedangTinggi 1 25 3,5

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 41 Kelembapan Udara RendahSedangTinggi 1 25 6 3,5

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 42 Kelembapan Udara RendahSedangTinggi 1 25 6 3,57

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 43 Kelembapan Udara RendahSedangTinggi 91 295 69 3,57

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 44 Kelembapan Udara RendahSedangTinggi 91 295 69 3,57 113,5

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 45 Kelembapan Udara RendahSedangTinggi 91 295 69 3,5712 113,5

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 46 Kelembapan Udara RendahSedangTinggi 91 295 69 3,5712 113,513

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 47 Kelembapan Udara RendahSedangTinggi 91 295 1469 3,5712 113,513

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik) Kelembapan Udara RendahSedangTinggi 5:074:128:25 4:285:074:40 6:394:495:07 4:354:585:43 5:164:356:14 48 Kelembapan Udara RendahSedangTinggi 9115 295 1469 3,5712 113,513

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Analisis Kruskal-Wallis H Test 49 Kelembapan Udara RendahSedangTinggi 9115 295 1469 3,5712 113,513 T 1 = 39,5T 2 = 26,5T 3 = 54 n = 5

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Analisis Kruskal-Wallis H Test 50 Kelembapan Udara RendahSedangTinggi T 1 = 39,5T 2 = 26,5T 3 = 54 n = 5 H = 12 N(N+1) Σ T2T2 n - 3(N+1)H = 12 15(16) + 39,5 2 5 - 3(16) + 26,5 2 5 54 2 5

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Analisis Kruskal-Wallis H Test H = 3,785Signifikan? Table B.8 Chi Square; df = k-1 df = k-1 = 2; critical value = 5,99 3,785 < 5,99  TIDAK Signifikan  Tidak Ada pengaruh kelembapan terhadap kecepatan mengetik

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   52 The FRIEDMAN Test An Alternative to The Repeated-Measures ANOVA

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Kruskal-Wallis Test Berbeda dengan analisis Wilcoxon T Test yang terbatas untuk membandingkan 2 kelompok (treatment) yang terpisah, analisis Friedman Test digunakan untuk mengevaluasi perbedaan urutan individu dari 3 treatment atau lebih dari satu kelompok (within subjects) 53

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   54 Ilustrasi Penelitian Tiara ingin melihat pengaruh pemberian pelatihan Empati terhadap keterampilan berkomunikasi karyawan. Seorang atasan diminta untuk meranking keterampilan berkomunikasi setiap karyawan pada ketiga pengukuran. Pengukuran keterampilan berkomunikasi dilakukan sebelum pemberian traning, 3 bulan setelah training, dan 6 bulan setelah training.

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Friedman Test  Apabila hasil pengukuran (original data) DV memiliki varians yang besar dan terdapat undetermined/infinite score, maka akan lebih tepat menggunakan analisis Friedman Test dibandingkan dengan Repeated-Measures ANOVA  Skala pengukuran DV merupakan skala ordinal (rank-order) atau data numerik (skala interval/rasio) diubah dalam bentuk rank-order (ordinal) 55

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Friedman Test Tujuan penelitiannya adalah untuk mengetahui apakah treatment yang satu akan memiliki ranking yang secara konsisten lebih tinggi (atau lebih rendah) dibandingkan treatment lainnya. 56

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Hipotesis Kruskal-Wallis Test  H 0 : tidak ada kecenderungan bahwa ranking pada treatment tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain. Dengan demikian, tidak ada perbedaan yang signifikan di antara treatment  H A : sedikitnya ranking pada satu treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain. Dengan demikian, terdapat perbedaan yang signifikan di antara treatment. 57

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Hipotesis Kruskal-Wallis Test  H 0 : tidak ada kecenderungan bahwa ranking pada treatment tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain. Dengan demikian, tidak ada perbedaan yang signifikan di antara treatment  H A : sedikitnya ranking pada satu treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain. Dengan demikian, terdapat perbedaan yang signifikan di antara treatment. 58

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Skor Keterampilan Berkomunikasi KaryawanSebelum3-bulan6-bulan A242230 B192528 C223430 D252834 E2029 F262433 59

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Rank Keterampilan Berkomunikasi KaryawanSebelum3-bulan6-bulan A213 B123 C132 D123 E12,5 F213 60

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Rank Keterampilan Berkomunikasi KaryawanSebelum3-bulan6-bulan A213 B123 C132 D123 E12,5 F213 Total811,516,5 61

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Analisis Friedman Test 62 Keterampilan Berkomunikasi RendahSedangTinggi R 1 = 8R 2 = 11,5R 3 = 16,5 n = 6 χ2r=χ2r= 12 nk(k+1) ΣT2ΣT2 - 3n(k+1) 12 (6)(3)(4) (8) 2 + (11,5) 2 + (16,5) 2 - (3)(6)(4) χ2r=χ2r=

© aSup-2007 STATISTICAL TECHNIQUES FOR ORDINAL DATA   Analisis Friedman Test χ 2 r = 6,08Signifikan? Table B.8 Chi Square; df = k-1 df = 2; critical value = 5,99 6,08 > 5,99  SIGNIFIKAN  Ada pengaruh pelatihan terhadap keterampilan berkomunikasi

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