Means: Difference between two dependent means (matched pairs)

The null hypothesis of this test is that the population means μ_x and μ_y of two matched samples x and y are identical. The sampling method leads to N pairs (x_i, y_i) of matched observations.

The null hypothesis that μ_x = μ_y can be reformulated in terms of the difference z_i = x_i − y_i. The null hypothesis is then given by μ_z = 0. The alternative hypothesis states that μ_z has a value different from zero.

H₀ : μ_z = 0
H₁ : μ_z = c, c ≠ 0

If the sign of μ_z cannot be predicted a priori then a two-sided test should be used. Otherwise you should use the one-sided test.

Effect size index

The effect size index d_z is defined as:

d_z = |μ_z |/σ_z = |μ_x − μ_y|/√(σ_x² + σ_y² − 2ρ_xy · σ_x · σ_y)

where μ_x and μ_y denote the population means, σ_x and σ_y denote the standard deviation in either population, and ρ_xy denotes the correlation between the two random variables. μ_z and σ_z are the population mean and standard deviation of the difference z.

A click on the Determine button to the left of the effect size label in the main window opens the effect size drawer. You can use this drawer to calculate d_z from the mean and standard deviation of the differences z_i. Alternatively you calculate d_z from the means μ_x and μ_y and the standard deviations σ_x and σ_y of the two random variables x and y as well as the correlation ρ_xy between the two random variables x and y.

Options

This test has no options..

Examples

Let us try to replicate the example in Cohen (1969, p. 48). The effect of two teaching methods on algebra achievements are compared between 50 IQ matched pairs of pupils (i.e., 100 pupils). The effect size that should be detected is d = (μ_x - μ_y)/σ = 0.4. Note that this is the effect size index representing differences between two independent means (two groups) . We want to use this effect size as a basis for a matched-pairs study. A sample estimate of the correlation between IQ-matched pairs in the population has been calculated to be r = 0.55. We thus assume ρ_xy = 0.55. What is the power of a two-sided test at an α level of 0.05?

To compute the effect size d_z we open the effect size drawer and choose From group parameters. We only know the ratio d = (μ_x - μ_y)/σ = 0.4. We are thus free to choose any values for the means and (equal) standard deviations that lead to this ratio. We set Mean group 1 = 0, Mean group 2 = 0.4, SD group 1 = 1, and SD group 2 = 1. Finally, we set the Correlation between groups to be 0.55. Pressing the Calculate and transfer to main window button copies the resulting effect size d_z = 0.421637 to the main window. We supply the remaining input values in the main window and press Calculate.

Select

Type of power analysis: Post hoc

Input

Tail(s): Two
Effect size dz: 0.421637
α err prob: 0.05
Total sample size: 50

Output

Noncentrality parameter δ: 2.981424
Critical t: 2.009575
df: 49
Power (1-β err prob): 0.832114

The computed power of 0.832114 is close to the value 0.84 estimated by Cohen using his tables. To estimate the increase in power due to the correlation between pairs (i.e., due to the shifting from a two-group design to a matched-pairs design), we enter Correlation between groups = 0 in the effect size drawer. This leads to d_z = 0.2828427. Repeating the above analysis with this effect size results in a power of only 0.500352.

How many subjects would we need to arrive at a power of about 0.832114 in a two-group design? We click X-Y plot for a range of values to open the Power Plot window. Let us plot (on the y axis) the power (with markers and displaying the values in the plot) as a function of the total sample size. We want to plot just 1 graph with the α err prob set to 0.05 and effect size dz fixed at 0.2828427. Clicking Draw plot yields a graph in which we can see  that we would need about 110 pairs, that is, 220 subjects. Thus, by matching pupils we cut down the required size of the sample by more than 50%.

Related tests

Difference between two independent means (two groups)
Means: Difference from constant (one-sample case)

Implementation notes

The H₀ distribution is the central Student t distribution t(N − 1, 0). The H₁ distribution is the noncentral Student t distribution t(N − 1, δ), with noncentrality parameter δ = d√N.

Validation

The results were checked against the values produced by GPower 2.0.

References

Cohen, J. (1969). Statistical power analysis for the behavioral sciences. New York, NY: Academic Press.