Sample size to estimate a binary outcome using simple random sampling

Description

Sample size to estimate a binary outcome using simple random sampling.

Usage

p: Expected prevalence of the outcome of interest in individuals that have been sampled.
π: Expected prevalence of the outcome of interest in the population.

References

Levy PS, Lemeshow S (1999). Sampling of Populations Methods and Applications. Wiley Series in Probability and Statistics, London, pp. 70 - 75.

Examples

Brucellosis

We want to estimate the seroprevalence of Brucella abortus in a population of cattle. An estimate of the unknown prevalence of B. abortus in this population is 0.15. We would like to be 95% certain that our estimate is within 20% of the true proportion of the population seropositive to B. abortus. Error is quoted in relative terms so the absolute error equals 0.15 × 0.20 = 0.03.

Minimum number to sample to be 95% confident sample prevalence is within 0.03 of the population prevalence: 545

Sample size to estimate a continuous outcome using simple random sampling

Description

Sample size to estimate a continuous outcome using simple random sampling.

Usage

x̄: Expected mean (average) of the outcome of interest.
μ: Expected mean (average) of the outcome of interest in the population.
σ: Expected standard deviation of the outcome of interest in the population.

References

Levy PS, Lemeshow S (1999). Sampling of Populations Methods and Applications. Wiley Series in Probability and Statistics, London, pp. 70 - 75.

Examples

The bodyweight of deer

We want to estimate the mean bodyweight of deer on a farm. We anticipate the mean body weight to be around 200 kg and the standard deviation of body weight to be 30 kg. We would like to be 95% certain that our estimate of bodyweight is within 10 kg of the true population mean. How many deer should be sampled?

x̄: Expected mean of the outcome of interest in individuals that have been sampled.
μ: Expected mean of the outcome of interest in the population.

Minimum number to sample to be 95% confident sample mean is within 10 unit(s) of the population mean: 35

Sample size to estimate a binary outcome using one-stage cluster sampling

Description

Sample size to estimate a binary outcome using one-stage cluster sampling.

Usage

p: Expected prevalence of the outcome of interest in individuals that have been sampled.
π: Expected prevalence of the outcome of interest in the population.
PSU: Primary sampling unit (e.g., herds, flocks, classes, households).
SSU: Secondary sampling unit (e.g., cows within herds, chickens within flocks, children within classes, individuals within households).

At least 25 primary sampling units are recommended for one-stage cluster sampling designs. If less than 25 clusters are returned by the function a warning is issued.

References

Levy PS, Lemeshow S (1999). Sampling of Populations Methods and Applications. Wiley Series in Probability and Statistics, London, pp. 258.

Machin D, Campbell MJ, Tan SB, Tan SH (2018). Sample Sizes for Clinical, Laboratory and Epidemiological Studies, Fourth Edition. Wiley Blackwell, London, pp. 195 - 214.

Otte J, Gumm I (1997). Intra-cluster correlation coefficients of 20 infections calculated from the results of cluster-sample surveys. Preventive Veterinary Medicine 31: 147 - 150.

Examples

Cook stoves

An aid project has distributed cook stoves in a single province in a resource-poor country. At the end of three years, the donors would like to know what proportion of households are still using their donated stove. A cross-sectional study is planned where villages in the province will be sampled and all households (approximately 75 per village) will be visited to determine whether or not the donated stove is still in use. A pilot study of the prevalence of stove usage in five villages showed that 0.46 of householders were still using their stove. The intra-class correlation for a study of this type is unknown, but thought to be relatively high (ρ = 0.20).

If the donor wanted to be 90% confident that the survey estimate of stove usage was within 10% of the true population value, how many villages (i.e., clusters, primary sampling unites) would need to be sampled? The relative error is 10%, equivalent to 0.10 × 0.46 = 4.6% absolute error.

Minimum total number of PSUs to sample if 75 SSUs sampled per PSU: 67
Minimum total number of SSUs to sample to be 90% confident sample prevalence is within 0.046 unit(s) of the population prevalence: 5019
Design effect: 15.8

Sample size to estimate a continuous outcome using one-stage cluster sampling

Description

Sample size to estimate a continuous outcome using one-stage cluster sampling.

Usage

Key:

x̄: Expected mean of the outcome of interest in individuals that have been sampled.
μ: Expected mean of the outcome of interest in the population.
σ: Expected standard deviation of the outcome of interest in the population.
PSU: Primary sampling unit (e.g., herds, flocks, classes, households).
SSU: Secondary sampling unit (e.g., cows within herds, chickens within flocks, children within classes, individuals within households).

At least 25 primary sampling units are recommended for one-stage cluster sampling designs. If less than 25 clusters are returned by the function a warning is issued.

References

Levy PS, Lemeshow S (1999). Sampling of Populations Methods and Applications. Wiley Series in Probability and Statistics, London, pp. 258.

Machin D, Campbell MJ, Tan SB, Tan SH (2018). Sample Sizes for Clinical, Laboratory ad Epidemiological Studies, Fourth Edition. Wiley Blackwell, London, pp. 195 - 214.

Otte J, Gumm I (1997). Intra-cluster correlation coefficients of 20 infections calculated from the results of cluster-sample surveys. Preventive Veterinary Medicine 31: 147 - 150.

Examples

Nurse services in retirement villages

A survey to estimate the average number of residents over 75 years of age that require the services of a nurse in a given retirement village is to be carried out using a one-stage cluster sampling strategy. There are five housing complexes in the village with 25 residents in each. We expect that there might be an average of 34 residents meeting this criteria (SD 5.5). We would like the estimated sample size to provide us with an estimate that is within 10% of the true value. Previous studies report an intracluster correlation for the number of residents requiring the services of a nurse in this retirement village housing complexes to be 0.10. How many housing complexes (clusters) should be sampled?

Minimum total number of PSUs to sample if 25 SSUs sampled per PSU: 2
Minimum total number of SSUs to sample if 95% confident sample mean is within 3.4 unit(s) of the population mean: 32
Design effect: 3.4

Sample size to estimate a binary outcome using two-stage cluster sampling

Description

Sample size to estimate a binary outcome using two-stage cluster sampling.

Usage

p: Expected prevalence of the outcome of interest in individuals that have been sampled.
π: Expected prevalence of the outcome of interest in the population.
PSU: Primary sampling unit (e.g., herds, flocks, classes, households).
SSU: Secondary sampling unit (e.g., cows within herds, chickens within flocks, children within classes, individuals within households).

The methodology used in this function follows the approach described by Bennett et al. (1991). At least 25 primary sampling units are recommended for two-stage cluster sampling designs. If less than 25 clusters are returned by the function a warning is issued.

As a rule of thumb, around 30 clusters will provide good estimates of the true population value with an acceptable level of precision (Binkin et al. 1992) when: (1) the true population value is between 10% and 90%; and (2) the desired absolute error is around 5%. For a fixed number of individuals selected per cluster (e.g., 10 individuals per cluster or 30 individuals per cluster), collecting information on more than 30 clusters can improve the precision of the final population estimate, however, beyond around 60 clusters the improvement in precision is minimal.

