P.Mean >> Statistics webinar >> P-values, confidence intervals, and the Bayesian alternative (presented 2009-09-14).

I will be hosting a free webinar (web seminar) on Wednesday, October 14, 10am-noon, CDT. The topic is "P-values, confidence intervals, and the Bayesian alternative." This is the handout I will use for that webinar.

I am also using this handout for a presentation to the Midwest Society for Pediatric Research on Thursday, October 8.

Abstract: P-values and confidence intervals are the fundamental tools used in most inferential data analyses. They are possibly the most commonly reported statistics in the medical literature. Unfortunately, both p-values and confidence intervals are subject to frequent misinterpretations. In this two hour webinar, you will learn the proper interpretation of p-values and confidence intervals, and the common abuses and misconceptions about these statistics. You will also see a simple application of Bayesian analysis which provides an alternative to p-values and confidence intervals.

In this seminar, you will learn how to:

If the free webinar is successful, then I will offer additional webinars for a fee. I have developed a list of possible topics for these webinars,

though this may change depending on feedback that I get. If you want to be notified about future webinars, the easiest way is to sign up for my newsletter, The Monthly Mean,

This newsletter will include publicity about the upcoming free webinar in the September issue, which I'm hoping to distribute in early October and future paid webinars in future issues. Future paid webinars will also be publicized on the www.pmean.com website.

Here's the outline of this talk

  1. Icebreaker
  2. Pop quiz
  3. Definitions
  4. What is a p-value?
  5. Practice exercises
  6. What is a confidence interval?
  7. Practice exercises
  8. A simple example of Bayesian data analysis.
  9. Repeat of pop quiz

This talk is based largely on a training class that I offered at Children's Mercy Hospital, Stats #22 : What Do All These Numbers Mean? Confidence Intervals and P-Values,

but also includes material from Chapter 6 of my book, Statistical Evidence in Medical Trials,

and material from the lead article of the July/August issue of The Monthly Mean,

Icebreaker

In your job, you may have had to calculate a statistic of one sort or another. It might have been a simple statistic like a mean or a percentage, it might have been more complicated, like a correlation coefficient, or it might have been something very difficult, like a Poisson regression model with an overdispersion parameter. Tell us about the most complex statistic that you have ever computed, either by hand or using a computer. Include only those statistics that you have calculated outside of your university training. Don't include any statistic that someone else calculated for you. Here's a list sorted (more or less) by the complexity of the statistic.

  1. percentage
  2. mean
  3. standard deviation
  4. t-test
  5. correlation coefficient
  6. linear regression model
  7. survival curve
  8. logistic regression model
  9. other

By the way, you won't need knowledge or familiarity with any of the above statistics to follow the content of this presentation. I am just trying to gauge the experience level of my audience.

Pop quiz

A research paper computes a p-value of 0.45. How would you interpret this p-value?

  1. Strong evidence for the null hypothesis
  2. Strong evidence for the alternative hypothesis
  3. Little or no evidence for the null hypothesis
  4. Little or no evidence for the alternative hypothesis
  5. More than one answer above is correct.
  6. I do not know the answer.

A research paper computes a confidence interval for a relative risk of 0.82 to 3.94. What does this confidence interval tell you.

  1. The result is statistically significant and clinically important.
  2. The result is not statistically significant, but is clinically important.
  3. The result is statistically significant, but not clinically important.
  4. The result is not statistically significant, and not clinically important.
  5. The result is ambiguous.
  6. I do not know the answer.

A Bayesian data analysis can incorporate subjective opinions through the use of

  1. Bayes rule.
  2. data shrinkage.
  3. a prior distribution.
  4. a posterior distribution.
  5. p-values.
  6. I do not know the answer.

Definitions

What is a population? A population is a collection of items of interest in research. The population represents a group that you wish to generalize your research to. Populations are often defined in terms of demography, geography, occupation, time, care requirements, diagnosis, or some combination of the above. Contrast this with a definition of a sample. An example of a population would be all infants born in the state of Missouri during the 1995 calendar year who have one or more visits to the Emergency room during their first year of life.

What is a sample? A sample is a subset of a population. A random sample is a subset where every item in the population has the same probability of being in the sample. Usually, the size of the sample is much less than the size of the population. The primary goal of much research is to use information collected from a sample to try to characterize a certain population. As such, you should pay a lot of attention to how representative the sample is of the population. If there are problems, with representativeness, consider redefining your population a bit more narrowly. For example, a sample of 85 smokers between the ages of 13 and 18 in Rochester, Minnesota who respond to an advertisement about participation in a smoking cessation program might not be considered representative of the population of all teenage smokers, because the participants selected themselves. The sample might be more representative if we restrict our population to those teenage smokers who want to quit.

