Sample Size Calculator

Partial correlation

Determines the sample size required to detect a specified partial correlation between two variables after controlling for one or more covariates, based on the anticipated effect size, significance level, statistical power, and number of covariates.

Key takeaways

  • What is calculated: Determine the smallest n at which the partial correlation can be distinguished from zero.
  • Quantities you must supply: Expected partial correlation; Number of variables controlled for; Significance level; Power.
  • Type of calculation: Power-based: it answers how much data is needed to detect a stated effect. Power 0.80 at α = 0.05 is the usual target; confirmatory studies use 0.90.

When to use

Use when the primary objective is to determine whether a specified association between two variables can be detected after controlling for one or more covariates with adequate statistical power.

Sample size

Type I error rate.

Probability of detecting the specified effect.

A two-sided alternative splits α between both tails; a one-sided alternative places all of α in one tail, needs a smaller n, and must be pre-specified.

Final n is inflated by 1/(1 − dropout).

Reset to defaults

Awaiting inputs

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Set the parameters and calculate.

Methodology — Partial correlation — Fisher's z with reduced degrees of freedom

Formula used Partial correlation — Fisher's z with reduced degrees of freedom

n=(z1−α/2+z1−βzr)2+3+k
Required sample size
  • z1−α/2(two-sided)in place ofz1−α(one-sided) Two-sided alternative selected: α is split between both tails, so the larger quantile is used and the requirement rises

where —

r_p
Partial correlation after adjustment, usually smaller than the unadjusted value. Must be non-zero
k
Covariates partialled out of both variables. 0 – 100
α
Probability of rejecting a true null hypothesis (Type I error). 0.0001 – 0.5; conventionally 0.05, or 0.025 one-sided for regulatory non-inferiority
1 − β
Probability of rejecting the null hypothesis when the specified alternative is true. 0.50 – 0.9999; conventionally 0.80 or 0.90
ℓ
Expected proportion of enrolled units lost before analysis. 0 – 0.95

How this method works

A partial correlation measures the association remaining after the linear effect of k covariates is removed from both variables. Each covariate consumes a degree of freedom, so the requirement rises by k.

Each covariate consumes a degree of freedom, so the direct cost is small — one subject each.

The indirect cost is larger: adjustment usually attenuates the correlation, and it is that reduced value which must be entered. Using an unadjusted correlation from the literature as if it were partial is the commonest error here.

Null hypothesis. H0: the partial correlation is zero.

Alternative hypothesis. Two-sided H1: it is non-zero. One-sided H1: it is positive or negative.

Calculation procedure

  1. State the partial correlation worth detecting — the association that remains after adjustment, not the raw one.
  2. State how many variables are being controlled for, q.
  3. Apply Fisher's transformation to the partial correlation.
  4. Evaluate n=(z1−α/2+z1−βzr)2+3+q: each control variable costs one degree of freedom.
  5. Round up, then inflate for dropout.

Exact or approximate

Approximate — a normal approximation on the transformed scale.

The formula shown always matches what was computed: when a one-sided test is selected the rendered quantile changes from z1−α/2 to z1−α, and where several methods exist, the formula follows the method selected in the calculator.

Advantages & limitations

Advantages

  • The direct cost of covariates is small — one subject each.
  • Matches the adjusted analysis that will actually be reported.
  • Same familiar transformation as the simple correlation.

Limitations

  • The indirect cost is larger: adjustment usually attenuates the correlation.
  • Requires an anticipated partial correlation, which is harder to obtain than an unadjusted one.
  • Assumes joint normality across all variables including covariates.

Assumptions

  • Observations are independent.
  • All variables, including covariates, are jointly normal.
  • Relationships with the covariates are linear.
  • Covariates are measured without substantial error.

Applicable adjustments

Supports dropout inflation.

Conclusion

Report n with the assumed partial correlation and the number of covariates. Confirm the entered value is genuinely adjusted, not borrowed from an unadjusted analysis.

References

  1. Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale: Lawrence Erlbaum; 1988.
  2. Chow S-C, Shao J, Wang H, Lokhnygina Y. Sample Size Calculations in Clinical Research. 3rd ed. Boca Raton: Chapman & Hall/CRC; 2017.
  3. Kim HY. Statistical notes for clinical researchers: covariance and correlation. Restor Dent Endod. 2018;43(1):e4. Link