DATASET INTRODUCTION 1

Dataset: Urine 2 From Cleveland Clinic

Outcome Variable: Categorical Variable  Calcium 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) 3

Other Variables (Covariates) QuantitativeVariables  Specific Gravity  pH  Osmolarity  Conductivity  Urea Concentration (millimoles/liter)  Calcium Concentration (millimoles/liter)  Cholesterol: serum cholesterol levels 4

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 5

6 Calcium Oxalate Crystal Presence NMean Std Dev MinQ1MedQ3Max No Yes

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 7

Comments  To facilitate our discussion of probability and classification tests  We will categorize the quantitative variable Calcium Concentration into four groups 1 = = = = 8 or More 8

BASIC PROBABILITY Part 1 (Unconditional Probability using Logic) 9

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? 10

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 11

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 sample Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

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 r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

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? Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

Let’s Practice! Basic Probability of an Event: Unions What is the probability the individual has calcium oxalate crystals present OR has a calcium concentration of 5 or more? Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

USING PROBABILITY RULES Part 1 16

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! 17

Continuing  We now illustrate a few formulas using the questions we have already answered using logic 18

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): Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

Let’s Practice Again! Addition Rule (Unions) What is the probability the individual has calcium oxalate crystals present OR has a calcium concentration of 5 or more? Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

Let’s Practice Again! Addition Rule (Unions) What is the probability the individual has calcium oxalate crystals present OR has a calcium concentration of 5 or more? 21

INDEPENDENCE Part 1 22

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) 23

Let’s Practice! Investigating Independence Part 1 We know the following from our sample 24 ?

Let’s Practice! Investigating Independence Part 1  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) 25

BASIC PROBABILITY Part 2: Conditional Probability (Logic & Formula) 26

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 27

Let’s Practice! CONDITIONAL Probability of an Event Given the individual has Calcium Oxalate Crystals present, what is the probability the individual’s calcium concentration is 5 or above? Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

Let’s Practice! CONDITIONAL Probability FORMULA Given the individual has Calcium Oxalate Crystals present, what is the probability the individual’s calcium concentration is 5 or above? 29

Let’s Practice! CONDITIONAL Probability of an Event Given the individual DOES NOT HAVE Calcium Oxalate Crystals present, what is the probability the individual’s calcium concentration is 5 or above? Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

MORE PRACTICE Conditional Probability 31

Let’s Verify! CONDITIONAL Probability of an Event 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 time Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

INDEPENDENCE Part 2 33

Let’s Practice! Investigating Independence Part 2 We know the following from our sample 34 ??

Comments Investigating Independence Part 2  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!! 35

GENERAL MULTIPLICATION RULE 36

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 37

General Multiplication Rule 38

REPEATED SAMPLING 39

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 40

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? Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

WANT TO LEARN MORE? READ THE FOLLOWING OPTIONAL MATERIAL The remaining slides are optional. They illustrate some more difficult probability rules along with additional examples of probability related to the health sciences 42

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 43

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 44

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. 45

Total Probability Rule 46

Bayes’ Rule  We want to find P(A|B) so that we will need to “rearrange” the formula swapping A’s and B’s 47

Bayes’ Rule 48

Let’s Verify! CONDITIONAL Probability of an Event 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 time Table of group by r group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency NoYesTotal or More 178 Total

SCREENING TESTS and ROC Curves 50

Screening Tests 51

Sensitivity & Specificity “Epi” Style Has Condition Does not have Condition Test Positive A TP B FP Total Positive Test (A+B) Test Negative C FN D TN Total Negative Test (C+D) Number with Condition (A+C) Number without Condition (B+D) 52

Sensitivity & Specificity Has Condition Does not have Condition NEGATIVE or more POSITIVE group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency YesNoTotal or More 718 Total

Sensitivity & Specificity Has Condition Does not have Condition NEGATIVE or more POSITIVE group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency YesNoTotal or More 718 Total

Sensitivity & Specificity Has Condition Does not have Condition NEGATIVE or more POSITIVE group (Calcium Concentration Group) r (Calcium Oxalate Crystal Presence) Frequency YesNoTotal or More 718 Total

Bayes’ Rule Has Condition Does not have Condition Negative Positive ≥ Here we Define: A = Disease B = Test Positive

Choosing Different Cut-Off Cut-pointSensitivitySpecificity 2 or more or more or more High Sensitivity but Low Specificity

Choosing Different Cut-Off Cut-pointSensitivitySpecificity 2 or more or more or more Specificity Increased But you reduce sensitivity (orange arrow)

Choosing Different Cut-Off Cut-pointSensitivitySpecificity 2 or more or more or more Very High Specificity Very Low Sensitivity (High False Negative Rate)

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 60

Receiver Operating Characteristic curve (ROC curve) Cut-pointSensitivitySpecificity 2 or more or more or more

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 62

Trapezoidal Rule (FYI) 63