Summaries

9th June 2010

Session 1.4

 

Marginal, Joint and Conditional Probabilities

 

Suppose that we have fair dice: d4 with face values {1,2,3,4}, d6 with face values {1,2,3,4,5,6} and d8 with face values {1,2,3,4,5,6,7,8}.  Our experiment consists of first randomly selecting one of the dice and then tossing that die and noting the face value.

 

The first stage probabilities:

 

Pr{Select d4} = 1/3( = P4)

Pr{Select d6} = 1/3( = P6)

Pr{Select d8} = 1/3( = P8)

 

The conditional probabilities:

 

Pr{1 shows | d4 selected} = 1/4

Pr{2 shows | d4 selected} = 1/4

Pr{3 shows | d4 selected} = 1/4

Pr{4 shows | d4 selected} = 1/4

 

Pr{1 shows | d6 selected} = 1/6

Pr{2 shows | d6 selected} = 1/6

Pr{3 shows | d6 selected} = 1/6

Pr{4 shows | d6 selected} = 1/6

Pr{5 shows | d6 selected} = 1/6

Pr{6 shows | d6 selected} = 1/6

 

Pr{1 shows | d8 selected} = 1/8

Pr{2 shows | d8 selected} = 1/8

Pr{3 shows | d8 selected} = 1/8

Pr{4 shows | d8 selected} = 1/8

Pr{5 shows | d8 selected} = 1/8

Pr{6 shows | d8 selected} = 1/8

Pr{7 shows | d8 selected} = 1/8

Pr{8 shows | d8 selected} = 1/8

 

The joint probabilities

 

Pr{1 shows} = Pr{1 shows | d4 selected}*Pr{d4 selected} + Pr{1 shows | d6 selected}*Pr{d6 selected} + Pr{1 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{2 shows} = Pr{2 shows | d4 selected}*Pr{d4 selected} + Pr{2 shows | d6 selected}*Pr{d6 selected} + Pr{2 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{3 shows} = Pr{3 shows | d4 selected}*Pr{d4 selected} + Pr{3 shows | d6 selected}*Pr{d6 selected} + Pr{3 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{4 shows} = Pr{4 shows | d4 selected}*Pr{d4 selected} + Pr{4 shows | d6 selected}*Pr{d6 selected} + Pr{4 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{5 shows} = Pr{5 shows | d6 selected}*Pr{d6 selected} + Pr{5 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(7/24) = 7/72 ≈ 0.0972

 

Pr{6 shows} = Pr{6 shows | d6 selected}*Pr{d6 selected} + Pr{6 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(7/24) = 7/72 ≈ 0.0972

 

Pr{7 shows} = Pr{7 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/8) = 3/72 ≈ 0.0417

 

Pr{8 shows} = Pr{8 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/8) = 3/72 ≈ 0.0417

 

Sample Tables

 

Compare sample proportions (p) to probabilities (P).

 

Sample 1

 

 

 

 

 

D8

Face Value

n

p

P

D6

Face Value

n

p

P

D4

Face

Value

n

p

P

Total

Face

Value

n

p

Joint

Probability

P

1

9

9/62

=

0.145

1/8

=

0.125

1

12

12/68

=

0.17647

1/6

≈

0.1667

1

13

13/70

=

0.186

1/4

=

0.25

1

9+12+13=34

34/200=0.17

0.1806

2

11

0.177

0.125

2

14

0.20588

0.1667

2

17

0.243

0.25

2

42

0.21

0.1806

3

8

0.129

0.125

3

11

0.16176

0.1667

3

23

0.329

0.25

3

42

0.21

0.1806

4

4

0.065

0.125

4

6

0.08824

0.1667

4

17

0.243

0.25

4

27

0.135

0.1806

5

10

0.161

0.125

5

15

0.22059

0.1667

 

 

 

5

25

0.125

0.0972

6

7

0.113

0.125

6

10

0.14706

0.1667

 

 

 

6

17

0.085

0.0972

7

3

0.333

0.125

 

 

 

 

 

 

 

7

3

0.015

0.0417

8

10

0.161

0.125

 

 

 

 

 

 

 

8

10

0.05

0.0417

Total

62

1.285

1

Total

68

1

1

Total

70

1

1

Total

200

1

 

Sample 2

 

 

 

 

 

Face Value

n

p

P

Face Value

n

p

P

Face Value

n

p

P

Total

Face

Value

N

p

Joint

Probability

P

1

3

0.047

0.125

1

8

0.11268

0.1667

1

16

0.246

0.25

1

27

0.135

0.1806

2

7

0.109

0.125

2

14

0.19718

0.1667

2

15

0.231

0.25

2

36

0.18

0.1806

3

14

0.219

0.125

3

16

0.22535

0.1667

3

19

0.292

0.25

3

49

0.245

0.1806

4

10

0.156

0.125

4

12

0.16901

0.1667

4

15

0.231

0.25

4

37

0.185

0.1806

5

8

0.125

0.125

5

14

0.19718

0.1667

 

 

 

5

22

0.11

0.0972

6

7

0.109

0.125

6

7

0.09859

0.1667

 

 

 

6

14

0.07

0.0972

7

9

0.141

0.125

 

 

 

 

 

 

 

7

9

0.045

0.0417

8

6

0.094

0.125

 

 

 

 

 

 

 

8

6

0.03

0.0417

Total

64

1

1

Total

71

1

1

Total

65

1

1

Total

200

1

 

Sample 3

 

 

 

 

 

Face Value

n

p

P

Face Value

n

p

P

Face Value

n

p

P

Total

Face

Value

n

p

Joint

Probability

P

1

7

0.108

0.125

1

17

0.21519

0.1667

1

9

0.153

0.25

1

33

0.162562

0.1806

2

5

0.077

0.125

2

15

0.18987

0.1667

2

15

0.254

0.25

2

35

0.172414

0.1806

3

5

0.077

0.125

3

8

0.10127

0.1667

3

20

0.339

0.25

3

33

0.162562

0.1806

4

9

0.138

0.125

4

9

0.11392

0.1667

4

15

0.254

0.25

4

33

0.162562

0.1806

5

12

0.185

0.125

5

13

0.16456

0.1667

 

 

 

5

25

0.123153

0.0972

6

10

0.154

0.125

6

17

0.21519

0.1667

 

 

 

6

27

0.133005

0.0972

7

11

0.169

0.125

 

 

 

 

 

 

 

7

11

0.054187

0.0417

8

6

0.092

0.125

 

 

 

 

 

 

 

8

6

0.029557

0.0417

Total

65

1

1

Total

79

1

1

Total

59

1

1

Total

203

1

 

Sample 4

 

 

 

 

 

Face Value

n

p

P

Face Value

n

p

P

Face Value

n

p

P

Total

Face

Value

n

p

Joint

Probability

P

1

7

0.093

0.125

1

10

0.13333

0.1667

1

15

0.3

0.25

1

32

0.16

0.1806

2

14

0.187

0.125

2

11

0.14667

0.1667

2

13

0.26

0.25

2

38

0.19

0.1806

3

13

0.173

0.125

3

20

0.26667

0.1667

3

14

0.28

0.25

3

47

0.235

0.1806

4

10

0.133

0.125

4

13

0.17333

0.1667

4

8

0.16

0.25

4

31

0.155

0.1806

5

10

0.133

0.125

5

10

0.13333

0.1667

 

 

 

5

20

0.1

0.0972

6

10

0.133

0.125

6

11

0.14667

0.1667

 

 

 

6

21

0.105

0.0972

7

7

0.093

0.125

 

 

 

 

 

 

 

