Pitman-Morgan Test for Paired Variances

The Pitman-Morgan test compares the variances of two paired measurements by testing whether the correlation between their difference and their sum is zero.

Pitman-Morgan 검정은 짝지은 두 측정의 차이와 합 사이 상관이 0 인지를 검정해 두 측정의 분산이 같은지 판정하는 방법입니다.

I measured the same subjects with two instruments; is one instrument's variance larger than the other's?

같은 대상을 두 기기로 측정했는데 한 기기의 측정값 분산이 더 큰지 어떻게 검정하나요?

For paired measurements X and Y, the covariance of D = X - Y and S = X + Y equals var(X) - var(Y), so the two variances are equal exactly when D and S are uncorrelated. The test uses t = r sqrt(n - 2) / sqrt(1 - r^2) on n - 2 degrees of freedom and also gives a confidence interval for the variance ratio.

When to use Pitman-Morgan Test for Paired Variances

• Two measurements are taken on the same units, such as before and after or two raters or instruments.
• You want to know whether variability changed between two paired conditions.
• You need a confidence interval for the ratio of two paired variances.

When to use something else

• The two samples are independent; use the F test or Levene's test.
• The data are strongly non-normal; use a rank-based or bootstrap comparison of spread.

Assumptions and requirements

• Pairs are independent of each other.
• The paired measurements are approximately bivariate normal.
• Both variables have non-zero variance.

What the analysis reports

• Variances of both measurements and their ratio
• Confidence interval for the variance ratio
• Correlation between difference and sum, t statistic, degrees of freedom and p value
• Shapiro-Wilk checks for each variable

How to report it

A Pitman-Morgan test showed that the variance of [variable 1] ([value]) differed from that of [variable 2] ([value]), variance ratio = [value], 95% CI [[lower], [upper]], t([df]) = [value], p = [value].

Related methods

• F Test for Equality of Two Variances
• Paired t-test
• Levene's Test

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