Newcombe Confidence Interval for Difference in Proportions

Usage

ci_prop_diff_nc(x, by, conf.level = 0.95, correct = FALSE, data = NULL)

Arguments

x: (binary/numeric/logical)
vector of a binary values, i.e. a logical vector, or numeric with values c(0, 1)
by: (string)
A character or factor vector with exactly two unique levels identifying the two groups to compare. Can also be a column name if a data frame provided in the data argument.
conf.level: (scalar numeric)
a scalar in (0,1) indicating the confidence level. Default is 0.95
correct: (logical)
apply continuity correction.
data: (data.frame)
Optional data frame containing the variables specified in x and by.

Value

An object containing the following components:

n: The number of responses for each group
N: The total number in each group
estimate: The point estimate of the difference in proportions
conf.low: Lower bound of the confidence interval
conf.high: Upper bound of the confidence interval
conf.level: The confidence level used
method: Anderson-Hauck Confidence Interval

Details

The Wilson (Score) confidence limits without continuity correction for each individual binomial proportion, $p_i = x_i / n_i$, for $i = 1, 2$, are given by:

$$ \frac{ (2 n_i \hat{p}_i + z^2) \pm z \sqrt{ 4 n_i \hat{p}_i (1 - \hat{p}_i) + z^2 } }{ 2 (n_i + z^2) } $$

Denote the lower and upper Wilson (Score) confidence limits for $p_i$ as $L_i$ and $U_i$, respectively.

Then, the Newcombe (Score) confidence limits for the difference in proportions ($p_1 - p_2$) are given by:

$$ \text{Lower limit: } (\hat{p}_1 - \hat{p}_2) - \sqrt{ (\hat{p}_1 - L_1)^2 + (U_2 - \hat{p}_2)^2 } $$

$$ \text{Upper limit: } (\hat{p}_1 - \hat{p}_2) + \sqrt{ (U_1 - \hat{p}_1)^2 + (\hat{p}_2 - L_2)^2 } $$

The confidence intervals with continuity correction for each individual binomial proportion are obtained using the Wilson (Score) confidence limits with continuity correction.

For each binomial proportion $p_i = x_i / n_i$, where $i = 1, 2$, the confidence intervals are given by:

$$ \frac{ 2 n_i \hat{p}_i + z^2 }{ 2 (n_i + z^2) } \; \pm \; \frac{ z }{ 2 (n_i + z^2) } \sqrt{ z^2 - \frac{2}{n_i} + 4 \hat{p}_i \left[ n_i (1 - \hat{p}_i) + 1 \right] } $$

References

Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions: Comparison of eleven methods. Statistics in Medicine, 17(8), 873–890. Constructing Confidence Intervals for the Differences of Binomial Proportions in SAS

Examples

responses <- expand(c(9, 3), c(10, 10))
arm <- rep(c("treat", "control"), times = c(10, 10))

# Calculate 95% confidence interval for difference in proportions
ci_prop_diff_nc(x = responses, by = arm)
#> 
#> ── Newcombe Confidence Interval without continuity correction ──────────────────
#> • 9/10 - 3/10
#> • Estimate: 0.6
#> • 95% Confidence Interval:
#>   (0.1705, 0.809)