7

7

0.035

0.0417

8

4

0.053

0.125

 

 

 

 

 

 

 

8

4

0.02

0.0417

Total

75

1

1

Total

75

1

1

Total

50

1

1

Total

200

1

 

Sample 5

 

 

 

 

 

Face Value

n

p

P

Face Value

n

p

P

Face Value

n

p

P

Total

Face

Value

n

p

Joint

Probability

P

1

14

0.209

0.125

1

7

0.10769

0.1667

1

26

0.382

0.25

1

47

0.235

0.1806

2

9

0.134

0.125

2

14

0.21538

0.1667

2

16

0.235

0.25

2

39

0.195

0.1806

3

8

0.119

0.125

3

11

0.16923

0.1667

3

10

0.147

0.25

3

29

0.145

0.1806

4

7

0.104

0.125

4

6

0.09231

0.1667

4

16

0.235

0.25

4

29

0.145

0.1806

5

5

0.075

0.125

5

17

0.26154

0.1667

 

 

 

5

22

0.11

0.0972

6

7

0.104

0.125

6

10

0.15385

0.1667

 

 

 

6

17

0.085

0.0972

7

8

0.119

0.125

 

 

 

 

 

 

 

7

8

0.04

0.0417

8

9

0.134

0.125

 

 

 

 

 

 

 

8

9

0.045

0.0417

Total

67

1

1

Total

65

1

1

Total

68

1

1

Total

200

1

 

Pooled

 

 

 

 

 

Face Value

n

p

P

Face Value

n

p

P

Face Value

n

p

P

Total

Face

Value

n

p

Joint

Probability

P

1

40

0.12

0.125

1

54

0.15084

0.1667

1

79

0.253

0.25

1

173

0.172483

0.1806

2

46

0.138

0.125

2

68

0.18994

0.1667

2

76

0.244

0.25

2

190

0.189432

0.1806

3

48

0.144

0.125

3

66

0.18436

0.1667

3

86

0.276

0.25

3

200

0.199402

0.1806

4

40

0.12

0.125

4

46

0.12849

0.1667

4

71

0.228

0.25

4

157

0.15653

0.1806

5

45

0.135

0.125

5

69

0.19274

0.1667

 

 

 

5

114

0.113659

0.0972

6

41

0.123

0.125

6

55

0.15363

0.1667

 

 

 

6

96

0.095713

0.0972

7

38

0.114

0.125

 

 

 

 

 

 

 

7

38

0.037886

0.0417

8

35

0.105

0.125

 

 

 

 

 

 

 

8

35

0.034895

0.0417

Total

333

1

1

Total

358

1

1

Total

312

1

1

Total

1003

1

 

 

 

 

Conditional Probability

 

Conditional = Joint / Prior

 

Pr{A|B} = Pr{A∩B} / Pr{B}

 

How much of B is tied up in A ?

 

Case Study 1.11

Conditional Probability

Case Study Description: Compute conditional probabilities associated with the color sequence experiment.

Suppose that we have a special box - each time we press a button on the box, it prints out a sequence of colors, in order - it prints four colors at a time. Suppose the box follows the following Probabilities for each Color Sequence:

 

 

 

 

Color Sequence

Probability CS Prints Out

BBBB

.10 = 10%

BGGB

.25 = 25%

RGGR

.05 = 05%

YYYY

.30 = 30%

BYRG

.15 = 15%

RYYB

.15 = 15%

Total

1.00 = 100%

 

Let's define the experiment: We push the button, and then the box prints out exactly one (1) of the above listed color sequences. We then note the resulting (printed out) color sequence.

Compute Pr{ blue shows 1st | blue shows 4th };

Pr{ B 1st and B 4th } = Pr{ exactly one of BBBB, BGGB shows } = Pr{ BBBB} + Pr{BGGB} =.10 + .25 = .35

 

Pr{ B 4th } = Pr{ exactly one of BBBB, BGGB, RYYB shows } = Pr{BBBB} + Pr{BGGB} +

Pr{RYYB} = .10+.25+.15 = .50

 

So, Pr{ B 1st | B 4th } = .35/ .50= .70

Compute Pr{ green shows 2nd or 3rd | yellow shows };

Pr{ G 2nd or 3rd and Y shows } = 0, since no sequences meet this requirement

 

Pr{ Y shows } = Pr{ exactly one of YYYY, BYRG, RYYB shows } = Pr{YYYY}+ Pr{BYRG}+ Pr{RYYB} = .30+.15+.15 = .60

 

So, Pr{ G 2nd or 3rd | Y shows } = 0 / .60= 0

Compute Pr{ yellow shows | red shows }.

Pr{ Y and R show } = Pr{ exactly one of BYRG, RYYB shows } = Pr{BYRG}+ Pr{RYYB } = .15 + .15 = .30

 

Pr{ R shows } = Pr{ exactly one of RGGR, BYRG, RYYB shows } = Pr{RGGR}+ Pr{BYRG}+

Pr{RYYB } = .05+.15+.15 = .35

 

So, Pr{ Y shows | R shows } = .30/.35 = 6/7 = .8571

 

Case Study 1.12

Conditional Probability II: Pair of Dice

Case Description: Compute conditional probabilities.

Suppose we have a pair of fair dice: d4(faces 1,2,3,4), d6(faces 1,2,3,4,5,6). In our experiment, we toss this pair of dice, and note the face value from each die. For simplicity, we write the outcome as (d4 result, d6 result). Assume that the dice operate independently and separately.

Case Objectives:

Identify the simple (basic) events. Compute (and justify) a probability for each simple event. 

As before, Pr{ ( any d4 face, any d6 face) } = Pr{ any d4 face }*Pr{ any d6 face } = (1/4)*(1/6) = 1/24

We have 24 equally likely pairs.

 

1

2

3

4

1

(1,1)

(2,1)

(3,1)

(4,1)

2

(1,2)

(2,2)

(3,2)

(4,2)

3

(1,3)

(2,3)

(3,3)

(4,3)

4

(1,4)

(2,4)

(3,4)

(4,4)

5

(1,5)

(2,5)

(3,5)

(4,5)

6

(1,6)

(2,6)

(3,6)

(4,6)

Suppose we observe the sum of the faces in the pair of dice. Identify the possible values of this sum, and compute (and justify) a probability for each value.

Now for the sums:

 

1

2

3

4

1

(1,1) @ 2

(2,1) @ 3

(3,1) @ 4

(4,1) @ 5

2

(1,2) @ 3

(2,2) @ 4

(3,2) @ 5

(4,2) @ 6

3

(1,3) @ 4

(2,3) @ 5

(3,3) @ 6

(4,3) @ 7

4

(1,4) @ 5

(2,4) @ 6

(3,4) @ 7

(4,4) @ 8

5

(1,5) @ 6

(2,5) @ 7

(3,5) @ 8

(4,5) @ 9

6

(1,6) @ 7

(2,6) @ 8

(3,6) @ 9

(4,6) @ 10

 

Compute the conditional probability Pr{Sum is Even|d4 shows Even}.