References

Bennett S, Woods T, Liyanage W, Smith D (1991). A simplified general method for cluster-sample surveys of health in developing countries. World Health Statistics Quarterly 44: 98 - 106.

Binkin N, Sullivan K, Staehling N, Nieburg P (1992). Rapid nutrition surveys: How many clusters are enough? Disasters 16: 97 - 103.

Machin D, Campbell MJ, Tan SB, Tan SH (2018). Sample Sizes for Clinical, Laboratory ad Epidemiological Studies, Fourth Edition. Wiley Blackwell, London, pp. 195 - 214.

Otte J, Gumm I (1997). Intra-cluster correlation coefficients of 20 infections calculated from the results of cluster-sample surveys. Preventive Veterinary Medicine 31: 147 - 150.

Examples

A cross-sectional study

We intend to conduct a cross-sectional study to determine the prevalence of disease X in a given country. The expected prevalence of disease is thought to be around 20%. Previous studies report an intracluster correlation coefficient for this disease to be 0.02. Suppose that we want to be 95% certain that our estimate of the prevalence of disease is within 5% of the true population value and that we intend to sample 20 individuals per cluster. How many clusters should be sampled to meet the requirements of the study?

Minimum total number of PSUs to sample if 20 SSUs sampled per PSU: 17
Minimum total number of SSUs to sample to be 95% confident sample prevalence is within 0.05 unit(s) of the population prevalence: 340
Design effect: 1.38

Sample size to estimate a continuous outcome using two-stage cluster sampling

Description

Sample size to estimate a continuous outcome using two-stage cluster sampling.

Usage

Key:

x̄: Expected mean of the outcome of interest in individuals that have been sampled.
μ: Expected mean of the outcome of interest in the population.
σ: Expected standard deviation of the outcome of interest in the population.
PSU: Primary sampling unit (e.g., herds, flocks, classes, households).
SSU: Secondary sampling unit (e.g., cows within herds, chickens within flocks, children within classes, individuals within households).

At least 25 primary sampling units are recommended for one-stage cluster sampling designs. If less than 25 clusters are returned by the function a warning is issued.

References

Levy PS, Lemeshow S (1999). Sampling of Populations Methods and Applications. Wiley Series in Probability and Statistics, London, pp. 292.

Machin D, Campbell MJ, Tan SB, Tan SH (2018). Sample Sizes for Clinical, Laboratory ad Epidemiological Studies, Fourth Edition. Wiley Blackwell, London, pp. 195 - 214.

Otte J, Gumm I (1997). Intra-cluster correlation coefficients of 20 infections calculated from the results of cluster-sample surveys. Preventive Veterinary Medicine 31: 147 - 150.

Examples

Nurse practitioners

We intend to conduct a survey of nurse practitioners to estimate the average number of patients seen by each nurse. There are approximately 1500 patients in the local health authority area, with patients clustered by health centre. We intend to sample two nurses from each health centre. We would like to be 95% confident that our estimate is within 30% of the true population value. We expect that the mean number of patients seen by each nurse is 28 (SD 12.6). Previous studies report an intracluster correlation for the number of patients seen per nurse to be 0.02. How many health centres should be sampled?

Minimum total number of PSUs to sample if 2 SSUs sampled per PSU: 5
Minimum total number of SSUs to sample: 9
Design effect: 1.02

Detectable prevalence ratio, cross-sectional study

Description

Sample size to determine the minimum detectable prevalence ratio, cross-sectional study.

Usage

p_{D+ | E+}: Expected prevalence of the outcome of interest in the exposed.
p_{D+ | E-}: Expected prevalence of the outcome of interest in the unexposed.
p_E+: Expected prevalence of exposure in the population.
r: The number in the exposed group divided by the number in the unexposed group.

References

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Mittleman MA (1995). Estimation of exposure prevalence in a population at risk using data from cases and an external estimate of the relative risk. Epidemiology 6: 551 - 553.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Q fever

A cross-sectional study is to be carried out to quantify the association between farm management type (intensive, extensive) and evidence of Q fever in dairy goat herds. The investigators would like to be 0.80 sure of being able to detect when the risk ratio of Q fever is 2.0 for intensively managed herds, using a 0.05 significance test. Previous evidence suggests that the prevalence of Q fever in extensively managed dairy goat herds is 5 per 100 herds at risk and the prevalence of intensively managed herds in the population (the prevalence of exposure) is around 0.20.

Assuming equal numbers of intensively managed and extensively managed herds will be sampled, how many herds need to be enrolled into the study? You estimate that there are around 60 dairy goat herds in the study area.

Total number of study subjects to sample: 58
Total number of subjects in the exposed group: 29
Total number of subjects in the unexposed group: 29
Power: 0.80
Minimum detectable prevalence ratio: 2.0
Minimum detectable odds ratio: 2.1

Detectable prevalence ratio, cross-sectional study

Description

Sample size to determine the minimum detectable prevalence ratio, cross-sectional study.

Usage

p_{D+ | E+}: Expected prevalence of the outcome of interest in the exposed.
p_{D+ | E-}: Expected prevalence of the outcome of interest in the unexposed.
p_E+: Expected prevalence of exposure in the population.
r: The number in the exposed group divided by the number in the unexposed group.

References

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Mittleman MA (1995). Estimation of exposure prevalence in a population at risk using data from cases and an external estimate of the relative risk. Epidemiology 6: 551 - 553.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Q fever

A cross-sectional study is to be carried out to quantify the association between farm management type (intensive, extensive) and evidence of Q fever in dairy goat herds. The investigators would like to be 0.80 sure of being able to detect when the risk ratio of Q fever is 2.0 for intensively managed herds, using a 0.05 significance test. Previous evidence suggests that the prevalence of Q fever in extensively managed dairy goat herds is 5 per 100 herds at risk and the prevalence of intensively managed herds in the population (the prevalence of exposure) is around 0.20.

Assuming equal numbers of intensively managed and extensively managed herds will be sampled, how many herds need to be enrolled into the study? You estimate that there are around 60 dairy goat herds in the study area.

Total number of study subjects to sample: 58
Total number of subjects in the exposed group: 29
Total number of subjects in the unexposed group: 29
Power: 0.80
Minimum detectable prevalence ratio: 2.0
Minimum detectable odds ratio: 2.1

Detectable incidence risk ratio, cohort study using count data

Description

Sample size to determine the minimum detectable incidence risk ratio for a cohort study using count data.

