Analisis Data Kategorik - STK654 (Materi UAS)

Dr. Kusman Sadik, M.Si

Program Studi Magister Statistika Terapan

Departemen Statistika IPB, Semester Ganjil 2019/2020

IPB University─ Bogor Indonesia ─ Inspiring Innovation with Integrity

Model Regresi Logit Ordinal(Peubah Respon Multikategori-Ordinal)

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The main feature of the ordinal logistic models is that

they predict the log odds, odds, or probability of a

response occurring at or below any given outcome

category.

For example, ordering the educational attainment

categories from lowest to highest (less than high

school, high school, junior college, bachelor’s degree,

graduate degree) we can use this model to predict the

probability of being (for example) at the bachelor’s

level or below from age at first marriage.

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.... (a)

(a)

4

5

a

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The slopes are assumed to be the same for all logits

and, under this assumption, the model is known as

the proportional odds model.

The underlying assumption of equivalent slopes

across all logits can, and should, be tested to verify

that this model is appropriate.

If this assumption appears to be violated, then one

could fit the nominal, or more complicated alternative

models.

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We use data from the 2006 GSS to predict a

respondent’s educational attainment level (degree),

measured as either less than high school, high school,

junior college, bachelor’s degree, or graduate degree,

from the respondent’s age when first married

(agewed).

The outcome variable (educational attainment level) is

treated as ordinal, so the proportional odds model is

used.

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# Model Logistik Ordinal untuk Data GSS (Azen, sub-bab 10.5)

# Data Respon : Harus Data Terurut

dataku

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# Pendugaan nilai peluang untuk tiap kategori

prediksi

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degree degree.order agewed

1 HIGH SCHOOL 2 22

2 HIGH SCHOOL 2 23

3 HIGH SCHOOL 2 24

4 HIGH SCHOOL 2 22

5 LT HIGH SCHOOL 1 28

6 LT HIGH SCHOOL 1 21

7 HIGH SCHOOL 2 29

8 LT HIGH SCHOOL 1 19

9 LT HIGH SCHOOL 1 28

10 LT HIGH SCHOOL 1 29

.

.

.

1158 HIGH SCHOOL 2 21

1159 HIGH SCHOOL 2 22

1160 BACHELOR 4 28

Catatan : yang dipakai “degree.order” bukan

“degree”, karena “degree” belum terurut.

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degree.order

degree 1 2 3 4 5

LT HIGH SCHOOL 195 0 0 0 0

BACHELOR 0 0 0 185 0

GRADUATE 0 0 0 0 104

HIGH SCHOOL 0 590 0 0 0

JUNIOR COLLEGE 0 0 86 0 0

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Coefficients:

Value Std. Error t value

agewed 0.05059 0.01031 4.908

Intercepts:

Value Std. Error t value

1|2 -0.4549 0.2431 -1.8711

2|3 1.9226 0.2501 7.6886

3|4 2.2940 0.2530 9.0670

4|5 3.5242 0.2682 13.1389

Residual Deviance: 3096.156

AIC: 3106.156

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degree.order agewed P.Y.1 P.Y.2 P.Y.3 P.Y.4 P.Y.5

1 2 22 0.172538473 0.51950698 0.07310156 0.1525423 0.08231071

2 2 23 0.165435593 0.51572572 0.07477403 0.1578513 0.08621336

3 2 24 0.158569083 0.51150679 0.07640620 0.1632351 0.09028283

4 2 22 0.172538473 0.51950698 0.07310156 0.1525423 0.08231071

5 1 28 0.133396465 0.49051546 0.08241161 0.1853397 0.10833676

6 1 21 0.179880581 0.52283962 0.07139470 0.1473156 0.07856954

7 2 29 0.127656350 0.48431357 0.08375223 0.1909568 0.11332101

8 1 19 0.195291690 0.52812158 0.06790088 0.1371355 0.07155037

9 1 28 0.133396465 0.49051546 0.08241161 0.1853397 0.10833676

10 1 29 0.127656350 0.48431357 0.08375223 0.1909568 0.11332101

11 4 30 0.122128426 0.47776346 0.08501689 0.1965871 0.11850410

12 2 21 0.179880581 0.52283962 0.07139470 0.1473156 0.07856954

13 4 24 0.158569083 0.51150679 0.07640620 0.1632351 0.09028283

14 2 18 0.203364066 0.53005548 0.06612507 0.1321935 0.06826190

15 4 52 0.043717274 0.28635273 0.08659973 0.2930019 0.29032839

16 4 26 0.145531789 0.50180589 0.07952560 0.1741929 0.09894384

17 1 29 0.127656350 0.48431357 0.08375223 0.1909568 0.11332101

18 2 19 0.195291690 0.52812158 0.06790088 0.1371355 0.07155037

19 1 25 0.151935686 0.50686240 0.07799206 0.1686853 0.09452453

20 4 18 0.203364066 0.53005548 0.06612507 0.1321935 0.06826190

21 1 16 0.220246789 0.53248085 0.06254204 0.1226288 0.06210152

22 2 20 0.187464338 0.52571395 0.06965925 0.1421779 0.07498452

.