Pr{ Sum is Even and d4 shows Even } = Pr{ exactly one of (2,2), (2,4), (2,6), (4,2), (4,4), (4,6) shows } = 6/24 = 1/4 = .25

Pr{ d4 shows Even } =

Pr{ exactly one of (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6) shows } = 12/24 = 1/2 = .50

So, Pr{ Sum is Even | d4 shows Even } = .25 / .50 = .50

Continuing,…

 

1

2

3

4

1

(1,1) @ 2

(2,1) @ 3

(3,1) @ 4

(4,1) @ 5

2

(1,2) @ 3

(2,2) @ 4

(3,2) @ 5

(4,2) @ 6

3

(1,3) @ 4

(2,3) @ 5

(3,3) @ 6

(4,3) @ 7

4

(1,4) @ 5

(2,4) @ 6

(3,4) @ 7

(4,4) @ 8

5

(1,5) @ 6

(2,5) @ 7

(3,5) @ 8

(4,5) @ 9

6

(1,6) @ 7

(2,6) @ 8

(3,6) @ 9

(4,6) @ 10

 

Compute the conditional probability Pr{Sum is Odd|d6 shows Odd}.

Pr{ Sum is Odd and d6 shows Odd } = Pr{ exactly one of (2,1), (2,3), (2,5), (4,1), (4,3), (4,5) shows } = 6/24 = 1/4 = .25

Pr{ d6 shows Odd } =

Pr{ exactly one of (1,1), (1,3), (1,5), (2,1), (2,3), (2,5), (3,1), (3,3), (3,5), (4,1), (4,3), (4,5) shows } = 12/24 = 1/2 = .50

So, Pr{ Sum is Odd | d6 shows Odd } = .25/.50 = 1/2 = .50

HR1 – Summer Version A, Case Three

 

Case Three | Color Slot Machine | Conditional Probabilities

 

Here is our slot machine – on each trial, it produces a 10-color sequence, using the table below:

 

Sequence*

Probability

RRBBRRYRRR

.10

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

1.00

*B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered from left to right: (1st 2nd 3rd 4th 5th6th7th 8th 9th 10th )

Compute the following conditional probabilities:

 

Pr{ Yellow Shows Exactly Twice | Blue Shows}

 

 

Sequence*

Probability

RRBBRRYRRR

.10

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

1.00

 

Pr{Blue Shows} = Pr{One of RRBBRRYRRR, RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =

Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} + Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} = .1+.1+.15+.1+.25+.1+.2 = 1.00

 

Sequence*

Probability

 

 

Total

0

 

 

Pr{ Yellow Shows Exactly Twice and  Blue Shows} = 0

 

 

Pr{ Yellow Shows Exactly Twice | Blue Shows} = Pr{ Yellow Shows Exactly Twice and Blue Shows}/Pr{Blue Shows} = 0/1 =0

 

Pr{ Green Shows | “BR” Shows }

 

Sequence*

Probability

RRBBRRYRRR

.10

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

Total

.45

 

Pr{ “BR” Shows } = Pr{One of RRBBRRYRRR, RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{RRBBRRYRRR}+ Pr{ RRGGRGBRRB}+

Pr{BBYYGGYGBR}+ Pr{GRRGGYBRGG} = .1+.1+.15+.1 = .45

 

 

Pr{Green Shows and “BR” Shows}

 

Sequence*

Probability

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

Total

.35

 

Pr{ Green Shows and “BR” Shows } = Pr{One of RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{ RRGGRGBRRB}+

Pr{BBYYGGYGBR}+ Pr{GRRGGYBRGG} =.1+.15+.1 = .35

 

Pr{ Green Shows | “BR” Shows } = Pr{ Green Shows and “BR” Shows }/Pr{ “BR” Shows } = .35/.45 = 7/9

 

 

Pr{ Red Shows | Green Shows}

 

Sequence*

Probability

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

.90

 

Pr{Green Shows} = Pr{One of RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =

Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} + Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} = .1+.15+.1+.25+.1+.2 = .90

 

Pr{ Red Shows and Green Shows}

 

Sequence*

Probability

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

.90

 

Pr{ Red and Green Show } = Pr{One of RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =

Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} + Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} = .1+.15+.1+.25+.1+.2 = .90

 

Pr{ Red Shows | Green Shows} = Pr{ Red Shows and Green Shows}/Pr{ Green Shows} = .90/.90 = 1

 

HR1 – Spring 2008, Case Four

 

Case Four: Color Slot Machine, Computation of Conditional Probabilities

 

Here is our slot machine – on each trial, it produces a 10-color sequence, using the table below:

 

Sequence*

Probability

RRBBR RYRRB

.10

RRGGRGBRRB

.10

BBYYRGYGBR

.15

GRRGRGBRGB

.10

BGYGYRYGYY

.25

RRGYGRRBBB

.10

YYGBYYBGRR

.20

Total

1.00

*B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered as 1st to 6th , from left to right: (1st 2nd 3rd 4th 5th6th7th 8th 9th 10th )

Compute the following conditional probabilities:

 

1. Pr{Red Shows Somewhere in the 1st ─ 4th slots | Yellow Shows Somewhere in the 7th ─ 10th slots}

 

Pr{Red Shows in the 1st – 4th slots|Yellow Shows in the 7th – 10th slots} =

 

Pr{Red Shows in the 1st – 4th slots and Yellow Shows in the 7th – 10th slots}/ Pr{ Yellow Shows in the 7th – 10th slots}

 

Sequence*

Probability

RRBBR RYRRB

.10

BBYYRGYGBR

.15

BGYGYRYGYY

.25

Total

0.50

 

Pr{ Yellow Shows in the 7th – 10th slots} = Pr{One of RRBBRRYRRB, BBYYRGYGBR, BGYGYRYGYY shows} =

Pr{RRBBRRYRRB}+ Pr{BBYYRGYGBR}+ Pr{BGYGYRYGYY} = .10+.15+.25 = .50

 

Sequence*

Probability

RRBBR RYRRB

.10

Total

0.10

 

Pr{ Red Shows in the 1st – 4th slots and Yellow Shows in the 7th – 10th slots } = Pr{One of RRBBRRYRRB shows} = .10

 

Pr{Red Shows in the 1st – 4th slots|Yellow Shows in the 7th – 10th slots} = .10/.50 = .20

 

2. Pr{Green Shows Anywhere  | “RB” Shows Anywhere}

 

Pr{Green Shows Anywhere|”RB” Shows Anywhere} =

 

Pr{ Green Shows Anywhere and ”RB” Shows Anywhere }/ Pr{”RB” Shows Anywhere}

 

Sequence*

Probability

RRBBR RYRRB

.10

RRGGRGBRRB

.10

RRGYGRRBBB

.10

Total

0.30

Pr{”RB” Shows Anywhere} = Pr{One of  RRBBRRYRRB, RRGGRGBRRB, RRGYGRRBBB Shows} = Pr{RRBBRRYRRB}+Pr{RRGGRGBRRB}+Pr{RRGYGRRBBB} =.10+.10+.10 = .30

 

Sequence*

Probability

RRGGRGBRRB

.10

RRGYGRRBBB

.10

Total

0.20

 

Pr{ Green Shows Anywhere and ”RB” Shows Anywhere } = Pr{One of  RRGGRGBRRB, RRGYGRRBBB Shows} = Pr{RRGGRGBRRB}+Pr{RRGYGRRBBB} =.10+.10 = .20

 

Pr{Green Shows Anywhere|”RB” Shows Anywhere} = .20/.30

 

3. Pr{Yellow Shows Anywhere | Blue Shows Anywhere}

 

Pr{Yellow Shows Anywhere | Blue Shows Anywhere} =

Pr{Yellow Shows Anywhere and Blue Shows Anywhere}/Pr{ Blue Shows Anywhere}

 

