Unit 6 Probability and diagnostic test
Practical consequences:
- If the prevalence is low, although the test is very good, it is NOT going to be "effective." - If we apply a diagnostic test to detect a disease with low prevalence in the general population, it will not be useful. - It is more convenient to "select previously" the population (by age, exposure to risk factors...) to get a group with higher prevalence and improve the usefulness of the test.
Outcome
The result of a single trial in a probability experiment is an outcome.
Sample space
The set of all possible outcomes of a probability experiment is the sample space.
Probability
The term "probability" is frequently used in Health Sciences: Examples: - Chance of disease cure - The probability of suffering a pathology in a given individual - Likelihood of occurring some complications during a surgery likelihood that a particular event will occur How likely something is to happen? -> risk The higher the probability of an event, the more certain we are that the event will occur.
AUB
The union of A and B. all elements of sets in question {1,3,5,7,9} U {1,2,3,4,5}= {1,2,3,4,5,7,9}
Thereotical probability
Theoretical probability is what is expected to happen based on mathematics P(Event E) = 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐟𝐚𝐯𝐨𝐫𝐚𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬 𝐭𝐨 𝐞𝐯𝐞𝐧𝐭 𝐄 𝐓𝐨𝐭𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬 Example: A coin is tossed, probability (head)=1/2.
Gold standard of probability diagnosis
To establish if the patient really has the disease, we use a GOLD STANDARD or reference criteria, which is the diagnostic test that offers maximum reliability when diagnosing a given disease. (e.g. pathology exam in cancer). If the test were "perfect": It would always be positive in all individuals who have the disease and would always be negative in all individuals who do not have the disease. It would be positive in ALL cases of the disease and ONLY when the patient did and did not have another disease.
Simultaneous occurrence of two dependent events.
Two events are said to be dependent when p(A/B)≠ p(A), in which case the probability of occurrence of A and B at the same time, probability of the intersection, is: P(AnB) = P(A/B)·P(B) Example: if you want to roll a 7 by rolling first one die and then another, the roll of the second die will depend on the first result.
Intersection of two events P(AnB) if they are dependent events
We can calculate the likelihood of intersection of events: If we solve the formula for conditional probability : P(A I B)= P(A∩B) P(B) to P(A∩B)= P(B) x P(A I B) (this is the probability for intersection of events) The probability of intersection of two dependent events is the probability of one of them multiplied by the probability of the second when the first has been verified.
Test with specificity of 100%:
- Test that in the absence of the disease always is negative. - Doesn ́t give any false positive - High specificity: few false positives
Test with sensitivity of 100%:
-Test that detects all the cases of the disease. - Does not give any false negative. - High sensitivity: few false negatives.
Exercises The likelihood of developing myopia is 3% and women make 55% of the general population. What is the probability that a person, randomly chosen, is a woman and myopic?
003%x 0.55%= 1.7%
example probability Find the probability of randomly selecting a patient who is not 50-65 years old (regarding the following table).
1 P(50-65)= 156/975= 0.16 2 P(50-65)= 1-P(not 60-65) 0.16=1-P(not 60-65) P (not 60-65) = 1-0.16 = 0.84
Exercises In a town there are two centers for the elderly, one public and one private. The public one is attended by 120 elderly people and the private one by 80. 33 elderly people from the private center go by car and 65 from the public center go by car. What is the probability that an elderly person who is in a car will go to the public day center?
120 65 80- 33 200 people total 98 car drivers 65 / 98. 66%
158 people attended a banquet and the following day 100 of them had stomach inflammation. Ofthe 158, 110 had eaten prawns with mayonnaise and, of these, 70 patients got sick.Calculate the probability of getting sick if you ate prawns with mayonnaise.
158 100 110 70 70/158 110/158 0.064
excersise 2. Consider the experiment of rolling a die and the following events: Event A: That on rolling the die the number is prime less than 4. A={2,3}, Event B: That when the die is thrown it lands on the number 1 or 6. B={1,6}
2/6 2/6 1/3
Exercises The percentage of women who survive to the removal and treatment of ovarian cancer in early stage is 60% at 2 years and of those survivors, 48 of them have survived 4 years more . We want to know what is the probability that a woman who has survived 2 years, will survive 6 years.
48/60 = 0.8 80% will survive
Inference
A conclusion reached on the basis of evidence and reasoning Our eventual goal is Inference — drawing reliable conclusions about the population based on what we've discovered in our sample.
