So in this section, we'll extend the idea of looking at confidence intervals for single quantities from single populations to look at confidence intervals for population comparison measures. So, to start, I'll give you an overview of the process we'll use throughout these lectures status and then we'll drill down on some of the details for the different measures. So, upon completion of this lecture section, you will be able to extend the concept of sampling distributions to conclude measures of association that compare two populations. Extend the principles of confidence interval estimation from a single population quantity to measures of association comparing two populations. Appreciate the confidence intervals for ratios need to be done on the natural log scale. So, we're bringing back this thing about ratios and logs again. I had promised you this would come up throughout the course, and once we've done their computations on the log scale to get the confidence interval for the log ratio, then the results can be then transferred back to the original ratio of scale. You'll be able to explain the concept of a null value, a value meaning "no association" for such measures of association for these population comparison measures, and what its absence or presence in a confidence interval signifies. So again, frequently in public health medicine science et cetera, researchers, practitioners are interested in comparing two or more outcomes between populations using data collected on samples from these populations. They were well aware of this, I'm just reviewing some ideas we passed on before. We've talked about how to estimate these comparisons. Now, we're going to finish the story by putting the uncertainty and of course such comparisons that we've use such comparisons these numerical comparisons to investigate questions like, how do salaries differ between males and females? How do cholesterol levels differ across weight groups? How does AZT impact the transmission of HIV from mother to child? How is a drug associated with survival among patients with a specific disease? It is not only important to estimate the magnitude of the difference in the outcome between the two groups being compared, but also to recognize the uncertainty in the estimate. These summary measures that we've developed thus far all sample-based and hence sample statistics. These are subject to sampling error just like single sample summary statistics were. So, one approach to quantifying the uncertainty in these estimates is to again create confidence intervals. So, we're going to consider two types of group comparisons at least when it comes to continuous outcomes. We're going to consider something called a paired design. But for now, we'll only consider this when we're doing continuous outcomes. Sometimes you'll see this in the literature, but we'll also talk about the issues with regards to substantive interpretation of the results from such a design there's some potential issues. So, it's not used all that often but it's an excellent launching point for the ideas. The paired design comes up when we actually have two groups, two samples from two populations, we want to compare but whether there's an inherent link between each member of one population and hence one sample and the other. So, example of such a study would be if I was doing a study on persons, some characteristic of persons before, and after they received an intervention. Then, within each sample persons from the population who are to receive the intervention, we have their sample before they've gotten the intervention and a sample of the same people after they've gotten the intervention. Each person in these samples are inherently linked to themselves in the two samples. So, there's these two samples are not completely independent, everybody who appears in the first sample appears in the second. We said that the observations are paired. Another example might be a study where we randomize persons to receive say a treatment or being in a control group but we randomized groups of siblings. So, for each sibling who gets the treatment, there's a corresponding sibling who's in the control. So, for each person who gets the treatment, we also have their sibling getting the control. Maybe they randomize the siblings. So, left the chance which sibling gets the treatment, which gets to control but our two samples are inherently linked by the familiar relationship of the siblings. So, when we can link one element of one sample to a specific element of our second sample, then we have a paired study design. The other type that's more commonly use is an unpaired study design, and a classic example with this would be a randomized controlled trial, where we start with a group of eligible participants, who ostensibly are randomly selected or representative of our population of interest, and they get randomized to a treatment or control group. So, there's no inherent relationship one to one relationship between the persons who get randomized to the treatment group and the persons who get randomized to control. We may not even have the same number of persons in each group. So, we can't draw a link between each person in the treatment group to a specific person in the control group in the unpaired design. Certainly, such studies such as observational cohort studies as well would fall into this criteria. So for example, if we looked at a group of eligible participants and then classified them as to their smoking status, smoker or non-smoker, and then follow them forward to see who developed a certain outcome. There's no inherent link between those who smoke and those who don't smoke in our samples. If we started with a group from some generalized population. Regardless of the study design employed though, the concepts and process will be pretty similar in terms of making confidence intervals for our group comparison measures. So in general, just extending the Central Limit Theorem to differences in two quantities, it turns out that the differences of two quantities whose distributions are approximately norma, l who's sampling distributions are approximately normal, have an approximately normal distribution as well. So, if we have a mean, a sample mean from one sample and its potential distribution across multiple samples from the same population is approximately normal. Similarly, we have a mean from a second sample taken from the second population and its theoretical sampling distribution cost multiple samples of the same size from his population is normal. Then the difference in means where we to replicate the study over and over again taking multiple random samples from population one and multiple random samples from population two, and look at the distribution of the mean differences. It would be normal or approximately normal across those study results. So as such, we can extend the basic principles of the central limit theorem to understand and quantify the sampling variability in mean differences between two independent populations and the difference in proportion between two