Tabel Kontingensi 2x2 (3)Rasio Odds dan Uji Kebebasan Khi-Kuadrat

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RASIO ODDS

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Rasio OddsExposure outcome

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Association

measure

Odds Ratio

• most commonly used in case-control studies,• can also be used in cross-sectional and cohort study designs as well (with some modifications and/or assumptions).

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Odds Ratioodds that an outcome will occur given a particular exposure

odds of the outcome occurring in the absence of that exposure

Rasio ODDS

Odds Sukses 1odds

• Odds bernilai positif• Nilai odss lebih besar dari satu, saat “sukses” lebih dipilih dibandingkan “gagal” • odds = 4.0, a success is four times as likely as a failure

“It occurs as a parameter in the most important type of model for categorical data”

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Rasio Odds Pada Tabel 2x2A1 A2

B1 π1 1-π1

B2 π2 1-π2 11

11odds

2221odds

Rasio OddsValues of θ farther from 1.0 in a given direction represent stronger association. 7

RASIO ODDS pada Study Cohort

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Develop Disease Do Not Develop DiseaseExposed a b

Non-Exposed c dThe Odds that an exposed person develop disease a

bThe Odds that a non exposed person develop disease c

d

Rasio Odds : Cohort• Odds ratio is the ratio of the odds of disease in the exposed to the odds of disease in the non-exposed

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odds that an exposed person develops the diseaseodds that a non exposed person develops the disease

abc d

OR

RASIO ODDS pada Study Case-Control

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Case ControlHistory of Exposure a b

No History of Exposure c d

The odds that a case was exposed ac

The odds that a control was exposed bd

Rasio Odds : Cohort

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odds that a case was exposedodds that a control was exposed

acbd

OR

Odds ratio (OR) is the ratio of the odds that a case was exposed to the odds that a control was exposed

Properties of OR• The odds ratio does not change value when the table orientation reverses so that the rows become the columns and the columns become the rows.• Thus, it is unnecessary to identify one classification as a response variable in order to estimate θ.• By contrast, the relative risk requires this, and its value also depends on whether it is applied to the first or to the second outcome category.

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Both variables are response variables

The odds ratio is also called the cross-product ratio, because it equals the ratio of the products π11π22 and π12π21 of cell probabilities from diagonally opposite cells.The sample odds ratio equals the ratio of the sample odds in the two rows,

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Ilustasi: kasus aspirin dan serangan jantung

11112

189 0.017410845nodds n

21222

104 0.009510933nodds n

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0.0174 1.8320.0095OddsOR Odds

This also equals thecross-product ratio (189 × 10, 933)/(10,845 ×104).

The estimated odds were 83% higher for the placebogroup. 14

Inferensia Rasio Odds dan Log Rasio Odds• Kecuali pada ukuran sampel sangat besar, sebaran percontohan dari OR sangat menceng (highly skewed).• Karena kemiringan ini, statistika inferensia untuk rasio odds menggunakan alternatif dengan ukuran yang setara -logaritma natural, log (θ). Dengan log (θ)=0.• Artinya =1 setara dengan log () dari 0.

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• Log(OR) simetrik di sekitar nilai 0.• Artinya, jika kita menukar posisi baris dan kolom akan

mengubah tandanya. Misal: log(2.0) = 0.7 dan log(0.5) = −0.7, kedua nilai ini mewakili kekuatan asosiasi yang sama

• Doubling a log odds ratio corresponds to squaring an odds ratio.

• Sebaran dari log() tidak terlalu menceng, menyerupai bentuk lonceng

• Sebaran log () mendekati sebaran normal dengan nilai tengah log() dan galat baku

16The SE decreases as the cell counts increase.

Selang Kepercayaan untuk log() 2ˆlog Z SE

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Ilustrasi: data aspirin • log(1.832) = 0.605• Galat baku =• SK 95% untuk log ()0.605 ± 1.96(0.123) (0.365, 0.846)

• SK 95% untuk [exp(0.365), exp(0.846)] = (e0.365, e0.846) = (1.44, 2.33)

• karena θ tidak mengandung 1, kemungkinan serangan jantung berbeda untuk kedua kelompok.

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Kita menduga bahwa odds serangan jantung setidaknya 44% lebih tinggi pada subjek yang mengkonsumsi placebo dibandingkan dengan subjek yang mengkonsumsi aspirin

Catatan• Bila terdapat nilai nij=0, maka perhitungan OR adalah

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Hubungan antara OR dan RR

Jika p1 dan p2 mendekati nol, maka nilai OR akan sama dgr RR

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This relationship between the odds ratio and the relative risk is useful.For some data sets direct estimation of the relative risk is not possible,yet one can estimate the odds ratio and use it to approximate therelative risk.

