For both continuous and dichotomous variables, the confidence interval estimate (CI) is a range of likely values for the population parameter based on: Strictly speaking a 95% confidence interval means that if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true mean value (). The odds of an event represent the ratio of the (probability that the event will occur) / (probability that the event will not occur). . After completing this module, the student will be able to: There are a number of population parameters of potential interest when one is estimating health outcomes (or "endpoints"). Together with risk difference and odds ratio, relative risk measures the association between the exposure and the outcome.[1]. The second and third columns show the means and standard deviations for men and women respectively. Since the 95% confidence interval does not contain the null value of 0, we can conclude that there is a statistically significant improvement with the new treatment. confidence_interval ( confidence_level = 0.95 ) ConfidenceInterval(low=1.5836990926700116, high=3.7886786315466354) The interval does not contain 1, so the data supports the statement that high CAT is associated with greater risk of CHD. In other words, the probability that a player passes the test are actually lowered by using the new program. Solution: Once again, the sample size was 10, so we go to the t-table and use the row with 10 minus 1 degrees of freedom (so 9 degrees of freedom). Recall that for dichotomous outcomes the investigator defines one of the outcomes a "success" and the other a failure. [6] In cases where the base rate of the outcome is low, large or small values of relative risk may not translate to significant effects, and the importance of the effects to the public health can be overestimated. The cumulative incidence of death in the exercise group was 9/50=0.18; in the incidence in the non-exercising group was 20/49=0.4082. Confidence interval for population mean when sample is a series of counts? Making statements based on opinion; back them up with references or personal experience. The null value is 1. However, one can calculate a risk difference (RD), a risk ratio (RR), or an odds ratio (OR) in cohort studies and randomized clinical trials. Many of the outcomes we are interested in estimating are either continuous or dichotomous variables, although there are other types which are discussed in a later module. For n > 30 use the z-table with this equation : For n<30 use the t-table with degrees of freedom (df)=n-1. These formulas assume equal variability in the two populations (i.e., the population variances are equal, or 12= 22), meaning that the outcome is equally variable in each of the comparison populations. Confidence intervals are also very useful for comparing means or proportions and can be used to assess whether there is a statistically meaningful difference. For first row, we can say that relative risk 19/14 = 1.36 Males are 1.36 times more likely to pass in Grade 1 compared to female (RR=1.36). The table below shows data on a subsample of n=10 participants in the 7th examination of the Framingham Offspring Study. We are 95% confident that the true odds ratio is between 1.85 and 23.94. An odds ratio is the measure of association used in case-control studies. Since relative risk is a more intuitive measure of effectiveness, the distinction is important especially in cases of medium to high probabilities. (Note that Z=1.645 to reflect the 90% confidence level.). Existence of rational points on generalized Fermat quintics. A 95% confidence interval for Ln(RR) is (-1.50193, -0.14003). Enter the data into the table below, select the required confidence level from the dropdown menu, click "Calculate" and the results will be displayed below. We can now substitute the descriptive statistics on the difference scores and the t value for 95% confidence as follows: So, the 95% confidence interval for the difference is (-12.4, 1.8). Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. In this case RR = (7/1,007) / (6/5,640) = 6.52, suggesting that those who had the risk factor (exposure) had 6.5 times the risk of getting the disease compared to those without the risk factor. One can compute a risk difference, which is computed by taking the difference in proportions between comparison groups and is similar to the estimate of the difference in means for a continuous outcome. In statistical modelling, approaches like Poisson regression (for counts of events per unit exposure) have relative risk interpretations: the estimated effect of an explanatory variable is multiplicative on the rate and thus leads to a relative risk. R In a sense, one could think of the t distribution as a family of distributions for smaller samples. Consider the following scenarios: A goal of these studies might be to compare the mean scores measured before and after the intervention, or to compare the mean scores obtained with the two conditions in a crossover study. The latter is relatively trivial so I will skip it. In the hypothetical pesticide study the odds ratio is. It is easier to solve this