Så används statistiska metoder i jordbruksförsök Svenska statistikfrämjandets vårkonferens den 23 mars 2012 i Alnarp Johannes Forkman, Fältforsk, SLU Agricultural field experiments Experimental treatments • • • • • Varieties Weed control treatments Plant protection treatments Tillage methods Fertilizers Experimental design Allocate Treatments A and B to eight plots... Option 1: A A A A B Option 2: A B A B A B A B Option 3: A B B A B B A B B B A Systematic error • The plots differ... • The treatments are not compared on equal terms. • There will be a systematic error in the comparison of A and B. Randomise the treatments. This procedure transforms the systematic error into a random error. R. A. Fisher Example Treatment Yield (kg/ha) Mean (kg/ha) 8165 A A 7792 8397 A B B B 7764 8483 8602 8641 8029.3 B 8783 8627.2 The difference is 598 A Randomisation test • The observed difference is 598 kg/ha. • There are 8!/(4! 4!) = 70 possible random arrangements. • The two most extreme differences are 598 and -598. • P-value = 2/70 = 0.029 t-test t 598 598 3.63 93438 15298 164.9 4 4 Compare with a t-distribution with 6 degrees of freedom P-value = 0.011 The randomisation model = + , E = cov() = ( − −1 ′ ) is the number of available plots. The approximate model = + , E = When is infinitely large cov() = . For statistical tests, we assume further that = MVN( , ). A crucial assumption Unit-Treatment additivity: • Variances and covariances do not depend on treatment Heterogeneity A B Inference about what?? • Randomisation model: The average if the treatment was given to all plots of the experiment. • The approximate model: The average if the treatment was given to infinitely many plots? Sample Population Variance in a difference When cov() = ( − −1 ′ ) , then var 1 − 2 = 1 1 + 1 2 2. + 1 2 2. When cov = , then var 1 − 2 = 1 1 Independent errors • Randomisation gives approximately independent error terms • Information about plot position was ignored • This information can be utilized B A B A A B B A Tobler’s law of geography “Everything is related to everything else, but near things are more related than distant things.” Waldo Tobler Random fields The random function Z(s) is a • stochastic process if the plots belong to a space in one dimension • random field, if the plots belong to a space in two or more dimensions Semivariogram (|| h ||) sill 95% practical range || h || Spatial modelling • Can improve precision. • Still rare in analysis of agricultural field experiments. • There are many possible spatial models and methods. • Can be used whether or not the treatments were randomized... • Which is the best design for spatial analysis? Randomised block design Gradient III E F C D G A H B II A C G B D E H F I G H D E F B C A Incomplete block design V VIII VII VI 2 M I A E L P D H G O C K J III IV II B N F I 1 I J K L P O M N F E H G A D C B Strata • Replicates • Blocks • Plots Ofullständiga block Split-plot design Replicate I 1 D A C B 3 Replicate II 2 B C A D C B D A Strata • Replicates • Plots • Subplots 2 B D C A 1 C B A D 3 B C A D Replicate I 1 3 Replicate II 2 2 1 3 Comparison 1a D A C B B C A D C B D A B D C A C B A D B C A D 1b D A C B B C A D C B D A B D C A C B A D B C A D 2a D A C B B C A D C B D A B D C A C B A D B C A D 2b D A C B B C A D C B D A B D C A C B A D B C A D 3 D A C B B C A D C B D A B D C A C B A D B C A D A design with several strata Each replicate: sown conventionally sown with no tillage cultivar 2 cultivar 1 cultivar 3 Mo applied Mo applied Bailey, R. A. (2008). Design of comparative experiments. Cambridge University Press. The linear mixed model y = Xb + Zu + e X: design matrix for fixed effects (treatments) Z: design matrix for random effects (strata) u is N(0, G) e is N(0, R) Bates about error strata “Those who long ago took courses in "analysis of variance" or "experimental design" that concentrated on designs for agricultural experiments would have learned methods for estimating variance components based on observed and expected mean squares and methods of testing based on "error strata". (If you weren't forced to learn this, consider yourself lucky.) It is therefore natural to expect that the F statistics created from an lmer model (and also those created by SAS PROC MIXED) are based on error strata but that is not the case.” Approximate t and F-tests = when L is one-dimensional, and var() = ′ var −1 rank var otherwise. The number of degrees of freedom is an issue. SAS: the Satterthwaite or the Kenward & Roger method. Likelihood ratio test Full model (FM): p parameters Reduced model (RM): q parameters L( FM | y) 2log L( RM | y) is asymptotically c2 with p – q degrees of freedom. Bayesian analysis y = Xb + Zu + e u is N(0, G) e is N(0, R) G is diag(Φ) R is diag(σ2) Independent priori distributions: p(b), p(Φ) Sampling from the posterior distribution: p(b,Φ | y) P-values in agricultural research • Only discuss statistically significant results • Do not discuss biologically insignificant results (although they are statistically significant). • “Limit statements about significance to those which have a direct bearing on the aims of the research”. (Onofri et al., Weed Science, 2009) Shrinkage estimators Galwey (2006). Introduction to mixed modelling. Wiley. Fixed or random varieties? Fixed varieties (BLUE) • Few varieties • Estimation of differences Random varieties (BLUP) • Many varieties • Ranking of varieties Conclusions based on a simulation study i. Modelling treatment as random is efficient for small block experiments. ii. A model with normally distributed random effects performs well, even if the effects are not normally distributed. iii. Bayesian methods can be recommended for inference about treatment differences. Summary • Fisher’s ideas about randomisation and blocking are still predominant. • Strong focus on p-values. • Linear mixed models are used extensively. • Spatial and Bayesian methods are used less often. • The question is what is random and fixed, and how to calculate p-values. Tack för uppmärksamheten!