Analysis of experiments in which the same experimental units (plots, animals, trees) are measured on several occasions: plant height or LAI at successive dates, soil moisture through the season, or body weight, heart rate or blood metabolites over time. Gives the univariate split-plot-in-time ANOVA, Mauchly's test of sphericity with Greenhouse-Geisser and Huynh-Feldt corrected p-values, a comparison of covariance structures by mixed models (compound symmetry, first-order autoregressive and unstructured), SE and CD for every comparison, letter groupings, and a separate analysis at each time.
Usage
anova_repeated(
data,
response,
treatment,
time,
subject = NULL,
block = NULL,
posthoc = "lsd",
alpha = 0.05,
covariance = TRUE,
verbose = TRUE
)Arguments
- data
Data frame in long format (one row per unit per time).
- response
Name of the response variable.
- treatment
Name of the treatment column.
- time
Name of the time (occasion) column. Times are taken in the order of their levels (numeric times are sorted).
- subject
Name of the column identifying the experimental unit that is measured repeatedly. If
NULLandblockis given, each block x treatment combination is one unit.- block
Name of the block column for units laid out in randomized complete blocks, or
NULL(completely randomized units).- posthoc
Method for the letters (default
"lsd").- alpha
Significance level. Default 0.05.
- covariance
Logical. If TRUE (default), fits mixed models with different covariance structures using nlme.
- verbose
Logical. If TRUE (default), prints the report.
Value
An object of class aridagri_repeated with
anova_table (univariate), sphericity (Mauchly's W and
p, GG and HF epsilons), corrected (within-unit tests with GG
and HF p-values), covariance_models (AIC/BIC table),
best_structure, mixed_anova, se_cd,
means (treatment x time table), treatment_means,
time_means, letters, by_time (ANOVA at each time)
and models.
Details
Univariate analysis. Treatment (and block) are tested against
the between-unit error, Error(a). Time and the treatment x time
interaction are tested against the within-unit error, Error(b). With
blocks, the block x time interaction is fitted as a separate
within-unit term, as in the univariate tests of SAS PROC GLM with a
REPEATED statement, so that the sphericity corrections apply to the
same error term. This is
valid when the covariance of the repeated measurements is spherical.
Mauchly's test checks this, and the Greenhouse-Geisser (1959) and
Huynh-Feldt (1976) epsilons, the latter with the correction of Lecoutre
(1991) as in anova.mlm(), scale the degrees of freedom of the
within-unit tests when it fails. A common rule is to use the
Greenhouse-Geisser p-value when epsilon is below 0.75 and the
Huynh-Feldt p-value otherwise.
Mixed models. Generalized least squares models with the same fixed effects are fitted by REML with compound symmetry (CS), first-order autoregressive (AR1, on the time order) and unstructured (UN, with separate variances per time) covariance, and compared by AIC and BIC. The marginal F-tests of the best structure are reported (with the residual degrees of freedom used by nlme, so tests of between-unit effects are approximate; the univariate test of treatment is exact for balanced data). Structures that fail to converge are skipped.
Comparisons. With \(r\) units per treatment, \(a\) treatments and \(t\) times, the SEd are as in a split-plot design with treatments in main plots and times in sub-plots: treatment means \(\sqrt{2E_a/(rt)}\), time means \(\sqrt{2E_b/(ra)}\), times within a treatment \(\sqrt{2E_b/r}\), and treatments at the same time \(\sqrt{2[(t-1)E_b + E_a]/(rt)}\) with the weighted t.
