The sign test for matched pairs offers a robust method for comparing two paired samples, often for the effects of an intervention in one of them. This method is very useful in situations where the underlying distribution of the data is unknown. The test compares two related samples—often pre- and post-treatment measurements on the same subjects—to determine if there are significant differences in their median values.
To conduct the sign test, we first calculate the differences in value between each pair of observations. The essence of the test lies in analyzing the signs (positive, negative, or zero) of these differences rather than their magnitudes. The process involves counting the number of positive and negative signs, disregarding any pairs where the difference is zero as they do not contribute to the conclusion of the test. In this case, the null hypothesis (H0) posits that the median difference between the matched pairs is zero, implying no effect from the treatment or intervention. Conversely, the alternative hypothesis (H1) suggests that the median difference is not equal to zero, indicating a significant effect of the treatment.
The outcome of the test is determined by the comparison of the count of the less frequent sign against a critical value from a pre-determined significance level, usually 0.05. If the count is less than or equal to the critical value, the result is statistically significant, leading to the rejection of the null hypothesis at a 95% confidence level. The sign test for matched pairs is particularly valuable for its simplicity and applicability to small sample sizes or when the normality of the distribution cannot be assumed, making it a versatile tool in statistical analysis.
Consider the data on hamsters' speed before and after administering a stimulant, which constitutes a matched pair. How does one test the claim that the stimulant impacts the hamster's speed?
Since the population distribution is unknown, the sign test—a nonparametric test—can be used.
The central concept of the sign test is that matched pairs with equal medians have approximately equal numbers of positive and negative signs.
First, convert the raw data into signs by subtracting the value of the second variable from the first variable. After excluding zeros, eight signs are obtained: three positive and five negative.
The null hypothesis posits that the median of the differences is zero, whereas the alternative hypothesis suggests otherwise.
Since n is less than 25, the test statistic x is the number of the less frequent sign. From the table, the critical value is zero at a significance level of 0.05.
Since the test statistic value is not less than or equal to the critical value, there is insufficient evidence to reject the claim that the median of the differences is zero.
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