Usage

p_{D+ | E+}: Expected incidence risk of the outcome of interest in the exposed.
p_{D+ | E-}: Expected incidence risk of the outcome of interest in the unexposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Coronary heart disease in middle aged men

A cohort study of smoking and coronary heart disease (CHD) in middle aged men is planned. A sample of men will be selected at random from the population and those that agree to participate will be asked to complete a questionnaire. The follow-up period will be 5 years. The investigators would like to be 0.90 sure of being able to detect when the risk ratio of CHD is 1.4 for smokers, using a 0.05 significance test. Previous evidence suggests that the incidence risk of death in non-smokers is 413 per 100,000 per year. Assuming equal numbers of smokers and non-smokers are sampled, how many men should be sampled overall?

Total number of study subject: 12130
Total number of subjects in the exposed group: 6065
Total number of subjects in the unexposed group: 6065
Power: 0.90
Minimum detectable incidence risk ratio: 1.4

Detectable incidence risk ratio, cohort study using count data

Description

Sample size to determine the minimum detectable incidence risk ratio for a cohort study using count data.

Usage

p_{D+ | E+}: Expected incidence risk of the outcome of interest in the exposed.
p_{D+ | E-}: Expected incidence risk of the outcome of interest in the unexposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Coronary heart disease in middle aged men

A cohort study of smoking and coronary heart disease (CHD) in middle aged men is planned. A sample of men will be selected at random from the population and those that agree to participate will be asked to complete a questionnaire. The follow-up period will be 5 years. The investigators would like to be 0.90 sure of being able to detect when the risk ratio of CHD is 1.4 for smokers, using a 0.05 significance test. Previous evidence suggests that the incidence risk of death in non-smokers is 413 per 100,000 per year. Assuming equal numbers of smokers and non-smokers are sampled, how many men should be sampled overall?

Total number of study subject: 12130
Total number of subjects in the exposed group: 6065
Total number of subjects in the unexposed group: 6065
Power: 0.90
Minimum detectable incidence risk ratio: 1.4

Detectable incidence rate ratio, cohort study using time at risk data

Description

Sample size to determine the minimum detectable incidence rate ratio for a cohort study using time at risk data.

Usage

r_{D+ | E+}: Expected incidence rate of the outcome of interest in the exposed.
r_{D+ | E-}: Expected incidence rate of the outcome of interest in the unexposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Lemeshow S, Hosmer D, Klar J, Lwanga S (1990). Adequacy of Sample Size in Health Studies. John Wiley and Sons, New York.

Lwanga S, Lemeshow S (1991). Sample Size Determination in Health Studies. World Health Organization, Geneva.

Examples

Long term effect of noise

As part of a study of the long-term effect of noise on workers in a particularly noisy industry, it is planned to follow up a cohort of people who were recruited into the industry during a given period of time and to compare them with a similar cohort of individuals working in a much quieter industry. Subjects will be followed up for the rest of their lives or until their hearing is impaired. The results of a previous small-scale survey suggest that the annual incidence rate of hearing impairment in the noisy industry may be as high as 25%. How many people should be followed up in each of the groups (which are to 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 impairment in the quieter industry is not more than the national average of about 10% (for people in the same age range), whereas in the noisy industry it differs from this.

An annuual incidence rate of 25% is equivalent to 25 cases of hearing impairment per 100 individuals per year.

Total number of study subjects: 520
Total number of subjects in the exposed group: 260
Total number of subjects in the unexposed group: 260
Power: 0.80
Minimum detectable incidence rate ratio: 2.5

Detectable incidence rate ratio, cohort study using time at risk data

Description

Sample size to determine the minimum detectable incidence rate ratio for a cohort study using time at risk data.

Usage

r_{D+ | E+}: Expected incidence rate of the outcome of interest in the exposed.
r_{D+ | E-}: Expected incidence rate of the outcome of interest in the unexposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Lemeshow S, Hosmer D, Klar J, Lwanga S (1990). Adequacy of Sample Size in Health Studies. John Wiley and Sons, New York.

Lwanga S, Lemeshow S (1991). Sample Size Determination in Health Studies. World Health Organization, Geneva.

Examples

Long term effect of noise

As part of a study of the long-term effect of noise on workers in a particularly noisy industry, it is planned to follow up a cohort of people who were recruited into the industry during a given period of time and to compare them with a similar cohort of individuals working in a much quieter industry. Subjects will be followed up for the rest of their lives or until their hearing is impaired. The results of a previous small-scale survey suggest that the annual incidence rate of hearing impairment in the noisy industry may be as high as 25%. How many people should be followed up in each of the groups (which are to 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 impairment in the quieter industry is not more than the national average of about 10% (for people in the same age range), whereas in the noisy industry it differs from this.

An annuual incidence rate of 25% is equivalent to 25 cases of hearing impairment per 100 individuals per year.

Total number of study subjects: 520
Total number of subjects in the exposed group: 260
Total number of subjects in the unexposed group: 260
Power: 0.80
Minimum detectable incidence rate ratio: 2.5

Detectable odds ratio, case-control study

Description

Sample size to determine the minimum detectable odds ratio for a case-control study.

Usage

OR: Expected odds ratio.
p_{E+ | D+}: Expected prevalence of exposure among cases.
p_{E+ | D-}: Expected prevalence of exposure among controls.
r: The number in the control group divided by the number in the case group.

This function implements the methodology to estimate a sample size for an unmatched case-control study described by Dupont (1988).

A value for p_{E+ | D+} (the expected prevalence of exposure among cases) is only required if Fleiss' correction is requested.

References

Dupont WD (1988). Power calculations for matched case-control studies. Biometrics 44: 1157 - 1168.

Fleiss JL, Levin B, Paik MC (2003). Statistical Methods for Rates and Proportions. John Wiley and Sons, New York.

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Smoking and coronary heart disease

A case-control study of the relationship between smoking and coronary heart disease (CHD) is planned. A sample of men with newly diagnosed CHD will be compared for smoking status with a sample of controls. Assuming an equal number of cases and controls, how many study subject are required to detect an odds ratio of 2.0 with 0.90 power using a two-sided 0.05 test? Previous surveys have shown that around 0.30 of males without CHD and 0.50 of males with CHD are smokers.

Total number of study subjects: 376
Total number of cases: 188
Total number of controls: 188
Power: 0.90
Minimum detectable odds ratio: 2.0

BCG vaccination

The efficacy of BCG vaccine in preventing childhood tuberculosis is in doubt and a study is designed to compare the immunisation coverage rates in a group of tuberculosis cases compared to a group of controls. Available information indicates that roughly 30% of the controls are not vaccinated, and we wish to have an 80% chance of detecting whether the odds ratio is significantly different from 1 at the 5% level. If an odds ratio of 2.0 would be considered an important difference between the two groups, how large a sample should be included in each study group?

Total number of study subjects: 282
Total number of cases: 141
Total number of controls: 141
Power: 0.80
Minimum detectable odds ratio: 2.0

Detectable odds ratio, case-control study

Description

Sample size to determine the minimum detectable odds ratio for a case-control study.

Usage

OR: Expected odds ratio.
p_{E+ | D+}: Expected prevalence of exposure among cases.
p_{E+ | D-}: Expected prevalence of exposure among controls.
r: The number in the control group divided by the number in the case group.

