Tabel Kontingensi 2x2 (3) Rasio Odds dan Uji Kebebasan KhiKuadrat
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RASIO ODDS 3
Rasio Odds Exposure
Association
outcome
measure
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Odds Ratio Odds Ratio
odds that an outcome will occur given a particular exposure odds of the outcome occurring in the absence of that exposure
• 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). 5
Rasio ODDS
“It occurs as a parameter in the most important type of model for categorical data”
Odds Sukses
odds
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• 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
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Rasio Odds Pada Tabel 2x2 B1 B2
A1
A2
π1
1-π1
odds1
π2
1-π2
odds2
1 1 1
2 1 2
Rasio Odds
Values of θ farther from 1.0 in a given direction represent stronger association.
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RASIO ODDS pada Study Cohort Develop Disease
Exposed Non-Exposed
a c
The Odds that an exposed person develop disease The Odds that a non exposed person develop disease
a b
c d
Do Not Develop Disease b d
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Rasio Odds : Cohort
• Odds ratio is the ratio of the odds of disease in the exposed to the odds of disease in the nonexposed odds that an exposed person develops the disease OR odds that a non exposed person develops the disease
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a c
b d
RASIO ODDS pada Study Case-Control History of Exposure
No History of Exposure
The odds that a case was exposed
The odds that a control was exposed
Case a c
Control b d
a c
b d
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Rasio Odds : Cohort
Odds ratio (OR) is the ratio of the odds that a case was exposed to the odds that a control was exposed
odds that a case was exposed OR odds that a control was exposed
a b
c d
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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. 12
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
n11 189 0.0174 n12 10845 n21 104 odds2 0.0095 n22 10933 odds1
This also equals the cross-product ratio (189 × 10, 933)/(10,845 × 104).
Odds1 0.0174 OR 1.832 Odds2 0.0095
The estimated odds were 83% higher for the placebo 14 group.
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
The SE decreases as the cell counts increase.
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Selang Kepercayaan untuk log() log ˆ Z SE
Ilustrasi: data aspirin • log(1.832) = 0.605
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• 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. 17
Kita menduga bahwa odds serangan jantung setidaknya 44% lebih tinggi pada subjek yang mengkonsumsi placebo dibandingkan dengan
subjek yang mengkonsumsi aspirin
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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 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 the relative risk.
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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. 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
• Each case was matched with two
control patients admitted to the same hospitals with other acute disorders.
• The controls fall in the second column of the table.
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• 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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Peubah penjelas
Peubah respon
Tidak bisa menghitung proporsi penderita MI pada kelompok smoker (atau non-smoker)
When the sampling design is
Karena untuk setiap penderita MI kita retrospective, we can constructpasangkan dengan 2 conditional distributions orang kontrol
for the explanatory variable, within levels of the fixed response.
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 23
• 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 were about 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 risk of MI or the difference of proportions suffering MI. • Binomial sample column, dependent because 1MI paired with 2 control 24
Bagaimana mengukur keeratan hubungan 2 peubah?? Korelasi
pearson
Hubungan linear
spearman
Data Nominal ? 25
Tahun 1900
Karl Pearson
Pearson chisquared statistic 26
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) •
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 29 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 are less than about 5.
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Menghitung Nilai Harapan Party Identification
Females Males Total
Dem Independent Republic Total ocrat an 762 327 468 1577 703,7
484
293
1246 566 1. 1246*1577= 1940022
477 945
2. 1940022/2757 = 703,7
1200 2757
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Ilustrasi: Data smoker-lung cancer Smoker Non Smoker Total
Lung Cancer Yes No 120 30 40 50
Total
160
240
80
150 90
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Hipotesis H0: Tidak ada asosiasi antara kebiasaan merokok dan penyakit kanker paru-paru H1: Ada asosiasi antara kebiasaan merokok dan penyakit kanker paru-paru )
(120 x50) 5 Nilai Rasio Odds (40 x30)
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Syntax SAS
Data aspirin; input smoking $ cancer $ frec ; cards; smoker yes 120 smoker no 30 non_smoker yes 40 non_smoker no 50 ; proc freq data=aspirin order=data; tables smoking*cancer/nopercent nocol norow expected; exact or chisq; weight frec; run; 35
Output
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Mengubah posisi tabel kontingensi
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Warning !!
Lebih dari 20% cell dengan nilai harapan > 5, kita tidak bisa menggunakan Chi Square test
Dua Solusi: 1. Menggabungkan kategori 2. Gunakan Exact Fisher test
Menggabungkan Kategori
Daya Listik
450 & 900 watt
1300 & 3500 watt Total
Penghasilan
>300.000750.000
> 1.000.0002.000.000
2
10
37 39
11 21
Total 48
12
50
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Uji Pasti Fisher ? Pertemuan Selanjutnya
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