Suppose we want to generate a 95% confidence interval estimate for an unknown population mean. Using algebra, we can rework this inequality such that the mean (μ) is the middle term, as shown below. In the health-related publications a 95% confidence interval is most often used, but this is an arbitrary value, and other confidence levels can be selected. With smaller samples (n< 30) the Central Limit Theorem does not apply, and another distribution called the t distribution must be used.

Suppose we wish to estimate the mean systolic blood pressure, body mass index, total cholesterol level or white blood cell count in a single target population.

Descriptive statistics on variables measured in a sample of a n=3,539 participants attending the 7th examination of the offspring in the Framingham Heart Study are shown below. The table below shows data on a subsample of n=10 participants in the 7th examination of the Framingham offspring Study. Suppose we compute a 95% confidence interval for the true systolic blood pressure using data in the subsample. Suppose we wish to estimate the proportion of people with diabetes in a population or the proportion of people with hypertension or obesity.

This formula is appropriate for large samples, defined as at least 5 successes and at least 5 failures in the sample. Specific applications of estimation for a single population with a dichotomous outcome involve estimating prevalence, cumulative incidence, and incidence rates.

The table below, from the 5th examination of the Framingham Offspring cohort, shows the number of men and women found with or without cardiovascular disease (CVD). There are many situations where it is of interest to compare two groups with respect to their mean scores on a continuous outcome. We could begin by computing the sample sizes (n1 and n2), means 1 and 2), and standard deviations (s1 and s2) in each sample. The confidence interval will be computed using either the Z or t distribution for the selected confidence level and the standard error of the point estimate. Where Sp is the pooled estimate of the common standard deviation (assuming that the variances in the populations are similar) computed as the weighted average of the standard deviations in the samples. Suppose we want to compare mean systolic blood pressures in men versus women using a 95% confidence interval.

The following table includes 95% confidence intervals for each characteristic, computed using the same formula we used for the confidence interval for the difference in mean systolic blood pressures.

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. We previously considered a subsample of n=10 participants attending the 7th examination of the Offspring cohort in the Framingham Heart Study. Suppose we wish to construct a 95% confidence interval for the difference in mean systolic blood pressures between men and women using these data.

The previous section dealt with confidence intervals for the difference in means between two independent groups. 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. One can compute a risk difference, which is computed by taking the difference in proportions between comparison groups and is similar to the estimate of the difference in means for a continuous outcome.

The risk ratio (or relative risk) is another useful measure to compare proportions between two independent populations and it is computed by taking the ratio of proportions. Note that this formula is appropriate for large samples (at least 5 successes and at least 5 failures in each sample). 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. A randomized trial is conducted among 100 subjects to evaluate the effectiveness of a newly developed pain reliever designed to reduce pain in patients following joint replacement surgery.

Using the data in the table below, compute the point estimate for the difference in proportion of pain relief of 3+ points.are observed in the trial. Compute the 95% confidence interval for the difference in proportions of patients reporting relief (in this case a risk difference, since it is a difference in cumulative incidence).

The risk difference quantifies the absolute difference in risk or prevalence, whereas the relative risk is, as the name indicates, a relative measure. The relative risk is a ratio and does not follow a normal distribution, regardless of the sample sizes in the comparison groups. A critically important aspect of any study is determining the appropriate sample size to answer the research question.

Studies should be designed to include a sufficient number of participants to adequately address the research question.

The formulas presented here generate estimates of the necessary sample size(s) required based on statistical criteria.

Provide examples demonstrating how the margin of error, effect size and variability of the outcome affect sample size computations. In practice we use the sample standard deviation to estimate the population standard deviation. In planning studies, we want to determine the sample size needed to ensure that the margin of error is sufficiently small to be informative. This formula generates the sample size, n, required to ensure that the margin of error, E, does not exceed a specified value. E is the margin of error that the investigator specifies as important from a clinical or practical standpoint. Example 1: An investigator wants to estimate the mean systolic blood pressure in children with congenital heart disease who are between the ages of 3 and 5. In order to ensure that the 95% confidence interval estimate of the mean systolic blood pressure in children between the ages of 3 and 5 with congenital heart disease is within 5 units of the true mean, a sample of size 62 is needed. An investigator wants to estimate the mean birth weight of infants born full term (approximately 40 weeks gestation) to mothers who are 19 years of age and under.