Rasio Odds pada studi case-control• Table 2.4 refers to a study that

investigated the relationship between smoking and myocardial infarction.

• The first column refers. • Each case was matched with two

control patients admitted to the samehospitals with other acute disorders.

• The controls fall in the second column of the table.

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to 262 young and middle-aged women (age < 69) admitted to 30 coronary care units in northern Italy with acute MI during a 5-year period

• All subjects were classified according to whether they had ever been smokers.

• The “yes” group consists of women who were current smokers or ex-smokers, whereas the “no” group consists of women who never were smokers.We refer to this variableas smoking status.

• The study, which uses a retrospective design to look into the past, is called a case–control study.

• Such studies are common in health-related applications, for instance to ensure a sufficiently large sample ofsubjects having the disease studied.

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Tidak bisa menghitung proporsi penderita MI pada kelompok smoker (atau non-smoker)

Karena untuk setiap penderita MI kita pasangkan dengan 2 orang kontrol

Untuk wanita penderita MI, proporsi yang merupakan perokok sebesalr172/262 = 0.656,Sedangkan untuk wanita bukan penderita MI, proporsi perokok sebesar 173/519 = 0.333

Peubah respon

Peubah

penje

las

When the sampling design is

levels of the fixed response.

When the sampling design is retrospective, we can construct conditional distributionsfor the explanatory variable, within levels of the fixed response.

• In Table 2.4, the sample odds ratio is [0.656/(1 − 0.656)]/[0.333/(1 − 0.333)] = (172 × 346)/(173 ×90) = 3.8. • The estimated odds of ever being a smoker wereabout 2 for the MI cases (i.e., 0.656/0.344) and about 1/2 for the controls (i.e.,0.333/0.667), yielding an odds ratio of about 2/(1/2) = 4.• For Table 2.4, we cannot estimate the relative riskof MI or the difference of proportions suffering MI.• Binomial sample column, dependent because 1MI paired with 2 control

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Bagaimana mengukur keeratan hubungan 2 peubah??

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Korelasi Hubungan linear

pearson spearman

Data Nominal ?

Tahun 1900

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Pearson chi-squared statisticKarl Pearson

Uji Kebebasan Khi - Kuadrat• Mengukur asosiasi antara dua peubah.• Korelasi Pearson and Spearman tidak dapat diterapkan pada data degan skala pengukuran nominal• Khi-kuadrat digunakan untuk data nominal dalam tabel kontingensi

A contingency table is a two-way table showing the contingency between two variables where the variables have been classified into mutually exclusive categories and the cell entries are frequencies.

Statistik Uji (pearson chi-squared & likelihood chi squared)

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• Pearson statistic X2 is a score statistic. (This means that X2 is based on a covariance matrix for the counts that is estimated under H0.)

• The Pearson X2 and likelihood-ratio G2 provide separate test statistics, but they share many properties and usually provide the same conclusions. • When H0 is true and the expected frequencies are large, the two statistics have the same chi-squared distribution, and their numerical values are similar.

• The convergence is quicker for X2 than G2.• The chi-squared approximation is often poor for G2 when some expected frequencies areless than about 5.

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Party IdentificationDemocrat

Independent Republican

TotalFemales 762 327 468 1577

Males 484 293 477 1200

Total 1246 566 945 2757

Menghitung Nilai Harapan

1. 1246*1577= 1940022 2. 1940022/2757 = 703,7

703,7

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Ilustrasi: Data smoker-lung cancerLung Cancer TotalYes No

Smoker 120 30 150Non Smoker 40 50 90Total 160 80 240

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HipotesisH0: Tidak ada asosiasi antara kebiasaan merokok dan penyakit kanker paru-paruH1: Ada asosiasi antara kebiasaan merokok dan penyakit kanker paru-paruNilai Rasio Odds

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(120 50) 5(40 30)xx )

Syntax SASData aspirin;input smoking $ cancer $ frec ;cards;smoker yes 120smoker no 30non_smoker yes 40non_smoker no 50;proc freq data=aspirin order=data;tables smoking*cancer/nopercent nocol norow expected;exact or chisq;weight frec;run;

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Output

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Mengubah posisi tabel kontingensi

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Warning !!

Lebih dari 20% cell dengan nilaiharapan > 5, kita tidak bisamenggunakan Chi Square testDua Solusi:1. Menggabungkan kategori2. Gunakan Exact Fisher test

Menggabungkan KategoriDaya Listik Penghasilan Total

>300.000-750.000 > 1.000.000-2.000.000450 & 900 watt 37 11 481300 & 3500 watt 2 10 12Total 39 21 50

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Uji Pasti Fisher ?Pertemuan Selanjutnya

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