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How to compute a confidence interval for a mean difference (matched pairs)

Description

Say we have two sets of data that are not independent of each other and come from a matched-pairs experiment, and we want to construct a confidence interval for the mean difference between these two samples. How do we make this confidence interval? Let’s assume we’ve chosen a confidence level of $\alpha$ = 0.05.

Related tasks:

Using NumPy and SciPy, in Python

View this solution alone.

We’ll use Numpy and SciPy to do some statistics later.

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import numpy as np
from scipy import stats

This example computes a 95% confidence interval, but you can choose a different level by choosing a different value for $\alpha$.

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alpha = 0.05

We have two samples of data, $x_1, x_2, x_3, \ldots, x_k$ and $x’_1, x’_2, x’_3, \ldots, x’_k$. We’re going to use some fake data below just as an example; replace it with your real data.

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sample1 = np.array([15, 10,  7, 22, 17, 14])
sample2 = np.array([ 9,  1, 11, 13,  3,  6])

And now the computations:

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diff_samples = sample1 - sample2                        # differences between the samples
n = len(sample1)                                        # number of observations per sample
diff_mean = np.mean(diff_samples)                       # mean of the differences
diff_variance = np.var( diff_samples, ddof=1 )          # variance of the differences
critical_val = stats.t.ppf(q = 1-alpha/2, df = n - 1)   # critical value
radius = critical_val*np.sqrt(diff_variance)/np.sqrt(n) # radius of confidence interval
( diff_mean - radius, diff_mean + radius )              # confidence interval

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(0.7033861582274517, 13.296613841772547)

Our 95% confidence interval for the mean difference is $[0.70338, 13.2966]$.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

Solution, in R

View this solution alone.

We have two samples of data, $x_1, x_2, x_3, \ldots, x_k$ and $x’_1, x’_2, x’_3, \ldots, x’_k$. We’re going to use some fake data below just as an example; replace it with your real data.

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sample.1 <- c(15, 10, 7, 22, 17, 14)
sample.2 <- c(9, 1, 11, 13, 3, 6)

The shortest way to create the confidence interval is with R’s t.test() function. It’s just one line of code (after we choose $\alpha$).

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alpha <- 0.05       # replace with your chosen alpha (here, a 95% confidence level)
t.test(sample.1, sample.2, paired = TRUE, conf.level = 1-alpha)

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	Paired t-test

data:  sample.1 and sample.2
t = 2.8577, df = 5, p-value = 0.0355
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
  0.7033862 13.2966138
sample estimates:
mean difference 
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If you need the lower and upper bounds later, you can save them as variables as follows.

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conf.interval <- t.test(sample.1, sample.2, paired = TRUE, conf.level = 1-alpha)
lower.bound <- conf.interval$conf.int[1]
upper.bound <- conf.interval$conf.int[2]

It’s also possible to do the computation manually, using the code below.

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diff.samples <- sample.1 - sample.2                # differences between the samples
n = length(sample.1)                               # number of observations per sample
diff.mean <- mean(diff.samples)                    # mean of the differences
diff.variance <- var( diff.samples )               # variance of the differences
critical.val <- qt(p = alpha/2, df = n - 1,
    lower.tail=FALSE)                              # critical value
radius <- critical.val*sqrt(diff.variance)/sqrt(n) # radius of confidence interval
c( diff.mean - radius, diff.mean + radius )        # confidence interval

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[1]  0.7033862 13.2966138

Either method gives the same result. Our 95% confidence interval is $[0.70338, 13.2966]$.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

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