7.3 – Conditional Probability and Evidence Based Medicine

Conditional probability and risk communication.

Conditional probabilities can be used to accurately and effectively convey the risk of specific health outcomes to patients and the public. Tools introduced in this chapter help move from an initial assessment of disease risk to a revised probability after a test result is returned. Although there are other interpretations, here we frame EBM — evidence-based-medicine — as a form of Bayesian decision-making.

This sub-chapter covers a lot of ground. We begin by addressing how probability of multiple events are calculated assuming each event is independent. The assumption of independence is then relaxed, and how to determine probability of an event happening given another event has already occurred, conditional probability, is introduced. Use of conditional probability to interpret results of a clinical test are also introduced, along with the concept of EBM. We also introduce Bayes’ rule and Bayesian probability.

Probability and independent events.

Probability distributions are mathematical descriptions of the probabilities of how often different possible outcomes occur. We also introduced basic concepts related to working with the probabilities involving more than one event.

For review, for independent events, you multiply the individual chance that each event occurs to get the overall probability.

Example of multiple, independent events.

Figure 1. Now that’s a box full of kittens. Creative Commons License, source: https://www.flickr.com/photos/83014408@N00/160490011.

What is the chance of five kittens in a litter of five to be of the same sex? In feral cat colonies, siblings in a litter share the same mother, but not necessarily the same father, superfecundation. Singleton births are independent events, thus the probability of the first kitten is female is 50%; the second kitten is female, also 50%; and so on. We can multiply the independent probabilities (hence, the multiplicative rule), to get our answer:

kittens <- c(0.5, 0.5, 0.5, 0.5, 0.5)
prod(kittens)
[1] 0.03125

Probabilistic risk analysis.

Risk analysis is the use of information to identify hazards and to estimate the risk. A more serious example. Consider the 1986 Space Shuttle Challenger Disaster (Hastings 2003). Among the crew killed was Ellison Onizuka, the first Asian American to fly in space (Fig 2, first on left back row). Onizuka was born and raised on Hawai`i and graduated from Konawaena High School in 1964.

Figure 2. STS-51-L crew: (front row) Michael J. Smith, Dick Scobee, Ronald McNair; (back row) Ellison Onizuka, Christa McAuliffe, Gregory Jarvis, Judith Resnik. Image by NASA – NASA Human Space Flight Gallery, Public Domain.

The shuttle was destroyed just 73 seconds after lift off (Fig 3).

Figure 3. Space Shuttle Challenger launches from launchpad 39B Kennedy Space Center, FL, at the start of STS-51-L. Hundreds of shorebirds in flight. Image by NASA – NASA Human Space Flight Gallery, Public Domain.

This next section relies on material and analysis presented in the Rogers Commission Report June 1986. NASA had estimated that the probability of one engine failure would be 1 in 100 or 0.01; two engine failures would mean the shuttle would be lost. Thus, the probability of two rockets failing at the same time was calculated as 0.01 X 0.01, which is 0.0001 or 0.01%.

NASA had planned to fly the fleet of shuttles 100 times per year, which would translate to a shuttle failure once in 100 years. The Challenger launch on January 28, 1986, represented only the 25th flight of the shuttle fleet.

One difference on launch day was that  the air temperature at Cape Canaveral was quite low for that time of year, as low as 22 °F overnight.

Attention was pointed at the large O-rings in the boosters (engines). In all, there were six of these O-rings. Testing suggested that, at the colder air temperatures, the chance that one of the rings would fail was 0.023. Thus, the chance of success was only 0.977. Assuming independence, what is the chance that the shuttle would experience O-ring failure?

shuttle <- c(0.977, 0.977, 0.977, 0.977, 0.977, 0.977)
#probability of success then was
prod(shuttle)
[1] 0.869

#and therefore probability of failure was
1 - prod(shuttle)
[1] 0.1303042

Conditional probability of non-independent events.

But in many other cases, independence of events cannot be assumed. The probability of an event given that another event has occurred is referred to as conditional probability. Conditional probability is used extensively to convey risk. We’ve touched on some of these examples already:

• the risk of subsequent coronary events given high cholesterol;
• the risk of lung cancer given a person smokes tobacco;
• the risk of mortality from breast cancer given that regular mammography screening was conducted.

There are many, many examples in medicine, insurance, you name it. It is even an important concept that judges and lawyers need to be able to handle (eg, discussion in Berry 2008).

A trigger warning: what follows is emotional, traumatic material, provided to emphasize the importance of these concepts, and the far-reaching implications understanding conditional probability can have for us.

A now famous example of conditional probability in the legal arena came from arguments over the chance that a husband or partner who beats his wife will subsequently murder her — this was an argument raised by the prosecution during pre-trial in the 1995 O.J. Simpson trial (The People of the State of California v. Orenthal James Simpson), and successfully argued by O.J. Simpson’s attorneys… judge ruled in favor of the defense and evidence of O.J. Simpson’s prior abuse were not included in trial. Gigerenzer (2002, following discussions by others, eg, Good 1995) and others (eg, Skorupski and Wainer 2015), have called this the Defendent’s Fallacy, the reverse of the Prosecutor’s Fallacy, where the more typical scenario is that the prosecution provides a list of probabilities about characteristics of the defendant, leaving the jury to conclude that no one else could have possibly fit the description (Thompson and Shumann 1987).

In the O.J. Simpson case, the argument went something like this. From the CDC we find that the chance of an average woman being murdered by a stranger or an unrelated party was about 1 in 20,000. CDC statistics indicate 1.3 million women are abused each year by their partner or former partner; each year about 1000 women are murdered. One thousand divided by 1.3 million is a small number, so even when there is abuse the argument goes, 99% of the time there is not murder. The Simpson judge ruled in favor of the defense and much of the evidence of prior abuse was excluded.

Something is missing from the defense’s argument. Nicole Brown Simpson did not belong to a population of battered women — she belonged to the population of murdered women. When we ask, if a woman is murdered, what is the chance that she knew her murderer, we find that more than 55% knew their murderer — and of that 55%, 93% were killed by a partner who had a history of prior abuse against the victim.

