statistical techniques for ordinal data

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STATISTICAL TECHNIQUES FOR ORDINAL DATA

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TOOLS WILL NEED Ordinal scales Probability (the unit normal table) Introduction to hypothesis testing Correlation

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Preview People have a passion for rankings thingsNile is the longest river in the worldThe Labrador retriever is the number one

registered dog in the USEnglish 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)

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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

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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

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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

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OBTAINING ORDINAL MEASUREMENT1. 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

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Rank the following scores3 4 4 7 9 9 9 12

14 3 4 0 3 5 14 3

Boy’s score 8, 17, 14, 21Girl’s score 18, 25, 23, 21, 34, 28, 32, 30, 13

LEARNING CHECK

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THE MANN-WHITNEY U TEST

An Alternative toThe Independent-Measures t Test

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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

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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

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The scores from the two samples are clustered at opposite ends of the rank ordering

1 2 173 4 5 6 14 167 8 9 10 11 12 1513 18

In this case, the data suggest a systematic difference between the two treatment (or

two samples)

Sample from treatment A

Sample from treatment B

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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

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The scores from the two samples are intermixed evenly along the scale

1 2 173 4 5 6 14 167 8 9 10 11 12 1513 18

In this case, the data indicating no consistent difference between the two

treatment (or two samples)

Sample from treatment A

Sample from treatment B

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THE NULL HYPOTHESIS FORTHE MANN-WHITNEY U TEST

Because the Mann-Whitney test compares two distributions (rather than two means), the hypotheses tend to be somewhat vague

H0 : 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

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CALCULATION OF THE MANN-WHITNEY U

Sample A : 27, 2, 9, 48, 6, 15Sample 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

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CALCULATION OF THE MANN-WHITNEY U

RANKORDERED SCORES

POINTS FOR SAMPLE A

POINTS FOR SAMPLE BSCORE SAMPLE

1 2 A 6 points

2 6 A 6 points

3 8 B 4 points

4 9 A 5 points

5 15 A 5 points

6 18 B 2 points

7 27 A 4 points

8 48 A 4 points

9 63 B 0 points

10 68 B 0 points

11 71 B 0 points

12 94 B 0 points

UA + UB = nAnB

30 + 6 = 6(6)

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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

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COMPUTING U FOR LARGE SAMPLES

RANKORDERED SCORES

SCORE SAMPLE

1 2 A

2 6 A

3 8 B

4 9 A

5 15 A

6 18 B

7 27 A

8 48 A

9 63 B

10 68 B

11 71 B

12 94 B

UA = nAnB +nA(nA+1)

2- Σ RA

UB = nAnB +nB(nB+1)

2- Σ RB

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THE MANN-WHITNEY U

… is the smaller USee Table B.9ATo be significant for any given nA and nB,

the obtained U must be equal to or less than the critical value in the table.

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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.

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HYPOTHESIS TESTS WITHTHE 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

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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, 20Test at the .01 level of significance

LEARNING CHECK

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The KRUSKAL-WALLIS Test

An Alternative toThe Independent-Measures ANOVA

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Kruskal-Wallis Test

Berbeda dengan analisis Mann-Whitney U Test yang terbatas untuk membandingkan

2 kelompok (treatment) yang terpisah, analisis Kruskal-Wallis T Test digunakan untuk mengevaluasi perbedaan urutan

individu dari 3 kelompok atau lebih yang independen (between subjects)

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Ilustrasi PenelitianSeorang 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.

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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)

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Kruskal-Wallis TestTujuan penelitiannya adalah untuk

mengetahui apakah kelompok treatment yang satu akan memiliki ranking yang secara

konsisten lebih tinggi (atau lebih rendah) dibandingkan kelompok lainnya.

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Hipotesis Kruskal-Wallis Test H0: 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

HA: 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.

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© aSup-2007


Hipotesis Kruskal-Wallis Test H0: 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

HA: 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. 30

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Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)

Kelembapan Udara

Rendah Sedang Tinggi

5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


Rank?

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


1

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


12

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


12

3,53,5

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


12 5

3,53,5

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


12 5

63,5

3,5

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


12 5

63,5 7

3,5

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


9 12 9 5

6 93,5 7

3,5

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


9 12 9 5

6 93,5 711 3,5

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


9 12 9 5

6 93,5 7 1211 3,5

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


9 12 9 5

6 93,5 7 1211 3,5 13

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


9 12 9 5

14 6 93,5 7 1211 3,5 13

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Kelembapan Udara


5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14

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Kelembapan Udara


9 1 152 9 5

14 6 93,5 7 1211 3,5 13

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Analisis Kruskal-Wallis T Test

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Kelembapan Udara


9 1 152 9 5

14 6 93,5 7 1211 3,5 13

T1 = 39,5 T2 = 26,5 T3 = 54

n = 5 n = 5 n = 5

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Analisis Kruskal-Wallis T Test

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Kelembapan Udara


T1 = 39,5 T2 = 26,5 T3 = 54

n = 5 n = 5 n = 5

T =12

N(N+1)

ΣT2

n-

3(N+1)

T =12

15(16)+

39,52

5- 3(16)+

26,52

5542

5

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Analisis Kruskal-Wallis T TestT = 3,785 Signifikan?

Table B.8 Chi Square; df = k-1

df = k-1 = 2; critical value = 5,993,785 < 5,99 TIDAK Signifikan

Tidak Ada pengaruh kelembapan terhadap kecepatan mengetik

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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)

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Ilustrasi PenelitianTiara 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.

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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)

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Friedman TestTujuan penelitiannya adalah untuk

mengetahui apakah treatment yang satu akan memiliki ranking yang secara konsisten lebih

tinggi (atau lebih rendah) dibandingkan treatment lainnya.

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© aSup-2007


Hipotesis Kruskal-Wallis Test H0: 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

HA: 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. 52

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Hipotesis Kruskal-Wallis Test H0: 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

HA: 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. 53

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Skor Keterampilan Berkomunikasi Karyawan Sebelum 3-bulan 6-bulan

A 24 22 30B 19 25 28C 22 34 30D 25 28 34E 20 29 29F 26 24 33

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Rank Keterampilan Berkomunikasi Karyawan Sebelum 3-bulan 6-bulan

A 2 1 3B 1 2 3C 1 3 2D 1 2 3E 1 2,5 2,5F 2 1 3

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Rank Keterampilan Berkomunikasi Karyawan Sebelum 3-bulan 6-bulan

A 2 1 3B 1 2 3C 1 3 2D 1 2 3E 1 2,5 2,5F 2 1 3

Total 8 11,5 16,5

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Analisis Friedman Test

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Keterampilan Berkomunikasi Rendah Sedang Tinggi

R1 = 8 R2 = 11,5 R3 = 16,5

n = 6 n = 6 n = 6

χ2r=

12nk(k+1)

ΣT2 - 3n(k+1)

H =12

(6)(3)(4)(8)2 + (11,5)2 + (16,5)2 - (3)(6)(4)

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Analisis Friedman Testχ2

r= 6,08 Signifikan?

Table B.8 Chi Square; df = k-1

df = 2; critical value = 5,996,08 > 5,99 SIGNIFIKAN

Ada pengaruh pelatihan terhadap keterampilan berkomunikasi

statistical techniques for ordinal data

Data & Analytics

ordinal data asup

statistical techniques

ordinal data ranks

ordinal scale asup

ordinal scale ordinal

ordinal measurement1

statistical analysis

score asup