I like statistics and I struggle with statistics. Often times I get frustrated when I don’t understand and I really struggled to make sense of Krushke’s Bayesian analysis of a split-plot, particularly because ‘it didn’t look like’ a split-plot to me.

Additionally, I have made a few posts discussing linear mixed models using several different packages to fit them. At no point I have shown what are the calculations behind the scenes. So, I decided to combine my frustration and an explanation to myself in a couple of posts. This is number one and the follow up is Split-plot 2: let’s throw in some spatial effects.

## How do linear mixed models look like

Linear mixed models, models that combine so-called fixed and random effects, are often represented using matrix notation as:

\( \boldsymbol{y} = \boldsymbol{X b} + \boldsymbol{Z a} + \boldsymbol{e}\)where \( \boldsymbol{y}\) represents the response variable vector, \( \boldsymbol{X} \mbox{ and } \boldsymbol{Z}\) are incidence matrices relating the response to the fixed (\( \boldsymbol{b}\) e.g. overall mean, treatment effects, etc) and random (\( \boldsymbol{a}\), e.g. family effects, additive genetic effects and a host of experimental design effects like blocks, rows, columns, plots, etc), and last the random residuals (\( \boldsymbol{e}\)).

The previous model equation still requires some assumptions about the distribution of the random effects. A very common set of assumptions is to say that the residuals are iid (identical and independently distributed) normal (so \( \boldsymbol{R} = \sigma^2_e \boldsymbol{I} \)) and that the random effects are independent of each other, so \( \boldsymbol{G} = \boldsymbol{B} \bigoplus \boldsymbol{M}\) is a direct sum of the variance of blocks (\( \boldsymbol{B} = \sigma^2_B \boldsymbol{I}\)) and main plots (\( \boldsymbol{M} = \sigma^2_M \boldsymbol{I}\)).

An interesting feature of this matrix formulation is that we can express all sort of models by choosing different assumptions for our covariance matrices (using different covariance structures). Do you have longitudinal data (units assessed repeated times)? Is there spatial correlation? Account for this in the \( \boldsymbol{R}\) matrix. Random effects are correlated (e.g. maternal and additive genetic effects in genetics)? Account for this in the \( \boldsymbol{G}\) matrix. Multivariate response? Deal with unstructured \( \boldsymbol{R}\) and \( \boldsymbol{G}\), or model the correlation structure using different constraints (and the you’ll need asreml).

By the way, the history of linear mixed models is strongly related to applications of quantitative genetics for the prediction of breeding values, particularly in dairy cattle. Charles Henderson developed what is now called Henderson’s Mixed Model Equations^{†} to simultaneously estimate fixed effects and predict random genetic effects:

\left [

\begin{array}{cc}

\boldsymbol{X}’ \boldsymbol{R}^{-1} \boldsymbol{X} & \boldsymbol{X}’ \boldsymbol{R}^{-1} \boldsymbol{Z} \\

\boldsymbol{Z}’ \boldsymbol{R}^{-1} \boldsymbol{X} & \boldsymbol{Z}’ \boldsymbol{R}^{-1} \boldsymbol{Z} + \boldsymbol{G}^{-1}

\end{array}

\right ]

\left [

\begin{array}{c}

\boldsymbol{b} \\

\boldsymbol{a}

\end{array}

\right ] =

\left [

\begin{array}{c}

\boldsymbol{X}’ \boldsymbol{R}^{-1} \boldsymbol{y} \\

\boldsymbol{Z}’ \boldsymbol{R}^{-1} \boldsymbol{y}

\end{array}

\right ] \)

The big matrix on the left-hand side of this equation is often called the \( \boldsymbol{C} \) matrix. You could be thinking ‘What does this all mean?’ It is easier to see what is going on with a small example, but rather than starting with, say, a complete block design, we’ll go for a split-plot to start tackling my annoyance with the aforementioned blog post.

## Old school: physical split-plots

Given that I’m an unsophisticated forester and that I’d like to use data available to anyone, I’ll rely on an agricultural example (so plots are actually physical plots in the field) that goes back to Frank Yates. There are two factors (oats variety, with three levels, and fertilization, with four levels). Yates, F. (1935) Complex experiments, Journal of the Royal Statistical Society Suppl. 2, 181–247 (behind pay wall here).

