DATASET INTRODUCTION 1. Dataset: Urine 2 From Cleveland Clinic 1981-1984

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DATASET INTRODUCTION

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Dataset: Urine

From Cleveland Clinic 1981-1984

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Outcome Variable:

Categorical VariableCalcium Oxalate Crystal Presence

• In this analysis, this variable will be our

• Outcome variable

• Response Variable

• Dependent Variable• Note: The dataset is coded directly as Yes/No (not 0/1 coding)

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Other Variables (Covariates)

QuantitativeVariables

Specific Gravity

pH

Osmolarity

Conductivity

Urea Concentration (millimoles/liter)

Calcium Concentration (millimoles/liter)

Cholesterol: serum cholesterol levels

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Discussion/Review Purpose of dataset: Determine which of the covariates

are related to the outcome. Covariates can also be called

• Independent Variables

• Predictors

• Explanatory Variables

Outcomes/Covariates can be categorical or quantitative

Can be more than one outcome and many covariates in a given study with any mixture of variable types

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Calcium Oxalate Crystal

PresenceN Mean

Std Dev

Min Q1 Med Q3 Max

No 42 2.69 1.90 0.17 1.22 2.16 3.93 8.48

Yes 31 5.92 3.59 0.27 3.10 6.19 7.82 14.34

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Discussion Clearly, those with calcium oxalate crystals present tend

to have higher calcium concentrations

Later we will learn to conduct hypothesis tests in such situations

Now we use this data to illustrate concepts of probability

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Comments To facilitate our discussion of probability and classification

tests

We will categorize the quantitative variable Calcium Concentration into four groups

1 = 0-1.992 = 2-4.993 = 5-7.994 = 8 or More

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BASIC PROBABILITYPart 1 (Unconditional Probability using Logic)

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Back to the Urine Dataset Suppose one individual is selected from our sample and

consider the following questions

• What is the probability that the individual has calcium oxalate crystals present?

• What is the probability that the individual has a calcium concentration of 5 or more?

• What is the probability the individual has calcium oxalate crystals present AND has a calcium concentration of 5 or more?

• What is the probability the individual has calcium oxalate crystals present OR has a calcium concentration of 5 or more?

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Comments All of these four probability questions relate to the

ENTIRE SAMPLE

We begin by answering the questions logically from the table we created using software

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Let’s Practice!

Basic Probability of an Event

• What is the probability that the individual has calcium oxalate crystals present? We will denote this event by A.

• = PREVALENCE of calcium oxalate crystals in our sampleTable of group by r

group (Calcium Concentration

Group)r (Calcium Oxalate Crystal Presence)

FrequencyNo Yes Total

0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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Let’s Practice!

Basic Probability of an Event

• What is the probability that the individual has a calcium concentration of 5 or more? We will denote this event by B.

Table of group by rgroup (Calcium Concentration



0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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Let’s Practice!

Basic Probability of an Event: Intersections

• What is the probability the individual has calcium oxalate crystals present AND has a calcium concentration of 5 or more?




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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Let’s Practice!

Basic Probability of an Event: Unions





0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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USING PROBABILITY RULESPart 1

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Probability Rules Rules are created and used for many reasons

The rules and properties stated previously are important and useful in probability and sometimes in statistics

Not always needed

• If you can determine the answer through logic alone you may not need a rule!

• If you are provided only pieces of the puzzle, sometimes a rule is faster than logic!

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Continuing We now illustrate a few formulas using the questions we

have already answered using logic

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Let’s Practice Again!

Complement Rule

• What is the probability that the individual DOES NOT have calcium oxalate crystals present?

• We could use logic and count the No’s instead of the Yes’s however knowing P(Yes)=P(A):




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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Addition Rule (Unions)





0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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Addition Rule (Unions)


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INDEPENDENCEPart 1

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Independent Events Two events are independent if knowing one event occurs

does not change the probability of the other

This is not the same as “disjoint” events which are separate in that they cannot occur together

These are two different concepts entirely

Independence is a statement about the equality of the probability of one event whether or not the other event occurs (or is occurring, or has occurred)

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Let’s Practice!

Investigating Independence Part 1

We know the following from our sample

?

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Let’s Practice!


From our sample we have:

This is clearly not equal to 0.247!!

In our sample the events are dependent (we can test this hypothesis about the population later)

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BASIC PROBABILITYPart 2: Conditional Probability (Logic & Formula)

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Conditional Probability So far, we have divided by the TOTAL

Sometimes, however, we have additional CONDITIONS that cause us to alter the denominator (bottom) of our probability calculation

Suppose, when choosing one person from the Urine data, we ask

• Given the individual has Calcium Oxalate Crystals present, what is the probability the individual’s calcium concentration is 5 or above?

“Conditional” refers to the fact that we have these additional conditions, restrictions, or other information

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Let’s Practice!

CONDITIONAL Probability of an Event





0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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Let’s Practice!

CONDITIONAL Probability FORMULA


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Let’s Practice!


• Given the individual DOES NOT HAVE Calcium Oxalate Crystals present, what is the probability the individual’s calcium concentration is 5 or above?




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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MORE PRACTICEConditional Probability

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Let’s Verify!


• Given the individual has a calcium concentration of 5 or above, what is the probability the individual has calcium oxalate crystals?

• We have a small amount of rounding error this timeTable of group by r




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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INDEPENDENCEPart 2

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Let’s Practice!


We know the following from our sample

? ?

