If another sampling method is used, a larger sample size is likely to be required due to the "design effect". How many children must be included in the sample to estimate the prevalence to within 5 percentage points of the true value with 95%. confidence if it is known that the true rate is unlikely to exceed 20%. a) Estimated population share (b) Confidence level. How many children must be studied if the 95% confidence estimate is to be within 10 percentage points of the true proportion.
How many children should be examined if the resulting 95% confidence estimate should fall within 10% (not 10 percentage points) of the true rate. How many children should be included in a new study designed to test for a reduction in caries prevalence if it is desired to be 90% confident of detecting a rate of 20% at the 5% significance level? a) Test caries rate (b) Expected caries rate (c) Significance level (d) Power of the test. e) Alternative hypothesis (one-sided test). How many patients should be examined to test the hypothesis that the success rate of the new treatment method is 70%, versus an alternative hypothesis that it is not 70% at the 5% significance level?
Thirty of the subjects with the disease (60%) and 25 of the controls (50%) were involved in fishing-related occupations.
Case-control studies
Estimating an odds ratio with specified relative precision
In a limited area where cholera is a serious public health problem, about 30% of the population is believed to be using water from contaminated sources. A case-control study of the association between cholera and exposure to contaminated water needs to be conducted in the area to estimate the odds ratio to within 25% of the true value, which is believed to be approximately 2, with 95% certainty. What sample sizes would be needed in the cholera and control groups. approximated by the total exposure rate) 30%.
Hypothesis tests for an odds ratio
What sample size would be needed in the cholera and control groups? approximated by the total exposure rate) 30%. provided that Pi is known:. The effectiveness of the BeG vaccine in preventing tuberculosis in children is questionable and a study was designed to compare the vaccination rates of a group of people with tuberculosis with that of a group of control subjects. The researchers want to have an 80% chance of detecting an odds ratio that is significantly different from 1 at the 5% level.
If an odds ratio of 2 would be considered an important difference between the two groups, how large a sample should be included in each study group. a) Test value of odds ratio.
Estimating a relative risk with specified relative precision
To determine the sample size from Table 9 when RRa > 1, the values of both Pz and RRa are needed. Two competing therapies for a given cancer will be evaluated through a cohort study in a multicenter clinical trial. Patients are randomized to treatment A or treatment B and are monitored for disease recurrence for 5 years after treatment.
Treatment A is a new therapy that will be widely used if it can be shown to halve the risk of recurrence in the first five years after treatment (i.e. recurrence of RRa is currently observed in patients who received treatment B How many patients should be treated? examined in each of the two treatment groups if the researchers want to be 90% confident that they can correctly reject the null hypothesis (RRo = 1) if it is false, and the test should be performed on a significance level of 5%. ) Test value of the relative risk.
Lot quality assurance sampling
In a large city, the local health authority aims to achieve 90% vaccination of all eligible children. In response to concerns about outbreaks of certain childhood diseases in certain parts of the city, a team of health authority investigators is planning a survey to identify areas where vaccination coverage is 50 percent or less so that appropriate action can be taken. What is the minimum number of children that should be studied in each area and what cut-off value should be used if the study is to test the hypothesis that the proportion of unvaccinated children is 50% or more at the 5% level of significance.
The investigators want to be 90% confident of recognizing areas where target vaccination coverage has been reached (ie where only 10% of children are not fully vaccinated). a) Test value of the population proportion. Because the error of accepting groups of children as adequately vaccinated when coverage is actually 50% or less is more important, Po=0.50 and Pa=O.lO. If more than 2 children in a sample are found to be insufficiently vaccinated, the batch (sample population) should be "rejected" and the health authority may take steps to improve vaccination coverage in that area.
However, if only 2 (or fewer) children are found to be inadequately vaccinated, the null hypothesis should be rejected and the group of children accepted as not an immediate priority for an intensified vaccination campaign.
Incidence-rate studies
Estimating an incidence rate with specified relative precision
Hypothesis tests for an incidence rate
Based on a 5-year follow-up study of a small number of people, the annual incidence of a particular disease is reported to be 40%. What minimum sample size would be needed to test the hypothesis that the population incidence is 40% at the 5% level of significance. The test is desired to have a power of 90% to detect a true annual incidence rate of 50%, and the investigators are only interested in rejecting the null hypothesis if the true rate is greater than 40%. a) Test value of the incidence rate (b) Expected incidence rate (c) Significance level (d) Power of the test. e) Alternative hypothesis (one-tailed test).
Hypothesis tests for two incidence rates in follow- up (cohort) studies
Example 22 As part of a study of the long-term effect of noise on workers in a particularly noisy industry, it is planned to follow a group of people who have been recruited into the industry over a period of time and compare them with a similar group individuals working in a much quieter industry. The results of a previous small-scale study suggest that the annual incidence rate of hearing damage in the noisy industry may be as high as 25%. How many people should be followed in each of the groups (which should be of equal size) to test the hypothesis that the incidence rates for hearing impairment in the two groups are the same, at the 5% level of significance and with a power of 80%.
The alternative hypothesis is that the annual incidence rate for hearing damage in the quietest industry is no more than the national average of about 10% (for people in the same age group), while in the noisy industry it differs from this. Solution (a) Test value of difference in incidence rates (b). c) Level of significance (d) Power of the test. e) Alternative hypothesis (two-sided test) (f) Duration of the study. A study similar to that described in example 22 will be undertaken, but the duration of the study will be limited to 5 years.
Solution (a) Test value of the difference in incidence rates (b) Expected incidence rates. e) Alternative hypothesis (two-sided test) (f) Duration of study.
Definitions of commonly used terms
The ratio of the odds of an event occurring under one set of circumstances to the odds of it occurring under another (see also page 9). In hypothesis testing, when the difference being tested is directional in advance (eg, when Xl < X 2 but not Xl > X 2 , testing against the null hypothesis Xl = X 2 ). The ratio of the risk (probability) of an outcome (for example, disease or death) among people exposed to a given factor to the risk among people not exposed.
Sampling procedure in which each unit of study has the same chance of being selected and each sample of the same size has the same chance of being chosen. Represent the number of standard errors relative to the mean; Zl-" and Zl-,/2 are functions of the confidence level and Z 1 _ p is a function of the power of the test. For a two-sided test for small proportions.. h) Significance level 5%, power 80%, two-sided test , small proportions.
For OR < 1, use the colur.m value corresponding to 1 lOR and the row value corresponding to Pi. For OR < 1, use the column value corresponding to 1/OR and the row value corresponding to Pi. For OR< 1, use the column value corresponding to 1 lOR and the row value corresponding to Pi.
For OR< 1, use the column value corresponding to 1/ OR and the row value corresponding to P~. For OR < 1, use the column value corresponding to 1/ OR and the row value corresponding to Pi. use the column value corresponding to 1/ OR and the row value corresponding to Pi. In this formula, the term 2PH1 - P~) is used instead of 2PH1 - P~) because the study population is likely to consist of many more controls than cases, and the exposure rate among controls is often known with a high degree of precision; under the null hypothesis, this is also the exposure rate for cases.
For ORa < 1, use the column value corresponding to 1/ORa and the row value corresponding to Pi. For RR < 1, use the column value equal to 1/ RR and the row value equal to P. For RR < 1, use the column value equal to 1/ RR and the row value equal to P,.
For RR< 1, use the column value corresponding to 1/ RR and the row value corresponding to P,. The value of d* is always rounded to the nearest whole number (for example 5.8 will become 5). Hypothesis tests for two incidence rates in follow-up (cohort) studies (study duration not established) F or a one-tailed test.
