This study introduces a correction to the approximation of effective df as proposed by Satterthwaite, specifically addressing scenarios where component df are small. The correction is grounded in analytical results concerning the moments of standard normal random variables. This modification is applicable to complex variance estimates that involve both small and large df, offering an enhanced approximation of the higher moments required by Satterthwaite’s framework. Additionally, this correction extends and partially validates the empirically derived adjustment by Johnson and Rust, as it is based on theoretical foundations rather than simulations used to derive empirical transformation constants. Finally, the proposed adjustment also provides a correction to the estimate of the total variance in cases missing data have been replaced by multiple imputations such as in the case of plausible values in national and international large scale assessments.
DelacreM.LakensD.LeysC. (2017). Why psychologists should by default use Welch’s t-test instead of Student’s t-test. International Review of Social Psychology, 30(1), 92–101. https://doi.org/10.5334/irsp.82
3.
EfronB. (1982). The jackknife, the bootstrap, and other resampling plans. CBMS-NSF Regional Conference Series in Applied Mathematics, Monograph 38. Society for Industrial and Applied Mathematics.
4.
FayR. E. (1989). Theory and application of replicate weighting for variance calculations. In Proceedings of the survey research methods section (pp. 212–217). American Statistical Association. http://www.asasrms.org/Proceedings/papers/1989_033.pdf
5.
JohnsonE. G.RustK. F. (1992a). Effective df for variance estimates from a complex sample survey. In Proceedings of the section on survey research methods. American Statistical Association. (Preprint copy obtained from K. Rust, September2024).
6.
JohnsonE. G.RustK. F. (1992b). Population inferences and variance estimation for NAEP data. Journal of Educational Statistics, 17(2), 175–190. https://doi.org/10.2307/1165168
7.
JohnsonE. G.RustK. F. (1993). Effective df for variance estimates from a complex sample survey. In 1993 Proceedings: Survey research methods section (pp. 863–866). American Statistical Association.
8.
LipsitzS.ParzenM.ZhaoL. P. (2002). A degrees-of-freedom approximation in multiple imputation. Journal of Statistical Computation and Simulation, 72(4), 309–318. https://doi.org/10.1080/00949650212848
9.
MislevyR. J.BeatonA. E.KaplanB.SheehanK. M. (1992). Estimating population characteristics from sparse matrix samples of item responses. Journal of Educational Measurement, 29(2), 133–161.
NedelmanJ. R.JiaX. (1998). An extension of Satterthwaite’s approximation applied to pharmacokinetics. Journal of Biopharmaceutical Statistics, 8(2), 317–328. https://doi.org/10.1080/10543409808835241
13.
RubinD. B. (1978). Multiple imputation in sample surveys: A phenomenological Bayesian approach to nonresponse. In Proceedings of the survey research methods section of the American Statistical Association (pp. 20–34). http://www.asasrms.org/Proceedings/papers/1978_004.pdf
14.
RubinD. B. (1987). Multiple imputation for nonresponse in surveys. John Wiley & Sons.
15.
RubinD. B.SchenkerN. (1986). Multiple imputation for interval estimation from simple random samples with ignorable nonresponse. Journal of the American Statistical Association, 81(394), 366–374. https://doi.org/10.2307/2289225
16.
RustK. F. (1985). Variance estimation for complex estimators in sample surveys. Journal of Official Statistics, 1(4), 381–397.
SatterthwaiteF. E. (1946). An approximate distribution of estimates of variance components. Biometrics Bulletin, 2(6), 110–114. https://doi.org/10.2307/3002019
19.
TukeyJ. W. (1958). Bias and confidence in not-quite large samples. Annals of Mathematical Statistics, 29(2), 614–623. https://doi.org/10.1214/aoms/1177706647
20.
QianJ. (1998). Estimation of the effective df in t-type tests for complex data. Proceedings of the Section on Survey Research Methods, American Statistical Association, 704–708.
WelchB. L. (1947). The generalization of “Student’s” problem when several different population variances are involved. Biometrika, 34(1/2), 28–35. https://doi.org/10.2307/2332510