In order to ensure that the 95% confidence interval estimate of the proportion of freshmen who smoke is within 5% of the true proportion, a sample of size 385 is needed.

Suppose that a similar study was conducted 2 years ago and found that the prevalence of smoking was 27% among freshmen. Example 3: An investigator wants to estimate the prevalence of breast cancer among women who are between 40 and 45 years of age living in Boston. A sample of size n=16,448 will ensure that a 95% confidence interval estimate of the prevalence of breast cancer is within 0.10 (or to within 10 women per 10,000) of its true value.

Thus, with n=5,000 women, a 95% confidence interval would be expected to have a margin of error of 0.0018 (or 18 per 10,000). If data are available on variability of the outcome in each comparison group, then Sp can be computed and used in the sample size formula.

Note that the formula for the sample size generates sample size estimates for samples of equal size.

A major issue is determining the variability in the outcome of interest (σ), here the standard deviation of HDL cholesterol.

Samples of size n1=250 and n2=250 will ensure that the 95% confidence interval for the difference in mean HDL levels will have a margin of error of no more than 3 units.

Again the issue is determining the variability in the outcome of interest (σ), here the standard deviation in pounds lost over 8 weeks.

Samples of size n1=56 and n2=56 will ensure that the 95% confidence interval for the difference in weight lost between diets will have a margin of error of no more than 3 pounds. Example 6: An investigator wants to estimate the impact of smoking during pregnancy on premature delivery.

Samples of size n1=508 women who smoked during pregnancy and n2=508 women who did not smoke during pregnancy will ensure that the 95% confidence interval for the difference in proportions who deliver prematurely will have a margin of error of no more than 4%.

In the module on hypothesis testing for means and proportions, we introduced techniques for means, proportions, differences in means, and differences in proportions.

Here we present formulas to determine the sample size required to ensure that a test has high power.

In designing studies most people consider power of 80% or 90% (just as we generally use 95% as the confidence level for confidence interval estimates).

The effect size represents the meaningful difference in the population mean - here 95 versus 100, or 0.51 standard deviation units different. In the planned study, participants will be asked to fast overnight and to provide a blood sample for analysis of glucose levels. We now substitute the effect size and the appropriate Z values for the selected α and power to compute the sample size.

If data are available on variability of the outcome in each comparison group, then Sp can be computed and used to generate the sample sizes. Example 9: An investigator is planning a clinical trial to evaluate the efficacy of a new drug designed to reduce systolic blood pressure. In order to compute the effect size, an estimate of the variability in systolic blood pressures is needed.

Samples of size n1=232 and n2= 232 will ensure that the test of hypothesis will have 80% power to detect a 5 unit difference in mean systolic blood pressures in patients receiving the new drug as compared to patients receiving the placebo. The investigator must enroll 258 participants to be randomly assigned to receive either the new drug or placebo. An investigator is planning a study to assess the association between alcohol consumption and grade point average among college seniors.

An investigator wants to evaluate the efficacy of an acupuncture treatment for reducing pain in patients with chronic migraine headaches. Then substitute the effect size and the appropriate Z values for the selected α and power to compute the sample size. Samples of size n1=324 and n2=324 will ensure that the test of hypothesis will have 80% power to detect a 30% difference in the proportions of students who develop flu between those who do and do not use the athletic facilities regularly.

Determining the appropriate design of a study is more important than the statistical analysis; a poorly designed study can never be salvaged, whereas a poorly analyzed study can be re-analyzed. Ramachandran V, Sullivan LM, Wilson PW, Sempos CT, Sundstrom J, Kannel WB, Levy D, D'Agostino RB. Many of the outcomes we are interested in estimating are either continuous or dichotomous variables, although there are other types which are discussed in a later module. In practice, however, we select one random sample and generate one confidence interval, which may or may not contain the true mean. This means that there is a 95% probability that the confidence interval will contain the true population mean. However, if the sample size is large (n > 30), then the sample standard deviations can be used to estimate the population standard deviation. Note that for a given sample, the 99% confidence interval would be wider than the 95% confidence interval, because it allows one to be more confident that the unknown population parameter is contained within the interval. The t distribution is similar to the standard normal distribution but takes a slightly different shape depending on the sample size. We select a sample and compute descriptive statistics including the sample size (n), the sample mean, and the sample standard deviation (s). The margin of error is very small (the confidence interval is narrow), because the sample size is large. Because the sample size is small, we must now use the confidence interval formula that involves t rather than Z. These diagnoses are defined by specific levels of laboratory tests and measurements of blood pressure and body mass index, respectively.