Sequence*

Probability

RRBBR RYRRB

.10

RRGGRGBRRB

.10

BBYYRGYGBR

.15

GRRGRGBRGB

.10

BGYGYRYGYY

.25

RRGYGRRBBB

.10

YYGBYYBGRR

.20

Total

1.00

Pr{ Blue Shows Anywhere} = Pr{one of RRBBRRYRRB, RRGGRGBRRB, BBYYRGYGBR, GRRGRGBRGB, BGYGYRYGYY, RRGYGRRBBB, YYGBYYBGRR Shows} =Pr{RRBBRRYRRB}+Pr{RRGGRGBRRB}+Pr{ BBYYRGYGBR}+Pr{GRRGRGBRGB}+Pr{BGYGYRYGYY}+Pr{RRGYGRRBBB}+Pr{YYGBYYBGRR} = .10+.10+.15+.10+.25+.10+.20 = 1.00

 

Sequence*

Probability

RRBBR RYRRB

.10

BBYYRGYGBR

.15

BGYGYRYGYY

.25

RRGYGRRBBB

.10

YYGBYYBGRR

.20

Total

0.80

 

 

Pr{Yellow Shows Anywhere and Blue Shows Anywhere} = Pr{one of RRBBRRYRRB, BBYYRGYGBR, BGYGYRYGYY, RRGYGRRBBB, YYGBYYBGRR Shows} =Pr{RRBBRRYRRB}+ Pr{BBYYRGYGBR}+Pr{BGYGYRYGYY}+Pr{RRGYGRRBBB}+Pr{YYGBYYBGRR} = .10+.15+.25+.10+.20 = .80

 

Pr{Yellow Shows Anywhere | Blue Shows Anywhere} = .80/1.00 = .80

 

 

Case Study 1.13

Conditional Probability

Case Description: Compute conditional probabilities for pairs of draws (without replacement).

Here is our bowl, in tabular form:

Color

# in Bowl

Proportion of Bowl

Blue

5

5/9

Green

3

3/9

Red

1

1/9

Total

9

1

 

Suppose that on each trial of this experiment that we make two (2) draws without replacement from the bowl.

Compute Pr{ green shows 2nd | red shows 1st };

Here is our bowl, after "red shows 1st", in tabular form:

 

Color

# in Bowl – Before 1st Draw

# in Bowl – After 1st Draw

Blue

5

5 – 0 = 5

Green

3

3 – 0 = 3

Red

1

1 – 1 = 0

Total

9

8

With the red chip out of the bowl, 3 of the 8 surviving chips are green. So, Pr{G 2nd | R 1st} = (3-0) / (9-1) = 3/8

Compute Pr{ red shows 2nd | red shows 1st };

Pr{R 2nd | R 1st} = (1-1) / (9-1) = 0/8 = 0. There are 1-1=0 surviving red chips after the first draw.

Compute Pr{ blue shows 2nd | blue shows 1st }.

Here is our bowl, after "blue shows 1st", in tabular form:

Color

# in Bowl – Before 1st Draw

# in Bowl – After 1st Draw

Blue

5

5 – 1 = 4

Green

3

3 – 0 = 3

Red

1

1 – 0 = 1

Total

9

8

Pr{B 2nd | B 1st} = (5-1)/(9-1) = 4/8. After the first draw, 4 of 8 surviving chips are blue.

HR1 – Fall 2004, Case Three

Case Three

Conditional Probability

Color Bowl/Draws without Replacement

 

We have a bowl containing the following colors and counts of balls (color@count):

 

Blue @ 5, Green @ 1, Red @ 2, Yellow @ 3

 

Each trial of our experiment consists of three (3) draws without replacement from the bowl.

 

Compute these directly.

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

 

Pr{ green shows 2nd | green shows 1st}

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

ß green shows 1st

Color

Count

B

5

G

0

R

2

Y

3

Total

10

 

Pr{ green shows 2nd  | green shows 1st} = 0/10

  

Pr{ yellow shows 3rd | green shows 1st, blue shows 2nd}

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

ß green shows 1st

Color

Count

B

5

G

0

R

2

Y

3

Total

10

ß blue shows 2nd

Color

Count

B

4

G

0

R

2

Y

3

Total

9

 

Pr{ yellow shows 3rd | green shows 1st, blue shows 2nd} = 3/9

 

Pr{ red shows 3rd | green shows 1st, red shows 2nd }

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

ß green shows 1st

Color

Count

B

5

G

0

R

2

Y

3

Total

10

ß red shows 2nd

Color

Count

B

5

G

0

R

1

Y

3

Total

9

 

Pr{ red shows 3rd | green shows 1st, red shows 2nd } = 1/9

 

Clinical Trial Worksheet

 

NIH Clinical Trial Database

 

Topical Search Interface

 

From here:

 

Case Two | Clinical Trial Sketch | Non-small Cell Lung Cancer (NSCLC)

 

A key ability of malignant cells is the ability to induce angiogensis, the formation of new blood supply. These cells can release a substance that stimulates the formation of new blood vessels. This ability is key in the ability of malignant tumors to survive and grow.Avastin is a monoclonal antibody that works by attaching to and inhibiting the action of vascular endothelial growth factor (VEGF) in laboratory experiments. VEGF is a substance that binds to certain cells to stimulate new blood vessel formation.  When VEGF is bound to Avastin, it cannot stimulate the formation and growth of new blood vessels. A number of cancers are driven by the derangement of cells composing the linings (epidermal cells) of various organs in the body. In particular, these cells lose control of their growth behaviors, leading to uncontrolled reproduction of cells. This deranged, accelerated cell reproduction is key to the ability of malignant tumors to grow.

 

Tarceva (erlotinib) is an oral anti-cancer drug under development by OSI Pharmaceuticals, Genentech and Roche. It is a member of the epidermal growth factor receptor (EGFR) inhibitor class of agents. Two general types of lung cancer exist: Non-Small Cell Lung Cancer (NSCLC) and small-cell lung cancer (SCLC). The most common type of lung cancer is NSCLC. Approximately 85% of all lung cancer cases are NSCLC. Three main types of NSCLC - General treatment options for each of these are the same: Squamous cell carcinoma. Most often related to smoking. These tumors may be found in the mucous membrane that lines the bronchi. Sometimes the tumor spreads beyond the bronchi. Coughing up blood may be a sign of squamous cell NSCLC. Adenocarcinoma (including bronchioloalveolar carcinoma). Most often found in nonsmokers and women. Cancer is usually found near the edge of the lung. Adenocarcinoma can enter the chest lining. When that happens, fluid forms in the chest cavity. This type of NSCLC spreads (metastasizes) early in the disease to other body organs. Large-cell undifferentiated carcinoma. Rare type of NSCLC. Tumors grow quickly and spread early in the disease. Tumors are usually larger than 1-1/2 inches.

 

First-line Treatments for NSCLC: Surgery: Removes the tumor. This can be done if the tumor is small and has not spread to other areas of your body. Radiation: Destroys any leftover cancer cells not removed by surgery. This may be done before surgery to make it easier to remove the tumor. Radiation can also be done after surgery. Chemotherapy may help slow the growth of cancer cells and destroy them. Chemotherapy may be used with radiation to help shrink the tumor before surgery. It may be used after surgery or radiation to destroy any cancer cells that may have been left behind.

 

Consider patients with locally advanced or metastatic Non-Small Cell Lung Cancer (NSCLC) after failure of at least one previous chemotherapy regimen. Consider two treatments: Avastin+Tarceva and Tarceva. Sketch a comparative clinical trial for Avastin+Tarceva versus Tarceva in the treatment of patients with locally advanced or metastatic Non-Small Cell Lung Cancer (NSCLC) after failure of at least one previous chemotherapy regimen.

 

We recruit subjects with with locally advanced or metastatic Non-Small Cell Lung Cancer (NSCLC) after failure of at least one previous chemotherapy regimen. Those who give informed consent and who qualify are enrolled in the trial.

