The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Confidence intervals vs. Asked 6 years, 6 months ago. Active 2 years, 11 months ago. Viewed 64k times.
Can someone shed some light on how they are different? Improve this question. Andy For analysis, we have samples from each of the comparison populations, and if the sample variances are similar, then the assumption about variability in the populations is reasonable. If not, then alternative formulas must be used to account for the heterogeneity in variances.
Next, we will check the assumption of equality of population variances. The ratio of the sample variances is Notice that for this example Sp, the pooled estimate of the common standard deviation, is 19, and this falls in between the standard deviations in the comparison groups i. Therefore, the confidence interval is 0. Our best estimate of the difference, the point estimate, is 1. The standard error of the difference is 0. Note that when we generate estimates for a population parameter in a single sample e.
In contrast, when comparing two independent samples in this fashion the confidence interval provides a range of values for the difference. In this example, we estimate that the difference in mean systolic blood pressures is between 0. In this example, we arbitrarily designated the men as group 1 and women as group 2. Had we designated the groups the other way i.
The table below summarizes differences between men and women with respect to the characteristics listed in the first column. The second and third columns show the means and standard deviations for men and women respectively. Men have lower mean total cholesterol levels than women; anywhere from The men have higher mean values on each of the other characteristics considered indicated by the positive confidence intervals.
The confidence interval for the difference in means provides an estimate of the absolute difference in means of the outcome variable of interest between the comparison groups. It is often of interest to make a judgment as to whether there is a statistically meaningful difference between comparison groups.
This judgment is based on whether the observed difference is beyond what one would expect by chance. If there is no difference between the population means, then the difference will be zero i. Zero is the null value of the parameter in this case the difference in means. If the confidence interval does not include the null value, then we conclude that there is a statistically significant difference between the groups. 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.
Note, however, that some of the means are not very different between men and women e. This means that there is a small, but statistically meaningful difference in the means. When there are small differences between groups, it may be possible to demonstrate that the differences are statistically significant if the sample size is sufficiently large, as it is in this example.
The following table contains descriptive statistics on the same continuous characteristics in the subsample stratified by sex. We will again arbitrarily designate men group 1 and women group 2. Since the sample sizes are small i. However,we will first check whether the assumption of equality of population variances is reasonable.
The ratio of the sample variances is 9. The solution is shown below. Note that again the pooled estimate of the common standard deviation, Sp, falls in between the standard deviations in the comparison groups i. Interpretation: Our best estimate of the difference, the point estimate, is The standard error of the difference is 6.
In this sample, the men have lower mean systolic blood pressures than women by 9. Again, the confidence interval is a range of likely values for the difference in means. Since the interval contains zero no difference , we do not have sufficient evidence to conclude that there is a difference. The previous section dealt with confidence intervals for the difference in means between two independent groups. 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. We also have a very interesting Normal Distribution Simulator. It helps us to understand how random samples can sometimes be very good or bad at representing the underlying true values. Now imagine we get to pick ALL the apples straight away, and get them ALL measured by the packing machine this is a luxury not normally found in statistics!
Each apple is a green dot, our observations are marked blue. Our result was not exact Each apple is a green dot, our observations are marked purple. That does not include the true mean. The standard deviation can provide a yardstick: If a data point is a few standard deviations away from the model being tested, this is strong evidence that the data point is not consistent with that model. However, how to use this yardstick depends on the situation.
John Tsitsiklis, the Clarence J. Of course, that also means that 5 percent of the time, the result would be outside the two-sigma range. Six sigmas can still be wrong Technically, the results of that experiment had a very high level of confidence: six sigma. In most cases, a five-sigma result is considered the gold standard for significance, corresponding to about a one-in-a-million chance that the findings are just a result of random variations; six sigma translates to one chance in a half-billion that the result is a random fluke.
Interestingly, a different set of results from the same CERN particle accelerator were interpreted quite differently. A possible detection of something called a Higgs boson — a theorized subatomic particle that would help to explain why particles weigh something rather than nothing — was also announced last year.
That result had only a 2. Yet because it fits what is expected based on current physics, most physicists think the result is likely to be correct, despite its much lower statistical confidence level.
0コメント