Instead of looking at all 1.3 million abused women, we isolate a representative cohort of 100,000 abused women to see what mathematically happens to them in a given year:

Group 1: Murdered by their abuser. According to the defense’s rate of 1 in 2,500:

Group 2: Murdered by someone else:

Given that an abused woman was murdered, what is the probability it was done by her abuser?

or 89%.

This is a huge difference from how the Defense portrayed the chance, where they wanted the judge to assume every single one of those 2500 abusers is equally likely to have committed the crime. In reality, we must condition the match on all other available evidence.

The key —  Nicole Simpson (and Ron Goldman) were murdered; O.J. Simpson was an ex-partner who had been guilty of assault against Nicole Simpson. Now, it goes from an impossibly small chance, to a much greater chance.  Conditional probability, and specifically Bayes’ rule, is used for these kinds of problems. The defense successfully masked a probability that was heavily stacked against their client.

Bayes’ rule provides guidance for how to update the probability of an event based on new evidence. Named after Thomas Bayes (1701- 1761), more generally, Bayes’ rule, described below, is used to quantify conditional probability. Bayesian probability, where probability is interpreted as reasonably expectation instead of frequency of some event. We discuss Bayesian and Frequentist approaches to hypothesis testing in Chapter 8.2.

Diagnosis from testing.

Let’s turn our attention to medicine. A growing practice in medicine is to claim that decision making in medicine should be based on approaches that give us the best decisions.  A search of PubMed texts for “evidence based medicine” found more than 91,944 (13 October 2021, and increase of thirteen thousand since last I looked (10 October, 2019). Evidence based medicine (EBM) is the  “conscientious, explicit, judicious and reasonable use of modern, best evidence in making decisions about the care of individual patients” (Masic et al 2008). By evidence, we may mean results from quantitative, systematic reviews, meta-analysis, of research on a topic of medical interest, eg, Cochrane Reviews.

As you know, no diagnostic test is 100% fool-proof. For many reasons, test results come back positive when the person truly does not have the condition — this is a false positive result. Correctly identifying individuals who do not have the condition, 100% – false positive rate, is called the specificity of a test.

Think of specificity in this way — provide the test 100 true negative samples (eg, 100 samples from people who do not have cancer) — how many times out of 100 does the test correctly return a “negative”? If 99 times out of 100, then the specificity rate for this test is 99%. Which is pretty good. But the test results mean more if the condition/disease is common; for rare conditions, even 99% is not good enough. Incorrect assignments are rare, we believe, in part because the tests are generally quite accurate. However, what we don’t consider is that detection and diagnosis from tests also depend on the base rate — how frequent the incidence of the condition is in the population. Paradoxically, the lower the base rate, the poorer diagnostic value even a sensitive test may have.

To summarize our jargon for interpreting a test or assay, so far we have

• True positive (a), the person has a disease and the test correctly identifies the person as having the disease.
• False positive (b), test result incorrectly identifies disease; the person does not have the disease, but the test classifies the person as having the disease.
• False negative (c), test result incorrectly classifies person does not have disease, but the person actually has the disease.
• True negative (d), the person does not have the disease and the test correctly categorizes the person as not having the disease.
• Positive predictive value (PPV), the proportion of patients with a positive test result who actually have the disease, .
  • Negative Predictive Value (NPV), the probability that a person who receives a negative test result really does not have the disease, .
• Sensitivity of test (Se), the proportion of persons who test positive and do have the disease (true positives):
  If a test has 75% sensitivity, then out of 100 individuals who do have the disease, then 75 will test positive (TP = true positive).
• Specificity of a test (Sp), refers to the rate that a test correctly classifies a person that does not have the disease (TN = true negatives): 
  If a test has 90% specificity, then out of 100 individuals who truly do not have the disease, then 90 will test negative (= true negatives).

A worked example. A 50 year old male patient is in the doctor’s office. The doctor is reviewing results from a diagnostic test, eg, a FOBT — fecal occult blood test — a test used as a screening tool for colorectal cancer (CRC). The doctor knows that the test has a sensitivity of about 75% and specificity of about 90%. Prevalence of CRC in this age group is about 0.2% (cf NCI SEER). The prevalence is also the prior probability — the baseline rate of the condition in the population before a test is taken. Figure 4 shows our probability tree using our natural numbers approach (Fig 4).

Figure 4. Probability tree for FOBT test; Good test outcomes shown in green: TP stands for true positive and TN stands for true negative. Poor outcomes of a test shown in red: FN stands for false negative and FP stands for false positive.

The associated probabilities for the four possible outcomes of these kinds of tests (eg, what’s the probability of a person who has tested positive in screening tests actually has the disease?) are shown in Table 1.

Table 1. A 2 X 2 table of possible outcomes of a diagnostic test.

We want the posterior probability — the chance that a positive test result is a true positive. Bayes’ rule is often given in probabilities,

Bayes’ rule, where truth is represented by either D (the person really does have the disease”) or ND (the person really does not have the disease”) and ⊕ is symbol for “exclusive or” and reads “not D” in this example.

An easier way to see this is to use frequencies and our table instead (Table 1, numbers in Figure 4).

Now, the formula is

Simplified Bayes’ rule, where a is the number of people who test positive and DO HAVE the disease and b is the number of people who test positive and DO NOT have the disease. Thus, , the posterior probability was

or 0.15%. Because the condition is so rare (0.2% prior) and the test generates a very high number of False Positives (9,998), getting a positive result actually lowers oour confidence slightly from the base rate (from 0.2% down to 0.15%). This occurs because the test is highly unreliable in screening this specific population — it’s known as the base rate fallacy. Thus, the chance that a positive FOBT test for our 50 year old male patient is 99.85% likely wrong.

Standardized million.