What is a Type I Error? In your research, you specify a null hypothesis (typically labeled H0) and an alternative hypothesis (typically labeled Ha, or sometimes H1). By tradition, the null hypothesis corresponds to no change. When you are using Statistics to decide between these two hypothesis, you have to allow for the possibility of error. Actually, if you are using any other procedure, you should still allow for the possibility of error, but we statisticians are the only ones honest enough to admit this. A Type I error is rejecting the null hypothesis when the null hypothesis is true. Example: Consider a new drug that we will put on the market if we can show that it is better than a placebo. In this context, H0 would represent the hypothesis that the average improvement (or perhaps the probability of improvement) among all patients taking the new drug is equal to the average improvement (probability of improvement) among all patients taking the placebo. A Type I error would be allowing an ineffective drug onto the market.

What is a Type II Error? A Type II error is accepting the null hypothesis when the null hypothesis is false. You should always remember that it is impossible to prove a negative. Some statisticians will emphasize this fact by using the phrase "fail to reject the null hypothesis" in place of "accept the null hypothesis." The former phrase always strikes me as semantic overkill. Many studies have small sample sizes that make it difficult to reject the null hypothesis, even when there is a big change in the data. In these situations, a Type II error might be a possible explanation for the negative study results. Example: Consider a new drug that we will put on the market if we can show that it is better than a placebo. In this context, H0 would represent the hypothesis that the average improvement (or perhaps the probability of improvement) among all patients taking the new drug is equal to the average improvement (probability of improvement) among all patients taking the placebo. A Type II error would be keeping an effective drug off the market.

What is a p-value?

Dear Professor Mean:  Can you give me a simple explanation of what a p-value is?

A p-value is a measure of how much evidence we have against the null hypothesis. The null hypothesis, traditionally represented by the symbol H0, represents the hypothesis of no change or no effect.

The smaller the p-value, the more evidence we have against H0. It is also a measure of how likely we are to get a certain sample result or a result “more extreme,” assuming H0 is true. The type of hypothesis (right tailed, left tailed or two tailed) will determine what “more extreme” means.

Much research involves making a hypothesis and then collecting data to test that hypothesis. In particular, researchers will set up a null hypothesis, a hypothesis that presumes no change or no effect of a treatment. Then these researchers will collect data and measure the consistency of this data with the null hypothesis.

The p-value measures consistency by calculating the probability of observing the results from your sample of data or a sample with results more extreme, assuming the null hypothesis is true. The smaller the p-value, the greater the inconsistency.

Traditionally, researchers will reject a hypothesis if the p-value is less than 0.05. Sometimes, though, researchers will use a stricter cut-off (e.g., 0.01) or a more liberal cut-off (e.g., 0.10). The general rule is that a small p-value is evidence against the null hypothesis while a large p-value means little or no evidence against the null hypothesis. Please note that little or no evidence against the null hypothesis is not the same as a lot of evidence for the null hypothesis.

It is easiest to understand the p-value in a data set that is already at an extreme. Suppose that a drug company alleges that only 50% of all patients who take a certain drug will have an adverse event of some kind. You believe that the adverse event rate is much higher. In a sample of 12 patients, all twelve have an adverse event.

The data supports your belief because it is inconsistent with the assumption of a 50% adverse event rate. It would be like flipping a coin 12 times and getting heads each time.

The p-value, the probability of getting a sample result of 12 adverse events in 12 patients assuming that the adverse event rate is 50%, is a measure of this inconsistency. The p-value, 0.000244, is small enough that we would reject the hypothesis that the adverse event rate was only 50%.

A large p-value should not automatically be construed as evidence in support of the null hypothesis. Perhaps the failure to reject the null hypothesis was caused by an inadequate sample size. When you see a large p-value in a research study, you should also look for one of two things:

  1. a power calculation that confirms that the sample size in that study was adequate for detecting a clinically relevant difference; and/or
  2. a confidence interval that lies entirely within the range of clinical indifference.

You should also be cautious about a small p-value, but for different reasons. In some situations, the sample size is so large that even differences that are trivial from a medical perspective can still achieve statistical significance.

Practice exercises

Read the following abstracts. Interpret each of the p-values presented in these abstracts.

