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the following contingency table: Table 2.1. Cross classification of Belief in Afterlife by gender. Belief in afterlife. Yes No/Undecided. Gender Female 509.
Typology: Schemes and Mind Maps
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I. Probability Structure of a 2-way Contingency Table
I.1 Contingency tables
Y 1 2 3 X 1 n 11 n 12 n 13 2 n 21 n 22 n 23
X 1 n 11 n 12 2 n 21 n 22 3 n 31 n 32
? In general, a 2 × 2 table from multinomial sampling Y 1 2 X 1 n 11 n 12 n1+ 2 n 21 n 22 n2+ n+1 n+2 n where (n 11 , n 12 , n 21 , n 22 ) are random variables that have a multinomial distribution with sample size n (n = n 11 + n 12 + n 21 + n 22 ) and probabilities Y 1 2 X 1 π 11 π 12 2 π 21 π 22 (π 11 , π 12 , π 21 , π 22 ) define the probability structure of the contingency table. Slide 43
The standard statistical model underlying analysis of contingency tables is to assume that (unconditional on the total count) the cell counts are independent Poisson random variables.
Once you impose a total cell count for the contingency table, or a row or column count, the resulting conditional distributions of the cell counts then become multinomial.
https://stats.stackexchange.com/questions/45479/pearsons-residuals
? πij ’s can be estimated by pij = nij /n. ? With multinomial sampling, we can estimate many relevant quantities:
P̂ [Y = 1] = n^11 +^ n^21 n
= n+ n P̂ [X = 1] = n^11 +^ n^12 n =^
n1+ n P̂ [Y = 1|X = 1] = n^11 n 11 + n 12 =^
n 11 n1+ P̂ [X = 1|Y = 1] = n^11 n 11 + n 21 =^
n 11 n+^ ... ? For afterlife example, we estimated that
P̂ [belief in afterlife] = 509 + 398 1127 = 80% P̂ [belief in afterlife|Female] = 509 509 + 116 = 81% P̂ [belief in afterlife|Male] = 398 398 + 104 = 79%...
Total 907 220 1,
? In general, the data looks like Y 1 2 3 X 1 n 11 n 12 n 13 n1+ 2 n 21 n 22 n 23 n2+ where n1+ and n2+, the sample sizes for X = 1 and X = 2, are fixed. (n 11 , n 12 , n 13 ) ⊥ (n 21 , n 22 , n 23 ) (n 11 , n 12 , n 13 ) ∼ multinom(n1+, (π 1 , π 2 , π 3 )), π 1 + π 2 + π 3 = 1 (n 21 , n 22 , n 23 ) ∼ multinom(n2+, (τ 1 , τ 2 , τ 3 )), τ 1 + τ 2 + τ 3 = 1 ? Since the likelihood of π’s and τ ’s is the product of the likelihood of π’s and the likelihood of τ ’s, this sampling scheme is called product-multinomial sampling on X. ? Clinical trials, cohort studies (prospective studies) all use this sampling scheme. Prospective study: Participants are enrolled into the study before they develop the disease or outcome in question.
? When X is also random (so has a distribution in the population), (π 1 , π 2 , π 3 )’s defines the conditional distribution of Y given X = 1 (τ 1 , τ 2 , τ 3 )’s defines the conditional distribution of Y given X = 2. ? With product-multinomial sampling on X, we can only estimate conditional probabilities of Y |X = x. Other probabilities are not estimable. For example, we cannot estimate P [Y = 1].
? We may consider a design such as the following one:
Lung Cancer Yes No Smoking Yes n 11 n 12 No n 21 n 22 n+1 = 100 n+2 = 200 All cell counts will not be small ⇒ efficient. n 11 ⊥ n 12 n 11 ∼ Bin(n+1, π 1 ), π 1 = P [smoking|case]. n 12 ∼ Bin(n+2, π 2 ), π 2 = P [smoking|control].
? We can still investigate the association between smoking and lung cancer using this design. ? This sampling scheme is product-multinomial on Y. ? The study is often called the case-control study.
? In general,
Lung Cancer Yes No Smoking Yes n 11 n 12 No n 21 n 22 n+1 n+ where n+1, n+2, are all fixed. n 11 ⊥ n 12 n 11 ∼ Bin(n+1, π 1 ), π 1 = P [smoking|case]. n 12 ∼ Bin(n+2, π 2 ), π 2 = P [smoking|control].
n 11 n 11 + n 21
π1=
π2=
n 12 n 12 + n 22
For example, if we have data from a multinomial sampling with sample size n: Y 1 2 X 1 n 11 n 12 2 n 21 n 22
X 1 π 11 π 12 2 π 21 π 22 Then we can view the data from product-multinomial sampling on X or product-multinomial sampling on Y. That is: n 11 |n1+ ∼ Bin(n1+, (^) π 11 π+^11 π 12 ) ⊥ n 21 |n 2 + ∼ Bin(n2+,π 21 π^21 +π 22 ) Or n 11 |n+1 ∼ Bin(n+1, (^) π 11 π+^11 π 21 ) ⊥ n 12 |n+ 2 ∼ Bin(n+2,π 12 π^12 +π 22 )
I.3 Sensitivity & Specificity in Diagnostic Tests
Y Positive Negative X Disease No Disease
Q: Find sensitivity and specificity.
The higher the sensitivity and specificity, the better the diagnostic test.
I.4 Independence of X and Y
⇔ P [ X =i , Y =j ] =P [ X =i ]*P [ Y =j ] f or i = 1 , 2 ,. .., I , j = 1 , 2 ,. .., J. ⇔ πij =πi+π+j f or i = 1 , 2 ,. .., I , j = 1 , 2 ,. .., J. (πi+ =πi 1 +πi 2 +. .. +πiJ , π+j =π 1 j +π 2 j +. .. +πIj ) ⇔ P [ Y =j |X =i ] =P [ Y =j |X =k] f or all i , j, k.
π 1 = (^) ππ^11 1+
= (^) ππ^21 2+
= π 2
Note that
p 1 − p 2 = n^11 n1+
− n^21 n2+
SE(p 1 − p 2 ) =
p 1 (1 − p 1 )/n1+ + p 2 (1 − p 2 )/n2+
p 1 − p 2 ± zα/ 2 SE(p 1 − p 2 ).
If this CI does not contain 0, we can reject H 0 : X ⊥ Y at significance level α.
Recall:
Critical value: Zα/ = qnorm(0.975) =1.
In a 5-yr study, 22,000+ physicians were randomized (blinded) to the placebo/aspirin (one tablet every other day) group: Myocardial infarction Yes No Treatment Placebo 189 10 , 845 11, Aspirin 104 10,933 11,
(on X)
Critical value: Zα/ = qnorm(0.975) =1.