Where did the 100,000 come from? We discussed this in chapter 2: it’s a simple trick to adjust rates to the same population size. We use this to work with natural numbers instead of percent or frequencies. You choose to correct to a standardized population based on the raw incidence rate. A rough rule of thumb:

Table 2. Relationship between standard population size and incidence rate.

The raw incident rate is simply the number of new cases divided by the total population.

Per capita rate.

Yet another standard manipulation is to consider the average incidence per person, or per capita rate. The Latin “per capita” translates to “by head” (Google translate), but in economics, epidemiology, and other fields it is used to reflect rates per person. Tuberculosis is a serious infectious disease of primarily the lungs. Incidence rates of tuberculosis in the United States have trended down since the mid 1900s: 52.6 per 100K in 1953 to 2.7 per 100K in 2019 (CDC). Corresponding per capita values are 5.26 x 10^-4 and 2.7 x 10^-5, respectively. Divide the rate by 100,000 to get per person rate.

Practice and introduce PPV and test decision criteria.

Let’s break these problems down, and in doing so, introduce some terminology common to the field of “risk analysis” as it pertains to evaluating assay and diagnotistc tests in biology and epidemiology. Our first example considers the fecal occult blood test, FOBT, test. Blood in the stool may (or may not) indicate polys or colon cancer (Park et al 2010).

A table like the one shown above (Table 1) will appear again and again throughout this and subsequent chapters, but in different forms. We start with Table 3.

Table 3. A 2 X 2 table of possible outcomes of FOBT test.

We want to know how good is the test, particularly if the goal is early detection? This is conveyed by the PPV, positive predictive value of the test. Unfortunately, the prevalence of a condition is also at play: the lower the prevalence, the lower the PPV must be, because most positive tests will be false when population prevalence is low.

Youden (1950) proposed a now widely adopted index that summarizes how effective a test is. Youden’s J is the sum of specificity and sensitivity minus one.

where Se stands for sensitivity of the test and Sp stands for sensitivity of the test.

Youden’s J takes on values between 0, for a terrible test, and 1, for a perfect test. For our FOBT example, Youden’s J was 0.65. This statistic looks like it’s independent of prevalence, but it’s use as a decision criterion (eg, a cutoff value, above which test is positive, below test is considered negative), assumes that the cost of misclassification (false positives, false negatives) are equal. Prevalence affects number of false positives and false negatives for a given diagnostic test, so any decision criterion based on Youden’s J will also be influenced by prevalence (Smits 2010).

Youden’s J is insensitive to base rates and therefore may be of little practical use in the field. Unsurprisingly, other tools are available, including Matthew’s Correlation Coefficient (MCC),

and the positive and negtive Clinical Utility Index (CUI+, CUI-),

or the F1-score,

epiR will calculate Youden’s J, but the others are either calculated by hand or by way of additional packages. So, we’ll provide a function.

R code to calculate testing decision criteria — code genereated with Gemini AI assistance.

run_screening_wizard <- function() {
# 1. Sequential user prompts starting with cell a
cat("--- 2x2 Matrix Entry Wizard ---\n")
a <- as.numeric(readline(prompt = "Enter cell a (True Positives, TP): "))
b <- as.numeric(readline(prompt = "Enter cell b (False Positives, FP): "))
c <- as.numeric(readline(prompt = "Enter cell c (False Negatives, FN): "))
d <- as.numeric(readline(prompt = "Enter cell d (True Negatives, TN): "))

# 2. Map cells to short variables for calculation
tp <- a
fp <- b
fn <- c
tn <- d

# 3. Construct and print the visual 2x2 matrix
user_matrix <- matrix(
c(tp, fp, fn, tn), 
nrow = 2, 
byrow = TRUE,
dimnames = list(Test = c("+", "-"), Disease = c("+", "-"))
)
cat("\n[+] Captured Matrix:\n")
print(user_matrix)

# 4. Run background metrics math
sensitivity <- tp / (tp + fn)
specificity <- tn / (tn + fp)
ppv <- tp / (tp + fp)
if (is.nan(ppv)) ppv <- 0 

youden_j <- sensitivity + specificity - 1
cui_pos <- sensitivity * ppv

f1_score <- 2 * ((ppv * sensitivity) / (ppv + sensitivity))
if (is.nan(f1_score)) f1_score <- 0 

mcc_num <- (tp * tn) - (fp * fn)
mcc_den <- sqrt(as.numeric(tp + fp) * as.numeric(tp + fn) * 
as.numeric(tn + fp) * as.numeric(tn + fn))
mcc <- if (mcc_den == 0) 0 else mcc_num / mcc_den

# 5. Compile into the copy/paste ready console matrix
output_matrix <- matrix(
c(round(youden_j, 4), "No", "Good (Overly optimistic baseline)",
round(mcc, 4), "Yes", "Poor (Correctly captures imbalance)",
round(cui_pos, 4), "Yes", "Unacceptable clinical utility",
round(f1_score, 4), "Yes", "Near-total prediction failure"),
nrow = 4, byrow = TRUE
)

dimnames(output_matrix) <- list(
c("Youden's J Index", "MCC", "CUI+", "F1-Score"),
c("Value", "Base_Rate_Dep", "Console_Interpretation")
)

final_df <- as.data.frame(output_matrix)

# 6. Print and invisibly return the dataframe
cat("\n[+] Resulting Metrics Table (Copy/Paste Ready):\n")
print(final_df)
return(invisible(final_df))
}

# Run the function at the R command prompt.
run_screening_wizard()

End R code.

Thus, using a metric sensitive to low base rate, our FOBT tests are (modified from R output)

Table 4. Decision criteria.