1. The Outcome of Extubation Failure in a Community Hospital Intensive Care Unit: A Cohort Study. Seymour CW, Martinez A, Christie JD, Fuchs BD. Critical Care 2004, 8:R322-R327 (20 July 2004) Introduction: Extubation failure has been associated with poor intensive care unit (ICU) and hospital outcomes in tertiary care medical centers. Given the large proportion of critical care delivered in the community setting, our purpose was to determine the impact of extubation failure on patient outcomes in a community hospital ICU. Methods: A retrospective cohort study was performed using data gathered in a 16-bed medical/surgical ICU in a community hospital. During 30 months, all patients with acute respiratory failure admitted to the ICU were included in the source population if they were mechanically ventilated by endotracheal tube for more than 12 hours. Extubation failure was defined as reinstitution of mechanical ventilation within 72 hours (n = 60), and the control cohort included patients who were successfully extubated at 72 hours (n = 93). Results: The primary outcome was total ICU length of stay after the initial extubation. Secondary outcomes were total hospital length of stay after the initial extubation, ICU mortality, hospital mortality, and total hospital cost. Patient groups were similar in terms of age, sex, and severity of illness, as assessed using admission Acute Physiology and Chronic Health Evaluation II score (P > 0.05). Both ICU (1.0 versus 10 days; P < 0.01) and hospital length of stay (6.0 versus 17 days; P < 0.01) after initial extubation were significantly longer in reintubated patients. ICU mortality was significantly higher in patients who failed extubation (odds ratio = 12.2, 95% confidence interval [CI] = 1.5–101; P < 0.05), but there was no significant difference in hospital mortality (odds ratio = 2.1, 95% CI = 0.8–5.4; P < 0.15). Total hospital costs (estimated from direct and indirect charges) were significantly increased by a mean of US$33,926 (95% CI = US$22,573–45,280; P < 0.01). Conclusion: Extubation failure in a community hospital is univariately associated with prolonged inpatient care and significantly increased cost. Corroborating data from tertiary care centers, these adverse outcomes highlight the importance of accurate predictors of extubation outcome.

2. Elevated White Cell Count in Acute Coronary Syndromes: Relationship to Variants in Inflammatory and Thrombotic Genes. Byrne CE, Fitzgerald A, Cannon CP, Fitzgerald DJ, Shields DC. BMC Medical Genetics 2004, 5:13 (1 June 2004) Background: Elevated white blood cell counts (WBC) in acute coronary syndromes (ACS) increase the risk of recurrent events, but it is not known if this is exacerbated by pro-inflammatory factors. We sought to identify whether pro-inflammatory genetic variants contributed to alterations in WBC and C-reactive protein (CRP) in an ACS population. Methods: WBC and genotype of interleukin 6 (IL-6 G-174C) and of interleukin-1 receptor antagonist (IL1RN intronic repeat polymorphism) were investigated in 732 Caucasian patients with ACS in the OPUS-TIMI-16 trial. Samples for measurement of WBC and inflammatory factors were taken at baseline, i.e. Within 72 hours of an acute myocardial infarction or an unstable angina event. Results: An increased white blood cell count (WBC) was associated with an increased C-reactive protein (r = 0.23, p < 0.001) and there was also a positive correlation between levels of β-fibrinogen and C-reactive protein (r = 0.42, p < 0.0001). IL1RN and IL6 genotypes had no significant impact upon WBC. The difference in median WBC between the two homozygote IL6 genotypes was 0.21/mm3 (95% CI = -0.41, 0.77), and -0.03/mm3 (95% CI = -0.55, 0.86) for IL1RN. Moreover, the composite endpoint was not significantly affected by an interaction between WBC and the IL1 (p = 0.61) or IL6 (p = 0.48) genotype. Conclusions: Cytokine pro-inflammatory genetic variants do not influence the increased inflammatory profile of ACS patients.