This function implements the methodology to estimate a sample size for an unmatched case-control study described by Dupont (1988).

A value for p_{E+ | D+} (the expected prevalence of exposure among cases) is only required if Fleiss' correction is requested.

References

Dupont WD (1988). Power calculations for matched case-control studies. Biometrics 44: 1157 - 1168.

Fleiss JL, Levin B, Paik MC (2003). Statistical Methods for Rates and Proportions. John Wiley and Sons, New York.

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Smoking and coronary heart disease

A case-control study of the relationship between smoking and coronary heart disease (CHD) is planned. A sample of men with newly diagnosed CHD will be compared for smoking status with a sample of controls. Assuming an equal number of cases and controls, how many study subject are required to detect an odds ratio of 2.0 with 0.90 power using a two-sided 0.05 test? Previous surveys have shown that around 0.30 of males without CHD and 0.50 of males with CHD are smokers.

Total number of study subjects: 376
Total number of cases: 188
Total number of controls: 188
Power: 0.90
Minimum detectable odds ratio: 2.0

BCG vaccination

The efficacy of BCG vaccine in preventing childhood tuberculosis is in doubt and a study is designed to compare the immunisation coverage rates in a group of tuberculosis cases compared to a group of controls. Available information indicates that roughly 30% of the controls are not vaccinated, and we wish to have an 80% chance of detecting whether the odds ratio is significantly different from 1 at the 5% level. If an odds ratio of 2.0 would be considered an important difference between the two groups, how large a sample should be included in each study group?

Total number of study subjects: 282
Total number of cases: 141
Total number of controls: 141
Power: 0.80
Minimum detectable odds ratio: 2.0

Comparison of binary outcomes

Description

Sample size to detect a difference in binary outcomes.

Usage

p_{D+ | E+}: Expected incidence or prevalence of the outcome of interest in the exposed.
p_{D+ | E-}: Expected incidence or prevalence of the outcome of interest in the unexposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Fleiss JL (1981). Statistical Methods for Rates and Proportions. Wiley, New York.

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

The prevalence of smoking

A government initiative has decided to reduce the prevalence of male smoking to, at most, 30%. A sample survey is planned to test, at the 0.05 level, the hypothesis that the percentage of smokers in the male population is 30% against the one-sided alternative that it is greater. The survey should be able to find a prevalence of 32%, when it is true, with 0.90 power. How many men need to be sampled?

Total number of study subjects: 3286
Total number in treatment (exposed) group: 1643
Total number in control (unexposed) group: 1643
Power: 0.80
Minimum detectable incidence risk ratio: 0.94

Comparison of binary outcomes

Description

Sample size to detect a difference in binary outcomes.

Usage

p_{D+ | E+}: Expected incidence or prevalence of the outcome of interest in the exposed.
p_{D+ | E-}: Expected incidence or prevalence of the outcome of interest in the unexposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Fleiss JL (1981). Statistical Methods for Rates and Proportions. Wiley, New York.

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

The prevalence of smoking

A government initiative has decided to reduce the prevalence of male smoking to, at most, 30%. A sample survey is planned to test, at the 0.05 level, the hypothesis that the percentage of smokers in the male population is 30% against the one-sided alternative that it is greater. The survey should be able to find a prevalence of 32%, when it is true, with 0.90 power. How many men need to be sampled?

Total number of study subjects: 3286
Total number in treatment (exposed) group: 1643
Total number in control (unexposed) group: 1643
Power: 0.80
Minimum detectable incidence risk ratio: 0.94

Comparison of continuous outcomes

Description

Sample size to detect a difference in continuous outcomes.

Usage

x̄_E+: Expected mean of the outcome of interest in the exposed.
x̄_E-: Expected mean of the outcome of interest in the unexposed.
σ: Expected standard deviation of the outcome of interest in the population.
r: The number in the exposed group divided by the number in the unexposed group.

References

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Cholesterol

Supposed we wish to test, at the 5% level of significance, the hypothesis that cholesterol means in a population are equal in two study years against the one-sided alternative that the mean is higher in the second of the two years. Suppose that equal sized samples will be taken in each year, but that these will not necessarily be from the same individuals (i.e., the two samples are drawn independently). Our test is to have a power of 0.95 at detecting a difference of 0.5 mmol/L. The standard deviation of serum cholesterol in humans is assumed to be 1.4 mmol/L. We expect serum cholesterol in the exposed and unexposed group to be 6.0 mmol/L and 5.5 mmol/L, respectively.

Total number of study subjects: 270
Total number in treatment (exposed) group: 135
Total number in control (unexposed) group: 135
Power: 0.90

Comparison of continuous outcomes

Description

Sample size to detect a difference in continuous outcomes.

Usage

x̄_E+: Expected mean of the outcome of interest in the exposed.
x̄_E-: Expected mean of the outcome of interest in the unexposed.
σ: Expected standard deviation of the outcome of interest in the population.
r: The number in the exposed group divided by the number in the unexposed group.

References

Kelsey JL, Thompson WD, Evans AS (1986). Methods in Observational Epidemiology. Oxford University Press, London, pp. 254 - 284.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Cholesterol

Supposed we wish to test, at the 5% level of significance, the hypothesis that cholesterol means in a population are equal in two study years against the one-sided alternative that the mean is higher in the second of the two years. Suppose that equal sized samples will be taken in each year, but that these will not necessarily be from the same individuals (i.e., the two samples are drawn independently). Our test is to have a power of 0.95 at detecting a difference of 0.5 mmol/L. The standard deviation of serum cholesterol in humans is assumed to be 1.4 mmol/L. We expect serum cholesterol in the exposed and unexposed group to be 6.0 mmol/L and 5.5 mmol/L, respectively.

Total number of study subjects: 270
Total number in treatment (exposed) group: 135
Total number in control (unexposed) group: 135
Power: 0.90

Comparison of time to event

Description

Sample size to detect a difference in time to event.

Usage

p_{D+ | E+}: Expected proportion outcome positive at the end of the follow-up period among those exposed.
p_{D+ | E-}: Expected proportion outcome positive at the end of the follow-up period among those not exposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Therneau TM, Grambsch PM (2001). Modelling Survival Data - Extending the Cox Model. Springer, London, pp. 61 - 65.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Survival probabilities

The 5-year survival probability of patients receiving a standard treatment is 0.30 and we anticipate that a new treatment will increase it to 0.45. Assume that a study will use a two-sided test at the 0.05 level with 0.90 power to detect this difference. How many events are required?

Total number of study subjects: 250
Total number in treatment (exposed) group: 125
Total number in control (unexposed) group: 125
Power: 0.90

Comparison of time to event

Description

Sample size to detect a difference in time to event.

Usage

p_{D+ | E+}: Expected proportion outcome positive at the end of the follow-up period among those exposed.
p_{D+ | E-}: Expected proportion outcome positive at the end of the follow-up period among those not exposed.
r: The number in the exposed group divided by the number in the unexposed group.