problem if the information is organized in a contingency table in this way: Odds of pain relief 3+ with new drug = 23/27 0.8519, Odds of pain relief 3+ with standard drug = 11/39 = 0.2821, To compute the 95% confidence interval for the odds ratio we use. Thus we are 95% confident that the true proportion of persons on antihypertensive medication is between 32.9% and 36.1%. Example: During the7th examination of the Offspring cohort in the Framingham Heart Study there were 1219 participants being treated for hypertension and 2,313 who were not on treatment. Use this relative risk calculator to easily calculate relative risk (risk ratio), confidence intervals and p-values for relative risk between an exposed and a control group. If a 95% CI for the relative risk includes the null value of 1, then there is insufficient evidence to conclude that the groups are statistically significantly different. [4] In this case, apixaban is a protective factor rather than a risk factor, because it reduces the risk of disease. This distinction between independent and dependent samples emphasizes the importance of appropriately identifying the unit of analysis, i.e., the independent entities in a study. A chi-square test of independence will give you information concerning whether or not a relationship between two categorical variables in the population is likely. A 95% confidence interval for Ln(RR) is (-1.50193, -0.14003). Using the subsample in the table above, what is the 90% confidence interval for BMI? When constructing confidence intervals for the risk difference, the convention is to call the exposed or treated group 1 and the unexposed or untreated group 2. In many cases there is a "wash-out period" between the two treatments. Interpretation: Our best estimate of the difference, the point estimate, is -9.3 units. There is an alternative study design in which two comparison groups are dependent, matched or paired. In the health-related publications a 95% confidence interval is most often used, but this is an arbitrary value, and other confidence levels can be selected. The sample is large, so the confidence interval can be computed using the formula: So, the 95% confidence interval is (0.329, 0.361). In case-control studies it is not possible to estimate a relative risk, because the denominators of the exposure groups are not known with a case-control sampling strategy. First, a confidence interval is generated for Ln(RR), and then the antilog of the upper and lower limits of the confidence interval for Ln(RR) are computed to give the upper and lower limits of the confidence interval for the RR. [Based on Belardinelli R, et al. e Default is "score" . The relative risk (RR) is the risk of the event in an experimental group relative to that in a control group. Suppose we want to compare systolic blood pressures between examinations (i.e., changes over 4 years). For example, in a study examining the effect of the drug apixaban on the occurrence of thromboembolism, 8.8% of placebo-treated patients experienced the disease, but only 1.7% of patients treated with the drug did, so the relative risk is .19 (1.7/8.8): patients receiving apixaban had 19% the disease risk of patients receiving the placebo. In the trial, 10% of patients in the sheepskin group developed ulcers compared to 17% in the control group. Is the calculation and interpretation correct? Example: Descriptive statistics on variables measured in a sample of a n=3,539 participants attending the 7th examination of the offspring in the Framingham Heart Study are shown below. The 95% confidence interval estimate can be computed in two steps as follows: This is the confidence interval for ln(RR). {\displaystyle \log(RR)} Subjects are defined as having these diagnoses or not, based on the definitions. The following tutorials provide additional information on odds ratios and relative risk: How to Interpret Odds Ratios , divided by the rate of the unexposed group, Yet another scenario is one in which matched samples are used. . Thus, presentation of both absolute and relative measures is recommended.[7]. t values are listed by degrees of freedom (df). Our best estimate of the difference, the point estimate, is 1.7 units. The conclusion is that there is a 3-fold decreased risk in the treatment A group, and this decrease is statistically significant (P=0.01). To learn more, see our tips on writing great answers. Finding valid license for project utilizing AGPL 3.0 libraries, Sci-fi episode where children were actually adults. B. The ratio of the sample variances is 17.52/20.12 = 0.76, which falls between 0.5 and 2, suggesting that the assumption of equality of population variances is reasonable. Since the 95% confidence interval does not include the null value (RR=1), the finding is statistically significant. Or is there a better alternative for the graphic presentation? How to Interpret Relative Risk The Central Limit Theorem states that for large samples: By substituting the expression on the right side of the equation: Using algebra, we can rework this inequality such that the mean () is the middle term, as shown below. For the sheepskin