Tabel Kontingensi 2x2 (3)
Rasio Odds
Exposure
outcome
4
Association
Rasio Odds
•
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).
Rasio Odds
odds that an outcome will occur given a particular exposure
Rasio ODDS
Odds Sukses
1
odds
•
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”
Rasio Odds Pada Tabel 2x2
A1
A2
B1
π
1
1-
π
1
B2
π
2
1-
π
2
1
1 1
1
odds
2
2 2
1
odds
Rasio OddsRASIO ODDS pada Study Cohort
8Develop
Disease
Do Not
Develop
Disease
Exposed
a
b
Non-Exposed
c
d
The Odds that an exposed person develop disease
a
b
The 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
odds that an exposed person develops the disease
odds that a non
exposed person develops the disease
a b c
d
OR
RASIO ODDS pada Study Case-Control
10
Case
Control
History of Exposure
a
b
No History of Exposure
c
d
The odds that a case was exposed
a
c
The odds that a control was exposed
b
d
Rasio Odds : Cohort
odds that a case was exposed
odds that a control was exposed
a c b
d
OR
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.
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.
Ilustasi:
kasus aspirin dan serangan jantung
11 1 12
189
0.0174
10845
n
odds
n
21 2 22104
0.0095
10933
n
odds
n
1 20.0174
1.832
0.0095
Odds
OR
Odds
This also equals the
cross-product ratio (189 × 10, 933)/(10,845 × 104).
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.
•
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
16
Selang Kepercayaan untuk log(
)
2
ˆ
log
Z
SE
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)] = (e
0.365, e
0.846) = (1.44, 2.33)
18
Kita menduga bahwa odds serangan
jantung setidaknya 44% lebih tinggi
pada subjek yang mengkonsumsi
placebo dibandingkan dengan
Catatan
Hubungan antara OR dan RR
Jika p1 dan p2 mendekati nol, maka nilai OR akan sama dgr RR
20
This relationship between the odds ratio and the relative risk is
useful.
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 same
hospitals with other acute disorders.
•
The controls fall in the second column
of the table.
•
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.
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,
Peubah respon P eu bah pe nje las
When the sampling design is
retrospective
, we can construct
conditional distributions
•
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
Bagaimana mengukur keeratan
hubungan 2 peubah??
Korelasi
Hubungan linear
pearson
spearman
Tahun 1900
26
Pearson
chi-squared statistic
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
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 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.
Party Identification
Dem
ocrat
Independent
Republic
an
Total
Females
762
327
468
1577
Males
484
293
477
1200
Total
1246 566
945
2757
Menghitung Nilai Harapan
Ilustrasi: Data smoker-lung cancer
Lung Cancer
Total
Yes
No
Smoker
120
30
150
Non
Smoker
40
50
90
Hipotesis
H
0
: Tidak ada asosiasi antara kebiasaan merokok
dan penyakit kanker paru-paru
H
1
: Ada asosiasi antara kebiasaan merokok dan
penyakit kanker paru-paru
Nilai Rasio Odds
34
(120 50)
5
(40 30)
x
x
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;
Output
Warning !!
Lebih dari 20% cell dengan nilai harapan > 5, kita tidak bisa
Dua Solusi:
Menggabungkan Kategori
Daya Listik
Penghasilan
Total
>300.000-750.000
>
1.000.000-2.000.000
450 & 900 watt
37
11
48
1300 & 3500 watt
2
10
12
Total
39
21
50
Uji Pasti Fisher ?