Now it is time to roll up our sleeves and use some code, getting the data and fitting the same model using `nlme`

(m1) and `asreml`

(m2), just for the fun of it. Anyway, `nlme`

and `asreml`

produce exactly the same results.

We will use the `oats`

data set that comes with MASS, although there is also an `Oats`

data set in `nlme`

and another version in the `asreml`

package. (By the way, you can see a very good explanation by Bill Venables of a ‘traditional’ ANOVA analysis for a split-plot here):

require(MASS) # we get the oats data from here require(nlme) # for lme function require(asreml) # for asreml function. This dataset use different variable names, # which may require renaming a dataset to use the code below # Get the oats data set and check structure data(oats) head(oats) str(oats) # Create a main plot effect for clarity's sake oats$MP = oats$V # Split-plot using NLME m1 = lme(Y ~ V*N, random = ~ 1|B/MP, data = oats) summary(m1) fixef(m1) ranef(m1) # Split-plot using ASReml m2 = asreml(Y ~ V*N, random = ~ B/MP, data = oats) summary(m2)$varcomp coef(m2)$fixed coef(m2)$random fixef(m1) # (Intercept) VMarvellous VVictory N0.2cwt # 80.0000000 6.6666667 -8.5000000 18.5000000 # N0.4cwt N0.6cwt VMarvellous:N0.2cwt VVictory:N0.2cwt # 34.6666667 44.8333333 3.3333333 -0.3333333 #VMarvellous:N0.4cwt VVictory:N0.4cwt VMarvellous:N0.6cwt VVictory:N0.6cwt # -4.1666667 4.6666667 -4.6666667 2.1666667 ranef(m1) # Level: B # (Intercept) # I 25.421511 # II 2.656987 # III -6.529883 # IV -4.706019 # V -10.582914 # VI -6.259681 # # Level: MP %in% B # (Intercept) # I/Golden.rain 2.348296 # I/Marvellous -3.854348 # I/Victory 14.077467 # II/Golden.rain 4.298706 # II/Marvellous 6.209473 # II/Victory -9.194250 # III/Golden.rain -7.915950 # III/Marvellous 10.750776 # III/Victory -6.063976 # IV/Golden.rain 5.789462 # IV/Marvellous -7.115566 # IV/Victory -1.001111 # V/Golden.rain 1.116768 # V/Marvellous -9.848096 # V/Victory 3.497878 # VI/Golden.rain -5.637282 # VI/Marvellous 3.857761 # VI/Victory -1.316009 |

Now we can move to implement the Mixed Model Equations, where probably the only gotcha is the definition of the \(\boldsymbol{Z}\) matrix (incidence matrix for random effects), as both `nlme`

and `asreml`

use ‘number of levels of the factor’ for both the main and interactions effects, which involves using the `contrasts.arg`

argument in `model.matrix()`

.

# Variance components varB = 214.477 varMP = 106.062 varR = 177.083 # Basic vector and matrices: y, X, Z, G & R y = matrix(oats$Y, nrow = dim(oats)[1], ncol = 1) X = model.matrix(~ V*N, data = oats) Z = model.matrix(~ B/MP - 1, data = oats, contrasts.arg = list(B = contrasts(oats$B, contrasts = F), MP = contrasts(oats$MP, contrasts = F))) G = diag(c(rep(varB, 6), rep(varMP, 18))) R = diag(varR, 72, 72) Rinv = solve(R) # Components of C XpX = t(X) %*% Rinv %*% X ZpZ = t(Z) %*% Rinv %*% Z XpZ = t(X) %*% Rinv %*% Z ZpX = t(Z) %*% Rinv %*% X Xpy = t(X) %*% Rinv %*% y Zpy = t(Z) %*% Rinv %*% y # Building C * [b a] = RHS C = rbind(cbind(XpX, XpZ), cbind(ZpX, ZpZ + solve(G))) RHS = rbind(Xpy, Zpy) blup = solve(C, RHS) blup # [,1] # (Intercept) 80.0000000 # VMarvellous 6.6666667 # VVictory -8.5000000 # N0.2cwt 18.5000000 # N0.4cwt 34.6666667 # N0.6cwt 44.8333333 # VMarvellous:N0.2cwt 3.3333333 # VVictory:N0.2cwt -0.3333333 # VMarvellous:N0.4cwt -4.1666667 # VVictory:N0.4cwt 4.6666667 # VMarvellous:N0.6cwt -4.6666667 # VVictory:N0.6cwt 2.1666667 # BI 25.4215578 # BII 2.6569919 # BIII -6.5298953 # BIV -4.7060280 # BV -10.5829337 # BVI -6.2596927 # BI:MPGolden.rain 2.3482656 # BII:MPGolden.rain 4.2987082 # BIII:MPGolden.rain -7.9159514 # BIV:MPGolden.rain 5.7894753 # BV:MPGolden.rain 1.1167834 # BVI:MPGolden.rain -5.6372811 # BI:MPMarvellous -3.8543865 # BII:MPMarvellous 6.2094778 # BIII:MPMarvellous 10.7507978 # BIV:MPMarvellous -7.1155687 # BV:MPMarvellous -9.8480945 # BVI:MPMarvellous 3.8577740 # BI:MPVictory 14.0774514 # BII:MPVictory -9.1942649 # BIII:MPVictory -6.0639747 # BIV:MPVictory -1.0011059 # BV:MPVictory 3.4978963 # BVI:MPVictory -1.3160021 |