	Value	Base rate dependent?	Interpretation
	0.65	No	Good
	0.0306	Yes	Poor
	0.0011	Yes	Unacceptable clinical utility
	0.0030	Yes	Near-total prediction failure

Another worked example.  A study on cholesterol-lowering drugs (statins) reported a relative risk reduction of death from cardiac event by 22%. This does not mean that for every 1000 people fitting the population studied, 220 people would be spared from having a heart attack. In the study, the death rate per 1000 people was 32 for the statin versus 41 for the placebo — recall that a placebo is a control treatment offered to overcome potential patient psychological bias (see Chapter 5.4). The absolute risk reduction due to statin is only 41 – 32 or 9 in 1000 or 0.9%. By contrast, relative risk reduction is calculated as the ratio of the absolute risk reduction (9) divided by the proportion of patients who died without treatment (41), which is 22% (LIPID Study Group 1998).

Note that risk reduction is often conveyed as a relative rather than as an absolute number. The distinction is important for understanding arguments based in conditional probability. Thus, the question we want to ask about a test is summarized by absolute risk reduction (ARR) and number needed to treat (NNT), and for problems that include control subjects, relative risk reduction (RRR). We expand on these topics in the next section, 7.4 – Epidemiology: Relative risk and absolute risk, explained.

Evidence Based Medicine.

One culture change in medicine is the explicit intent to make decisions based on evidence (Masic et al 2008). Of course, the joke then is, well, what were doctors doing before, diagnosing without evidence? The comic strip xkcd offers one possible answer (Fig 5).

Figure 5. A summary of “evidence based medical” decisions, perhaps? “Watson Medical Algorithm,” https://xkcd.com/1619/.

As you can imagine, there’s considerable reflection about the EBM movement (see discussions in response to Accad and Francis 2018, eg, Goh 2018). More practically, our objective is for you to be able to work your way through word problems involving risk analysis. You can expect to be asked to calculate, or at least set up for calculation, any of the statistics listed above (eg, False negative, false positive, etc.). Practice problems are listed at the end of this section, and additional problems are provided to you (Homework 4). You’ll also want to check your work, and in any real analysis, you’d most likely want to use R.

Software.

R has several epidemiology packages, and with some effort, can save you time. Another option is to run your problems in OpenEpi, a browser-based set of tools. OpenEpi is discussed with examples in the next section, 7.4.

Here, we illustrate some capabilities of the epiR package, expanded more also in the next section, 7.4. We’ll use the example from Table 3.

R code

library(epiR) 
Table3 <- matrix(c(15, 5, 9998, 89982), nrow = 2, ncol = 2)
epi.tests(Table3)

R output

         Outcome + Outcome -   Total
Test +          15      9998   10013
Test -           5     89982   89987
Total           20     99980  100000

Point estimates and 95% CIs:
--------------------------------------------------------------
Apparent prevalence *                      0.10 (0.10, 0.10)
True prevalence *                          0.00 (0.00, 0.00)
Sensitivity *                              0.75 (0.51, 0.91)
Specificity *                              0.90 (0.90, 0.90)
Positive predictive value *                0.00 (0.00, 0.00)
Negative predictive value *                1.00 (1.00, 1.00)
Positive likelihood ratio                  7.50 (5.82, 9.67)
Negative likelihood ratio                  0.28 (0.13, 0.59)
False T+ proportion for true D- *          0.10 (0.10, 0.10)
False T- proportion for true D+ *          0.25 (0.09, 0.49)
False T+ proportion for T+ *               1.00 (1.00, 1.00)
False T- proportion for T- *               0.00 (0.00, 0.00)
Correctly classified proportion *          0.90 (0.90, 0.90)
--------------------------------------------------------------
* Exact CIs

Oops! I wanted PPV, which by hand calculation was 0.15%, but R reported “0.00?” This is a significant figure reporting issue. The simplest solution is to submit options(digits=6) before the command, then save the output from epi.tests() to an object and use summary(). For example

options(digits=6)
myEpi <- epi.tests(Table3)
summary(myEpi)

And R returns

     statistic           est        lower        upper
1           ap  0.1001300000 0.0982761568  0.102007072
2           tp  0.0002000000 0.0001221693  0.000308867
3           se  0.7500000000 0.5089541283  0.913428531
4           sp  0.9000000000 0.8981238085  0.901852950
5      diag.ac  0.8999700000 0.8980937508  0.901823014
6      diag.or 27.0000000000 9.8110071871 74.304297826
7         nndx  1.5384615385 1.2265702376  2.456532054
8       youden  0.6500000000 0.4070779368  0.815281481
9       pv.pos  0.0014980525 0.0008386834  0.002469608
10      pv.neg  0.9999444364 0.9998703379  0.999981958
11      lr.pos  7.5000000000 5.8193604069  9.666010707
12      lr.neg  0.2777777778 0.1300251423  0.593427490
13      p.rout  0.8998700000 0.8979929278  0.901723843
14       p.rin  0.1001300000 0.0982761568  0.102007072
15      p.tpdn  0.1000000000 0.0981470498  0.101876192
16      p.tndp  0.2500000000 0.0865714691  0.491045872
17      p.dntp  0.9985019475 0.9975303919  0.999161317
18      p.dptn  0.0000555636 0.0000180416  0.000129662

There we go — pv.pos reported as 0.0014980525, which, after turning to a percent and rounding, we have 0.15%. Note also the additional statistics provided — a good rule of thumb — always try to save the output to an object, then view the object, eg, with summary(). Refer to help pages for additional details of the output (?epi.tests).

What about R Commander menus?

Rcmdr has a plugin that will calculate ARR, RRR and NNT. The plugin is called RcmdrPlugin.EBM (Leucuta et al 2014) and it would be downloaded as for any other package via R.

Download the package from your selected R mirror site, then start R Commander.

install.packages("RcmdrPlugin.EBM")

From within R Commander (Fig 6), select

Tools → Load Rcmdr plug-in(s)…

Figure 6. To install an Rcmdr plugin, first go to Rcmdr → Tools → Load Rcmdr plug-in(s)…

Next, select from the list the plug-in you want to load into memory, in this case, RcmdrPlugin.EBM (Fig 7).

Figure 7. Select the Rcmdr plugin, then click the “OK” button to proceed.