3. Is There a Clinically Significant Gender Bias in Post-Myocardial Infarction Pharmacological Management in the Older (>60) Population of a Primary Care Practice? Di Cecco R, Patel U, Upshur REG. BMC Family Practice 2002, 3:8 (3 May 2002) Background: Differences in the management of coronary artery disease between men and women have been reported in the literature. There are few studies of potential inequalities of treatment that arise from a primary care context. This study investigated the existence of such inequalities in the medical management of post myocardial infarction in older patients. Methods: A comprehensive chart audit was conducted of 142 men and 81 women in an academic primary care practice. Variables were extracted on demographic variables, cardiovascular risk factors, medical and non-medical management of myocardial infarction. Results: Women were older than men. The groups were comparable in terms of cardiac risk factors. A statistically significant difference (14.6%: 95% CI 0.048–28.7 p = 0.047) was found between men and women for the prescription of lipid lowering medications. 25.3% (p = 0.0005, CI 11.45, 39.65) more men than women had undergone angiography, and 14.4 % (p = 0.029, CI 2.2, 26.6) more men than women had undergone coronary artery bypass graft surgery. Conclusion: Women are less likely than men to receive lipid-lowering medication which may indicate less aggressive secondary prevention in the primary care setting.

What is a confidence interval?

Dear Professor Mean:  Can you give me a simple explanation of what a confidence interval is?

We statisticians have a habit of hedging our bets. We always insert qualifiers into our reports, warn about all sorts of assumptions, and never admit to anything more extreme than probable. There's a famous saying: "Statistics means never having to say you're certain."

We qualify our statements, of course, because we are always dealing with imperfect information. In particular, we are often asked to make statements about a population (a large group of subjects) using information from a sample (a small, but carefully selected subset of this population). No matter how carefully this sample is selected to be a fair and unbiased representation of the population, relying on information from a sample will always lead to some level of uncertainty.

Short Explanation

A confidence interval is a range of values that tries to quantify this uncertainty. Consider it as a range of plausible values. A narrow confidence interval implies high precision; we can specify plausible values to within a tiny range. A wide interval implies poor precision; we can only specify plausible values to a broad and uninformative range.

Consider a recent study of homoeopathic treatment of pain and swelling after oral surgery (Lokken 1995). When examining swelling 3 days after the operation, they showed that homoeopathy led to 1 mm less swelling on average. The 95% confidence interval, however, ranged from -5.5 to 7.5 mm. From what little I know about oral surgery, this appears to be a very wide interval. This interval implies that neither a large improvement due to homoeopathy nor a large decrement could be ruled out.

Generally when a confidence interval is very wide like this one, it is an indication of an inadequate sample size, an issue that the authors mention in the discussion section of this paper.

How to Interpret a Confidence Interval

When you see a confidence interval in a published medical report, you should look for two things. First, does the interval contain a value that implies no change or no effect? For example, with a confidence interval for a difference look to see whether that interval includes zero. With a confidence interval for a ratio, look to see whether that interval contains one.

Here's an example of a confidence interval that contains the null value. The interval shown below implies no statistically significant change.

Figure 2.1

Here's an example of a confidence interval that excludes the null value. If we assume that larger implies better, then the interval shown below would imply a statistically significant improvement.

Figure 2.2 (1222 bytes)

Here's a different example of a confidence interval that excludes the null value. The interval shown below implies a statistically significant decline.

Figure 2.3 (1214 bytes)

Practical Significance

You should also see whether the confidence interval lies partly or entirely within a range of clinical indifference. Clinical indifference represents values of such a trivial size that you would not want to change your current practice. For example, you would not recommend a special diet that showed a one year weight loss of only five pounds. You would not order a diagnostic test that had a predictive value of less than 50%.

Clinical indifference is a medical judgment, and not a statistical judgment. It depends on your knowledge of the range of possible treatments, their costs, and their side effects. As statistician, I can only speculate on what a range of clinical indifference is. I do want to emphasize, however, that if a confidence interval is contained entirely within your range of clinical indifference, then you have clear and convincing evidence to keep doing things the same way (see below).

Figure 2.4 (1558 bytes)

One the other hand, if part of the confidence interval lies outside the range of clinical indifference, then you should consider the possibility that the sample size is too small (see below).

Figure 2.5 (1553 bytes)

Some studies have sample sizes that are so large that even trivial differences are declared statistically significant. If your confidence interval excludes the null value but still lies entirely within the range of clinical indifference, then you have a result with statistical significance, but no practical significance (see below).

Figure 2.6 (1548 bytes)

Finally, if your confidence interval excludes the null value and lies outside the range of clinical indifference, then you have both statistical and practical significance (see below).

Figure 2.7 (1550 bytes)

Practice exercises

Read the abstracts presented above. Interpret the confidence intervals presented in these abstracts.

A simple example of Bayesian data analysis.