.

.

1160 4 28 0.133396465 0.49051546 0.08241161 0.1853397 0.10833676

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Output SAS : Bandingkan dengan Output R

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Output SAS : Bandingkan dengan Output R

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Perbedaan Model antara R, SPSS, dan SAS

R dan SPSS

SAS

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Perbedaan Model antara R, SPSS, dan SAS

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Interpretasi dan Pengujian Parameter

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Ilustrasi Interpretasi Parameter (Output R)

Coefficients:

Value Std. Error t value

agewed 0.05059 0.01031 4.908

Intercepts:

Value Std. Error t value

1|2 -0.4549 0.2431 -1.8711

2|3 1.9226 0.2501 7.6886

3|4 2.2940 0.2530 9.0670

4|5 3.5242 0.2682 13.1389

Nilai negatif dari β

Misal untuk Y = 1:

ln𝑃(𝑌≤1)

𝑃(𝑌>1)=ln

𝑃 𝑌≤1

1−𝑃 𝑌≤1

= −0.455 − 0.051(𝑎𝑔𝑒𝑤𝑒𝑑)

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Penentuan Nilai Peluang Kumulatif

Penjabaran untuk Y = 1:

ln𝑃(𝑌≤1)

𝑃(𝑌>1)=ln

𝑃 𝑌≤1

1−𝑃 𝑌≤1= −0.455 − 0.051𝑥

⇔𝑃 𝑌≤1

1−𝑃 𝑌≤1= 𝑒−0.455−0.051𝑥

⇔ 𝑃 𝑌 ≤ 1 = (1 − 𝑃 𝑌 ≤ 1 )𝑒−0.455−0.051𝑥

⇔ 𝑃 𝑌 ≤ 1 =𝑒−0.455−0.051𝑥

1 + 𝑒−0.455−0.051𝑥

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𝑃 𝑌 ≤ 1 =𝑒−0.455−0.051𝑥

1 + 𝑒−0.455−0.051𝑥=

𝑒−0.455−0.051(20)

1 + 𝑒−0.455−0.051(20)=

0.2288

1.2288= 0.1862

𝑃 𝑌 ≤ 2 =𝑒1.923−0.051𝑥

1 + 𝑒1.923−0.051𝑥=

𝑒1.923−0.051(20)

1 + 𝑒1.923−0.051(20)=

2.4670

3.4670= 0.7116

dengan cara yang sama dapat dihitung 𝑃 𝑌 ≤ 3 dan 𝑃 𝑌 ≤ 4

Penentuan Nilai Peluang Kumulatif

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𝑃 𝑌 = 4 = 𝑃 𝑌 ≤ 4 − 𝑃 𝑌 ≤ 3

dengan cara yang sama dapat dihitung peluang setiap kategori 𝑌: 𝑃 𝑌 = 1 ,

𝑃 𝑌 = 2 , 𝑃 𝑌 = 3 , 𝑃 𝑌 = 4 , dan 𝑃 𝑌 = 5 .

Penentuan Nilai Peluang Setiap Kategori

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Kesimpulan

Berdasarkan nilai dugaan peluang tiap kategori tersebut,

jika seseorang diketahui berumur 20 tahun saat menikah,

apa dugaan tingkat pendidikan terakhir orang tersebut?

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1 Gunakan Program R untuk data Mental Impairment (Agresti, sub-

bab 7.2.4, hlm. 279 ) .

a. Bandingkan hasilnya dengan output SAS pada buku Agresti

tersebut serta berikan interpretasi pada tiap nilai dugaan

parameter model.

b. Berdasarkan hasil pada poin (a) di atas, tentukan nilai

dugaan P(Y = 1), P(Y = 3), dan P(Y > 2).

c. Tentukan model terbaik.

d. Misalkan seorang individu diketahui bahwa Life Events (x1 =

8) dan SES (x2 = 1), berdasarkan model pada poin (c)

tentukan dugaan “Mental Impairment”.

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2 Kerjakan Problem 10.7 (Azen, 2011)

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3 Kerjakan Problem 10.8 (Azen, 2011)

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Pustaka

1. Azen, R. dan Walker, C.R. (2011). Categorical Data

Analysis for the Behavioral and Social Sciences.

Routledge, Taylor and Francis Group, New York.

2. Agresti, A. (2002). Categorical Data Analysis 2nd. New

York: Wiley.

3. Pustaka lain yang relevan.

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kusmansadik.wordpress.com

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