????(Probability)%20(Chapter%204)

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????(Probability) (Chapter 4)

• ?????
• ?????????????????
• ??????????????????
• ?????????????????
• jpliu_at_ntu.edu.tw

?????????,???????? ??CC ????-?????-?????? 3.0
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• ????(Concept of Probability)
• ???????(Sample Space and Events)
• ??????(Elementary Probability Rules)
• ???????(Conditional Probability)
• ?????(Applications)

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??(Experiment)

• ????????(Outcome)???????,
• ???????????(Outcome)
• ? ?????
• ??????????(H)???(T)
• ???????????
• ???????????????????
• ???????????????

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??(Probability)

• 1.?????0?1??
• 2.?????????1
• ? ?????(fair coin)??
• ????????0.5
• ????????0.5
• ??0?1??
• ????????????
• ?0.50.51

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????(Sample Space)

• ????(Sample Space)
• ???????????
• ??????
• H,T
• ?????
• 1,2,3,4,5,6
• ?????????
• ??,??,??,??
• BB, GG, BG, GB

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??(Event)

• ???????????
• ???????????????
• GG,BG,GB
• ??????????3
• 4,5,6

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????(Probability of an Event)

• ????????????????
• ???????????????
• ??E????
• ??P(E)??????

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????(Probability of an Event)

• ?
• ?????????
• ????BB,GG,BG,GB
• ???????4
• ???????
• EGG,BG,GB
• ????????3
• P(E)3/40.75

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??????

• 1.??E????Ec????
• P(Ec)1-P(E)
• P(E) P(Ec)1
• ??????????
• E???????
• GG,BG,GB
• Ec?????? BB
• P(Ec)1/41-P(E)1-3/4

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??????

• 2.????
• A?B??????(Intersection)AnB
• ????A???B?????
• A?B??????(Union)A?B
• ????A???B?????

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• P(A?B)P(A)P(B)-P(AnB)

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• ?
• Alt20? B??
• P(A)P(lt20?)35/500.7
• P(B)P(??)30/500.6
• P(AnB)P(lt20????)21/500.42
• P(A?B)P(lt20????)0.700.60-0.420.88

?? ?? ?? ??
?? lt20? gt20? ?
? 14 6 20
? 21 9 30
? 35 15 50
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????(Mutually Exclusive Events)

• ????(Mutually Exclusive Events)
• ?A???B?????????
• AnB? P(AnB)0
• P(A?B)P(A)P(B)

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????(Mutually Exclusive Events)

• ??????????
• A???J
• B???Q
• C?????(?????)
• AnB? ? P(AnB)0
• P(A?B)P(A)P(B)
• P(A?C)P(A)P(C)-P(AnC)

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????(Conditional Probability)

• ????
• Alt20? B??
• ???lt20???????
• P(BA)21/350.6

?? ?? ?? ??
?? lt20? gt20? ?
? 14 6 20
? 21 9 30
? 35 15 50
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????(Conditional Probability)

• A, B???
• A???B?????????

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????(Conditional Probability)

• ????
• P(AnB)P(BA)?P(A)
• P(AB)?P(B)
• ?????????????
• P(BA)P(B)
• P(AnB)P(BA)?P(A)
• P(B)?P(A)
• P(AB)?P(B)
• P(A) ?P(B)

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????(Conditional Probability)

• ??????

??? ??? ??? ???
??? H T ?
H 1 1 2
T 1 1 2
2 2 4
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• A??????
• B??????
• P(A)P(??????)2/41/2
• P(B)P(??????)2/41/2
• P(BA)P(????????????)
• ???????????????
• P(AnB)1/4(1/2)(1/2)P(A)?P(B)

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????-?????????

• ?????(measure of belief)
• ???????????????????
• (?P(?????), P(?????????))
• Critics ??!???????

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2019/6/19
Jen-pei Liu, PhD
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????

• ???????? ??????
• ??,????????????!
• ?????????personal belief,??????????????,??????????
  ????????
• ?????????????????????????

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2019/6/19
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????-????

• ?????(Bayesian statistics) ???
• ???? ?? ??????
• Proposed early in 1700

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2019/6/19
Jen-pei Liu, PhD
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????-????