Restart Rcmdr again (Fig 8),

Figure 8. Select “Yes” to restart R Commander and finish installation of the plug-in.

and the menu “EBM” should be visible in the menu bar (Fig 9).

Figure 9. After restart of R Commander the EBM plug-in is now visible in the menu.

Note that you will need to repeat these steps each time you wish to work with a plug-in, unless you modify your .RProfile file. See

Rcmdr → Tools → Save Rcmdr options…

Clicking on the EBM menu item brings up the template for the Evidence Based Medicine module. We’ll mostly work with 2 X 2 tables (eg, see Table 1) , so select the “Enter two-way table…” option to proceed (Fig 10).

Figure 10. Select “Enter two-way table…”.

And finally, Figure 11 shows the two-way table entry cells along with options. We’ll try a problem by hand then use the EBM plugin to confirm and gain additional insight.

Figure 11. Two-way table Rcmdr EBM plug-in.

For assessing how good a test or assay is, use the Diagnosis option in the EBM plugin. For situations with treated and control groups, use Therapy option. For situations in which you are comparing exposure groups (eg, smokers vs non-smokers), use the Prognosis option.

Example.

Here’s a simple one (problem from Gigerenzer 2002).

About 0.01% of men in Germany with no known risk factors are currently infected with HIV. If a man from this population actually has the disease, there is a 99.9% chance the tests will be positive. If a man from this population is not infected, there is a 99.9% chance that the test will be negative. What is the chance that a man who tests positive actually has the disease?

Start with the reference, or base population (Figure 12). It’s easy to determine the rate of HIV infection in the population if you use numbers. For 10,000 men in this group, exactly one man is likely to have HIV (0.0001X10,000), whereas 9,999 would not be infected.

For the man who has the disease it’s virtually certain that his results will be positive for the virus (because the sensitivity rate = 99.9%). For the other 9,999 men, one will test positive (the false positive rate = 1 – specificity rate = 0.01%).

Thus, for this population of men, for every two who test positive, one has the disease and one does not, so the probability even given a positive test is only 100*1/2 = 50%. This would also be the test’s Positive Predictive Value.

Note that if the base rate changes, then the final answer changes! For example, if the base rate was 10%

It also helps to draw a tree to help you determine the numbers (Fig 12)

Figure 12. Draw a probability tree to help with the frequencies.

From our probability tree in Figure 12 it is straight-forward to collect the information we need.

• Given this population, how many are expected to have HIV? Two.
• Given the specificity and sensitivity of the assay for HIV, how many persons from this population will test positive? Two.
• For every positive test result, how many men from this population will actually have HIV? One.

Thus, given this population with the known risk associated, the probability that a man testing positive actually has HIV is 50% (=1/(1+1)).

Use the EBM plugin. Select two-way table, then enter the values as shown in Fig 13.

Figure 13. EBM plugin with data entry.

Select the “Diagnosis” option — we are answering the question: How probable is a positive result given information about sensitivity and specificity of a diagnosis test. The results from the EBM functions are given below

Rcmdr> .Table
      Yes No
   +  1    1
   -  0 9998

Rcmdr> fncEBMCrossTab(.table=.Table, .x='', .y='', .ylab='', .xlab='', 
Rcmdr+ .percents='none', .chisq='1', .expected='0', .chisqComp='0', .fisher='0', 
Rcmdr+ .indicators='dg', .decimals=2)

# Notations for calculations
        Disease + Disease -
Test +      "a"         "b" 
Test -      "c"         "d"

# Sensitivity (Se) = 100 (95% CI 2.5 - 100) %. Computed using formula: a / (a + c)
# Specificity (Sp) = 99.99 (95% CI 99.94 - 100) %. Computed using formula: d / (b + d)
# Diagnostic accuracy (% of all correct results) = 99.99 (95% CI 99.94 - 100) %. Computed using formula: (a + d) / (a + b + c + d)
# Youden's index = 1 (95% CI 0.02 - 1). Computed using formula: Se + Sp - 1
# Likelihood ratio of a positive test = 9999 (95% CI 1408.63 - 70976.66). Computed using formula: Se / (Sp - 1)
# Likelihood ratio of a negative test = 0 (95% CI 0 - NaN). Computed using formula: (1 - Se) / Sp
# Positive predictive value = 50 (95% CI 1.26 - 98.74) %. Computed using formula: a / (a + b)
# Negative predictive value = 100 (95% CI 99.96 - 100) %. Computed using formula: d / (c + d)
# Number needed to diagnose = 1 (95% CI 1 - 40.91). Computed using formula: 1 / [Se - (1 - Sp)]

Note that the formulas used to calculate Sensitivity, Specificity, etc., follow our Table 1 (compare to “Notations for calculations”). The use of EBM provides calculations of our confidence intervals.

Questions.

1. The sensitivity of the fecal occult blood test (FOBT) is reported to be 0.68. What is the False Negative Rate?
2. The specificity of the fecal occult blood test (FOBT) is reported to be 0.98. What is the False Positive Rate?
3. For men between 50 and 54 years of age, the rate of colon cancer is 61 per 100,000. If the false negative rate of the fecal occult blood test (FOBT) is 10%, how many persons who have colon cancer will test negative?
4. For men between 50 and 54 years of age, the rate of colon cancer is 61 per 100,000. If the false positive rate of the fecal occult blood test (FOBT) is 10%, how many persons who do not have colon cancer will test positive?
5. A study was conducted to see if mammograms reduced mortality
   

   Mammogram	Deaths/1000 women
   No	4
   Yes	3

   data from Table 5-1 p. 60 Gigerenzer (2002)
   What is the RRR?

6. A study was conducted to see if mammograms reduced mortality
   

   Mammogram	Deaths/1000 women
   No	4
   Yes	3

   data from Table 5-1 p. 60 Gigerenzer (2002)
   What is the NNT?