References

Therneau TM, Grambsch PM (2001). Modelling Survival Data - Extending the Cox Model. Springer, London, pp. 61 - 65.

Woodward M (2014). Epidemiology Study Design and Data Analysis. Chapman & Hall/CRC, New York, pp. 295 - 329.

Examples

Survival probabilities

The 5-year survival probability of patients receiving a standard treatment is 0.30 and we anticipate that a new treatment will increase it to 0.45. Assume that a study will use a two-sided test at the 0.05 level with 0.90 power to detect this difference. How many events are required?

Total number of study subjects: 250
Total number in treatment (exposed) group: 125
Total number in control (unexposed) group: 125
Power: 0.90

Parallel equivalence trial, binary outcome

Description

Sample size for a parallel equivalence trial, binary outcome.

Usage

p_{D+ | E+}: Expected proportion of exposure positive (`new treatment') subjects that are outcome positive.
p<sub>D+ | E-</sub>: Expected proportion of exposure negative (`standard treatment' or control) subjects that are outcome positive.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For an equivalence trial the null hypothesis is:

H₀: |p_{D+ | E-} - p_{D+ | E+}| ≥ δ

The alternative hypothesis is:

H₁: |p_{D+ | E-} - p_{D+ | E+}| < δ

An equivalence trial is used if want to prove that two treatments produce the same clinical outcomes. The value of the maximum acceptable difference δ is chosen so that a patient will not detect any change in effect when replacing the standard treatment with the new treatment.

References

Bennett J, Dismukes W, Duma R, Medoff G, Sande M, Gallis H, Leonard J, Fields B, Bradshaw M, Haywood H, McGee Z, Cate T, Cobbs C, Warner J, Alling D (1979). A comparison of amphotericin B alone and combined with flucytosine in the treatment of cryptoccal meningitis. New England Journal of Medicine 301, 126 - 131.

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 91.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Combination chemotherapy

Bennett, Dismukes, Duma et al. (1979) designed a clinical trial to test whether combination chemotherapy for a shorter period would be at least as good as conventional therapy for patients with cryptococcal meningitis. They recruited 39 patients to each treatment arm and wished to conclude that a difference of less than 20% in response rate between the treatments would indicate equivalence. Assuming a one-sided test size of 10% and a power of 80%, what would be an appropriate sample size if the trial were to be repeated?

Total number of study subjects: 166
Total number in treatment group: 83
Total number in control group: 83
Power: 0.80
Equivalence limit: 0.20

Parallel equivalence trial, binary outcome

Description

Sample size for a parallel equivalence trial, binary outcome.

Usage

p_{D+ | E+}: Expected proportion of exposure positive (`new treatment') subjects that are outcome positive.
p<sub>D+ | E-</sub>: Expected proportion of exposure negative (`standard treatment' or control) subjects that are outcome positive.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For an equivalence trial the null hypothesis is:

H₀: |p_{D+ | E-} - p_{D+ | E+}| ≥ δ

The alternative hypothesis is:

H₁: |p_{D+ | E-} - p_{D+ | E+}| < δ

An equivalence trial is used if want to prove that two treatments produce the same clinical outcomes. The value of the maximum acceptable difference δ is chosen so that a patient will not detect any change in effect when replacing the standard treatment with the new treatment.

References

Bennett J, Dismukes W, Duma R, Medoff G, Sande M, Gallis H, Leonard J, Fields B, Bradshaw M, Haywood H, McGee Z, Cate T, Cobbs C, Warner J, Alling D (1979). A comparison of amphotericin B alone and combined with flucytosine in the treatment of cryptoccal meningitis. New England Journal of Medicine 301, 126 - 131.

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 91.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Combination chemotherapy

Bennett, Dismukes, Duma et al. (1979) designed a clinical trial to test whether combination chemotherapy for a shorter period would be at least as good as conventional therapy for patients with cryptococcal meningitis. They recruited 39 patients to each treatment arm and wished to conclude that a difference of less than 20% in response rate between the treatments would indicate equivalence. Assuming a one-sided test size of 10% and a power of 80%, what would be an appropriate sample size if the trial were to be repeated?

Total number of study subjects: 166
Total number in treatment group: 83
Total number in control group: 83
Power: 0.80
Equivalence limit: 0.20

Parallel equivalence trial, continuous outcome

Description

Sample size for a parallel equivalence trial, continuous outcome.

Usage

x̄_{D+ | E+}: Expected mean of exposure positive (`new treatment') subjects that are outcome positive.
x̄<sub>D+ | E-</sub>: Expected mean of exposure negative (`standard treatment' or control) subjects that are outcome positive.
σ: Expected standard deviation of the outcome of interest in the population.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For an equivalence trial the null hypothesis is:

H₀: |x̄_{D+ | E-} - x̄_{D+ | E+}| ≥ δ

The alternative hypothesis is:

H₁: |x̄_{D+ | E-} - x̄_{D+ | E+}| < δ

An equivalence trial is used if want to prove that two treatments produce the same clinical outcomes. The value of the maximum acceptable difference δ is chosen so that a patient will not detect any change in effect when replacing the standard treatment with the new treatment.

References

Bennett J, Dismukes W, Duma R, Medoff G, Sande M, Gallis H, Leonard J, Fields B, Bradshaw M, Haywood H, McGee Z, Cate T, Cobbs C, Warner J, Alling D (1979). A comparison of amphotericin B alone and combined with flucytosine in the treatment of cryptoccal meningitis. New England Journal of Medicine 301, 126 - 131.

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 91.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Blood pressure

It is anticipated that patients on a particular drug have a mean diastolic blood pressure of 96 mmHg, as against 94 mmHg on an alternative. It is also anticipated that the standard deviation of diastolic BP is approximately 8 mmHg. If one wishes to confirm that the difference is likely to be less than 5 mmHg, that is, one wishes to show equivalence, how many patients are needed to be enrolled in the trial? Assume 80% power and 95% significance.

Total number of study subjects: 244
Total number in treatment group: 122
Total number in control group: 122
Power: 0.80
Equivalence limit: 5

Parallel equivalence trial, continuous outcome

Description

Sample size for a parallel equivalence trial, continuous outcome.

Usage

x̄_{D+ | E+}: Expected mean of exposure positive (`new treatment') subjects that are outcome positive.
x̄<sub>D+ | E-</sub>: Expected mean of exposure negative (`standard treatment' or control) subjects that are outcome positive.
σ: Expected standard deviation of the outcome of interest in the population.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For an equivalence trial the null hypothesis is:

H₀: |x̄_{D+ | E-} - x̄_{D+ | E+}| ≥ δ

The alternative hypothesis is:

H₁: |x̄_{D+ | E-} - x̄_{D+ | E+}| < δ

An equivalence trial is used if want to prove that two treatments produce the same clinical outcomes. The value of the maximum acceptable difference δ is chosen so that a patient will not detect any change in effect when replacing the standard treatment with the new treatment.

