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... AAAABBBB ABABABAB ABBAABBA Option 1: Option 2: Option 3:

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 TreatmentYield (kg/ha)Mean (kg/ha) A8165 A7792 A8397 A B8483 B8602 B8641 B The difference is 598

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 P-value = 2/70 = 0.029

t-test Compare with a t-distribution with 6 degrees of freedom P-value = 0.011

The randomisation model

The approximate model

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

Independent errors Randomisation gives approximately independent error terms Information about plot position was ignored This information can be utilized BABAABBA

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

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

Incomplete block design Strata Replicates Blocks Plots

Ofullständiga block

DACB BCAD CBDABDCACBAD BCAD Replicate I Replicate II Strata Replicates Plots Subplots Split-plot design

DACB BCAD CBDABDCACBAD BCA D Replicate I Replicate II DA C BBCAD C BDABD C A C BADBCAD D A C BB C A DC B D AB DC A C BA D B C A D DACBBCAD C BD A BD CA CBADBCAD D A CBBCAD C BDABD C ACB A DBCAD 1a 1b 2a 2b 3 Comparison

sown conventionally sown with no tillage cultivar 2 cultivar 1 cultivar 3 Mo applied Each replicate: A design with several strata Bailey, R. A. (2008). Design of comparative experiments. Cambridge University Press.

The linear mixed model y = X  + 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 The number of degrees of freedom is an issue. SAS: the Satterthwaite or the Kenward & Roger method. when L is one-dimensional, and otherwise.

Likelihood ratio test Full model (FM): p parameters Reduced model (RM): q parameters is asymptotically  2 with p – q degrees of freedom.

Bayesian analysis y = X  + Zu + e u is N(0, G)e is N(0, R) G is diag(Φ) R is diag(σ 2 ) Independent priori distributions: p(  ), p(Φ) Sampling from the posterior distribution: p(  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.

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