The precise statistical definition of the 95 percent confidence interval is that if the telephone poll were conducted times, 95 times the percent of respondents favoring Bob Dole would be within the calculated confidence intervals and five times the percent favoring Dole would be either higher or lower than the range of the confidence intervals.
Instead of 95 percent confidence intervals, you can also have confidence intervals based on different levels of significance, such as 90 percent or 99 percent. Level of significance is a statistical term for how willing you are to be wrong.
With a 95 percent confidence interval, you have a 5 percent chance of being wrong. With a 90 percent confidence interval, you have a 10 percent chance of being wrong. A 99 percent confidence interval would be wider than a 95 percent confidence interval for example, plus or minus 4. A 90 percent confidence interval would be narrower plus or minus 2. It also tells you about how stable the estimate is.
If you repeated the experiment, that value wouldn't change and you still wouldn't know what it is. Therefore it isn't strictly correct to ask about the probability that the population mean lies within a certain range. In contrast, the confidence interval you compute depends on the data you happened to collect. If you repeated the experiment, your confidence interval would almost certainly be different. So it is OK to ask about the probability that the interval contains the population mean. This means that the researcher can only estimate the parameters i.
Therefore, a confidence interval is simply a way to measure how well your sample represents the population you are studying. The probability that the confidence interval includes the true mean value within a population is called the confidence level of the CI. To calculate the confidence interval, start by computing the mean and standard error of the sample.
Remember, you must calculate an upper and low score for the confidence interval using the z-score for the chosen confidence level see table below. For the lower interval score divide the standard error by the square root on n, and then multiply the sum of this calculation by the z-score 1. Finally, subtract the value of this calculation from the sample mean. There is an alternative study design in which two comparison groups are dependent, matched or paired.
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. Yet another scenario is one in which matched samples are used. For example, we might be interested in the difference in an outcome between twins or between siblings.
Once again we have two samples, and the goal is to compare the two means. However, the samples are related or dependent. In the first scenario, before and after measurements are taken in the same individual. In the last scenario, measures are taken in pairs of individuals from the same family.
When the samples are dependent, we cannot use the techniques in the previous section to compare means. Because the samples are dependent, statistical techniques that account for the dependency must be used. These techniques focus on difference scores i. This distinction between independent and dependent samples emphasizes the importance of appropriately identifying the unit of analysis, i. Again, the first step is to compute descriptive statistics. We compute the sample size which in this case is the number of distinct participants or distinct pairs , the mean and standard deviation of the difference scores , and we denote these summary statistics as n, d and s d , respectively.
The appropriate formula for the confidence interval for the mean difference depends on the sample size. The formulas are shown in Table 6. When samples are matched or paired, difference scores are computed for each participant or between members of a matched pair, and "n" is the number of participants or pairs, is the mean of the difference scores, and S d is the standard deviation of the difference scores.
In the Framingham Offspring Study, participants attend clinical examinations approximately every four years. Suppose we want to compare systolic blood pressures between examinations i. Since the data in the two samples examination 6 and 7 are matched, we compute difference scores by subtracting the blood pressure measured at examination 7 from that measured at examination 6 or vice versa. Notice that several participants' systolic blood pressures decreased over 4 years e. We now estimate the mean difference in blood pressures over 4 years.
This is similar to a one sample problem with a continuous outcome except that we are now using the difference scores. The calculations are shown below.
Difference - Mean Difference. Difference - Mean Difference 2. The null or no effect value of the CI for the mean difference is zero.
Crossover trials are a special type of randomized trial in which each subject receives both of the two treatments e.
Participants are usually randomly assigned to receive their first treatment and then the other treatment. In many cases there is a "wash-out period" between the two treatments.
Outcomes are measured after each treatment in each participant. When the outcome is continuous, the assessment of a treatment effect in a crossover trial is performed using the techniques described here. A crossover trial is conducted to evaluate the effectiveness of a new drug designed to reduce symptoms of depression in adults over 65 years of age following a stroke. Symptoms of depression are measured on a scale of with higher scores indicative of more frequent and severe symptoms of depression.
Patients who suffered a stroke were eligible for the trial. The trial was run as a crossover trial in which each patient received both the new drug and a placebo. Patients were blind to the treatment assignment and the order of treatments e. After each treatment, depressive symptoms were measured in each patient. The difference in depressive symptoms was measured in each patient by subtracting the depressive symptom score after taking the placebo from the depressive symptom score after taking the new drug.
A total of participants completed the trial and the data are summarized below. The mean difference in the sample is Since the sample size is large, we can use the formula that employs the Z-score. Because we computed the differences by subtracting the scores after taking the placebo from the scores after taking the new drug and because higher scores are indicative of worse or more severe depressive symptoms, negative differences reflect improvement i.
It is common to compare two independent groups with respect to the presence or absence of a dichotomous characteristic or attribute, e. When the outcome is dichotomous, the analysis involves comparing the proportions of successes between the two groups. There are several ways of comparing proportions in two independent groups.
Generally the reference group e. The risk ratio is a good measure of the strength of an effect, while the risk difference is a better measure of the public health impact, because it compares the difference in absolute risk and, therefore provides an indication of how many people might benefit from an intervention.
An odds ratio is the measure of association used in case-control studies. It is the ratio of the odds or disease in those with a risk factor compared to the odds of disease in those without the risk factor.
When the outcome of interest is relatively uncommon e. The odds are defined as the ratio of the number of successes to the number of failures. All of these measures risk difference, risk ratio, odds ratio are used as measures of association by epidemiologists, and these three measures are considered in more detail in the module on Measures of Association in the core course in epidemiology. Confidence interval estimates for the risk difference, the relative risk and the odds ratio are described below.
A risk difference RD or prevalence difference is a difference in proportions e. The point estimate is the difference in sample proportions, as shown by the following equation:.
The sample proportions are computed by taking the ratio of the number of "successes" or health events, x to the sample size n in each group:. The formula for the confidence interval for the difference in proportions, or the risk difference, is as follows:. Note that this formula is appropriate for large samples at least 5 successes and at least 5 failures in each sample.
If there are fewer than 5 successes events of interest or failures non-events in either comparison group, then exact methods must be used to estimate the difference in population proportions. The following table contains data on prevalent cardiovascular disease CVD among participants who were currently non-smokers and those who were current smokers at the time of the fifth examination in the Framingham Offspring Study.
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. Here smoking status defines the comparison groups, and we will call the current smokers group 1 and the non-smokers group 2.
A confidence interval for the difference in prevalent CVD or prevalence difference between smokers and non-smokers is given below. In this example, we have far more than 5 successes cases of prevalent CVD and failures persons free of CVD in each comparison group, so the following formula can be used:.
The null value for the risk difference is zero. A randomized trial is conducted among subjects to evaluate the effectiveness of a newly developed pain reliever designed to reduce pain in patients following joint replacement surgery.
The trial compares the new pain reliever to the pain reliever currently used the "standard of care". Patients are randomly assigned to receive either the new pain reliever or the standard pain reliever following surgery.
The patients are blind to the treatment assignment. Before receiving the assigned treatment, patients are asked to rate their pain on a scale of with high scores indicative of more pain. Each patient is then given the assigned treatment and after 30 minutes is again asked to rate their pain on the same scale.
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