independent populations is also a difference in quantities with normally distributed are approximately normally distributed sampling behavior. So, the sampling distributions of differences are approximately normally distributed and centered at the true difference. Again, "large" samples so this will just be business as usual in the sense that the true difference we're trying to estimate will be the center of our sampling distribution whether it be a difference in means or proportions and the estimates we can get from any one study taking samples, a respective sizes N1, and N2, from the two populations. The estimates we can get in using sample mean differences are the difference in sample proportions were we to replicate the study would be normally distributed roughly normally distributed around this unknown truth. So again, we can use the same logic we did for single means and single proportions to create a confidence interval for the difference in these quantities. Ratios are a bit different, but pretty easy to handle in terms of sampling distributions. The distribution of ratios and the ratios stay at scale where you to replicate a study over and over again and plot the distribution of the resulting relative risk or incidence rate ratios it would not be perfectly normal. In fact, it would be somewhat right skewed and this comes from the fact that we have seen that the allowable range of valuables values for ratios indicating a smaller numerator than denominator is between zero and one, and the allowable range for values with a larger denominator than numerator go from one to positive infinity. Well, it turns out you may remember that we equalize those ranges by putting things on the log scale, and it turns out on the log scale, the sampling behavior ratios is approximately normal. That is, if we were to replicate a study, where we took samples from two populations that computed, for example, the relative risk, we did this thousands of times and got thousands of relative risk estimates. If we did a histogram of the relative risks, it wouldn't be necessarily approximately normal. But on the log scale, if we took the log of each relative risk and plotted those, it would be approximately normal. Another way to think about this is, this goes back to the idea that differences are normally distributed of normally distributed quantities and on the log scale for any ratio A over B, if I take the natural log of A over B, that's numerically equal to the log of A, the numerator, minus the log of B, the denominator. So, the log of a ratio can be expressed as a difference, and differences tend to have normal sampling behavior. So, again, as such the sampling distribution for the natural log of ratio is approximately normally distributed and centered at the natural log of the true population value of the ratio being estimated. We'll certainly operationalize this with numbers to take it out of a theoretical but again, we have a situation where we know something about the potential variability over estimate, this time on the log scale, around it's unknown truth, but it's approximately normal. And we'll get one estimate under this distribution and most of the estimates we get will be a plus or minus standard two standard errors of the truth. So, it's really not a big deal the confidence intervals for ratios but, like I said in the introduction, it is a two-step process. We'll first have to look at things on the log ratio scale. So, we'll take the log over observe ratio, we'll see how to estimate the standard error of the log ratio, then we'll take our log ratio estimate plus or minus two of those standard errors to get a confidence interval for the log ratio, but of course we don't want a confidence interval for the log ratio per se. We want a confidence interval for the ratio of interests. All we have to do, once we have the confidence interval endpoints for the log ratio, is the antilog or exponentiate those endpoints back to the ratio scale. The null value for a measure of association comparing two populations is the value of this measure if both population outcome qualities being compared are equal. And hence, there is no association between this outcome and the populations. So, no difference between treatment and control. No difference for smokers to non-smokers, etc. So, if I was comparing systolic blood pressures for smokers to non-smokers and there were no difference in those systolic blood pressures in the populations, then the expected difference in population means would be zero because the two underlying populations means are equal. If I did a clinical trial where I randomized people to treatment or control and the underlying true proportion of persons who have the outcome of interest is the same regardless of whether they've treated or not, then that difference in proportions would be zero. For ratios, if the underlying two things being compared are the same, so, you cannot align proportion of persons for example who respond as the same, then the corresponding ratio would be one. So, the null value for ratios, things on the ratio scale is one, and for differences is zero, and just to put this in context with log ratios, remember the log of a ratio can be expressed as a difference. So, on the ratio scale, the null value is one and the log ratio, the null value is zero. If the null value does not appear in the confidence interval for a measure of association, then no association between the grouping factor and the outcome of interest is not within the range of possible population level associations, and hence can be ruled out. That sounds overly rovosed, but we'll see, that sort of language will be using. So, if the number zero does not occur in a confidence interval for a study comparing mean blood pressures between groups, then we've ruled out zero as a possibility for the true mean difference and hence concluded that all possibilities for the mean difference are non-zero and there is an association between that outcome and the grouping factor. So, if this null value does not appear in the confidence interval for a measure of association, the finding is called statistically significant and we'll back this up with another approach to the thing called Hypothesis Testing coming in lecture sets 9 and 10. So, in summary, the sampling behavior of differences in quantities that both have approximately normal sampling distributions like means or proportions, have an approximately normal sampling distribution. The sampling distributions of ratios is approximately normal on the log scale. So, the approach to getting 95 percent confidence intervals for the above quantities at the population level will be again, business as usual, taking our estimate and adding, subtracting two standard errors of our estimate. We'll drill down more on this idea of the null values and what their presence means in the confidence interval or their lack of seeing the value in the confidence interval what that means in the IDF statistical significance as we go through these lecture sets and start looking at some actual data based examples.