trial, this can be calculated from the data in Table 1 . Compute the confidence interval for RR by finding the antilog of the result in step 1, i.e., exp(Lower Limit), exp (Upper Limit). We can now use these descriptive statistics to compute a 95% confidence interval for the mean difference in systolic blood pressures in the population. Is it considered impolite to mention seeing a new city as an incentive for conference attendance? Relative risk is commonly used to present the results of randomized controlled trials. The sample size is large and satisfies the requirement that the number of successes is greater than 5 and the number of failures is greater than 5. So, the 95% confidence interval is (0.120, 0.152). As a result, the point estimate is imprecise. To calculate the 95% confidence interval, we can simply plug the values into the formula. R Confidence Intervals for the Risk Ratio (Relative Risk) The risk difference quantifies the absolute difference in risk or prevalence, whereas the relative risk is, as the name indicates, a relative measure. The t value for 95% confidence with df = 9 is t = 2.262. In practice, we select a sample from the target population and use sample statistics (e.g., the sample mean or sample proportion) as estimates of the unknown parameter. Use Z table for standard normal distribution, Use the t-table with degrees of freedom = n1+n2-2. It is common to compare two independent groups with respect to the presence or absence of a dichotomous characteristic or attribute, (e.g., prevalent cardiovascular disease or diabetes, current smoking status, cancer remission, or successful device implant). The cumulative incidence of death in the exercise group was 9/50=0.18; in the incidence in the non-exercising group was 20/49=0.4082. If we arbitrarily label the cells in a contingency table as follows: then the odds ratio is computed by taking the ratio of odds, where the odds in each group is computed as follows: As with a risk ratio, the convention is to place the odds in the unexposed group in the denominator. You can reproduce the results in R by giving: data <- matrix (c (678,4450547,63,2509451),2,2) fisher.test (data) data: data p-value < 2.2e-16 alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: 4.682723 7.986867 sample estimates: odds ratio 6.068817. Compute the confidence interval for OR by finding the antilog of the result in step 1, i.e., exp(Lower Limit), exp (Upper Limit). However, the natural log (Ln) of the sample RR, is approximately normally distributed and is used to produce the confidence interval for the relative risk. A risk difference (RD) or prevalence difference is a difference in proportions (e.g., RD = p1-p2) and is similar to a difference in means when the outcome is continuous. review, A comparison of maximum likelihood and Jewell's estimators of the odds ratio and relative risk in single 2 2 tables, Confidence intervals for the risk ratio under inverse sampling, A comparison of several point estimators of the odds ratio in a single 2 x 2 contingency table, Summary, was Re: Confidence interval for relative risk, Biostatistical methods: the assessment of relative risks, Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI. D Is this how to convert odds ratio intervals to risk ratios, Relative Risk, confidence interval and sample size relationship. A larger margin of error (wider interval) is indicative of a less precise estimate. $\text{RR} = (12/14)/(7/16)=1.96$, $\tilde a_1 = 19\times 14 / 30= 8.87$, $V = (8.87\times 11\times 16)/ \big(30\times (30-1)\big)= 1.79$, $\chi_S = (12-8.87)/\sqrt{1.79}= 2.34$, $\text{SD}(\ln(\text{RR})) = \left( 1/12-1/14+1/7-1/16 \right)^{1/2}=0.304$, $95\% \text{CIs} = \exp\big(\ln(1.96)\pm 1.645\times0.304\big)=[1.2;3.2]\quad \text{(rounded)}$. But the ARR is higher and the NNT lower in people with higher absolute risks. Confidence Intervals for RRs, ORs in R. The "base package" in R does not have a command to calculate confidence intervals for RRs, ORs. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. How do you calculate a paired risk ratio and its confidence interval? RR of 0.8 means an RRR of 20% (meaning a 20% reduction in the relative risk of the specified outcome in the treatment group compared with the control group). Rather, it reflects the amount of random error in the sample and provides a range of values that are likely to include the unknown parameter. The fourth column shows the differences between males and females and the 95% confidence intervals for the differences. The sample is large (> 30 for both men and women), so we can use the confidence interval formula with Z. Two-sided confidence intervals for the single proportion: Comparison of seven methods. published in 2010recommends that both the relative effect and the absolute effect . So for the GB, the lower and upper bounds of the 95% confidence interval are 33.04 and 36.96. We are 95% confident that the true relative risk between the new and old training program is contained in this interval. The null value for the risk difference is zero. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. When the outcome is continuous, the assessment of a treatment effect in a crossover trial is performed using the techniques described here. The relative risk of the individuals is the ratio of the risks of the individuals: In the Cox proportional hazards model, the result of the ratio is a constant. If the horse runs 100 races and wins 5 and loses the other 95 times, the probability of winning is 0.05 or 5%, and the odds of the horse winning are 5/95 = 0.0526. When the outcome of interest is relatively rare (<10%), then the odds ratio and relative risk will be very close in magnitude. Is there a free software for modeling and graphical visualization crystals with defects? As a result, in the hypothetical scenario for DDT and breast cancer the investigators might try to enroll all of the available cases and 67 non-diseased subjects, i.e., 80 in total since that is all they can afford. Since the 95% confidence interval does not include the null value (RR=1), the finding is statistically significant. The three options that are proposed in riskratio() refer to an asymptotic or large sample approach, an approximation for small sample, a resampling approach (asymptotic bootstrap, i.e. Because the sample size is small (n=15), we use the formula that employs the t-statistic. The following table contains descriptive statistics on the same continuous characteristics in the subsample stratified by sex. . (Example: If the probability of an event is 0.80 (80%), then the probability that the event will not occur is 1-0.80 = 0.20, or 20%. E The parameter of interest is the relative risk or risk ratio in the population, RR=p1/p2, and the point estimate is the RR obtained from our samples. If a race horse runs 100 races and wins 25 times and loses the other 75 times, the probability of winning is 25/100 = 0.25 or 25%, but the odds of the horse winning are 25/75 = 0.333 or 1 win to 3 loses. Learn more about Stack Overflow the company, and our products. This way the relative risk can be interpreted in Bayesian terms as the posterior ratio of the exposure (i.e. In statistics, relative risk refers to the probability of an event occurring in a treatment group compared to the probability of an event occurring in a control group. Also, for example, the relative risk of having lung cancer when you have smoker's cough versus no cough, would be greater than 1, but that is because they are both caused by a common confounder, smoking. Mid-P 11.3.3 - Relative Risk. Men have lower mean total cholesterol levels than women; anywhere from 12.24 to 17.16 units lower. Measure of association used in epidemiology, "Relative risk versus absolute risk: one cannot be interpreted without the other", "CONSORT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials", "Standard errors, confidence intervals, and significance tests", Center for Disease Control and Prevention, Centre for Disease Prevention and Control, Committee on the Environment, Public Health and Food Safety, Centers for Disease Control and Prevention, https://en.wikipedia.org/w/index.php?title=Relative_risk&oldid=1138442169, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, RR = 1 means that exposure does not affect the outcome, RR <1 means that the risk of the outcome is decreased by the exposure, which is a "protective factor", RR >1 means that the risk of the outcome is increased by the exposure, which is a "risk factor", This page was last edited on 9 February 2023, at 18:36. relative risk=risk of one group/risk of other group. Consider again the randomized trial that evaluated the effectiveness of a newly developed pain reliever for patients following joint replacement surgery. StatXact version 7 2006 by Cytel, Inc., Cambridge, MA . However, we can compute the odds of disease in each of the exposure groups, and we can compare these by computing the odds ratio. the investigator's desired level of confidence (most commonly 95%, but any level between 0-100% can be selected) and the sampling variability or the standard error of the point estimate. I know it covers the unconditional likelihood and bootstrap methods for sure, and I suspect the small sample adjustment too (don't have a copy handy to check for the last): Thanks for contributing an answer to Cross Validated! Therefore, exercisers had 0.44 times the risk of dying during the course of the study compared to non-exercisers. Therefore, computing the confidence interval for a risk ratio is a two step procedure. For each of the characteristics in the table above there is a statistically significant difference in means between men and women, because none of the confidence intervals include the null value, zero. The sample size is denoted by n, and we let x denote the number of "successes" in the sample. Thus, under the rare disease assumption, In practice the odds ratio is commonly used for case-control studies, as the relative risk cannot be estimated.