References

Bennett J, Dismukes W, Duma R, Medoff G, Sande M, Gallis H, Leonard J, Fields B, Bradshaw M, Haywood H, McGee Z, Cate T, Cobbs C, Warner J, Alling D (1979). A comparison of amphotericin B alone and combined with flucytosine in the treatment of cryptoccal meningitis. New England Journal of Medicine 301, 126 - 131.

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 91.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Blood pressure

It is anticipated that patients on a particular drug have a mean diastolic blood pressure of 96 mmHg, as against 94 mmHg on an alternative. It is also anticipated that the standard deviation of diastolic BP is approximately 8 mmHg. If one wishes to confirm that the difference is likely to be less than 5 mmHg, that is, one wishes to show equivalence, how many patients are needed to be enrolled in the trial? Assume 80% power and 95% significance.

Total number of study subjects: 244
Total number in treatment group: 122
Total number in control group: 122
Power: 0.80
Equivalence limit: 5

Parallel superiority trial, binary outcome

Description

Sample size for a parallel superiority trial, binary outcome.

Usage

p_{D+ | E+}: Expected proportion of exposure positive ('new treatment') subjects that are outcome positive.
p_{D+ | E-}: Expected proportion of exposure negative ('standard treatment' or control) subjects that are outcome positive.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

Consider a clinical trial comparing two groups, a standard treatment (s) and a new treatment (n). A proportion of subjects in the standard treatment group experience the outcome of interest P_s and a proportion of subjects in the new treatment group experience the outcome of interest P_n. We specify the absolute value of the maximum acceptable difference between P_s and P_n as δ. For a superiority trial the value entered for δ must be greater than or equal to zero.

For a superiority trial the null hypothesis is:

H₀: |p_{D+ | E-} - p_{D+ | E+}| = 0

The alternative hypothesis is:

H₁: p_{D+ | E-} - p_{D+ | E+} ≠ 0

References

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 90.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Pocock SJ (1983). Clinical Trials: A Practical Approach. Wiley, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Antimicrobial agents

Suppose that a pharmaceutical company is interested in conducting a clinical trial to compare the efficacy of two antimicrobial agents when administered orally once daily in the treatment of patients with skin infections. In what follows, we consider the situation where the intended trial is for testing superiority of the test drug over the active control drug. For this purpose, the following assumptions are made. First, sample size calculation will be performed for achieving 80% power at the 5% level of significance. Assume the true mean cure rates of the treatment agents and the active control are 85% and 65%, respectively. Assume the superiority margin is 5%.

Total number of study subjects: 144
Total number in treatment group: 72
Total number in control group: 72
Power: 0.80
Equivalence limit: 0.05

Parallel superiority trial, binary outcome

Description

Sample size for a parallel superiority trial, binary outcome.

Usage

p_{D+ | E+}: Expected proportion of exposure positive ('new treatment') subjects that are outcome positive.
p_{D+ | E-}: Expected proportion of exposure negative ('standard treatment' or control) subjects that are outcome positive.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

Consider a clinical trial comparing two groups, a standard treatment (s) and a new treatment (n). A proportion of subjects in the standard treatment group experience the outcome of interest P_s and a proportion of subjects in the new treatment group experience the outcome of interest P_n. We specify the absolute value of the maximum acceptable difference between P_s and P_n as δ. For a superiority trial the value entered for δ must be greater than or equal to zero.

For a superiority trial the null hypothesis is:

H₀: |p_{D+ | E-} - p_{D+ | E+}| = 0

The alternative hypothesis is:

H₁: p_{D+ | E-} - p_{D+ | E+} ≠ 0

References

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 90.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Pocock SJ (1983). Clinical Trials: A Practical Approach. Wiley, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Antimicrobial agents

Suppose that a pharmaceutical company is interested in conducting a clinical trial to compare the efficacy of two antimicrobial agents when administered orally once daily in the treatment of patients with skin infections. In what follows, we consider the situation where the intended trial is for testing superiority of the test drug over the active control drug. For this purpose, the following assumptions are made. First, sample size calculation will be performed for achieving 80% power at the 5% level of significance. Assume the true mean cure rates of the treatment agents and the active control are 85% and 65%, respectively. Assume the superiority margin is 5%.

Total number of study subjects: 144
Total number in treatment group: 72
Total number in control group: 72
Power: 0.80
Equivalence limit: 0.05

Parallel superiority trial, continuous outcome

Description

Sample size for a parallel superiority trial, continuous outcome.

Usage

x̄_{D+ | E+}: Expected mean of exposure positive ('new treatment') subjects.
x̄_{D+ | E-}: Expected mean of exposure negative ('standard treatment' or control) subjects.
σ: Expected standard deviation of the outcome of interest in the population.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For a superiority trial the null hypothesis is:

H₀: |x̄_{D+ | E-} - x̄_{D+ | E+}| = 0

The alternative hypothesis is:

H₁: x̄_{D+ | E-} - x̄_{D+ | E+} ≠ 0

References

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 61.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Pocock SJ (1983). Clinical Trials: A Practical Approach. Wiley, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Cholesterol

A pharmaceutical company is interested in conducting a clinical trial to compare two cholesterol lowering agents for treatment of patients with coronary heart disease (CHD) using a parallel design. The primary efficacy parameter is the concentration of high density lipoproteins (HDL). We consider the situation where the intended trial is to test superiority of the test drug over the active control agent. Sample size calculations are to be calculated to achieve 80% power at the 5% level of significance.

In this example, we assume that if treatment results in a 5 unit (i.e., δ = 5) increase in HDL it is declared to be superior to the active control. Assume the standard deviation of HDL is 10 units and the HDL concentration in the treatment group is 20 units and the HDL concentration in the control group is 20 units.

Total number of study subjects: 100
Total number in treatment group: 50
Total number in control group: 50
Power: 0.80
Equivalence limit: 5

Parallel superiority trial, continuous outcome

Description

Sample size for a parallel superiority trial, continuous outcome.

Usage

x̄_{D+ | E+}: Expected mean of exposure positive ('new treatment') subjects.
x̄_{D+ | E-}: Expected mean of exposure negative ('standard treatment' or control) subjects.
σ: Expected standard deviation of the outcome of interest in the population.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For a superiority trial the null hypothesis is:

H₀: |x̄_{D+ | E-} - x̄_{D+ | E+}| = 0

The alternative hypothesis is:

H₁: x̄_{D+ | E-} - x̄_{D+ | E+} ≠ 0

References

Chow S, Shao J, Wang H (2017). Sample Size Calculations in Clinical Research. Chapman & Hall/CRC Biostatistics Series, pp. 61.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Pocock SJ (1983). Clinical Trials: A Practical Approach. Wiley, New York.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Examples

Cholesterol

A pharmaceutical company is interested in conducting a clinical trial to compare two cholesterol lowering agents for treatment of patients with coronary heart disease (CHD) using a parallel design. The primary efficacy parameter is the concentration of high density lipoproteins (HDL). We consider the situation where the intended trial is to test superiority of the test drug over the active control agent. Sample size calculations are to be calculated to achieve 80% power at the 5% level of significance.