For example, we might be interested in comparing mean systolic blood pressure in men and women, or perhaps compare BMI in smokers and non-smokers. 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.

The sample is large (> 30 for both men and women), so we can use the confidence interval formula with Z.

Next we substitute the Z score for 95% confidence, Sp=19, the sample means, and the sample sizes into the equation for the confidence interval. The following table contains descriptive statistics on the same continuous characteristics in the subsample stratified by sex. There is an alternative study design in which two comparison groups are dependent, matched or paired.

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.

When the outcome is dichotomous, the analysis involves comparing the proportions of successes between the two groups. 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. 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.

However, the natural log (Ln) of the sample RR, is approximately normally distributed and is used to produce the confidence interval for the relative risk. This module will focus on formulas that can be used to estimate the sample size needed to produce a confidence interval estimate with a specified margin of error (precision) or to ensure that a test of hypothesis has a high probability of detecting a meaningful difference in the parameter. Studies that have either an inadequate number of participants or an excessively large number of participants are both wasteful in terms of participant and investigator time, resources to conduct the assessments, analytic efforts and so on.

However, in many studies, the sample size is determined by financial or logistical constraints. Note that there is an alternative formula for estimating the mean of a continuous outcome in a single population, and it is used when the sample size is small (n<30).

When we use the sample size formula above (or one of the other formulas that we will present in the sections that follow), we are planning a study to estimate the unknown mean of a particular outcome variable in a population. The mean birth weight of infants born full-term to mothers 20 years of age and older is 3,510 grams with a standard deviation of 385 grams. How many freshmen should be involved in the study to ensure that a 95% confidence interval estimate of the proportion of freshmen who smoke is within 5% of the true proportion? If the investigator believes that this is a reasonable estimate of prevalence 2 years later, it can be used to plan the next study.

This is a situation where investigators might decide that a sample of this size is not feasible. The investigators must decide if this would be sufficiently precise to answer the research question.

If a study is planned where different numbers of patients will be assigned or different numbers of patients will comprise the comparison groups, then alternative formulas can be used. The plan is to enroll participants and to randomly assign them to receive either the new drug or a placebo. Before presenting the formulas to determine the sample sizes required to ensure high power in a test, we will first discuss power from a conceptual point of view.

The figure below shows the distributions of the sample mean under the null and alternative hypotheses. A statistical test is much more likely to reject the null hypothesis in favor of the alternative if the true mean is 98 than if the true mean is 94. The inputs for the sample size formulas include the desired power, the level of significance and the effect size. A cross-sectional study is planned to assess the mean fasting blood glucose levels in people who drink at least two cups of coffee per day.

Based on prior experience, the investigators hypothesize that 10% of the participants will fail to fast or will refuse to follow the study protocol. How many patients should be studied to ensure that the power of the test is 90% to detect a 5% difference in the proportion with elevated LDL cholesterol?

During the manufacturing process, approximately 10% of the stents are deemed to be defective. If a study is planned where different numbers of patients will be assigned or different numbers of patients will comprise the comparison groups, then alternative formulas can be used (see Howell3 for more details). Analysis of data from the Framingham Heart Study showed that the standard deviation of systolic blood pressure was 19.0. However, the investigators hypothesized a 10% attrition rate (in both groups), and to ensure a total sample size of 232 they need to allow for attrition. The plan is to categorize students as heavy drinkers or not using 5 or more drinks on a typical drinking day as the criterion for heavy drinking. The formulas are organized by the proposed analysis, a confidence interval estimate or a test of hypothesis.

C-reactive protein, the metabolic syndrome and prediction of cardiovascular events in the Framingham Offspring Study. Relative importance of borderline and elevated levels of coronary heart disease risk factors. The parameters to be estimated depend not only on whether the endpoint is continuous or dichotomous, but also on the number of groups being studied. Recall that sample means and sample proportions are unbiased estimates of the corresponding population parameters.

In a sense, one could think of the t distribution as a family of distributions for smaller samples. The formulas for confidence intervals for the population mean depend on the sample size and are given below. Both of these situations involve comparisons between two independent groups, meaning that there are different people in the groups being compared. The standard error of the point estimate will incorporate the variability in the outcome of interest in each of the comparison groups.