7. Does supplemental Vitamin C decrease risk of stroke in Type II diabetic women? A study conducted on 1923 women, a total of 57 women had a stroke, 14 in the normal Vitamin C level and 32 in the high Vitamin C level. What is the NNT between normal and high supplemental Vitamin C groups?
8. Sensitivity of a test is defined as
   A. False Positive Rate
   B. True Positive Rate
   C. False Negative Rate
   D. True Negative Rate
9. Specificity of a test is defined as
   A. False Positive Rate
   B. True Positive Rate
   C. False Negative Rate
   D. True Negative Rate
10. In thinking about the results of a test of a null hypothesis, Type I error rate is equivalent to
    A. False Positive Rate
    B. True Positive Rate
    C. False Negative Rate
    D. True Negative Rate
11. During the Covid-19 pandemic, number of reported cases each day were published. For example, 155 cases were reported for 9 October 2020 by Department of Health. What is the raw incident rate?

Quiz Chapter 7.3

Conditional Probability and Evidence Based Medicine

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Chapter 7 contents

• Introduction
• Epidemiology definitions
• Epidemiology basics
• Conditional Probability and Evidence Based Medicine
• Epidemiology: Relative risk and absolute risk, explained
• Odds ratio
• Confidence intervals
• References and suggested readings

6.1 – Some preliminaries

Introduction

OK, you say, I get it: statistics is important, and if I am to go on as a biologist, I should learn statistics. Let’s get on with it, start with the equations and the problems already!

Before we review probability theory and introduce risk analysis I want to spend some time to emphasize that at issue is critical thinking, so please bear with me (Fig 1).

Figure 1. “Slides,” https://xkcd.com/365/.

How likely?

As we start our journey in earnest, we start with the foundations of statistics, probability. Probability in statistics means the chance that a specific outcome will occur. Probability is something we measure, or estimate, or draw from theoretical models, about whether something, a specific outcome from a class of outcomes (events — see Chapter 7.1), will occur. We speak about how likely an outcome is to occur, and this is quantified by the probability. Probabilities are given a value between 0 and 1: at 0% chance, the event will not happen; 100% chance, the event is certain to happen. Remember, probability applies to predicting future events; obviously, when viewed in the past, the event either did (100%), or did not (0%) happen.

If probability is the chance that an outcome will occur, then risk is the probability of a particular outcome occurring over a specified period of time. Related, epidemiologists define exposure risk as the potential for harm (or benefit) from contact with or vulnerability to a hazard (or treatment) or risk factor.

I introduce our discussion of probability through a risk analysis — the process of evaluating the probability and potential impact of health risks to individuals or populations. I like to start this discussion by relaying something I overheard while a number of us were standing on a lava field on the slopes of Kilauea back in November of 1998 (Kilauea erupted with lava flow more or less continually between 1983 and 2018; 5 Jan 2023, started up again).

Figure 2. View of Kamokuna Lava Bench, eruption of Pu`u `O`o, Kilauea, November 1998. Photo by S. Dohm.

In 1998, Volcanoes National Park hadn’t established barricades at the end of Chain of Craters Road, and people were walking to see new lava flows. The night we went, we met a park ranger who pleaded with us that the Park Service believed it was unsafe for us to walk out to see new lava flows because the area, called a lava delta, was unstable. Someone (not in my group), snapped back, “Oh, what are the chances that that will happen?” Of course, the ranger couldn’t quote a chance between zero and 100% for that particular evening. The ranger was saying the risk had increased, based on their subjective, but experienced opinion (see Kinds of probability, below).

As you can see in Figure 2, we went anyway. I have been thinking about her question ever since. We were lucky — some of the same area collapsed two weeks later (SOEST information, eruption about episode 55).

Cool picture though.

While the terms are used interchangeably in everyday language, they have distinct mathematical and statistical definitions. Prospective: You use a known model and its parameters to predict the probability of future data.

Multiple events

Whether or not a newly formed lava delta collapses, that’s a singular or simple event. For cases with more than one one outcome, that’s a multiple event.

I have two coins, a dime and a quarter, in my pocket; when I place the coins on the table, what are the chances that both coins will show heads? A blood sample from a crime scene was typed for two Combined DNA Index System (CODIS) Short Tandem Repeat (STR) loci, THO1 (allele 9.3) and TPOX (allele 8), the same allele types for the defendant. What are the chances of a match, someone other than the defendant has the same genetic profile? In both examples I describe a random process. With randomness, we can predict the probability of each kind of outcome, or series of outcomes, because we know the total possible number of outcomes. If there are just two outcomes, then the probability is equal — 50% that either will occur. If, however, the process is not random, then even if there are just two possible outcomes, the probability can’t be 50% — the outcome depends on some other process than random chance.

What is the chance that two events occur? For two or more independent events we can get the answers by using the product rule.

The two coins are independent, therefore the chance that both are placed heads up is 25% — we would expect to see this combined event one out of every four times. This is an illustration of the counting rule, aka fundamental counting principle: if there are n ways to do one thing (n elements in set A), and m ways to do another thing (m elements in set B), then there are ways to do both things (combination of elements of A and B sets).

For the DNA profile CODIS problem (cf. Chapter 4, National Research Council 1996), the two alleles are both the most common observed in US Caucasian population at 30.45% and 54.7%, respectively (Moretti et al 2016). Assuming the individual was homozygous at both loci (ie, THO1_9.3,9.3 and TPOX_8,8), then the genotype frequencies (p²) are:

Since THO1 is located on the p arm of chromosome 11, and TPOX is on the p arm of chromosome 2, the two loci are independent and therefore should be in linkage disequilibrium. We can use the product rule to get the probability of the DNA profile for the sample, 2.8%

If two events are not independent, then the product rule cannot be used. For example, CODIS STR D5S818 and CSF1PO are both located on the q arm of chromosome 5 and are therefore linked and not independent (the recombination frequency is about 0.25).  The common allele for D5S818 is 11 at 40.84% and for CSF1PO the allele is 12 at 34.16%. Thus the chance that the two most common alleles is not simply the product rule result of 14%, instead we need to view this problem as one of dependent events.