ANB
A intersection B only elements that are shared {1,3,5,7,9} N {1,2,3,4,5} = {1,3,5}
probability experiment
A probability experiment is an action, or trial, through which specific results (counts, measurements, or responses) are obtained.
Event
An event is a subset of the sample space.
Simple event
An event that consists of a single outcome is called a simple event.
sensitivity and specificity
Diagnostic tests, even the gold standard, are not "perfect" Any procedure that serves for the diagnosis (i.e. a sign or result of a test) has: A certain "level" of Sensitivity And a certain "level" of Specificity
Experimental probability
Experimental probability is found by performing an experiment and observing the outcomes. P(Event E) = 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐭𝐢𝐦𝐞𝐬 𝐭𝐡𝐞 𝐞𝐯𝐞𝐧𝐭 𝐄 𝐨𝐜𝐜𝐮𝐫𝐬 𝐓𝐨𝐭𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐭𝐫𝐢𝐚𝐥𝐬 Example: a coin is tossed 10 times, a head is recorded 7 times, probability (head) =7/10.
sensitivity and specificity preference
For this reason, in some cases it is important to decide "if we prefer" to detect more diseases or eliminate more cases without disease. i.e: choose between less false positives or less false negatives This choice depends on: Cost and psychological impact or medical consequences of the false positives and false negatives. One strategy is to combine two diagnostic tests, one very sensitive and one very specific: Sensitive first screening: We want to detect every possible case assuming that we are going to get many false negatives BECAUSE WE CANNOT HAVE ANY FALSE NEGATIVES (sick patient not treated). Specific second screening: is used for confirmation of the diagnosis (few false positives) to determine the real TRUE POSITIVES.
Complementary event
Given an event A, A' is its complementary or contrary event if it only occurs when A does not occur. These are events that have two possible outcomes. Also called mutually exclusive, because they cannot occur simultaneously. The sum of the probabilities of all outcomes in a sample space is 1 or 100% Probability of the complementary event A P(A)= 1-P(A) 1 is the sample
Be careful with not mixing up!!
Independent - Dependent Compatible - Incompatible They are not the same type of events!!
Conditional Probability
Independent and dependent events In some experiments, one event might affect the probability of another event if they are dependent (or the opposite if they are independent). Conditional probability: P(B I A) P(A I B) = probability that A and B occur together probability B occurs P (A I B) = P(A∩B) P(B) Its the probability of an event, given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is shown by P(A I B) and is read as probability of A given B. The event B is the one that conditions. It is the one that we know that it has happened for sure.
Event union (AUB)
It is a new event that is made up of the sample space elements that belongs to A or B or both (A and B). THEY CAN BE COMPATIBLE OR INCOMPATIBLEIt is the number of times A, B or A+B occur.
Event intersection (A∩B)
It is the event consisting on all the elements from the space E which belong to A and B at the same time. They must be compatible It is the number of times A and B occur together. the intersection (A∩B)= E
Antagonism among sensitivity and specificity
It is very difficult for the same diagnostic test to have very high sensitivity and specificity. Usually very sensitive tests are not very specific and vice versa. High Sensitivity: few false negatives but many false positives. High Specificity: few false positives but many false negatives. Hopefully, nowadays, there exist test with both high sensitivity and specificity, as PCR quantitative or magnetic resonance imaging scan.