The men have higher mean values on each of the other characteristics considered (indicated by the positive confidence intervals). This judgment is based on whether the observed difference is beyond that expected by chance. 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. Here smoking status defines the comparison groups, and we will call the current smokers group 1 and the non-smokers group 2.

Because the 95% confidence interval includes zero, we conclude that the difference in prevalent CVD between smokers and non-smokers is not statistically significant. Patients are randomly assigned to receive either the new pain reliever or the standard pain reliever following surgery. These situations can also be viewed as unethical as participants may have been put at risk as part of a study that was unable to answer an important question.

For example, suppose a study is proposed to evaluate a new screening test for Down Syndrome.

It involves a value from the t distribution, as opposed to one from the standard normal distribution, to reflect the desired level of confidence.

We conduct a study and generate a 95% confidence interval as follows 125 + 40 pounds, or 85 to 165 pounds. The formula above gives the number of participants needed with complete data to ensure that the margin of error in the confidence interval does not exceed E.

The investigator plans on using a 95% confidence interval (so Z=1.96) and wants a margin of error of 5 units.

Because the estimates of the standard deviation were derived from studies of children with other cardiac defects, it would be advisable to use the larger standard deviation and plan for a study with 62 children. How many women 19 years of age and under must be enrolled in the study to ensure that a 95% confidence interval estimate of the mean birth weight of their infants has a margin of error not exceeding 100 grams? Here we are planning a study to generate a 95% confidence interval for the unknown population proportion, p.

Using this estimate of p, what sample size is needed (assuming that again a 95% confidence interval will be used and we want the same level of precision)? Suppose that the investigators thought a sample of size 5,000 would be reasonable from a practical point of view. Note that the above is based on the assumption that the prevalence of breast cancer in Boston is similar to that reported nationally. HDL cholesterol will be measured in each participant after 12 weeks on the assigned treatment.

Suppose one such study compared the same diets in adults and involved 100 participants in each diet group.

How many women should be enrolled in the study to ensure that the 95% confidence interval for the difference in proportions has a margin of error of no more than 4%? The effect size is the difference in the parameter of interest that represents a clinically meaningful difference.

To facilitate interpretation, we will continue this discussion with as opposed to Z. Notice also in this case that there is little overlap in the distributions under the null and alternative hypotheses.

The effect size is selected to represent a clinically meaningful or practically important difference in the parameter of interest, as we will illustrate.

The formulas shown below produce the number of participants needed with complete data, and we will illustrate how attrition is addressed in planning studies. Similar to the issue we faced when planning studies to estimate confidence intervals, it can sometimes be difficult to estimate the standard deviation.

Therefore, a total of 35 participants will be enrolled in the study to ensure that 31 are available for analysis (see below). The manufacturer wants to test whether the proportion of defective stents is more than 10%. Recall from the module on Hypothesis Testing that, when we performed tests of hypothesis comparing the means of two independent groups, we used Sp, the pooled estimate of the common standard deviation, as a measure of variability in the outcome.

Systolic blood pressures will be measured in each participant after 12 weeks on the assigned treatment.

Mean grade point averages will be compared between students classified as heavy drinkers versus not using a two independent samples test of means. Each will be asked to rate the severity of the pain they experience with their next migraine before any treatment is administered. Each student will be asked if they used the athletic facility regularly over the past 6 months and whether or not they had the flu. The sample size must be large enough to adequately answer the research question, yet not too large so as to involve too many patients when fewer would have sufficed.

Gestational weight gain and pregnancy outcome in terms of gestation at delivery and infant birth weight: a comparison between adolescents under 16 and adult women.

When the outcome of interest is dichotomous like this, the record for each member of the sample indicates having the condition or characteristic of interest or not.

If we assume equal variances between groups, we can pool the information on variability (sample variances) to generate an estimate of the population variability. However,we will first check whether the assumption of equality of population variances is reasonable.

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. 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. A confidence interval for the difference in prevalent CVD (or prevalence difference) between smokers and non-smokers is given below.