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Comments


These probabilities are clearly unequal in our sample, our eventual question might be if this is also true for our population

In this sample, these events are dependent

From our analysis so far, it seems likely they may be dependent in our population (we can test later)

Knowing whether or not the person has calcium oxalate crystals present CHANGES the probability of having a calcium concentration of 5 or above!!

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GENERAL MULTIPLICATION RULE

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General Multiplication Rule

This formula comes from rearranging the definition of conditional probability

To achieve the second formulation on the right consider the formula below for P(A|B) instead and note that the numerator is unchanged

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General Multiplication Rule

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REPEATED SAMPLING

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Repeated Sampling Often we consider problems in which we draw multiple

individuals from a set of individuals

• Drawing parts from a box where some are defective

• Choosing multiple people from a certain population

The formulas we have investigated can be used to calculate probabilities in these situations

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Let’s Practice! If we select two subjects at random from our sample, what

is the probability that both have a calcium concentration of 8 or more?




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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WANT TO LEARN MORE?

READ THE FOLLOWING OPTIONAL MATERIALThe remaining slides are optional. They illustrate some more difficult probability rules along with additional examples of probability related to the health sciences

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Optional Content: Read About Relative Risk

Total Probability Rule

Bayes Rule

Screening Tests

• Sensitivity/Specificity

• PV+/PV-

• False Positive and False Negative Rates

ROC Curves

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Relative Risk Relative risk is

• the risk of an “event” relative to an “exposure”

• the ratio of the probability of the event occurring among “exposed” versus “non-exposed”

• If A and B are independent, the relative risk is 1

In our rule B is the EVENT and A is the EXPOSURE

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Let’s Practice! Find the Relative Risk of High Calcium Concentration

Given Calcium Oxalate Crystal Presence

• Note: this is the reverse of what we probably want in this case, consider that for more practice!

• INTERPRET RR: Having a calcium concentration of 5 or more is around 4 times more likely among those with calcium oxalate crystals than among those without.

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Total Probability Rule

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Bayes’ Rule

We want to find P(A|B) so that we will need to “rearrange” the formula swapping A’s and B’s

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Bayes’ Rule

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Let’s Verify!


• Given the individual has a calcium concentration of 5 or above, what is the probability the individual has calcium oxalate crystals?

• We have a small amount of rounding error this timeTable of group by r




0-1.99 19 4 23

2-4.99 17 9 26

5-7.99 5 11 16

8 or More 1 7 8

Total 42 31 73

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SCREENING TESTSand ROC Curves

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Screening Tests

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Sensitivity & Specificity

“Epi” StyleHas

Condition

Does not have

Condition

Test Positive

ATP

BFP

Total Positive

Test (A+B)

TestNegative

CFN

DTN

Total Negative

Test (C+D)

Number with

Condition(A+C)

Number without

Condition(B+D)

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Has Condition

Does not have

Condition

0-1.99NEGATIVE 4 19

2 or morePOSITIVE 27 23

31 42



FrequencyYes No Total

0-1.99 4 19 23

2-4.99 9 17 26

5-7.99 11 5 16

8 or More 7 1 8

Total 31 42 73

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Has Condition

Does not have

Condition

0-4.99NEGATIVE 13 36


31 42




0-1.99 4 19 23

2-4.99 9 17 26

5-7.99 11 5 16

8 or More 7 1 8

Total 31 42 73

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Has Condition

Does not have

Condition

0-7.99NEGATIVE 24 41


31 42




0-1.99 4 19 23

2-4.99 9 17 26

5-7.99 11 5 16

8 or More 7 1 8

Total 31 42 73

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Bayes’ Rule

Has Condition

Does not have

Condition

Negative0- 4.99 24 41

Positive ≥ 8 7 1

31 42

Here we Define: A = Disease B = Test Positive

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Choosing Different Cut-Off

2 or more

Cut-point Sensitivity Specificity

2 or more 0.87 0.45

5 or more 0.58 0.86

8 or more 0.23 0.98

High Sensitivity but Low Specificity

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5 or more


2 or more 0.87 0.45

5 or more 0.58 0.86

8 or more 0.23 0.98

Specificity IncreasedBut you reduce sensitivity

(orange arrow)

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8 or more


2 or more 0.87 0.45

5 or more 0.58 0.86

8 or more 0.23 0.98

Very High SpecificityVery Low Sensitivity (High

False Negative Rate)

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What happens when We assign all individuals a positive test result?

• Sensitivity = P(Test+|Disease) = 1

• Specificity = P(Test-|No Disease) = 0

• 1 – Specificity = 1

We assign all individuals a negative test result?

• Sensitivity = P(Test+|Disease) = 0

• Specificity = P(Test-|No Disease) =1

• 1 – Specificity = 0

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Receiver Operating Characteristic curve (ROC curve)


2 or more 0.87 0.45

5 or more 0.58 0.86

8 or more 0.23 0.980.000.100.200.300.400.500.600.700.800.901.00

0.00 0.20 0.40 0.60 0.80 1.00

True

Pos

itive

Rat

e (S

ensi

tivity

)

False Positive Rate (1-Specificity)

ROC Curve for Calcium Oxalate

2

5

8

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ROC Curves

Area under the curve = probability that for a randomly selected pair of normal and abnormal subjects, the test will correctly identify the normal subject given the “measurement”

Area = 0.89 for the example on the left

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Trapezoidal Rule (FYI)

Documents

DATASET INTRODUCTION 1. Dataset: Urine 2 From Cleveland Clinic 1981-1984