Bayesian data analysis seems hard, and it is. Even for me, I struggle with understanding Bayesian data analysis. In fairness, I must admit that much of my discomfort is just lack of experience with Bayesian methods. In fact, I've found that in some ways, Bayesian data analysis is simpler than classical data analysis. You, too, can understand Bayesian data analysis, even if you'll never be an expert at it. There's a wonderful example of Bayesian data analysis at work that is simple and fun. It's taken directly from an article by Jim Albert in the Journal of Statistics Education (1995, vol. 3 no. 3) which is available on the web at www.amstat.org/publications/jse/v3n3/albert.html.

I want to use his second example, involving a comparison of ECMO to conventional therapy in the treatment of babies with severe respiratory failure. In this study, 28 of 29 babies assigned to ECMO survived and 6 of 10 babies assigned to conventional therapy survived. Refer to the Albert article for the source of the original data. Before I show how Jim Albert tackled a Bayesian analysis of this data, let me review the general paradigm of Bayesian data analysis.

Wikipedia gives a nice general introduction to the concept of Bayesian data analysis with the following formula:

P (H|E) = P(E|H) P(H) / P(E)

where H represents a particular hypothesis, and E represents evidence (data). P, of course, stands for probability. I don't like to present a lot of formulas in this newsletter, but this one is not too complicated. If you follow this formula carefully, you will see there are four steps in a typical Bayesian analysis.

The first step is to specify P(H), which is called the prior probability. Specifying the prior probability is probably the one aspect of Bayesian data analysis that causes the most controversy. The prior probability represents the degree of belief that you have in a particular hypothesis prior to collection of your data. The prior distribution can incorporate data from previous related studies or it can incorporate subjective impressions of the researcher. What!?! you're saying right now. Aren't statistics supposed to remove the need for subjective opinions? There is a lot that can be written about this, but I would just like to note a few things.

First, it is impossible to totally remove subjective opinion from a data analysis. You can't do research without adopting some informal rules. These rules may be reasonable, they may be supported to some extent by empirical data, but they are still applied in a largely subjective fashion. Here are some of the subjective beliefs that I use in my work:

  1. you should always prefer a simple model to a complex model if both predict the data with the same level of precision.
  2. you should be cautious about any subgroup finding that was not pre-specified in the research protocol.
  3. if you can find a plausible biological mechanism, that adds credibility to your results.

Advocates of Bayesian data analysis will point out that use of prior distributions will force you to explicit some of the subjective opinions that you bring with you to the data analysis.

Second, the use of a range of prior distributions can help resolve controversies involving conflicting beliefs. For example, an important research question is whether a research finding should "close the book" to further research. If data indicates a negative result, and this result is negative even using an optimistic prior probability, then all researchers, even those with the most optimistic hopes for the therapy, should move on. Similarly, if the data indicates a positive result, and this result is positive even using a pessimistic prior probability, then it's time for everyone to adopt the new therapy. Now, you shouldn't let the research agenda be held hostage by extremely optimistic or pessimistic priors, but if any reasonable prior indicates the same final result, then any reasonable person should close the book on this research area.

Third, while Bayesian data analysis allows you to incorporate subjective opinions into your prior probability, it does not require you to incorporate subjectivity. Many Bayesian data analyses use what it called a diffuse or non-informative prior distribution. This is a prior distribution that is neither optimistic nor pessimistic, but spreads the probability more or less evenly across all hypotheses.

Here's a simple example of a diffuse prior that Dr. Albert used for the ECMO versus conventional therapy example. Let's assume that the true survival rate could be either 0, 10%, 20%, ..., 100% in the ECMO group and similarly for the conventional therapy group. This is not an optimal assumption, but it isn't terrible either, and it allows us to see some of the calculations in action. With 11 probabilities for ECMO and 11 probabilities for conventional therapy, we have 121 possible combinations. How should we arrange those probabilities? One possibility is to assign half of the total probability to combinations where the probabilities are the same for ECMO and conventional therapy and the remaining half to combinations where the probabilities are different. Split each of these probabilities evenly over all the combinations.

If you split 0.50 among the eleven combinations where the two survival rates are equal, you get 0.04545. Splitting 0.50 among the 110 combinations where the two survival rates are unequal, you get 0.004545.