By convention we typically regard the unexposed (or least exposed) group as the comparison group, and the proportion of successes or the risk for the unexposed comparison group is the denominator for the ratio. First, a confidence interval is generated for Ln(RR), and then the antilog of the upper and lower limits of the confidence interval for Ln(RR) are computed to give the upper and lower limits of the confidence interval for the RR. Studies that are much larger than they need to be to answer the research questions are also wasteful. In sample size computations, investigators often use a value for the standard deviation from a previous study or a study done in a different, but comparable, population. We will illustrate how attrition is addressed in planning studies through examples in the following sections. The standard deviation of systolic blood pressure is unknown, but the investigators conduct a literature search and find that the standard deviation of systolic blood pressures in children with other cardiac defects is between 15 and 20.

The equation to determine the sample size for determining p seems to require knowledge of p, but this is obviously this is a circular argument, because if we knew the proportion of successes in the population, then a study would not be necessary! The standard deviation of the outcome variable measured in patients assigned to the placebo, control or unexposed group can be used to plan a future trial, as illustrated below. In order to ensure that the total sample size of 500 is available at 12 weeks, the investigator needs to recruit more participants to allow for attrition. Each child will then be randomly assigned to either the low fat or the low carbohydrate diet.

In order to ensure that the total sample size of 112 is available at 8 weeks, the investigator needs to recruit more participants to allow for attrition. The first is called a Type I error and refers to the situation where we incorrectly reject H0 when in fact it is true. Similar to the margin of error in confidence interval applications, the effect size is determined based on clinical or practical criteria and not statistical criteria. We compute the sample mean and then must decide whether the sample mean provides evidence to support the alternative hypothesis or not. The figure below shows the same components for the situation where the mean under the alternative hypothesis is 98. If a sample mean of 97 or higher is observed it is very unlikely that it came from a distribution whose mean is 90.

In sample size computations, investigators often use a value for the standard deviation from a previous study or a study performed in a different but comparable population. How many patients should be enrolled in the study to ensure that the power of the test is 80% to detect this difference? If the process produces more than 15% defective stents, then corrective action must be taken. The standard deviation of the outcome variable measured in patients assigned to the placebo, control or unexposed group can be used to plan a future trial, as illustrated. Based on prior experience with similar trials, the investigator expects that 10% of all participants will be lost to follow up or will drop out of the study.

Pain will be recorded on a scale of 1-100 with higher scores indicative of more severe pain. A test of hypothesis will be conducted to compare the proportion of students who used the athletic facility regularly and got flu with the proportion of students who did not and got flu.

The determination of the appropriate sample size involves statistical criteria as well as clinical or practical considerations. The sample should be representative of the population, with participants selected at random from the population. In contrast, when comparing two independent samples in this fashion the confidence interval provides a range of values for the difference.

If a 95% confidence interval includes the null value, then there is no statistically meaningful or statistically significant difference between the groups. Based on this interval, we also conclude that there is no statistically significant difference in mean systolic blood pressures between men and women, because the 95% confidence interval includes the null value, zero.

When the samples are dependent, we cannot use the techniques in the previous section to compare means.

Before receiving the assigned treatment, patients are asked to rate their pain on a scale of 0-10 with high scores indicative of more pain. To be informative, an investigator might want the margin of error to be no more than 5 or 10 pounds (meaning that the 95% confidence interval would have a width (lower limit to upper limit) of 10 or 20 pounds). The sample size computation is not an application of statistical inference and therefore it is reasonable to use an appropriate estimate for the standard deviation.

Suppose the investigator wants the estimate to be within 10 per 10,000 women with 95% confidence.

This is done by computing a test statistic and comparing the test statistic to an appropriate critical value.

Therefore, the manufacturer wants the test to have 90% power to detect a difference in proportions of this magnitude. If the new drug shows a 5 unit reduction in mean systolic blood pressure, this would represent a clinically meaningful reduction. How many college seniors should be enrolled in the study to ensure that the power of the test is 80% to detect a 0.25 unit difference in mean grade point averages? Sample size determination involves teamwork; biostatisticians must work closely with clinical investigators to determine the sample size that will address the research question of interest with adequate precision or power to produce results that are clinically meaningful. One can use published tables of random numbers and simply assign subjects based on the next number listed on the table, or one can use a random number generated by a computer, such as the random number function in Excel. In generating estimates, it is also important to quantify the precision of estimates from different samples. Rather, it reflects the amount of random error in the sample and provides a range of values that are likely to include the unknown parameter. Just as with large samples, the t distribution assumes that the outcome of interest is approximately normally distributed. If the confidence interval does not include the null value, then we conclude that there is a statistically significant difference between the groups. Because the samples are dependent, statistical techniques that account for the dependency must be used. Each patient is then given the assigned treatment and after 30 minutes is again asked to rate their pain on the same scale.

