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Tabel Kontingensi 2x2 (3) Rasio Odds dan Uji Kebebasan KhiK d t Kuadrat
Rasio ODDS “It occurs as a parameter in the most important type of model for categorical data”
Odds Sukses
odds =
π
(1 − π )
• 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 A1
A2
B1
π1
1-π1
odds1 =
π1 (1 − π1 )
B2
π2
1-π2
odds2 =
π2 (1 − π 2 )
Rasio Odds Values of θ farther from 1.0 in a given direction represent stronger association. 3
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. 4
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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 rows,
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Ilustasi: kasus aspirin dan serangan jantung
This also equals the cross-product ratio (189 × 10, 933)/(10,845 × 104). )
n11 189 = = 0.0174 n12 10845 n 104 odds2 = 21 = = 0.0095 n22 10933 odds1 =
OR = θ =
Odds1 0.0174 = = 1.832 Odds2 0.0095 The estimated odds were 83% higher for the placebo 6 group.
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Inferensia Rasio Odds dan Log Rasio Odds • Kecuali pada ukuran sampel sangat besar, sebaran percontohan dari OR (hi hl skewed). k d) sangat menceng (highly • Karena kemiringan ini, statistika inferensia untuk rasio odds menggunakan alternatif dengan ukuran yang setara - logaritma natural, log (θ). Dengan log (θ)=0. • Artinya θ =1 setara dengan log (θ) 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, k d nilai kedua il i iinii mewakili kili kkekuatan k t asosiasi i i yang sama • Doubling a log odds ratio corresponds to squaring an odds ratio. • Sebaran dari log(θ) tidak terlalu menceng, menyerupai bentuk lonceng • Sebaran log (θ) mendekati sebaran normal dengan nilai tengah log(θ) dan galat baku The SE decreases as the cell counts increase.
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Selang Kepercayaan untuk log(θ) log θˆ ± Zα ( SE ) 2
Ilustrasi: data aspirin • log(1.832) = 0.605 • Galat baku = • SK 95% untuk log (θ) 0.605 ± 1.96(0.123) Ù (0.365, 0.846) (0 365 0 846) • SK 95% untuk θ Ù[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. 9
Kita menduga bahwa odds serangan jjantungg setidaknya y 44% lebih tinggi gg pada subjek yang mengkonsumsi placebo dibandingkan dengan subjek yang mengkonsumsi aspirin
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Catatan • Bila terdapat nilai nij=0, maka perhitungan OR d l h 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. 12
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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. to 262 young and middle-aged women (age < 69) admitted to 30 coronary care units in northern Italy with acute MI during a 5 5-year year period
• The first column refers. • Each case was matched with two control patients admitted to the same hospitals with other acute disorders. • The controls fall in the second
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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. 14
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Peubah penjelas
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Peubah respon
When the sampling design is
Tidak bisa menghitung proporsi penderita MI pada kelompok smoker (atau non-smoker)
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 15
• 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 off ever being g 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. MI • Binomial sample Æ column, dependent because 1MI paired with 2 control 16
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Types of Observational study TTugas!! !! Cari tahu macam2 tipe studi observasi beserta penjelasan dan contohnya!!
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Bagaimana mengukur keeratan hubungan 2 peubah?? K l i Korelasi
Hubungan linear
Data Nominal ? pearson
spearman 18
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Tahun 1900
Karl Pearson
Pearson chisquared statistic 19
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.
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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 22 chi-squared distribution, and their numerical values are similar.
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• 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
Dem Independent Republic Total an ocrat 762 327 468 1577 703,7
Males Total
484
293
1246 566 1. 1246*1577= 1940022
477 945
1200 2757
2. 1940022/2757 = 703,7
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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
Nilai Rasio Odds θ =
(120 x50) =5 (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 p q data=aspirin p order=data; tables smoking*cancer/nopercent nocol norow expected; exact or chisq; weight frec; run; 28
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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
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Menggabungkan Kategori Daya Listik
Penghasilan
450 & 900 watt 1300 & 3500 watt Total
300 000 >300.000750.000 37 2 39
1 000 000 > 1.000.0002.000.000 11 10 21
Total
48 12 50
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Uji Pasti Fisher ? Pertemuan Selanjutnya
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