References
Greenhouse, S.W. and Geisser, S. (1959). On methods in the analysis of profile data. Psychometrika, 24(2), 95-112. doi:10.1007/BF02289823
Lecoutre, B. (1991). A correction for the epsilon approximate test in repeated measures designs with two or more independent groups. Journal of Educational Statistics, 16(4), 371-372. doi:10.3102/10769986016004371
Huynh, H. and Feldt, L.S. (1976). Estimation of the Box correction for degrees of freedom from sample data in randomized block and split-plot designs. Journal of Educational Statistics, 1(1), 69-82. doi:10.3102/10769986001001069
Littell, R.C., Henry, P.R. and Ammerman, C.B. (1998). Statistical analysis of repeated measures data using SAS procedures. Journal of Animal Science, 76(4), 1216-1231. doi:10.2527/1998.7641216x
Examples
data(mungbean_repeated)
rm <- anova_repeated(mungbean_repeated, response = "spad",
treatment = "treatment", time = "das", block = "block")
#>
#> ======================================================================
#> REPEATED MEASURES ANOVA (SPLIT-PLOT IN TIME)
#> ======================================================================
#> Response : spad
#> Treatment : treatment
#> Time : das
#> Units : 16
#> Times : 4
#>
#> Univariate analysis of variance
#> Source Df Sum Sq Mean Sq F value Pr(>F)
#> block 3 48.65 16.22 5.124 0.0244 *
#> treatment 3 87.91 29.30 9.259 0.0041 **
#> Error(a) 9 28.48 3.16
#> das 3 1203.91 401.30 417.333 0.0000 ***
#> block:das 9 14.89 1.65 1.721 0.1326 NS
#> treatment:das 9 40.66 4.52 4.698 0.0008 ***
#> Error(b) 27 25.96 0.96
#> Total 63 1450.47
#> Signif. codes: *** p < 0.001, ** p < 0.01, * p < 0.05, NS not significant
#>
#> Grand mean = 38.202 CV(a) = 4.66% CV(b) = 2.57%
#>
#> Mauchly's test of sphericity: W = 0.5414, p = 0.4518
#> Greenhouse-Geisser epsilon = 0.7684 Huynh-Feldt epsilon = 1.0000
#>
#> Within-unit tests with corrected p-values
#> Source Df F_value p_uncorrected p_GG p_HF recommended
#> das 3 417.333 1.013e-22 6.414e-18 1.013e-22 1.013e-22
#> treatment:das 9 4.698 8.219e-04 2.793e-03 8.219e-04 8.219e-04
#>
#> Mixed models: covariance structures (lower AIC is better)
#> Structure Parameters AIC BIC logLik
#> CS 21 238.70 276.63 -98.35
#> AR1 21 230.72 268.66 -94.36
#> UN 29 238.50 290.90 -90.25
#>
#> F-tests with the AR1 structure (residual df; for treatment, the univariate test above is exact)
#> numDF F-value p-value
#> block 3 3.971 1.357e-02
#> treatment 3 7.060 5.465e-04
#> das 3 349.707 3.510e-31
#> treatment:das 9 3.268 3.867e-03
#>
#> SEm, SEd and CD (alpha = 0.05)
#> Comparison SEm SEd CD Error
#> treatment means 0.445 0.629 1.423 E(a)
#> das means 0.245 0.347 0.711 E(b)
#> treatment levels at the same or different das 0.615 0.870 1.880 E(a)+E(b)
#> das levels at the same treatment 0.490 0.693 1.423 E(b)
#>
#> Treatment x time means
#> 25 35 45 55
#> Control 31.50 38.03 41.33 38.48
#> Water spray 30.62 38.00 41.58 37.48
#> Urea 2% 31.60 39.35 45.98 42.67
#> ZnSO4 0.5% 31.40 38.85 43.80 40.58
#>
#> Analysis at each time
#> Time Mean F_value p_value Sig CV SEm CD
#> 25 31.28 0.446 0.72600 NS 4.26 0.666 2.132
#> 35 38.56 2.108 0.16900 NS 2.36 0.455 1.455
#> 45 43.17 10.450 0.00273 ** 3.12 0.673 2.154
#> 55 39.80 13.088 0.00124 ** 3.21 0.639 2.044
rm$sphericity
#> Mauchly_W p_value GG_epsilon HF_epsilon lower_bound
#> 1 0.5413907 0.4518438 0.7683995 1 0.3333333
rm$covariance_models
#> Structure Parameters AIC BIC logLik
#> 1 CS 21 238.6950 276.6349 -98.34750
#> 2 AR1 21 230.7225 268.6625 -94.36127
#> 3 UN 29 238.5023 290.8955 -90.25116