Kinds of probability.

So, how does one go about estimating the likelihood that a particular event, whether it is collapse of a lava delta, or that a person will have a heart attack? The probability of lava delta collapse or of heart attack are examples of empirical probability, as opposed to a theoretical probability. Despite many years of effort, we have no applicable theory that we can apply to say, if a person does this, and that, then a heart attack will happen. But we do have a body of work documenting how often heart attacks occur, and when they occur in association with certain risk factors. Similarly, progress is being made to determine markers of risk of lava field collapse (Di Traglia et al. 2018). Analogously, this is the essential goal of risk analysis in epidemiology. We know of associations between cholesterol and heart attack risk, for example, but we also know that high cholesterol does not raise the probability of the event (heart attack) to 100%. How is this uncertainty part of statistics? Or perhaps you are a molecular scientist in training and have learned about how to assess results of a Western blot where typically the results are scored as “yes” or “no.” How is this relevant to statistics and probability?

A misconception about statistics and statistical thinking

There’s a long history of skepticism of conclusions from health studies, in part because it seems the advice flips. For example,

• Coffee is bad for you (Medical News Today January 2008)
• Coffee is good for you (NBC News July 2018)
• Even light drinking can be harmful to health (Science Daily, January 2022)
• Seven science-backed reasons beer is good for you (NBC News August 2017)
• Meat and cheese may be as bad for you as smoking (Science Daily, March 2014)
• Cheese actually isn’t bad for you (WIRED, February 2021)

The common thread is these studies are assessment of risk: about studies that seem to conclude “only: with statements of probability.

Perhaps you may have heard …?  “There are lies and then there is statistics“. The full quote reads as follows:

Figures often beguile me, particularly when I have the arranging of them myself; in which case the remark attributed to Disraeli would often apply with justice and force: “There are three kinds of lies: lies, damned lies and statistics.” – Autobiography of Mark Twain (www.twainquotes.com/Lies.html).

Figure 3. Mark Twain. Image from The Miriam and Ira D. Wallach Division of Art, Prints and Photographs: Photography Collection, The New York Public Library. “Mark Twain in Middle Life” The New York Public Library Digital Collections. 1860 – 1920. https://digitalcollections.nypl.org/items/510d47d9-baec-a3d9-e040-e00a18064a99

Twain attributed the remarks to Benjamin Disraeli, the prime minister of Britain during much of Queen Victoria’s reign, but others have not been able to document this utterance to that effect. Still others believe that the “3-lies” quote belongs to Leonard Courtney, an English mathematician and statistician (1847-1929) (see University of York web site for more).

What is meant by this quote? Tossing aside cynicism — or healthy skepticism of authority one should have, whether that authority is a scientist or a politician(within reason, please!) — what this quote means is that it seems that results of very similar studies are in conflict. To some, this is part of the replication crisis in science (Baker 2016). There’s a perception that one can say just about anything with a number. Partly this is a matter of semantics, but also there is legitimacy to this concern. However, it is not necessarily the case that statistics have been intentionally done to mislead, rather, there is evidence that researchers are not always using proper statistical procedures.

One word, several meanings

We use the term statistics in multiple ways, all correct, but not all equal. For example, a statistic may refer to a number used to describe a population characteristic. From the 2010 U.S. census, we learn that the racial (self-reported) make-up of the U.S. population (then at 303 million) was 72.4% “white” and 12.6% “black” self-reported. In this sense, a statistic is something you calculate as a description. Do you recall the distinction between “statistic” and “statistics” discussed in Chapter 2?

Secondly, two studies essentially about the same topic, yet reaching seemingly different conclusions, may differ in the assumptions employed. It should be obvious that if different assumptions are used, researchers may reach different outcomes.

Finally, how we communicate statistics can be misleading. For example, use of percentages in particular can be confusing, especially in communication of the chance that some event may happen to us (eg, incidence of disease: number of new cases in a specific time period, compared to prevalence of a disease: number of cases of a disease in a specific period of time). On the one hand, percentages seem easy. A percentage is simply a proportion multiplied by 100%, and takes any value between 0% and 100%.

When a product says that it kills 99.99% of all germs on contact, do you feel better? Here’s a cartoon to consider as you think about that statement (Fig 4).

Figure 4. “Hand sanitizer,” https://imgs.xkcd.com/comics/hand_sanitizer.png

Of course numbers cannot be used to justify simultaneously mutually exclusive conclusions, but being able to recognize careless (or deliberate) miscommunication with numbers, well, this needs to be part of your skill set. Here’s a little exercise from transportation safety to ponder as you reflect on the subject of lies and statistics. rather, as you read this next section I ask that you consider

• are the correct statistical descriptors in use?
• what assumptions are being made?
• does the reporting of percentages lead to clear conclusions?

Some concluding thoughts about “lies and statistics”

Statistics is tricky because there are assumptions to be made. And you have to be clear in your thinking.

If the assumptions hold true, then we aren’t lying, and Twain had it wrong.

But if we disagree on the assumptions, then we will necessarily have to disagree on the conclusions drawn from the calculated numbers (the statistics). Risk analysis in particular, but statistics in general, is a tricky business because many assumptions need to be made, and we won’t necessarily have all of the relevant information available to make sure our assumptions are truthful. But it is the assumptions that matter: if we agree with the assumptions that are made, then we have confidence in the conclusions drawn from the statistics.

In a typical statistics course, we would spend a bunch of time on probability. We will here as well, but in the context of risk analysis and in the other contexts, in a less than formal presentation on the subject of probability. For example, in talking about inference, the testing of null hypotheses and estimating the probability that the null hypothesis is correct, I will say things like, “Imagine we repeated the experiment a million times — how many times by chance would we think a correct null hypothesis would none the less be rejected?”

There’s no real substitute for a formal course in probability theory and you should be aware that this foundation is pretty important if you go forward with biostatistics and epidemiology. For now, I will simply refer you to chapters 1, 2 and 3 of a really nice online book on probability from one of the masters, Richard Jeffrey (1926 – 2002; click here to go to Wikipedia). Much of what I will present to you follows from similar discussions.

