In this short section, we'll look at two more examples to look at the use of confidence intervals in the literature and their computation as well. So, the first thing I want to look at it as an article that was published in 2008 and it's a really neat example of behavioral medicine intervention that was very effective, so we'll look at that. So, the title of the article was titled 'Effects of Lower Targets for Blood Pressure and LDL cholesterol level on atherosclerosis in diabetes', the sands Randomized Trial. This was published in what is formerly known as the Journal of the American Medical Association, now known just as jama which was published in 2008. So, I'm pulling the text from the abstract of the article because it does a nice job of conveying the results. So let's first talk about what they wanted to do in this study. Their objective was to compare the progression of subclinical atherosclerosis in adults with type two diabetes treated to reach aggressive targets. A low-density lipoprotein cholesterol level of 70 milligrams per deciliter or lower systolic blood pressure of 115 millimeters of mercury or lower versus the standard targets used in type two diabetics of LDL cholesterol level of 100 milligrams per deciliter or lower and systolic blood pressure of 130 millimeters of mercury or lower. This was a randomized, open labeled, blinded to endpoint, three year trial from April 2003 to July 2007, in four clinical centers in Oklahoma, Arizona, and South Dakota, three US states and there were 499 American Indian men and women. So, it was looking at the population of American Indian men and women age 40 years or older with type two diabetes who had no prior cardiovascular disease events. So patients were randomized to either the aggressive or standard targets 252 were randomized to the aggressive, 247 to the standard treatment groups with step treatment algorithms defined for both. In other words, definitions and how these were implemented. So, here are the results and here's where we see some confidence intervals that are really illuminating. So the mean target LDL cholesterol level and systolic blood pressure levels for both groups were reached and maintained. The mean and 95% confidence interval levels for the LDL cholesterol level in the last 12 months were 72 with a confidence interval 69 to 75 milligrams per deciliter. And 104 with the confidence interval of 101 to 106 milligrams per deciliter and SBP levels were 117 with the confidence interval 115-118 millimeters of mercury. And 129 with a confidence interval of 128 to 130 millimeters of mercury in the aggressive versus standard groups respectively. So, the phrasing on that's a little tricky so let's let's just write a summary of what they did. So, this is the results for the aggressive and the standard and we'll do the ending systolic blood pressure and confidence interval and the ending LDL means and confidence interval. So in the aggressive group, the ending systolic blood pressure we're 117 with the confidence interval of 115 to 118. The LDL was 72 with a confidence interval had everyone been given the aggressive treatment, a range of possible values for the ending mean LDL 69 to 75 milligrams per deciliter. Compare this within the standard group. An ending mean blood pressure of 129, confidence interval 120 to 130 millimeters of mercury, and in LDL cholesterol level of 104 milligrams per deciliter with a confidence interval of 101 to 106. So we can see very clearly from these data and these results including, the confidence intervals that account for the uncertainty in your estimates that the results were strikingly different in the aggressive group versus the standard group even after accounting for the uncertainty. Certainly, they're finishing average systolic blood pressure was 12 millimeters of mercury lower and when you look at these confidence intervals, they're relatively tight around their estimates and they do not overlap. So soon, we'll get to, in the next section, we'll show how we can do a head to head comparison of these two mean values and actually create a confidence interval for the true difference between the two populations. Similarly, in the LDL cholesterol results the scores were notably lower, 27 points lower, 27 milligrams per deciliter or lower in the aggressive group than the standard group with tight confidence intervals that also do not overlap. So, this is pretty strong results showing that the aggressive targeting was effective at really reducing those endpoints on average. Just FYI, this articles rife with a bunch of other comparisons between the results in the after three years in the study that also quantifies the difference and put confidence limits on that. We'll see how to do that soon. This table is hard to read especially when it's been condensed to fit this slide but I just want to impress on you that confidence intervals are used very often in the scientific literature and there's a large number of confidence intervals appearing in this table. If you look and work through it you can actually find the confidence intervals we just quoted in the previous slide for their main findings. Let's look again at computing some confidence intervals for incidence rates, let's go back to our Pennsylvania lung cancer data and look at the separation we'd done before comparing the rates of lung cancer. We have done the estimated rates, now let's throw confidence intervals on them between men and women in the state of Pennsylvania in 2002. Here were the respective observed for sample incidence rates in that year. So, for females there were 4,587 diagnoses in the year and 6.35 million women living in the state for that year or estimated to live in the state for that year. The incidence rate among females was 0.00072 cases per female person year, compared with males. There were fewer males living in Pennsylvania but there were more lung cancer diagnoses and the resulting incidence rate among males in the same state in the same year was 0.00096 cases per person a year. So notably, higher this estimate based on a large number of people. But if we think of this one year is a sample from a process unfolding over time, in the distribution of cases to man and woman. We want to account for the uncertainty in these, so we can estimate the standard errors of these incidence rate ratio simply by taking the square root of the number of cases in each sex group and again dividing by the person year. So, if you do the math on this, your estimated standard error for the incidence rate in females is on the order of 0.000011 case per person years. It's slightly higher among males and that's partially because the number of cases is larger and also because the number of men in state is lesser than females, is 0.00013 cases per person year, that's the standard error for males. So, if we create the confidence intervals, if when we take that incidence rate estimate for each sex and add and subtract two standard errors, we get the following confidence intervals for females. The estimate was 0.00072 cases per person years and that can range from 0.0007 to 0.00774. After counting for sampling variability in males, the estimate was notably higher as were the endpoints of the confidence interval. If we prorate these cases so that they have an integer form, a whole number here, this means there were 72 cases per 100,000 persons years in females with the confidence interval of 70 to 74 cases. And 96 cases per 100,000 person years among the males with the confidence interval of 93 to 99 cases. So not only is the estimated rate for males much larger in terms of cases per year or cases per 100,000 person years, but the confidence interval values are shifted up. The lowest possibility for males is much larger than the largest possibility for females, after we've accounted for the sampling variability in both of these estimates.