 

Enrolled subjects are randomly assigned to either Tarceva + Avastin (TA) or to Tarceva + PalceboAvastin (T) with double blinding, so that neither the subjects nor the trial workers know the actual treatment status of the subjects.

 

Subjects are followed for safety and toxicity, including kidney or liver damage.

 

Subjects are followed for their cancer status – has the cancer stabilized? Has it spread further? Has it receded? Is the cancer more treatable?

 

Subjects are followed for mortality and time-to-death. Do fewer subjects die in the TA group relative to the T group? Do those who die live longer in the TA group relative to the T group?

 

Subjects are followed for quality of life – are subjects in the TA group better able to live normally and to manage their pain than subjects in the T group?

 

From here:

 

Case Six | Clinical Trial Sketch | Study of Tamoxifen and Raloxifene (STAR) for the Prevention of Breast Cancer

 

The purpose of this study is to examine the performance of the drug Raloxifene (relative to the drug Tamoxifen) in reducing the incidence of breast cancer in postmenopausal women1 who are at increased risk of the disease2.

 

1. Postmenopausal women at increased risk for developing invasive breast cancer, who meet one of the following criteria: At least 12 months since spontaneous menstrual bleeding; Prior documented hysterectomy and the surgical removal of fallopian tubes and ovaries; At least 55 years of age with prior hysterectomy with or without surgical removal of the ovaries; Aged 35 to 54 years with a prior hysterectomy without surgical removal of the ovaries or with a status of ovaries unknown with documented follicle-stimulating hormone level demonstrating elevation in postmenopausal range.

 

2. Women without prior breast cancer, but who are at elevated risk for breast cancer: Histologically confirmed lobular carcinoma in situ treated by local excision only or at least 1.66% probability of invasive breast cancer within 5 years using Breast Cancer Risk Assessment Profile; No clinical evidence of malignancy on physical exam within the past 180 days; No evidence of suspicious or malignant disease on bilateral mammogram within the past year; No bilateral or unilateral prophylactic mastectomy and No prior invasive breast cancer or intraductal carcinoma in situ

Objectives: Determine whether Raloxifene is more or less effective than Tamoxifen in significantly reducing the incidence rate of invasive breast cancer in postmenopausal women; Evaluate the effects of Tamoxifen and Raloxifene on the incidence of intraductal carcinoma in situ, lobular carcinoma in situ, endometrial cancer, ischemic heart disease, fractures of the hip and spine, or Colles' fractures of the wrist in these participants; Evaluate the toxic effects of these regimens in these participants and Determine the effect of these regimens on the quality of life of these participants.

Sketch a comparative clinical trial to evaluate the drug Raloxifene (relative to the drug Tamoxifen) in reducing the incidence of breast cancer in postmenopausal women1 who are at increased risk of the disease2.

 

http://www.cancer.gov/star

Solution

Purpose of Treatment: The purpose of this study is to examine the performance of the drug Raloxifene (relative to the drug Tamoxifen) in reducing the incidence of breast cancer in postmenopausal women1 who are at increased risk of the disease2.

 

Eligible subjects are: 1. postmenopausal women at increased risk for developing invasive breast cancer, who meet one of the following criteria: At least 12 months since spontaneous menstrual bleeding; Prior documented hysterectomy and the surgical removal of fallopian tubes and ovaries; At least 55 years of age with prior hysterectomy with or without surgical removal of the ovaries; Aged 35 to 54 years with a prior hysterectomy without surgical removal of the ovaries or with a status of ovaries unknown with documented follicle-stimulating hormone level demonstrating elevation in postmenopausal range.

 

2. Women without prior breast cancer, but who are at elevated risk for breast cancer: Histologically confirmed lobular carcinoma in situ treated by local excision only or at least 1.66% probability of invasive breast cancer within 5 years using Breast Cancer Risk Assessment Profile; No clinical evidence of malignancy on physical exam within the past 180 days; No evidence of suspicious or malignant disease on bilateral mammogram within the past year; No bilateral or unilateral prophylactic mastectomy and No prior invasive breast cancer or intraductal carcinoma in situ. The eligible patients are briefed as to the details and potential risks and benefits of study participation, and those who give informed consent and who meet all inclusion and exclusion requirements are enrolled in the trial.

Study treatments include Raloxifene and Tamoxifen. Enrolled subjects are randomly assigned either to Raloxifene with PlaceboTamoxifen or to Tamoxifen with PlaceboRalixifene. Double-blinding is employed, so that neither the subjects nor the clinical workers know the actual individual treatment assignments.

Subjects are then followed for: Incidence of invasive breast cancer in postmenopausal women; Incidence of intraductal carcinoma in situ, Incidence of lobular carcinoma in situ, Incidence of endometrial cancer, Incidence of ischemic heart disease, Incidence of fractures of the hip and spine, and Incidence of  Colles' fractures of the wrist, Toxic effects of the medications, and Quality of Life.

Case Study - Gastric Adenocarcinoma

Case Study - Preeclampsia I

Case Study - Preeclampsia II

Case Study - Myocardial Infarction

Case Study - Traumatic Brain Injury

Case Study - Carbon Monoxide Intoxication

Case Study - Ocular Hypertension

From http://clinicaltrials.gov: Study Phase (FDA Clinical Trials)

Most clinical trials are designated as phase I, II, or III, based on the type of questions that study is seeking to answer:

In Phase I clinical trials, researchers test a new drug or treatment in a small group of people (20-80) for the first time to evaluate its safety, determine a safe dosage range, and identify side effects.

In Phase II clinical trials, the study drug or treatment is given to a larger group of people (100-300) to see if it is effective and to further evaluate its safety.

In Phase III studies, the study drug or treatment is given to large groups of people (1,000-3,000) to confirm its effectiveness, monitor side effects, compare it to commonly used treatments, and collect information that will allow the drug or treatment to be used safely.

These phases are defined by the Food and Drug Administration in the Code of Federal Regulations.

Comparative Clinical Trial

Advanced Stomach Cancer (Gastric Adenocarcinoma)

Use of Combination Chemotherapy

The purpose of this trial is to test the combination of Gleevec® (also known as imatinib mesylate) and Taxotere (also known as docetaxel) in patients with incurable stomach cancer. This study is being performed to see if the combination of Gleevec and Taxotere is an effective treatment for incurable stomach cancer with minimal side effects.

Gleevec is a pill form of chemotherapy and is indicated for the treatment of adult patients with chronic myeloid leukemia (CML) and gastrointestinal stromal tumors (GIST). It is considered experimental for the treatment of stomach cancer.

Taxotere is a chemotherapy which is injected into the vein. It is approved for breast and lung cancer but has been shown to shrink many different types of tumors. Taxotere has been shown to shrink stomach cancer in about 20% - 30% of patients treated with Taxotere.

An adenocarcinoma is a cancer that develops in the glandular lining of an organ. A gastric adenocarcinoma is a cancer that that develops in the glandular lining of the stomach.

Define advanced gastric adenocarcinoma for the purposes of this clinical trials as surgically inoperable gastric adenocarcinoma.

This study is being performed to see if the combination of Gleevec and Taxotere is an effective treatment for advanced stomach cancer.

Sketch a comparative clinical trial for Gleevec+Taxotere versus Taxotere in the treatment of advanced gastric adenocarcinoma.

Condition of Interest: advanced gastric adenocarcinoma, defined as surgically inoperable gastric adenocarcinoma.

Subjects: Adult patients diagnosed with advanced gastric adenocarcinoma.