My aim is to teach you what you need about probability theory by the doing. In the next couple of days we will deal with an aspect of risk analysis, namely a consideration of conditional probability and Baye’s Theorem that will help you evaluate claims such as the one made for airline safety. Risk analysis is tricky, but it is not a subject above and beyond our abilities; by applying some of the rules of statistical reasoning, we can check claims based on statistics. A healthy degree of skepticism is part of becoming a scientist. Do try this at home!

Some examples to consider: what is the relative risk for the following scenarios?

1. Drug testing at the workplace: risk of a worker, who does not use illegal drugs, registering positive (false positives in drug testing);

2. Positive HIV from blood sample from USA male with no associated risks (eg, intravenous drug user), false positives in HIV testing; false positives with mammography;

3. Benefits versus risks of taking a statin drug (drug that reduces serum cholesterol levels) to a person with no history of heart disease.

4. Is it safer to travel by car or by airliner? We’ll break this problem down in the next section.

What we are looking for is the probability of an occurrence of a particular event: that a person who does not use illegal drugs may nonetheless test positive; we are looking for a way to make rational decisions and understanding probability is the foundation.

Questions

1. What do you make of the claim (joke) that “There are lies and then there are statistics?”
2. Distinguish between empirical and theoretical probability; use examples.

Quiz Chapter 6.1

Some preliminaries

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Chapter 6 contents

• Introduction
• Some preliminaries
• Ratios and proportions
• Combinations and permutations
• Types of probability
• Discrete probability distributions
• Continuous distributions
• Normal distribution and the normal deviate (Z)
• Moments
• Chi-square distribution
• t distribution
• F distribution
• References and suggested readings

6 – Probability, Distributions

Introduction

Probability is how likely something, an event, is likely to occur, ie, a prediction of the future. Related, we can ask of the likelihood of an event, which is like working backwards. The event has already occurred and we ask how well a model (parameters) explain the observed data.

Working with probability allows us to contextualize the inherent uncertainty in any measure, which in turn allows us to interpret results from hypothesis tests of observational or experimental data sets. Thus, an important concept to appreciate is that in many cases, like R.A. Fisher’s Lady tasting tea analogy, we can count in advance all possible outcomes of an experiment.

For many more experiments, we cannot count all possible outcomes of the sample space, either because they are too numerous or simply unknowable. In such cases, applying theoretical probability distributions allow us to circumvent the countability problem. Whereas empirical probability distributions are  frequency counts of observations, theoretical probabilities are based on mathematical formulas.

At this stage of your training in statistics, you would spend time with the foundations of probability. Scenarios of likelihood of single events regardless of other events are called marginal probability, whereas the union refers to at least one event occuring. The likelihood of two or more events occuring at the same time is called joint probability. Worked examples are presented in the chapter, although the bulk of this chapter concerns conditonal probability, the likelihood of one event occuring given another event already happened. Working with conditional probability is essential for risk analysis, which we introduce in Chapter 7 – Probability, Risk Analysis and expand in Chapter 7.3 – Conditional Probability and Evidence Based Medicine and Chapter 7.4 – Epidemiology: Relative risk and absolute risk, explained.

Probability distributions are key to the null hypothesis significance testing framework for statistical inference. Given assumptions about the data, probability distributions are used to evaluate the likelihood of observing data under the null hypothesis, eg, no difference between means of a control group compared to a treatment group. Much of classical inferential statistics, especially the kind one finds in introductory courses like ours, are built on probability distributions. ANOVA, t-tests, linear regression, etc., are parametric tests and assume errors are distributed according to a particular type of distribution, the normal or Gaussian distribution.

A probability distribution is a list of probabilities for each possible outcome of a discrete random variable in an entire population. Depending on the data type, there are many classes of probability distributions. In contrast, probability density functions are used to for continuous random variables. This chapter begins with basics of probability then gently introduces discrete and continuous probability distributions. In the other sections of this chapter we describe several probability density functions. Emphasis is placed on the normal distribution, which underlies most parametric statistics.

Homework to go with this topic

Homework 3: Distributions & Probability in Mike’s Workbook for Biostatistics.

Quizzes in this chapter

A total of 88 questions among the several subchapters, a mix of true or false and multiple choice question format.

Chapter 6 contents

• Introduction
• Some preliminaries
• Ratios and proportions
• Combinations and permutations
• Types of probability
• Discrete probability distributions
• Continuous distributions
• Normal distribution and the normal deviate (Z)
• Moments
• Chi-square distribution
• t distribution
• The F distribution
• References and suggested readings

7 – Probability, Risk Analysis

Introduction

The public health applications of epidemiology, the branch of medicine concerned with identifying patterns and potential causes of disease and health in populations, were every day in the news during the Covid-19 pandemic. From contact tracing to reproductive rate of the SARS-Cov-2 virus to numbers of hospital beds and nurses available in ICU units across the country, to the discussions and debates over how the virus is spread, no doubt you have learned much about the critical role epidemiology continues to play.

Figure 1. “Health data,” https://xkcd.com/2620/.

This chapter is about probability and introduced you to risk analysis (Fig 1), used to “… characterize the nature and magnitude of risks to human health for various populations…“, a foundational topic in biostatistics and epidemiology. The epiR package will be introduced and code examples provided for descriptive epidemiology and again for statistical inference (Chapter 9).

Homework to go with this topic

Homework 4: Risk in Mike’s Workbook for Biostatistics.

Quizzes in this chapter

A total of 51 questions among the several subchapters, a mix of true or false and multiple choice question format.

Chapter 7 contents

• Introduction
• Epidemiology definitions
• Epidemiology basics
• Conditional Probability and Evidence Based Medicine
• Epidemiology: Relative risk and absolute risk, explained
• Odds ratio
• Confidence intervals
• References and suggested readings