Not surprisingly, we get the same results, except that we start assuming the variance components from the previous analyses, so we can avoid implementing the code for restricted maximum likelihood estimation as well. By the way, given that \( \boldsymbol{R}^{-1} \) is in all terms it can be factored out from the MME, leaving terms like \( \boldsymbol{X}’ \boldsymbol{X} \), i.e. without \( \boldsymbol{R}^{-1}\), making for simpler calculations. In fact, if you drop the \( \boldsymbol{R}^{-1} \) it is easier to see what is going on in the different components of the \( \boldsymbol{C} \) matrix. For example, print \( \boldsymbol{X}’ \boldsymbol{X} \) and you’ll get the sum of observations for the overall mean and for each of the levels of the fixed effect factors. Give it a try with the other submatrices too!

XpXnoR = t(X) %*% X XpXnoR # (Intercept) VMarvellous VVictory N0.2cwt N0.4cwt N0.6cwt #(Intercept) 72 24 24 18 18 18 #VMarvellous 24 24 0 6 6 6 #VVictory 24 0 24 6 6 6 #N0.2cwt 18 6 6 18 0 0 #N0.4cwt 18 6 6 0 18 0 #N0.6cwt 18 6 6 0 0 18 #VMarvellous:N0.2cwt 6 6 0 6 0 0 #VVictory:N0.2cwt 6 0 6 6 0 0 #VMarvellous:N0.4cwt 6 6 0 0 6 0 #VVictory:N0.4cwt 6 0 6 0 6 0 #VMarvellous:N0.6cwt 6 6 0 0 0 6 #VVictory:N0.6cwt 6 0 6 0 0 6 # VMarvellous:N0.2cwt VVictory:N0.2cwt VMarvellous:N0.4cwt #(Intercept) 6 6 6 #VMarvellous 6 0 6 #VVictory 0 6 0 #N0.2cwt 6 6 0 #N0.4cwt 0 0 6 #N0.6cwt 0 0 0 #VMarvellous:N0.2cwt 6 0 0 #VVictory:N0.2cwt 0 6 0 #VMarvellous:N0.4cwt 0 0 6 #VVictory:N0.4cwt 0 0 0 #VMarvellous:N0.6cwt 0 0 0 #VVictory:N0.6cwt 0 0 0 # VVictory:N0.4cwt VMarvellous:N0.6cwt VVictory:N0.6cwt #(Intercept) 6 6 6 #VMarvellous 0 6 0 #VVictory 6 0 6 #N0.2cwt 0 0 0 #N0.4cwt 6 0 0 #N0.6cwt 0 6 6 #VMarvellous:N0.2cwt 0 0 0 #VVictory:N0.2cwt 0 0 0 #VMarvellous:N0.4cwt 0 0 0 #VVictory:N0.4cwt 6 0 0 #VMarvellous:N0.6cwt 0 6 0 #VVictory:N0.6cwt 0 0 6 |

I will leave it here and come back to this problem as soon as I can.

† Incidentally, a lot of the theoretical development was supported by Shayle Searle (a Kiwi statistician and Henderson’s colleague in Cornell University).