• P(B) P(AC?B) P(A?B)
• By Conditional probability
• P(A?B) P(BA)P(A)
• and
• P(AC?B) P(BAc)P(Ac)
• ?P(B) P(A?B) P(AC?B)
• P(BA)P(A) P(BAc)P(Ac)

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????-????
???? ???? Pr(A) ???? Pr(AB) ?????personal
probability
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Applications

• Diagnosis of Diseases
• Classification
• Pattern Recognition
• Estimation of Survival Function

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Diagnosis of Diseases
Contingency Table
True Condition Status True Condition Status True Condition Status True Condition Status
Test Results Present (S2) Absent (S1) Total
Positive (R2) a b ab
Negative (R1) c d cd
Total ac bd
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Indices of Diagnostic Accuracy

• Sensitivity (True Positive rate) Capacity for
  making a correct diagnosis in subjects with the
  disease
• Estimated Sensitivity P(R2S2)
• P(R2S2) 100 x a/(ac)
• Specificity (True Negative rate) Capacity for
  making a correct diagnosis in subjects without
  disease
• Estimated Specificity
• P(R1S1) 100 x d/(bd)

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Indices of Diagnostic Accuracy

• Positive Predictive Value (Positive Predictive
  Accuracy) the proportion of subjects with the
  disease given the positive results.
• P(S2R2) 100 x a/(ab)
• Negative Predictive Value (Negative Predictive
  Accuracy) the proportion of subjects without the
  disease given the negative results.
• P(S1R1) 100 x d/(cd)
• False positive rate given the positive results
  ,the proportion of subjects without the disease
• P(S1R2) 1 positive predictive value
  100 x b/(ab)
• False negative rate given the negative results,
  the proportion of subjects with the disease
• P(S2R1) 1 negative predictive value
  100 x c/(cd)

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????-???????
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Example 2 (Feinstein, 2002)
New Maker Test Result Diseased Cases Non-diseased Control Total
Positive Negative 46 4 2 48 48 52
Total 50 50 100
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Indices of Diagnostic Accuracy

• Data from Example 2 (Feinstein, 2002)
• Sensitivity 100 x 46/50 92.0
• Specificity 100 x 48/50 96.0
• Prevalence 100 x 50/100 50.0
• Positive Predictive Value
• 100 x 46/48 95.8
• (0.92x0.5)/0.92x0.5 (10.96)x(10.5)
• Negative Predictive Value
• 100 x 48/52 92.3
• False Positive Rate 100 x 2/48 4.2
• False Negative Rate 100 x 4/52 7.7

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Example 3 (Feinstein, 2002)
New Maker Test Result Diseased Cases Non-diseased Control Total
Positive Negative 46 4 38 912 84 916
Total 50 950 1000
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Indexes of Diagnostic Accuracy

• Example 3 (Feinstein, 2002)
• Sensitivity 100 x 46/50 92.0
• Specificity 100 x 912/950 96.0
• Prevalence 100 x 50/1000 5.0
• Positive Predictive Value 100 x 46/84 54.8
• 0.92x0.05/0.92x0.05 (10.96)x(10.05)
• Negative Predictive Value
• 100 x 912/916 99.6
• False Positive Rate 100 x 38/84 45.2
• False Negative Rate 100 x 4/916 0.4

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Indexes of Diagnostic Accuracy

• Type of Diagnostic Tests (Feinstein, 1977)
• Screening or discovery tests mammogram, fasting
  blood sugar - required high sensitivity gt high
  false positive rate.
• Exclusion tests to rule out the presence of the
  disease such as colonoscopic examination gt
  require extremely high sensitivity
• Confirmation test to verify the suspicion of the
  presence of the disease such as biopsy for lung
  cancer gt require extremely high specificity with
  very few false positive.

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??????

• Odds (??) p/(1-p)
• ????????????,????
• P(?????)/P(??????)5
• ???????odds?5
• ?P(?????)5/(15)

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??????

• ??????????????????999
• ? P(????)/P(????)999,
• ? P(?????????)0.001

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2019/6/19
Jen-pei Liu, PhD
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??????-?????????????
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??????-?????????????

• ???????AD ??????,?BC ???????????,???????
• ????????1,????????????????????????????????????????
  ????????,??????????????????????????

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Jen-pei Liu, PhD
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??????-?????????????
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Computation of Kaplan-Meier Estimate of Survival
(Actuarial Estimate)

• Time point t1, t2, and t3
• E1 event of surviving from 0 to t1
• E2 event of surviving from t1 to t2
• E3 event of surviving from t2 to t3
• E1?E2 ?E3 event of surviving from 0 to t3
• By conditional probability
• P(E1?E2 ?E3) P(E3 E1?E2)P(E1?E2)
• P(E3 E1?E2)P(E2E1)P(E1)

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Computation of Kaplan-Meier Estimate of Survival
(Actuarial Estimate)

• Divide the time into intervals by the time points
  where the pre-defined event (death) occurred.
• For each interval, count the number of the
  patients who were alive at the beginning of the
  interval and the number of the patients who were
  still alive at the end of the interval.
• Compute the survival rate for each interval as
  the number of the patients still alive at the end
  of interval divided by the number of the patients
  alive at the beginning of the interval.
• For the time point where pre-defined event
  occurred, the Kaplan-Meier estimate is the
  product of survival rate of the preceding
  intervals and present one.