In this example, we assume that if treatment results in a 5 unit (i.e., δ = 5) increase in HDL it is declared to be superior to the active control. Assume the standard deviation of HDL is 10 units and the HDL concentration in the treatment group is 20 units and the HDL concentration in the control group is 20 units.

Total number of study subjects: 100
Total number in treatment group: 50
Total number in control group: 50
Power: 0.80
Equivalence limit: 5

Parallel non-inferiority trial, binary outcome

Description

Sample size for a parallel non-inferiority trial, binary outcome.

Usage

p_{D+ | E+}: Expected proportion of exposure positive ('new treatment') subjects that are outcome positive.
p_{D+ | E-}: Expected proportion of exposure negative ('standard treatment' or control) subjects that are outcome positive.
δ: The equivalence limit: the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

Consider a clinical trial comparing two groups, a standard treatment (s) and a new treatment (n). A proportion of subjects in the standard treatment group experience the outcome of interest P_s and a proportion of subjects in the new treatment group experience the outcome of interest P_n. We specify the absolute value of the maximum acceptable difference between P_n and P_s as δ. For a non-inferiority trial the value entered for δ must be greater than or equal to zero.

For a non-inferiority trial the null hypothesis is:

H₀: p_{D+ | E-} - p_{D+ | E+} ≥ δ

The alternative hypothesis is:

H₁: p_{D+ | E-} - p_{D+ | E+} < δ

The aim of a non-inferiority trial is show that a new treatment is not (much) inferior to a standard treatment. Showing non-inferiority can be of interest because: (a) it is often not ethically possible to do a placebo-controlled trial; (b) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is safer; (c) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is cheaper to produce or easier to administer; and (d) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints in clinical trial, but compliance will be better outside the clinical trial and hence efficacy better outside the trial.

References

Blackwelder WC (1982). Proving the null hypothesis in clinical trials. Controlled Clinical Trials 3: 345 - 353.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Scott IA (2009). Non-inferiority trials: determining whether alternative treatments are good enough. Medical Journal of Australia 190: 326 - 330.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Zhong B (2009). How to calculate sample size in randomized controlled trial? Journal of Thoracic Disease 1: 51 - 54.

Examples

Skin infections

A pharmaceutical company would like to conduct a clinical trial to compare the efficacy of two antimicrobial agents when administered orally to patients with skin infections. Assume the true mean cure rate of the treatment is 0.85 and the true mean cure rate of the control is 0.65. We consider the proportion cured in the treatment group minus the proportion cured in the control group (i.e., delta) of 0.10 or less to be of no clinical significance.

Assuming a one-sided test size of 5% and a power of 80% how many subjects should be included in the trial?

Total number of study subjects: 50
Total number in treatment group: 25
Total number in control group: 25
Power: 0.80
Equivalence limit: 0.10

Parallel non-inferiority trial, binary outcome

Description

Sample size for a parallel non-inferiority trial, binary outcome.

Usage

p_{D+ | E+}: Expected proportion of exposure positive ('new treatment') subjects that are outcome positive.
p_{D+ | E-}: Expected proportion of exposure negative ('standard treatment' or control) subjects that are outcome positive.
δ: The equivalence limit: the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

Consider a clinical trial comparing two groups, a standard treatment (s) and a new treatment (n). A proportion of subjects in the standard treatment group experience the outcome of interest P_s and a proportion of subjects in the new treatment group experience the outcome of interest P_n. We specify the absolute value of the maximum acceptable difference between P_n and P_s as δ. For a non-inferiority trial the value entered for δ must be greater than or equal to zero.

For a non-inferiority trial the null hypothesis is:

H₀: p_{D+ | E-} - p_{D+ | E+} ≥ δ

The alternative hypothesis is:

H₁: p_{D+ | E-} - p_{D+ | E+} < δ

The aim of a non-inferiority trial is show that a new treatment is not (much) inferior to a standard treatment. Showing non-inferiority can be of interest because: (a) it is often not ethically possible to do a placebo-controlled trial; (b) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is safer; (c) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is cheaper to produce or easier to administer; and (d) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints in clinical trial, but compliance will be better outside the clinical trial and hence efficacy better outside the trial.

References

Blackwelder WC (1982). Proving the null hypothesis in clinical trials. Controlled Clinical Trials 3: 345 - 353.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Scott IA (2009). Non-inferiority trials: determining whether alternative treatments are good enough. Medical Journal of Australia 190: 326 - 330.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Zhong B (2009). How to calculate sample size in randomized controlled trial? Journal of Thoracic Disease 1: 51 - 54.

Examples

Skin infections

A pharmaceutical company would like to conduct a clinical trial to compare the efficacy of two antimicrobial agents when administered orally to patients with skin infections. Assume the true mean cure rate of the treatment is 0.85 and the true mean cure rate of the control is 0.65. We consider the proportion cured in the treatment group minus the proportion cured in the control group (i.e., delta) of 0.10 or less to be of no clinical significance.

Assuming a one-sided test size of 5% and a power of 80% how many subjects should be included in the trial?

Total number of study subjects: 50
Total number in treatment group: 25
Total number in control group: 25
Power: 0.80
Equivalence limit: 0.10

Parallel non-inferiority trial, continuous outcome

Description

Sample size for a parallel non-inferiority trial, continuous outcome.

Usage

x̄_{D+ | E+}: Expected mean of exposure positive (`new treatment') subjects.
x̄<sub>D+ | E-</sub>: Expected mean of exposure negative (`standard treatment' or control) subjects.
σ: Expected standard deviation of the outcome of interest in the population.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For a non-inferiority trial the null hypothesis is:

H₀: x̄_{D+ | E-} - x̄_{D+ | E+} ≥ δ

The alternative hypothesis is:

H₁: x̄_{D+ | E-} - x̄_{D+ | E+} < δ

The aim of a non-inferiority trial is show that a new treatment is not (much) inferior to a standard treatment. Showing non-inferiority can be of interest because: (a) it is often not ethically possible to do a placebo-controlled trial; (b) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is safer; (c) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is cheaper to produce or easier to administer; and (d) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints in clinical trial, but compliance will be better outside the clinical trial and hence efficacy better outside the trial.

References

Blackwelder WC (1982). Proving the null hypothesis in clinical trials. Controlled Clinical Trials 3: 345 - 353.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Scott IA (2009). Non-inferiority trials: determining whether alternative treatments are good enough. Medical Journal of Australia 190: 326 - 330.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Zhong B (2009). How to calculate sample size in randomized controlled trial? Journal of Thoracic Disease 1: 51 - 54.