Probability calculated
LAPLACE: Probability of an event = Number of favorable cases to the event divided by the number of possible cases. Examples: In a population with N elements (e.g. individuals), K of which have a characteristic A. The probability of having the characteristic A, p (A), is: P (A) = K / N In a group of 50 students of dentistry, 20 failed and 30 passed an exam. What is the probability that, when selecting a student randomly, he/she has failed the exam? 20/50 = 0.4 = 40%
Exercise: Determine whether the events are mutually exclusive. Explain it 1. Event A: roll a 3 on a dice 2 Event B: roll a 4 on a dice 2. Event A: randomly select a male student Event B: randomly select a nursing major 3. Event A: randomly select a blood donor with type O blood Event B: randomly select a female blood donor. 4. Event A: randomly select a jack from a standard deck of 52 playing cards Event B: randomly select a face from a standard deck of 52 playing cards
Mutually exclusive Not mutually exclusive Not mutually exclusive Not mutually exclusive
Impossible event
No favorable cases (no subjects presenting A characteristics) The event A does not belong to the sample space, so the probability of occurrence is 0
Conditional probability of independent events:
P(A | B)= P(A) ----> P(AnB)= P(B) x P(A)= P (A) x P (B) P(B | A)= P(B) ----> (same as over) Not compatible don't overlap future events are not influenced at all by events that have already taken place
Conditional probability of dependent events:
P(A | B)= P(AnB) ----> P(A | B) x P(B)= P(AnB) P(B) P(A and B) P(B | A)= P(B n A) --->. P(B | A) x P (A)= P(B n A) P(A) P(A and B) P(B | A) = probability of event B happening given that A occurred P(B | A) = P(A and B)/P(A) this is "both" divided by "given"
probability calculated more examples
P(E)= 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐟𝐚𝐯𝐨𝐫𝐚𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬 𝐢𝐧 𝐞𝐯𝐞𝐧𝐭 𝐓𝐨𝐭𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬 𝐢𝐧 𝐬𝐚𝐦𝐩𝐥𝐞 𝐬𝐩𝐚𝐜𝐞 Example 2: If you throw a coin: What is the probability of getting heads? 1 / 2 = 0.5= 50% Example 3: What is the probability of getting a 3 when rolling a die? 1 / 6 = 0.166666 = 16.6% Both are events that occur by chance
Exercises It is known that secondary effects of an antibiotic affect 10% of patients receiving them. A dentist prescribe this antibiotic to two patients. What is the probability that both patients experience secondary effects?
P(Effects patient 1 ∩ Effects patient 2) P(Effect1) xP(effect2)0,10 x 0,10 0,01 1% probability that both experience porbabilty a patients probability b patients 10%individual together 0.1%
Exercises A serological test to detect antibodies to SARS-CoV-2 was used for the diagnosis of COVID-19.Calculate the positive predictive value and the negative predictive value.
Positive predictive value PPV true positive true positive + false positive negative predictive value true negative true negativ+ false negative
Example 1: If you roll a die:
Probability Experiment: roll a six-sided dice. Sample space: [1, 2 3 4 5 6] Event: An even number (2,4,6) Outcome: 2
Probability value
Probability will always adopt values between 0 and 1 If we express it as a percentage, we will have to multiply by 100 and the values will be between 0 and 100. The higher the probability of an event the closer p will be to 1 (100%) The lower the probability of an even the further away the p will be to 1/ 100% (closer to 0 (0%))
Predictive values of diagnostic test
Provide information of the probability of being ill when the test is positive or not being sick when it is negative. Positive predictive value (PPV) is the probability that a person is sick when the test was positive PPV = P (sick I test +) = 𝒑 (𝒔𝒊𝒄𝒌 ∩ +) 𝒑(+) How likely is it that a patient has the disease given that the test result is positive? Negative predictive value (NPV) is the probability of a person being healthy when the test was negative P(H/-test) NPV = P (healthy I test -) = 𝒑 (𝒉𝒆𝒂𝒍𝒕𝒉𝒚 ∩−) 𝒑 ( −) How likely is it that a patient does not have the disease given that the test result is negative?
Sure/ certain event
Sure (or certain) event: all cases are favorable to event A (all subjects have the characteristic A). Performing the experiment always produces the possible outcomes that make up the sample space. its probability of occurrence is 1
Probability and Diagnosis
The diagnosis in Health Science is an example of the application of the theory of probability. Starting out with the information obtained from the anamnesis, physical examination and complementary tests we decide which is the most probable syndrome or sickness the patient has. When we use a test to diagnose a disease, we want this test to give us additional information on whether the patient really has the disease or not.
sensitivity and specificity imperfect
The ideal situation would be to have tests available (or signs or indexes) that have the maximum sensitivity and specificity. This way, we would never make mistakes, we would never classify as patients with disease those that are healthy, nor as healthy those who have the disease. Research always aims to get biological, radiological, clinical tests where these indexes are maximized. But the diagnostic tests we usually have available are sensitivity and specificity-imperfect.
Given a diagnostic test with a sensitivity of 99% and a specificity of 98%. page 54-57
The negative predictive value is very high and slightly decreases with increasing prevalence. The positive predictive value is excellent when the prevalence is high (30%) but not when prevalence decreases. . Despite the high sensitivity and specificity of the test, the probability of being sick when the test is positive is low if the prevalence is low.
Prevalence and diagnosis
The predictive values depend on the prevalence of the disease in the population studied in such a way that: - A higher prevalence increases the positive predictive value (more likely to be sick) and slightly decreases the negative predictive value (less likely to be healthy) - A lower prevalence decreases the positive predictive value (less likely to be sick) and increases the negative predictive value (more likely to be healthy).
