A single sample of participants and each participant is measured twice, once before and then after an intervention.

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. For example, we might be interested in the difference in an outcome between twins or between siblings. In the one sample and two independent samples applications participants are the units of analysis. However, with two dependent samples application,the pair is the unit (and not the number of measurements which is twice the number of units). In the Framingham Offspring Study, participants attend clinical examinations approximately every four years. We can now use these descriptive statistics to compute a 95% confidence interval for the mean difference in systolic blood pressures in the population.

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. However, even if we draw a random sample of the population we cannot be certain that it is a true mirror image.

Our next step is to imagine that we conducted several opinion polls (for example, one million polls) asking 1000 persons each time whether or not they would vote for Obama.

95 percent of our estimates of Obamas percentage will fall within 1.96 standard deviations of the true population mean of people who will vote for him.

But how can we calculate our confidence interval when we have not performed more than one opinion poll (the one where we found that 54 percent will vote Obama)? There are in fact several ways to calculate the confidence interval, and the normal approximation interval is the simplest formula.

We see that we have a confidence interval of ±3.09 percent for our estimated percentage of 54. From the formula we see that the range of the confidence interval will vary depending on the percentage who said they will vote for Obama. We can also note that the confidence intervals decreases when our sample increases, so if we interview 1200 persons we will get better estimates than when we ask 600. Now that you know how to interpret the statistical uncertainty associated with random sampling, we will move on to the other things you as a critical reader should be concerned with when viewing the opinion polls before an election.

An additional concern when it comes to opinion polls is whether or not you are drawing a sample from the correct population.

If this bias was not corrected for, the Democratic Party in the United States would do considerably better in the polls than in the elections, and vice versa for the Republican Party.

There are also instances when such an effect comes into play with regard to unwillingness to even admit to voting for mainstream candidates (for example, voting for conservatives over social democrats, or a white candidate over a black candidate). Political opinion polls are meant to give a picture of the strength of candidates or parties at a given time before an election.

There is an alternative study design in which two comparison groups are dependent, matched or paired. For example, a trial was conducted in patients with asthma to compare laser acupuncture with a sham procedure.

Participants are usually randomly assigned to receive their first treatment and then the other treatment. Symptoms of depression are measured on a scale of 0-100 with higher scores indicative of more frequent and severe symptoms of depression.

According to the central limit theorem the distributions of the results of all of these polls will center round the true population percentage of people who favor Obama, and the distribution will follow the shape of a bell curve (normal distribution). The standard deviation is a measure of the mean distance of each opinion poll from the true population mean.

Well, we can insert a standard deviation using formulas determined by statistical inference.

This calculation is based on the central limit theorem, and applies poorly when the distribution is less than 30, or if the percentage who say they will vote for Obama is close to 0 or 100. This is illustrated in the table below, where different intervals (95 %) are calculated for different percentages, all with a sample size of 1000 (remember that the larger the sample size the more precise estimates we will get). The larger our sample the closer we can assume our sample is to the true values of the population. One can easily argue that when trying to determine who will win an election one should not be interested in a mirror image of the voting population, but rather with a mirror image of those that are actually going to vote. If an opinion poll has not taken this into account, the results will be skewed in favor of certain political parties or candidates. The former pertains to the fact that some people are unwilling to either answer the phone or refuses to answer the poll. This can actually work both in the direction of overrepresentation of both more extreme and more moderate views. Patients were randomized either to acupuncture for five weeks (first period) then sham for five weeks (second period), or sham then acupuncture. 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 sd, respectively. The data below are systolic blood pressures measured at the sixth and seventh examinations in a subsample of n=15 randomly selected participants.

This is similar to a one sample problem with a continuous outcome except that we are now using the difference scores.

Because the 95% confidence interval for the mean difference does not include zero, we can conclude that there is a statistically significant difference (in this case a significant improvement) in depressive symptom scores after taking the new drug as compared to placebo.

In 1936 George Gallup introduced the concept of representative sampling, which means that the people asked should be a mirror image of the population under scrutiny. To understand the logic of this uncertainty we need to get acquainted with what is known as the central limit theorem. What we are doing here is calculating what is known as a binomial proportion confidence interval.

The true value of 56 percent falls within this interval (but 5 percent of the times it will be further away). Remember that this calculation does not work well for N<30 and values close to 0 and 100. One way of determining this is to ask the respondent if he or she intends to vote in the upcoming election.

Parties and candidates with a strong party identity and a voter base with strong resources will thus be underestimated in the polls, as they are more probable to actually vote than people without party identity or from a less privileged background. Whether this effect favors the extreme or the moderate differs from one country to another. The appropriate formula for the confidence interval for the mean difference depends on the sample size. 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. The trial was run as a crossover trial in which each patient received both the new drug and a placebo.

Let us say that we conduct a survey of 1000 persons and ask if they prefer the candidate Obama or one of the other candidates.

These may include whether the person has voted in the previous election, if they know where to vote, and if they have given much thought to the election.

They have certain characteristics separating them from those that answer the phone and states who they will vote for.

It is important to take these polls with a grain of salt, and to think about the points mentioned in this article.

When the samples are dependent, we cannot use the techniques in the previous section to compare means. This results in the remaining sample not being a true mirror image of neither the electorate nor those that are likely voters.

Because the samples are dependent, statistical techniques that account for the dependency must be used.

Naturally, one should always strive for the highest possible response rate in order to minimize this bias. I am contented to find a number of valuable information and facts right here while in the distribute, we want work out extra techniques normally made available, many thanks for revealing.

When the outcome is continuous, the assessment of a treatment effect in a crossover trial is performed using the techniques described here. 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.

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