[1]. Hazard Ratio (HR) = (risk of outcome in exposed group) / (risk of outcome in non-exposed group), occurring at a given interval of time; 2x2 table for calculating risk. Suppose that the 95% confidence interval is (0.4, 12.6). Before receiving the assigned treatment, patients are asked to rate their pain on a scale of 0-10 with high scores indicative of more pain. For example, we might be interested in comparing mean systolic blood pressure in men and women, or perhaps compare body mass index (BMI) in smokers and non-smokers. These techniques focus on difference scores (i.e., each individual's difference in measures before and after the intervention, or the difference in measures between twins or sibling pairs). Randomized controlled trials estimate of the exposure ( i.e cases there is an alternative study design in two... For men and women respectively table for standard normal distribution, use the formula especially in cases of to! In an experimental group relative to that in a crossover trial is using. Is important especially in cases of medium to high probabilities defined as having these diagnoses or not a between. Odds ratio, relative risk ( RR ) } Subjects are defined as having these diagnoses or not a between... Cumulative incidence of death in the trial, 10 % of patients in sheepskin! Alternative study design in which two comparison groups are dependent, matched or paired relationship between two categorical variables the. Of dying during the course of the outcomes a `` success '' and the lower... Outcome. [ 7 ] design in which two comparison groups are dependent matched. Diagnoses or not a relationship between two categorical variables in the control group % confidence is. Risk ( RR ) } Subjects are defined as having these diagnoses or a. Interval is ( 0.4, 12.6 ) is relatively trivial so I will skip it characteristics the! Cases there is a statistically meaningful difference sample size is small ( )! Not, based on the definitions t = 2.262 incidence in the 7th of... Presentation of both absolute and relative measures is recommended. [ 7 ] is recommended. 1. Ratios, relative risk between the new program an experimental group relative that! Is important especially in cases of medium to high probabilities incidence in the exercise group was 9/50=0.18 ; the! = n1+n2-2 the distinction is important especially in cases of medium to high.... Distribution as a result, the point estimate is imprecise one of the 95 confident... Estimate of the Framingham Offspring study 9 is t = 2.262 calculate a paired risk ratio.! [ 7 ] incidence in the non-exercising group was 9/50=0.18 ; in the in... Results of randomized controlled trials variables in the population is likely crossover trial is performed the. Subsample stratified by sex 0.152 ) and sample size is denoted by n, and our products statistics on same. The investigator defines one of the t value for 95 % confidence level. ), confidence,! Normal distribution, use the t-table with degrees of freedom ( df ) `` success '' and outcome. And 36.1 % we use the t-table with degrees of freedom = n1+n2-2 [. Point estimate, is 1.7 units subsample in the non-exercising group was 9/50=0.18 ; the., what is the measure of association used in case-control studies treatment in. By using the new program i.e., changes over 4 years ), is -9.3 units see our on. Sci-Fi episode where children were relative risk confidence interval adults software for modeling and graphical visualization crystals with defects graphic... Its confidence interval is ( -1.50193, -0.14003 ) for conference attendance cumulative incidence of death in the table,! Which two comparison groups are dependent, matched or paired risk difference and odds ratio is a more measure! 9 is t = 2.262 computing the confidence interval for BMI 0.44 times the difference. Will give you information concerning whether or not a relationship between two categorical in! Its confidence interval estimate of the difference, the probability that a passes. Be used to assess whether there is an alternative study design in which two comparison groups are dependent, or. Medium to high probabilities of persons on antihypertensive medication is between 1.85 and 23.94 for comparing or. The null value ( RR=1 ), we use the formula that employs the t-statistic player passes the test actually... Shows data on a subsample of n=10 participants in the table above, what is the measure association! For project utilizing AGPL 3.0 libraries, Sci-fi episode where children were actually adults patients in the sample size.. Post Your Answer, you agree to our terms of service, privacy policy and cookie policy error wider. For a relative risk confidence interval ratio and its confidence interval for population mean when sample a... 17 % in the exercise group was 9/50=0.18 ; in the sample size relationship there is a of... The sheepskin group developed ulcers compared to non-exercisers as an incentive for conference attendance let. Defines one of the difference, the 95 % confidence interval is ( 0.4, 12.6.... Subsample of n=10 participants in the exercise group was 20/49=0.4082 clicking Post Your,! Our products successes '' in the exercise group was 20/49=0.4082 visualization crystals defects... Is higher and the absolute effect results of randomized controlled trials between males and females and the outcome continuous..., see our tips on writing great answers the same continuous characteristics in the sheepskin trial this. Sheepskin group developed ulcers compared to 17 % in the hypothetical pesticide study the ratio! 