You can arrange these prior probabilities into a rectangular grid where the columns represent a specific survival rate with ECMO and the rows represent a specific survival rate with conventional therapy. To simplify the display, we multiplied each probability by 1000 and rounded the result. So we have 121 hypotheses, ranging from ECMO and conventional therapy both having 0% survival rates to ECMO having 100% survival and conventional therapy having 0% survival rates to ECMO having 0% and conventional therapy having 100% survival rates to both therapies having 100% surivival rates. Each hypothesis has a probability assigned to it. The probability for ECMO 90% and conventional therapy 60% has a probability of roughly 5 in a thousand and the probability for ECMO 80% and conventional therapy 80% has a probability of roughly 45 in a thousand.

The second step in a Bayesian data analysis is to calculate P(E | H), the probability of the observed data under each hypothesis. If the ECMO survival rate is 90% and the conventional therapy survival rate is 60%, then the probability of observed 28 out of 29 survivors in the ECMO group is 152 out of one thousand, the probability of observing 6 out of 10 survivors in the conventional therapy group is 251 out of one thousand. The product of those two probabilities is 38,152 out of one million which we can round to 38 out of one thousand. If you've forgotten how to calculate probabilities like this, that's okay. It involves the binomial distribution, and there are functions in many programs that will produce this calculation for you. In Microsoft Excel, for example, you can use the following formula.

The calculation under different hypotheses will lead to different probabilities. If both ECMO and conventional therapy have a survival probability of 0.8, Then the probability of 28 out of 29 for ECMO is 11 out of one thousand, the probability of 6 out of 10 for conventional therapy is 88 out of one thousand. The product of these two probabilities is 968 out of one million, which we round to 1 out of one thousand.

The table above shows the binomial probabilities under each of the 121 different hypotheses.  Many of the probabilities are much smaller than one out of one thousand. The likelihood of seeing 28 survivals out of 29 babies in the ECMO survivals is very small when the hypothesized survival rate is 10%, 30%, or even 50%. Very small probabilities are represented by zeros.

Now multiply the prior probability of each hypothesis by the likelihood of the data under each hypothesis. For ECMO=0.9, conventional therapy=0.6, this product is 5 out of a thousand times 38 out of a thousand, which equals 190 out of a million (actually it is 173 out of a million when you don't round the data so much). For ECMO=conventional=0.8, the product is 45 out of a thousand times 1 out of a thousand, or 45 out of a million.

This table shows the product of the prior probabilities and the likelihoods. We're almost done, but there is one catch. These numbers do not add up to 1 (they add up to 794 out of a million), so we need to rescale them. We divide by P(E) which is defined in the wikipedia article as

P(E) = P(E|H1) P(H1) + P(E|H2) P(H2) + ...

In the example shown here, this calculation is pretty easy: add up the 121 cells to get 794 out of a million and then divide each cell by that sum. For more complex setting, this calculation requires some calculus, which should put some fear and dread into most of you. It turns out that even experts in Calculus will find it impossible to get an answer for some data analysis settings, so often Bayesian data analysis requires computer simulations at this point.

Here's the table after standardizing all the terms so they add up to 1.

This table is the posterior probabilities, P(H | E). You can combine and manipulate these posterior probabilities far more easily than classical statistics would allow. For example, how likely are we to believe the hypothesis that ECMO and conventional therapy have the same survival rates? Just add the cells along the diagonal (0+0+...+5+57+97+0) to get 159 out of a thousand. Prior to collecting the data, we placed the probability that the two rates were equal at 500 out of a thousand, so the data has greatly (but not completely) dissuaded us from this belief. You can calculate the probability that ECMO is exactly 10% better than conventional therapy (0+0+...+1+13+84+0 = 98 out of a thousand), that ECMO is exactly 20% better (0+0+...+13+218+0 = 231 out of a thousand), exactly 30% better (0+0+...+7+178+0 = 185 out of a thousand), and so forth.

Here's something fun that Dr. Albert didn't show. You could take each of the cells in the table, compute a ratio of survival rates and then calculate the median of these ratios as 1.5 (see above for details). You might argue that 1.33 is a "better" median because 448 is closer to 500 than 666, and I wouldn't argue too much with you about that choice.

Dr. Albert goes on to show an informative prior distribution. There is a fair amount of data to indicate that the survival rate for the conventional therapy is somewhere between 10% and 30%, but little or no data about the survival rates under ECMO.

The table above shows this informative prior distribution. Recall that the rows represent survival rates under conventional therapy. This prior distribution restricts the probabilities for survival rates in the conventional therapy to less than 70%. There is no such absolute restriction for ECMO, though the probabilities for survival rates of 70% and higher are fairly small.

Repeat of pop quiz

Review the pop quiz presented earler. Do you feel more confident in your answers?

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