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Computation of Kaplan-Meier Survival

• Sy(k) P(?y(k)??)
• P(??y(1),y(2),......,y(k-1),y(k)?
  ??)
• P(?y(k)?? ??y(1),y(2),,y(k-1),
  ???)
• P(??y(1),y(2),,y(k-1)???)
• P(?y(k)????y(1),y(2),......,y(k-1),
  ???)
• P(?y(k-1)????y(1),y(2),,y(k-2)
  ???)
• ......P(?y(2)????y(1)??)
  P(?y(1)??)?

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Time in Months to Progression of the Patients with Stage?or?A Ovarian Carcinoma by Low-grade or Well-differentiated Cancer Time in Months to Progression of the Patients with Stage?or?A Ovarian Carcinoma by Low-grade or Well-differentiated Cancer Time in Months to Progression of the Patients with Stage?or?A Ovarian Carcinoma by Low-grade or Well-differentiated Cancer Time in Months to Progression of the Patients with Stage?or?A Ovarian Carcinoma by Low-grade or Well-differentiated Cancer
Patient Number Time in Months Death (non-censored) Cell Grade
1 0.92 Yes Low Grade
2 2.93 Yes Low Grade
3 5.76 Yes Low Grade
4 6.41 Yes Low Grade
5 10.16 Yes Low Grade
6 12.40 No Low Grade
7 12.93 No Low Grade
8 13.85 No Low Grade
9 14.70 No Low Grade
10 15.20 Yes Low Grade
11 23.32 No Low Grade
12 24.47 No Low Grade
13 25.33 No Low Grade
14 36.38 No Low Grade
15 39.67 No Low Grade
16 1.12 Yes High Grade
17 2.89 Yes High Grade
18 4.51 Yes High Grade
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19 6.55 Yes High Grade
20 9.21 Yes High Grade
21 9.57 Yes High Grade
22 9.84 No High Grade
23 9.87 No High Grade
24 10.16 Yes High Grade
25 11.55 Yes High Grade
26 11.78 Yes High Grade
27 12.14 Yes High Grade
28 12.14 Yes High Grade
29 12.17 Yes High Grade
30 12.34 Yes High Grade
31 12.57 Yes High Grade
32 12.89 Yes High Grade
33 14.11 Yes High Grade
34 14.84 Yes High Grade
35 36.81 No High Grade
Source Fleming, et al. (1980) Source Fleming, et al. (1980) Source Fleming, et al. (1980) Source Fleming, et al. (1980)
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Data Layout for Computation of Kaplan-Meier Estimates of Survival Function Data Layout for Computation of Kaplan-Meier Estimates of Survival Function Data Layout for Computation of Kaplan-Meier Estimates of Survival Function Data Layout for Computation of Kaplan-Meier Estimates of Survival Function Data Layout for Computation of Kaplan-Meier Estimates of Survival Function Data Layout for Computation of Kaplan-Meier Estimates of Survival Function
Ordered Distinct Event Time Ordered Distinct Event Time Number of Events Number of Censored in y(k), y(k1) Number in Risk Set S(y)
Y(0) 0 Y(0) 0 d0 0 m0 n0 1
Y(1) Y(1) d1 m1 n1 1- d1/n1
Y(2) Y(2) d2 m2 n2 (1- d1/n1) (1- d2/n2)

Y(k) Y(k) dk mk nk (1- d1/n1)(1- d2/n2)(1- dk/nk)
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Computation of Kaplan-Meier Estimates of Survival Function for Patients with Low-grade Cancer Computation of Kaplan-Meier Estimates of Survival Function for Patients with Low-grade Cancer Computation of Kaplan-Meier Estimates of Survival Function for Patients with Low-grade Cancer Computation of Kaplan-Meier Estimates of Survival Function for Patients with Low-grade Cancer Computation of Kaplan-Meier Estimates of Survival Function for Patients with Low-grade Cancer
Ordered Distinct Progression Time Number of Events Number of Censored in y(k), y(k1) Number in Risk Set S(y)
0 0 0 15 1
0.92 1 0 15 0.9333
2.93 1 0 14 0.8667
5.76 1 0 13 0.8000
6.41 1 0 12 0.7333
10.16 1 4 11 0.6667
15.20 1 1 6 0.5556
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Example

• K-M estimate of survival at 15.20 months or
  longer
• S(15.20 months or longer)
• 0.5556

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??(Summary)

• ??
• ????
• ???????
• 0?P(E) ?1
• ????
• P(A?B)P(A)P(B)-P(AnB)
• ????P(AnB)0

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??(Summary)

• ?? ????
• P(AB)P(AnB)/P(B)
• ????P(AB)P(A)
• P(BA)P(B)
• P(AnB)P(A)?P(B)
• ????
• ??
• ???????????

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?? ?? ???? ??/??
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