Recruitment and Informed Consent: We recruit volunteer candidates who have been diagnosed with advanced gastric adenocarcinoma, and who meet all requirements for study inclusion, We exclude all candidates presenting one or more conditions for exclusion. The volunteers are briefed as to the requirements, details, potential benefits and risk associated with trial participation. Those who give informed consent agree to participate and are enrolled in the trial.

Assignment to Treatment: Enrolled subjects are randomly assigned to either Gleevec+Taxotere or to Placebo+Taxotere, where Placebo represents a placebo version of Gleevec. Double-blinding is employed in the trial, so that neither the study subjects nor their clinical personnel know the actual assignment status of any subject.

Endpoints and Follow-up: Subjects are followed for toxicity, safety, effect and quality-of-life.

Toxicity involves severe events such as anaphylaxis (shock), kidney or liver failure/damage, and the like.

Adverse Events involve lesser events like the things you read in the package inserts: rashes, “dry mouth”, gastrointestinal effects, nausea, and the like.

Effect involves the actual effect of the treatment, measured as change in disease status or progression. In this case, we’re dealing with progression and stage of the cancer: tumor size, metastasis (spreading) and the like. We also consider survival time,vital status and stomach function.

Quality of Life: We consider pain control, basic body function, ability to work, live and play, ability to maintain cogent consciousness, ability to live independently or with minimal assistance.

We compare the performance of each treatment group in these results: Toxicity, Safety, Effect and QoL.

Comparative Clinical Trial

Pre-eclampsia

Magnesium Sulfate versus Nimodipine

Determine the effectiveness of nimodipine versus magnesium sulfate in the prevention of eclamptic seizures in patients with severe pre-eclampsia.

Nimodipine: Patients receive nimodipine by mouth every 4 hours. Treatment is continued until 24 hours post-partum.

Magnesium Sulfate: Patients receive a loading dose of magnesium sulfate IV for 20 minutes, followed by continuous infusion of magnesium sulfate. Treatment is continued until 24 hours post-partum.

Severe pre-eclampsia involves the onset of hypertension (high blood pressure) in the late stages of pregnancy, as well as proteinuria (excessive levels of protein in the urine), thrombocytopenia (deficiency of blood platelets) and swelling (edema).

This study is being performed to compare the effectiveness of Magnesium Sulfate and Nimodipine in the treatment of severe pre-eclampsia.

Sketch a comparative clinical trial for the comparison of Magnesium Sulfate and Nimodipine in the treatment of severe pre-eclampsia.

Condition of Interest: severe pre-eclampsia.

Subjects: Pregnant patients diagnosed with pre-eclampsia.

Recruitment and Informed Consent: We recruit volunteer candidates who have been diagnosed with severe pre-eclampsia, and who meet all requirements for study inclusion, We exclude all candidates presenting one or more conditions for exclusion. The volunteers are briefed as to the requirements, details, potential benefits and risk associated with trial participation. Those who give informed consent agree to participate and are enrolled in the trial.

Assignment to Treatment: Enrolled subjects are randomly assigned to either magnesium sulfate + placebo{ Nimodipine} or to Nimodipine +placebo{Magnesium Sulfate}. Double-blinding is employed in the trial, so that neither the study subjects nor their clinical personnel know the actual assignment status of any subject.

Endpoints and Follow-up: Subjects are followed for toxicity, safety, effect and quality-of-life.

Toxicity involves severe events such as anaphylaxis (shock), kidney or liver failure/damage, and the like.

Adverse Events involve lesser events like the things you read in the package inserts: rashes, “dry mouth”, gastrointestinal effects, nausea, and the like.

Effect involves the actual effect of the treatment, measured as change in disease status or progression. In this case, we’re dealing with the frequency and severity of eclamptic seizures in the pregnant woman. We will also track the other aspects of pre-eclampsia: hypertension (high blood pressure) in the late stages of pregnancy, proteinuria (excessive levels of protein in the urine), thrombocytopenia (deficiency of blood platelets) and swelling (edema).  

We compare the performance of each treatment group in these results: Toxicity, Safety, Effect.

Basic Clinical Trial

Pre-eclampsia

Sildenafil Citrate

To determine the efficacy and safety of sildenafil citrate in the treatment of established pre-eclampsia.

Sildenafil Citrate: Better known as Viagra, this drug is a vaso-dilator. The medication causes blood vessels to dilate, enabling a drop in blood pressure.

Pre-eclampsia involves the onset of hypertension (high blood pressure) in the late stages of pregnancy, as well as proteinuria (excessive levels of protein in the urine), thrombocytopenia (deficiency of blood platelets) and swelling (edema).

This study is being performed to see if the Sildenafil Citrate is an effective treatment for pre-eclampsia.

Sketch a basic clinical trial for Sildenafil Citrate in the treatment of Pre-eclampsia.

Condition of Interest: Pre-eclampsia.

Subjects: Pregnant patients diagnosed with pre-eclampsia.

Recruitment and Informed Consent: We recruit volunteer candidates who have been diagnosed with pre-eclampsia, and who meet all requirements for study inclusion, We exclude all candidates presenting one or more conditions for exclusion. The volunteers are briefed as to the requirements, details, potential benefits and risk associated with trial participation. Those who give informed consent agree to participate and are enrolled in the trial.

Assignment to Treatment: Enrolled subjects are randomly assigned to either Sildenafil Citrate or to Placebo. Double-blinding is employed in the trial, so that neither the study subjects nor their clinical personnel know the actual assignment status of any subject.

Endpoints and Follow-up: Subjects are followed for toxicity, safety, effect and quality-of-life.

Toxicity involves severe events such as anaphylaxis (shock), kidney or liver failure/damage, and the like.

Adverse Events involve lesser events like the things you read in the package inserts: rashes, “dry mouth”, gastrointestinal effects, nausea, and the like.

Effect involves the actual effect of the treatment, measured as change in disease status or progression. In this case, we’re dealing equally with all aspects of pre-eclampsia. We track the frequency and severity of eclamptic seizures in the pregnant woman. We track all aspects of pre-eclampsia: hypertension (high blood pressure) in the late stages of pregnancy, proteinuria (excessive levels of protein in the urine), thrombocytopenia (deficiency of blood platelets) and swelling (edema).  

We compare the performance of each treatment group in these results: Toxicity, Safety, Effect.

Old Case Studies

Case Study - Simvastatin and Heart Attacks

Heart Attacks

The heart continuously pumps blood enriched with oxygen and vital nutrients through a network of arteries to all parts of the body's tissues. The heart muscle itself needs a plentiful supply of oxygen-rich blood, which is provided through a network of coronary arteries. These arteries carry oxygen-rich blood to the heart's muscular walls (the myocardium). Coronary artery disease is the most common cause of heart attacks, which occurs when blood flow to the myocardium is interrupted. Heart attack occurs when blood flow is blocked and tissue death occurs from loss of oxygen, severely damaging the heart. Coronary artery disease is the end result of a complex process commonly called "hardening of the arteries"). This causes blockage of arteries and prevents oxygen-rich blood from reaching the heart.