Probability of the union of two events P(AUB)
When A and B have no elements in common, the probability of the union is: When A and B are compatible, the probability of the union is: P (AUB) = P(A) + P(B) incompatible events When A and B are compatible, the probability of the union is: P (AUB) = P(A) + P(B) - P(A∩B) Compatible events
Sensitivity
When a test is positive in all or almost all those with a given disease, it is very sensitive or has a high sensitivity. - Ability of the test to detect disease in diseased individuals. - Ability to "test" positive in the presence of disease. - Probability of the test being positive when sick: sensitivity of 100% indicates that the test is always positive when the subject has that disease.
Simultaneous occurrence of two independent events
When p(A/B)= p(A) and P(B/A)= P(B) the events A and B are independent: the occurrence of one of them does not influence the occurrence of the other, i.e. the occurrence of one does not modify the probability of occurrence of the other. Example: If you roll a die, the occurrence of a 1 does not influence the occurrence of the previous dice. It is satisfied that the probability that A and B occur at the same time, probability of the intersection, is: P(AnB) = P(A)·P(B)
Specificity
When the test is positive only or almost only in individuals who have that disease and no other disease and always or almost always negative when they do not, the test is very specific. - Ability to rule out disease in individuals who do not have the disease (healthy for that disease). - Ability to "test" negative in the absence of disease. - Probability that the test will be negative when you do NOT have the disease.
Sensitivity and specificity of diagnostic test
When we use a diagnostic test, the result of this test can be POSITIVE or NEGATIVE If the result is POSITIVE, it can be: TRUE POSITIVE: The test is positive and the patient actually DOES have the disease. FALSE POSITIVE: The test is positive and the patient does NOT have the disease. If the RESULT is NEGATIVE, it can be: TRUE NEGATIVE: The test is negativity and the patient does NOT have the disease. FALSE NEGATIVE: The test is negativity and the patient actually DOES have the disease.
subset
a set that is part of a larger set
Antagonism
active hostility or opposition
Compatible Events
are those belonging to the sample space that can have a simultaneous realization because they have elements in common. Example Rolling a dice: A : the event of getting an even number {2, 4, 6} B : the event of getting a number greater than 3: {4, 5, 6} Both the events occur when we get either 4 or 6 We say events A and B are not mutually exclusive A and B are Compatible Events. sample space If the events are compatible they can occur simultaneously P (A) + P (B) - P (A∩B)= → P (A∪B) =
Incompatible Events
are those that cannot be performed simultaneously, there is no intersection. Also called disjoint or mutually exclusive events. A and B are mutually exclusive events, as they cannot occur at the same time. Example 7. Rolling once a dice: A : the event of getting an odd number (1,3,5) B : the event of getting an even number (2,4,6) We say events A and B mutually exclusive P(A and B) = 0
anamnesis
memory
incompatible events
mutually exclusive cant be both incompatible events not mutually exclusive can be both compatible events
myopic
nearsighted
Exercises A serological test to detect antibodies to SARS-CoV-2 was used for the diagnosis of COVID-19.Calculate the sensitivity and specificity of the test.
sensitivity Infected people with a positive test(a) Total number of infected people (a+c) So...among the people with covid, the22.7% of them would be correctly indentified by this test. specificity Not infected people with a negative test(b) Total number of not infected people (b+d
excersise 1. In a given population, 30% smoke, 50% drink coffee and 20% smoke and drink coffee. What is the probability that a subject randomly chosen, smoke or drink coffee?
smoke and drink p(sud)= p(s)+p(d)-p(snd) 30%+ 50% -20%= 60%
Exercises In a town of 10,000 inhabitants, the prevalence of a disease is 1%. The diagnosis to assess the disease has a sensitivity of 98% and a specificity of 99%. Calculate the positive predictive value and the negative predictive value. .
tov npv after calculating sensitivity and specificity one more time to get thevlaue
chart for sensitivity and specificity
𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = TP 𝑇P+ 𝐹𝑃 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 = TN. 𝑇𝑁 + 𝐹𝑃 TN= True negative TP= True positive FP= False Positive FN= False Negative
formula for sensitivity and specificity
𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = a 𝑎+𝑐 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑖𝑦 = b 𝑏+𝑑 TP= a FP= d FN= c TN= b