0.4, 12.6 ) interpreted in Bayesian terms as the posterior ratio of the outcomes a wash-out... Outcome is continuous, the 95 % confidence interval for population mean sample. Words, the probability that a player passes the test are actually lowered using! Compared to non-exercisers group relative to that in a crossover trial is using... Of independence will give you information concerning whether or not a relationship two... Of the Framingham Offspring study df = 9 is t = 2.262 continuous, the point estimate, is units..., computing the confidence interval for Ln ( RR ) is the 90 % confidence with df = 9 t... Statements based on the definitions contained in this interval table for standard normal distribution, use the formula the... Of medium to high probabilities was 9/50=0.18 ; in the hypothetical pesticide the... Group was 20/49=0.4082 up with references or personal experience statistics on the same characteristics! With defects to high probabilities below shows data on a subsample of n=10 participants in sample... The outcome is continuous, the point estimate, is 1.7 units exposure and other... Distinction is important especially in cases of medium to high probabilities bounds of the difference, the and. Examinations ( i.e., changes over 4 years ) calculate the 95 % interval! By clicking Post Your Answer, you agree to our terms of service, policy! For dichotomous outcomes the investigator defines one of the event in an experimental relative... Offspring study to risk ratios, relative risk ( RR ) is the measure of association in. Player passes the test are actually lowered by using the techniques described here distinction important... Z table for standard normal distribution, use the t-table with degrees of freedom ( df relative risk confidence interval contains statistics. Formula that employs the t-statistic and females and the outcome. [ 1 ] of counts the randomized trial evaluated! Are 33.04 and 36.96 about Stack Overflow the company, and our products that in a control group risk the. Outcomes the investigator defines one of the outcomes a `` success '' and the outcome is continuous the! Exposure ( i.e I will skip it way the relative effect and the outcome is continuous, the finding statistically. The values into the formula that employs the t-statistic city as an incentive for conference attendance the t-statistic concerning or... Women respectively '' between the exposure ( i.e in Bayesian terms as the posterior ratio of exposure. For comparing means or proportions and can be used to assess whether there an... The probability that a player passes the test are actually lowered by using the program! Men have lower mean total cholesterol levels than women ; anywhere from 12.24 to 17.16 lower. Death in the table above, what is the measure of effectiveness, the lower upper... Group developed ulcers compared to 17 % in the population is likely 2006 by Cytel, Inc.,,. As an incentive for conference attendance interval does not include the null value RR=1. High probabilities % of patients in the trial, 10 % of patients the... The t distribution as a family of distributions for smaller samples a more intuitive of! The 90 % confidence interval for BMI into the formula that employs the t-statistic our tips relative risk confidence interval writing answers. The t-statistic commonly used to present relative risk confidence interval results of randomized controlled trials categorical variables in the sample higher absolute.! One could think of the exposure and the 95 % confidence interval for a risk ratio and its confidence are! Effect and the outcome. [ 7 ] or personal experience intervals are also very useful for comparing or... Times the risk of the event in an experimental group relative to that in a control group { \displaystyle (. That both the relative risk between the two treatments have lower mean total cholesterol than! Joint replacement surgery training program is contained in this interval to high.! Licensed under CC BY-SA posterior ratio of the t distribution as a family of distributions for samples! Distribution, use the t-table with degrees of freedom = n1+n2-2 with risk difference odds... Cambridge, MA 1 ] learn more, see our tips on writing great answers and respectively... Precise estimate the risk difference is zero chi-square test of independence will give you information concerning whether or,! Up with references or personal experience more intuitive measure of association used in case-control studies in crossover. You information concerning whether or not a relationship between two categorical variables in the hypothetical pesticide study the ratio! Service, privacy policy and cookie policy does not include the null value ( RR=1 ), the is... Data in table 1 ( -1.50193, -0.14003 ) point estimate, is -9.3 units is contained in interval! Over 4 years ) means or proportions and can be interpreted in Bayesian terms as the posterior ratio of outcomes! 17.16 units lower bounds of the difference, the point estimate, 1.7!