Cholesterol and Lipoproteins. The story begins with cholesterol and sphere shaped bodies called lipoproteins that transport cholesterol. Cholesterol is a white, powdery nutrient that is found in all animal cells and in animal-based foods. The lipoproteins that transport cholesterol are referred to by their size. The most commonly known are low-density lipoproteins (LDL) and high density lipoproteins (HDL). In heart disease, free radicals are released in artery linings and oxidize low-density lipoproteins (LDL). The oxidized LDL is the basis for cholesterol build-up on the artery walls. The injuries to the arteries during oxidation signal the immune system to release white blood cells (particularly those called neutrophils and macrophages) at the site. These factors initiate the inflammatory response. Macrophages literally "eat" foreign debris, in this case oxidized LDL cholesterol. The process converts LDL cholesterol into foamy cells that attach to the smooth muscle cells of the arteries. The cholesterol becomes mushy and accumulates on artery walls. Over time the cholesterol dries and forms a hard plaque, which causes further injury to the walls of the arteries. Eventually these calcified (hardened) arteries become narrower. As this narrowing and hardening process continues, blood flow slows and prevents sufficient oxygen-rich blood from reaching the heart.

Simvastatin is a drug that interferes in the early stages of cholesterol. This drug actively lowers the levels of serum cholesterol, and it is thought that this effect may afford protection against heart attacks.

Sketch a basic clinical trial of simvastatin that evaluates the effectiveness of simvastatin in preventing heart attacks.

Describe the treatments, and the outcome(s) by which the treatments will be evaluated.

The active treatment is simvastatin. A basic clinical trial is indicated, so there will be a placebo version of simvastatin.

Do we want a basic, or comparative trial ?

A basic clinical trial, which uses a placebo for comparison.

Identify the subject population for this trial.

For the purpose of this trial, we might focus on subjects who are free of previous heart attacks, but who do show elevated serum cholesterol. Subjects eligible for the trial must volunteer and give informed consent in order to participate in the trial.

Discuss the assignment of subjects to the treatments in the trial.

Enrolled subjects are randomly assigned to either simvastatin or to its placebo version. Neither the subjects nor the clinical workers will know which drug has been assigned – this is called double blinding.

We will track the trial subjects in both treatment groups for a number of outcomes:

 

Safety – any adverse reactions to Taxol

Heart Attack – do the subjects present heart attacks (MI) during follow-up?

Time to Event – how long does it take the subjects to present MI?

Survival Status – do the subjects die during followup? Do they die of MI?

Cholesterol Levels – do subjects show decreased serum cholesterol?

 

Case Study - Corticosteroids and Traumatic Brain Injury (TBI)

Traumatic Brain Injury

Traumatic Brain Injury (TBI) involves the injury of the brain when it involves sudden or intense physical force resulting in the presence of Concussion, Skull Fracture, or Bleeding and Tissue Damage (Contusions, Lacerations, Hemorrhaging) involving the brain. Tissue damage to the brain results from the traumatic force of injury, swelling (inflammation) and bleeding. Consequences of TBI include death, intellectual impairment, social and emotional impairment and physical disability.

Inflammation

Inflammation is the response of living tissue to damage. The acute inflammatory response has 3 main functions. The affected area is occupied by a transient material called the acute inflammatory exudate. The exudate carries proteins, fluid and cells from local blood vessels into the damaged area to mediate local defenses. The damaged tissue can be broken down and partially liquefied, and the debris removed from the site of damage.

The cause of acute inflammation may be due to physical damage, chemical substances, micro-organisms or other agents. The inflammatory response consist of changes in blood flow, increased permeability of blood vessels and escape of cells from the blood into the tissues. The changes are essentially the same whatever the cause and wherever the site.

Acute inflammation is short-lasting, lasting only a few days. If it is longer lasting however, then it is referred to as chronic inflammation.

Corticosteroids

Corticosteroids are drugs that reduce inflammation. Corticosteroids, often referred to as steroids, are related to cortisol, a naturally produced hormone that controls many important body functions. In normal amounts, corticosteroids play an important role in the regulation of blood sugar levels, salt and water, and in metabolism and growth. They also reduce the activity of the body's immune system and act to suppress allergic reactions. Corticosteroids are used to decrease the inflammation that causes the pain, redness and swelling associated with inflammatory diseases.

Sketch a basic clinical trial of corticosteroids that evaluates the effectiveness of corticosteroids in reducing death and disability following TBI.

Describe the treatments, and the outcome(s) by which the treatments will be evaluated.

The active treatment is corticosteroid (CS). A basic clinical trial is indicated, so there will be a placebo version of CS.

Do we want a basic, or comparative trial ?

A basic clinical trial, which uses a placebo for comparison.

Identify the subject population for this trial.

Subjects who qualify for this trial have just suffered a traumatic brain injury (TBI). Subjects eligible for the trial must volunteer and give informed consent in order to participate in the trial. Given the altered consciousness that goes with brain injuries, this trial will utilize the appropriate proxy consent, in which the subject’s medical agent gives consent.

Discuss the assignment of subjects to the treatments in the trial.

Enrolled subjects are randomly assigned to either CS or to its placebo version. Neither the subjects nor the clinical workers will know which drug has been assigned – this is called double blinding.

We will track the trial subjects in both treatment groups for a number of outcomes:

 

Safety – any adverse reactions to corticosteroid (CS)?

Physical Effects – how does post treatment brain tissue damage compare?

Cognitive Effects – how well does the patient recover cognitive function:

            Memory Function

            Coordination

            Speech

            Thought

            Emotional Stability

            Impulse Control

Life Effects – how well does the patient recover life function:

            Career/Job Function

            Social Function

            Family Function

            Psycho/Sexual Function

Mortality – how do death rates compare for each treatment group?

Case Study - Acute Carbon Monoxide Intoxication

Normal Oxygen versus Normal Oxygen + Hyperbaric Oxygen

 

Carbon Monoxide and Hemoglobin

 

Hemoglobin is a protein that is carried by red cells. Heme is the prosthetic group that mediates reversible binding of oxygen by hemoglobin. This mechanism allows the red blood cells to transport oxygen to the cells of the body. Globin is the protein that surrounds and protects the heme molecule. It picks up oxygen in the lungs and delivers it to the peripheral tissues to maintain the viability of cells. Carbon monoxide quickly binds with hemoglobin with an affinity 200 to 250 times greater than that of oxygen. The resulting bonding of carbon monoxide and hemoglobin is called carboxyhemoglobin (COHb).

 

Effects of Carbon Monoxide Intoxication

 

Carbon monoxide inhibits the blood's ability to carry oxygen to body tissues including vital organs such as the heart and brain. When CO is inhaled, it combines with the oxygen carrying hemoglobin of the blood to form carboxyhemoglobin. Once combined with the hemoglobin, that hemoglobin is no longer available for transporting oxygen.

 

Symptoms of carbon monoxide intoxication vary with the degree of intoxication, and the nature of damage caused to affected organs. For the purposes of this trial, let us focus on the neurological aspects: Cognitive Skills, Memory Impairment, Coordination, Headaches.

 

Normal Oxygen Therapy

 

A nonrebreather mask supplies 100% oxygen at the usual atmospheric pressure to quickly clear COHb from the blood. This frees up the hemoglobin for oxygen uptake and transport.

 

Hyperbaric Oxygen

 

Hyperbaric oxygen involves delivering oxygen to a patient under higher levels of atmospheric pressure. Once a patient with acute carbon monoxide poisoning has received initial treatment and is in stable condition, the physician must decide whether to initiate hyperbaric oxygen therapy. Hyperbaric oxygen may allow more rapid clearance of COHb.

 

Sketch a comparative clinical trial of normal versus enhanced oxygen therapies in the treatment of acute carbon monoxide intoxication.

Describe the treatments, and the outcome(s) by which the treatments will be evaluated.

The standard treatment is oxygen therapy (OT). The experimental treatment is OT followed by hyper-baric oxygen therapy (OT+HOT).

Do we want a basic, or comparative trial ?

A comparative clinical trial, which compares the oxygen treatments.

Identify the subject population for this trial.

