This results in an expected value of the numerator being approximately equal to that of the denominator if the Lake effect is very small. In this example, the mean square for Lake is divided by the mean square for the Supplement*Lake interaction. Therefore, Minitab uses an approximate F-test. Notice, however, that for a very small Lake effect, there are no mean squares such that the expected value of the numerator equals the expected value of the denominator. If the effect for Supplement is very small, the expected value of the numerator equals the expected value of the denominator. The F-statistic for Supplement is the mean square for Supplement divided by the mean square for the Supplement*Lake interaction. Therefore, a high F-statistic indicates a significant Screen effect.įor example, suppose you performed an ANOVA with the fixed factor Supplement and the random factor Lake, and the got following output for the expected mean squares (EMS): To calculate the F-statistic for Screen, the mean square for Screen is divided by the mean square for Screen*Tech so that the expected value of the numerator (EMS for Screen = (4) + 2.0000(3) + Q ) differs from the expected value of the denominator (EMS for Screen*Tech = (4) + 2.0000(3) ) only by the effect due to the Screen (Q). Q equals (b*n * sum((coefficients for levels of Screen)**2)) divided by (a - 1), where a and b are the number of levels of Screen and Tech, respectively, and n is the number of replicates. Mean Square Within Groups: MSW SSW / (N k) F-Statistic (or F-ratio): F MSB / MSW. The EMS for Screen is the effect of the error term plus two times the effect of the Screen*Tech interaction plus a constant times the effect of Screen. For example, Q is the fixed effect of Screen. Therefore, a high F-statistic indicates a significant Screen*Tech interaction.Ī number with Q indicates the fixed effect associated with the term listed beside the source number. To calculate the F-statistic for Screen*Tech, the mean square for Screen*Tech is divided by the mean square of the error so that the expected value of the numerator (EMS for Screen*Tech = (4) + 2.0000(3)) differs from the expected value of the denominator (EMS for Error = (4)) only by the effect of the interaction (2.0000(3)). In addition, the EMS for Screen*Tech is the effect of the error term plus two times the effect of the Screen*Tech interaction. The EMS for Error is the effect of the error term. (2) represents the random effect of Tech, (3) represents the random effect of the Screen*Tech interaction, and (4) represents the random effect of Error. Moreover, the KR method also adjusts the SEs of the fixed effect estimates based on the uncertainty of the variance-covariance estimates.Suppose you performed an ANOVA with the fixed factor Screen and the random factor Tech, and get the following output for the EMS:Ī number with parentheses indicates a random effect associated with the term listed beside the source number. The KR method does take the uncertainty of the variance-covariance estimates into account in the df calcuation. Another way to look at this: The traditional df calculation methods do not account for the fact that the variances and covariances are estimated. ), F (or t) statistic is only approximately distributed as F (or t) under H0 the approximation is best when the denominator df are calculated based on the model and the estimated G and R. With mixed models (with random effects, correlations. The idea is this: to test a null hypothesis, F (or t) statistic should have an F (or t) statistical distribution when the null hypothesis is true. Actually, I recommend the ddfm=KR with any repeated measure. (G for random effects, R for repeated measures). You are using the Satterthwaite df calcuation method, which is an estimation based on the estimated variance-covariance G and R matrices. In the world of mixed models, one has to drop old pre-mixed-model concepts, such as integer degrees of freedom.
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