Examples

Cholesterol lowering agents

A pharmaceutical company is interested in conducting a clinical trial to compare two cholesterol lowering agents for treatment of patients with coronary heart disease (CHD) using a parallel design. The primary efficacy parameter is the low density lipoproteins (LDL). In what follows, we will consider the situation where the intended trial is for testing non-inferiority of mean responses in LDL. Assume that 80% power is required at a 5% level of significance.

In this example we assume a -0.05 unit change in LDL is a clinically meaningful difference. Assume the standard deviation of LDL is 0.10 units and the LDL concentration in the treatment group is 0.20 units and the LDL concentration in the control group is 0.20 units.

Total number of study subjects: 100
Total number in treatment group: 50
Total number in control group: 50
Power: 0.80
Equivalence limit: 0.05

Parallel non-inferiority trial, continuous outcome

Description

Sample size for a parallel non-inferiority trial, continuous outcome.

Usage

x̄_{D+ | E+}: Expected mean of exposure positive (`new treatment') subjects.
x̄<sub>D+ | E-</sub>: Expected mean of exposure negative (`standard treatment' or control) subjects.
σ: Expected standard deviation of the outcome of interest in the population.
δ: The equivalence limit --- the absolute change in the outcome of interest that represents a clinically meaningful difference.
r: The number in the exposure negative (control) group divided by the number in the exposure positive (treatment) group.
α: The desired alpha level.

For a non-inferiority trial the null hypothesis is:

H₀: x̄_{D+ | E-} - x̄_{D+ | E+} ≥ δ

The alternative hypothesis is:

H₁: x̄_{D+ | E-} - x̄_{D+ | E+} < δ

The aim of a non-inferiority trial is show that a new treatment is not (much) inferior to a standard treatment. Showing non-inferiority can be of interest because: (a) it is often not ethically possible to do a placebo-controlled trial; (b) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is safer; (c) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints, but is cheaper to produce or easier to administer; and (d) the new treatment is not expected to be better than the standard treatment on primary efficacy endpoints in clinical trial, but compliance will be better outside the clinical trial and hence efficacy better outside the trial.

References

Blackwelder WC (1982). Proving the null hypothesis in clinical trials. Controlled Clinical Trials 3: 345 - 353.

Ewald B (2013). Making sense of equivalence and non-inferiority trials. Australian Prescriber 36: 170 - 173.

Julious SA (2004). Sample sizes for clinical trials with normal data. Statistics in Medicine 23: 1921 - 1986.

Julious SA (2009). Estimating Samples Sizes in Clinical Trials. CRC, New York.

Machin D, Campbell MJ, Tan SB, Tan SH (2009). Sample Size Tables for Clinical Studies. Wiley Blackwell, New York.

Scott IA (2009). Non-inferiority trials: determining whether alternative treatments are good enough. Medical Journal of Australia 190: 326 - 330.

Wang B, Wang H, Tu X, Feng C (2017). Comparisons of superiority, non-inferiority, and equivalence trials. Shanghai Archives of Psychiatry 29, 385 - 388.

Zhong B (2009). How to calculate sample size in randomized controlled trial? Journal of Thoracic Disease 1: 51 - 54.

Examples

Cholesterol lowering agents

A pharmaceutical company is interested in conducting a clinical trial to compare two cholesterol lowering agents for treatment of patients with coronary heart disease (CHD) using a parallel design. The primary efficacy parameter is the low density lipoproteins (LDL). In what follows, we will consider the situation where the intended trial is for testing non-inferiority of mean responses in LDL. Assume that 80% power is required at a 5% level of significance.

In this example we assume a -0.05 unit change in LDL is a clinically meaningful difference. Assume the standard deviation of LDL is 0.10 units and the LDL concentration in the treatment group is 0.20 units and the LDL concentration in the control group is 0.20 units.

Total number of study subjects: 100
Total number in treatment group: 50
Total number in control group: 50
Power: 0.80
Equivalence limit: 0.05

Sample size to achieve a desired surveillance system sensitivity assuming simple random sampling

Description

Sample size to achieve a desired surveillance system sensitivity assuming simple random sampling. The term `surveillance system sensitivity' in this context can be interpreted as the probability of detecting disease in a population if it is, in fact, present.

Usage

Se: Sensitivity.

References

Cannon RM (2001). Sense and sensitivity --- designing surveys based on an imperfect test. Preventive Veterinary Medicine 49: 141 - 163.

Dohoo I, Martin W, Stryhn H (2009). Veterinary Epidemiologic Research. AVC Inc, Charlottetown, Prince Edward Island, Canada, pp. 54.

MacDiarmid S (1988). Future options for brucellosis surveillance in New Zealand beef herds. New Zealand Veterinary Journal 36, 39 - 42.

Examples

Dairy herds

We would like to confirm the absence of disease in a single 1000-cow dairy herd. We expect the prevalence of disease in the herd to be 5% (if disease is present). We intend to use a single test with a sensitivity of 0.90 and a specificity of 1.00. How many cows should be tested to be 95% confident of detecting at least one that is disease positive assuming the diagnostic sensitivity and specificity of the test we're using is 0.95 and 1.00, respectively?

Surveillance units available for sampling: 1000
Surveillance units to sample to be 95% confident of detecting disease if surveillance unit Se is 90%: 65

Sample size to achieve a desired surveillance system sensitivity assuming two-stage sampling

Description

Sample size to achieve a desired surveillance system sensitivity assuming two-stage sampling. The term `surveillance system sensitivity' in this context can be interpreted as the probability of detecting disease in a population if it is, in fact, present.

Usage

Key:

PSU: Primary sampling unit (e.g., herds, flocks, classes, households).
SSU: Secondary sampling unit (e.g., cows within herds, chickens within flocks, children within classes, individuals within households).
Se: Sensitivity.

References

Cannon RM (2001). Sense and sensitivity — designing surveys based on an imperfect test. Preventive Veterinary Medicine 49: 141 - 163.

Dohoo I, Martin W, Stryhn H (2009). Veterinary Epidemiologic Research. AVC Inc, Charlottetown, Prince Edward Island, Canada, pp. 54.

MacDiarmid S (1988). Future options for brucellosis surveillance in New Zealand beef herds. New Zealand Veterinary Journal 36, 39 - 42.

Examples

Coxiellosis exposure in sheep

We would like to confirm the absence of coxiellosis exposure in flocks of sheep in a single farm. Sheep on the farm of interest are managed in 11 mobs (flocks). The average number of sheep within each flock is around 350. How many flocks and how many sheep within each flock should be tested to be 95% confident of detecting at least one animal that shows evidence of (exposure to) coxiellosis assuming the prevalence of coxiellosis-exposed flocks is 0.01 and, for flocks that are coxiellosis exposed, the within-flock prevalence of exposure is 0.15. Assume the diagnostic sensitivity and specificity of the test we're using is 0.95 and 1.00, respectively.

PSUs to sample to be 95% confident of detecting disease at the PSU level: 11
SSUs to sample within each PSU to be 95% confident of detecting disease at the SSU level: 21
Total SSUs to sample: 231