RESPONSE TIME LAB - RSTUDIO - LINK TO GOOGLE DOC - PLEASE MAKE YOUR OWN COPY AND EDIT THAT COPY106. The amniocentesis is included as the gold standard and the plan is to compare the results of the screening test to the results of the amniocentesis. In order to determine the sample size needed, the investigator must specify the desired margin of error. The estimate can be derived from a different study that was reported in the literature; some investigators perform a small pilot study to estimate the standard deviation. The research team, with input from clinical investigators and biostatisticians, must carefully evaluate the implications of selecting a sample of size n = 5,000, n = 16,448 or any size in between.

Because we purposely select a small value for α , we control the probability of committing a Type I error. If the null hypothesis is true (μ=90), then we are likely to select a sample whose mean is close in value to 90.

While a better test is one with higher power, it is not advisable to increase α as a means to increase power.

How many patients should be enrolled in the trial to ensure that the power of the test is 80% to detect this difference? On their next migraine (post-treatment), each patient will again be asked to rate the severity of the pain. The investigators feel that a 30% increase in flu among those who used the athletic facility regularly would be clinically meaningful. The unpredictability means that, if a sufficiently large number of subjects are randomly assigned to treatment groups, the groups will have similar distributions of all characteristics.

Another way of thinking about a confidence interval is that it is the range of likely values of the parameter (defined as the point estimate + margin of error) with a specified level of confidence (which is similar to a probability).

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. Since the interval contains zero (no difference), we do not have sufficient evidence to conclude that there is a difference. 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.

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.

Suppose that the collection and processing of the blood sample costs $250 per participant and that the amniocentesis costs $900 per participant. It is important to note that this is not a statistical issue, but a clinical or a practical one.

Based on data reported from diet trials in adults, the investigator expects that 20% of all children will not complete the study. The second type of error is called a Type II error and it is defined as the probability we do not reject H0 when it is false. However, it is also possible to select a sample whose mean is much larger or much smaller than 90. How many students should be enrolled in the study to ensure that the power of the test is 80% to detect this difference in the proportions? As a result both known and unknown confounders will tend to be equally distributed among the study groups. Note also that this 95% confidence interval for the difference in mean blood pressures is much wider here than the one based on the full sample derived in the previous example, because the very small sample size produces a very imprecise estimate of the difference in mean systolic blood pressures. Confidence interval estimates for the risk difference, the relative risk and the odds ratio are described below. These financial constraints alone might substantially limit the number of women that can be enrolled. For example, suppose we want to estimate the mean birth weight of infants born to mothers who smoke cigarettes during pregnancy.

Data from the participants in the pilot study can be used to compute a sample standard deviation, which serves as a good estimate for σ in the sample size formula.

Consequently, if there is no information available to approximate p, then p=0.5 can be used to generate the most conservative, or largest, sample size. A 95% confidence interval will be estimated to quantify the difference in weight lost between the two diets and the investigator would like the margin of error to be no more than 3 pounds.

Just as it is important to consider both statistical and clinical significance when interpreting results of a statistical analysis, it is also important to weigh both statistical and logistical issues in determining the sample size for a study. Birth weights in infants clearly have a much more restricted range than weights of female college students. How many patients should be involved in the study to ensure that the test has 80% power to detect a difference of 10 units on the pain scale? 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. Therefore, we would probably want to generate a confidence interval for the mean birth weight that has a margin of error not exceeding 1 or 2 pounds. If that is unsuccessful, the infection has been treated by switching to another antibiotic. There have been sporadic reports of successful treatment by infusing feces from healthy donors into the duodenum of patients suffering from C. Additional Normal Probability Reference Page - Good Overview of Normal Probability Plots20. Using TI-83 to create a Normal Probability Plot (PAGE 10) - Also, seems to be a GREAT TI-83 reference21.

The efficacy of this approach was tested in a randomized clinical trial reported in the New England Journal of Medicine (Jan. In order to estimate the sample size that would be needed, the investigators assumed that the feces infusion would be successful 90% of the time, and antibiotic therapy would be successful in 60% of cases. How many subjects will be needed in each group to ensure that the power of the study is 80% with a level of significance α = 0.05?

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