Subjects who qualify for this trial have just suffered acute carbon monoxide intoxication. Subjects eligible for the trial must volunteer and give informed consent in order to participate in the trial. Given the altered consciousness that goes with brain injuries, this trial may utilize both direct consent and proxy consent, in which the subject’s medical agent gives consent, depending on the state of the subject.

Discuss the assignment of subjects to the treatments in the trial.

Enrolled subjects are randomly assigned to either OT or to OT+HOT. Neither the subjects nor the clinical workers will know which drug has been assigned – this is called double blinding.

We will track the trial subjects in both treatment groups for a number of outcomes:

 

Safety – any adverse reactions to either treatment?

Physical Effects – how does post treatment brain tissue damage compare?

Cognitive Effects – how well does the patient recover cognitive function:

            Memory Function

            Coordination

            Speech

            Thought

            Emotional Stability

            Impulse Control

Physical Effects – how often do patients persist in certain effects after treatment?

Headaches

            Balance/Coordination

Mortality – how do death rates compare for each treatment group?

Case Study - Nephrogenic Diabetes Insipidus

Nephrogenic Diabetes Insipidus is a disease in which the patient’s kidneys are resistant to the diuretic hormone vasopressin. Vasopressin is a hormone produced by the hypothalamus, and among other things, stimulates the kidneys to preserve water and concentrate urine. In NDI, the kidneys are not responsive to normal amounts of vasopressin.

 

Symptoms of NDI include:

 

Excessive Thirst – polydipsia

Excessive and Dilute Urine – polyuria

 

Complications of NDI include:

 

Acute Hyperosmolar Dehydration – excessively high blood plasma concetration

Low Blood Pressure – hypotension

Shock

Poor Nutrition and Growth

 

In NDI, the problem isn’t a lack of vasopressin, it is a lack of response to vasopressin. Suppose that we have a new treatment for NDI cases who have normal levels of vasopressin, but whose kidneys do not respond adequately to the vasopressin – let’s call it ActiVasex. The purpose of ActiVasex is to enable the kidneys to respond to the body’s levels of vasopressin. Suppose further that the only effective intervention for cases of NDI is that of hydration – maintaining a steady supply of water to replace the outgoing urine. Assume that all subjects will continue to drink as much water as they need, regardless of treatment group.

Sketch a basic clinical trial that evaluates the experimental treatment ActiVasex in the treatment of NDI cases, following the examples from class and in the course files. For full credit, discuss completely.

Solution:

 

Population of Interest: Cases of Nephrogenic Diabetes Insipidus;

 

Treatments: ActiVasex, and Placebo* ;

 

We begin with a set of possible subjects for our study. Those who present with NDI are briefed as to the particulars of the study, including information about the possible treatments to be assigned, the methods of assigning treatments and the potential risks and benefits of the treatments. Those who give informed consent and join the trial are then randomly assigned to either Activasex or to Placebo. Neither the assigned subjects nor their clinical workers are aware of the treatment assignments (double blinding).

 

The subjects are then tracked for the following:

 

Degree of Thirst

Frequency of Urination

Concentration of Urine

Acute Hyperosmolar Dehydration – excessively high blood plasma concentration

Low Blood Pressure – hypotension

Shock

Poor Nutrition and Growth

Medication Toxicity or Allergic Reactions

We also track the occurrence of side effects and toxicity. 

Case Study - Ocular Hypertension / Early Prevention of Glaucoma

The eye is filled with a fluid – there are mechanisms, which provide for the replacement and draining of this fluid. There is a certain amount of intra-ocular pressure exerted by the fluid in the eye. A condition called ocular hypertension (OHT) involves excessive pressures exerted by the fluid in the eye – sustained OHT can cause damage to the optic nerve, which can then cause the onset of glaucoma. Glaucoma involves loss of visual acuity and visual fields due to optic nerve damage. These losses include loss of visual acuity and loss of peripheral vision.

It is thought that individuals with OHT are at high risk of developing glaucoma. The purpose of this clinical trial is the early prevention of glaucoma in individuals who are glaucoma-free but exhibit ocular hypertension. There is a standard suite of medications that are used in treating OHT in glaucoma patients. The purpose of this trial is the evaluation of this suite of medications in the early prevention of glaucoma.

Sketch a basic clinical trial that evaluates the standard OHT suite in the early prevention of glaucoma in OHT subjects, following the examples from class and in the course files. For full credit, discuss completely.

Solution:

The treatments:

Placebo/Close Observation  – Placebo version of standard OHT medication suite. Watch these subjects for progression of OHT and Glaucoma.

Standard Suite of Glaucoma/OHT Drugs  – The usual suite of meds given to glaucoma patients in reducing intra-ocular hypertension.

Primary Outcome to be observed is the progression of glaucoma from OHT. The basic issues are whether the OHT case progresses to Glaucoma, and the extent to which the onset of Glaucoma is delayed.

Secondary Outcomes to be observed are Adverse Events and Toxicity

We require individuals who are currently free of Glaucoma, but who exhibit excessive intra-ocular pressure – Ocular Hypertension (OHT).

Subjects who meet all requirements for study admission and who give informed consent are then randomly assigned to either Placebo/Observation  or Standard Glaucoma/OHT Suite. Double blinding is employed – neither the subjects nor the clinical workers know the treatment status of the subjects. 

We also track the occurrence of side effects and toxicity.

Case Study - Nephrogenic Diabetes Insipidus

Nephrogenic Diabetes Insipidus is a disease in which the patient’s kidneys are resistant to the diuretic hormone vasopressin. Vasopressin is a hormone produced by the hypothalamus, and among other things, stimulates the kidneys to preserve water and concentrate urine. In NDI, the kidneys are not responsive to normal amounts of vasopressin.

Symptoms of NDI include:

Excessive Thirst – polydipsia

Excessive and Dilute Urine – polyuria

Complications of NDI include:

Acute Hyperosmolar Dehydration – excessively high blood plasma concetration

Low Blood Pressure – hypotension

Shock

Poor Nutrition and Growth

In NDI, the problem isn’t a lack of vasopressin, it is a lack of response to vasopressin. Suppose that we have a new treatment for NDI cases who have normal levels of vasopressin, but whose kidneys do not respond adequately to the vasopressin – let’s call it ActiVasex. The purpose of ActiVasex is to enable the kidneys to respond to the body’s levels of vasopressin. Suppose further that the only effective intervention for cases of NDI is that of hydration – maintaining a steady supply of water to replace the outgoing urine. Assume that all subjects will continue to drink as much water as they need, regardless of treatment group.

Sketch a basic clinical trial that evaluates the experimental treatment ActiVasex in the treatment of NDI cases, following the examples from class and in the course files. For full credit, discuss completely.

Solution:

Population of Interest: Cases of Nephrogenic Diabetes Insipidus;

Treatments: ActiVasex, and Placebo* ;

We begin with a set of possible subjects for our study. Those who present with NDI are briefed as to the particulars of the study, including information about the possible treatments to be assigned, the methods of assigning treatments and the potential risks and benefits of the treatments. Those who give informed consent and join the trial are then randomly assigned to either Activasex or to Placebo. Neither the assigned subjects nor their clinical workers are aware of the treatment assignments (double blinding).

The subjects are then tracked for the following:

Degree of Thirst

Frequency of Urination

Concentration of Urine

Acute Hyperosmolar Dehydration – excessively high blood plasma concentration

Low Blood Pressure – hypotension

Shock

Poor Nutrition and Growth

Medication Toxicity or Allergic Reactions

We also track the occurrence of side effects and toxicity.