Wireworms controlled by fumigants in a latin square

Format

A data frame with 25 observations on the following 4 variables.

row: row
col: column
trt: fumigant treatment, 5 levels
worms: count of wireworms per plot

Details

Plots were approximately 22 cm by 13 cm. Layout of the experiment was a latin square. The number of wireworms in each plot was counted, following soil fumigation the previous year.

Source

W. G. Cochran (1938). Some difficulties in the statistical analysis of replicated experiments. Empire Journal of Experimental Agriculture, 6, 157--175.

References

Ron Snee (1980). Graphical Display of Means. The American Statistician, 34, 195-199. https://www.jstor.org/stable/2684060 https://doi.org/10.1080/00031305.1980.10483028

W. Cochran (1940). The analysis of variance when experimental errors follow the Poisson or binomial laws. The Annals of Mathematical Statistics, 11, 335-347. https://www.jstor.org/stable/2235680

G W Snedecor and W G Cochran, 1980. Statistical Methods, Iowa State University Press. Page 288.

Examples

# \dontrun{

library(agridat)
data(cochran.wireworms)
dat <- cochran.wireworms

libs(desplot)
desplot(dat, worms ~ col*row,
        text=trt, cex=1, # aspect unknown
        main="cochran.wireworms")

# Trt K is effective, but not the others.  Really, this says it all.
libs(lattice)
bwplot(worms ~ trt, dat, main="cochran.wireworms", xlab="Treatment")

# Snedecor and Cochran do ANOVA on sqrt(x+1).
dat <- transform(dat, rowf=factor(row), colf=factor(col))
m1 <- aov(sqrt(worms+1) ~ rowf + colf + trt, data=dat)
anova(m1)
#> Analysis of Variance Table
#> 
#> Response: sqrt(worms + 1)
#>           Df Sum Sq Mean Sq F value  Pr(>F)  
#> rowf       4 2.3579 0.58947  1.8044 0.19285  
#> colf       4 0.7787 0.19467  0.5959 0.67248  
#> trt        4 5.7616 1.44041  4.4092 0.02014 *
#> Residuals 12 3.9202 0.32668                  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Instead of transforming, use glm
m2 <- glm(worms ~ trt + rowf + colf, data=dat, family="poisson")
anova(m2)
#> Analysis of Deviance Table
#> 
#> Model: poisson, link: log
#> 
#> Response: worms
#> 
#> Terms added sequentially (first to last)
#> 
#> 
#>      Df Deviance Resid. Df Resid. Dev
#> NULL                    24     64.555
#> trt   4  26.5294        20     38.026
#> rowf  4  15.6955        16     22.331
#> colf  4   2.8225        12     19.508

# GLM with random blocking.
libs(lme4)
m3 <- glmer(worms ~ -1 +trt +(1|rowf) +(1|colf), data=dat, family="poisson")
summary(m3)
#> Generalized linear mixed model fit by maximum likelihood (Laplace
#>   Approximation) [glmerMod]
#>  Family: poisson  ( log )
#> Formula: worms ~ -1 + trt + (1 | rowf) + (1 | colf)
#>    Data: dat
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>    124.2    132.7    -55.1    110.2       18 
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -1.3342 -0.7364 -0.1740  0.4505  2.7797 
#> 
#> Random effects:
#>  Groups Name        Variance  Std.Dev. 
#>  rowf   (Intercept) 8.389e-02 0.2896419
#>  colf   (Intercept) 1.451e-08 0.0001205
#> Number of obs: 25, groups:  rowf, 5; colf, 5
#> 
#> Fixed effects:
#>      Estimate Std. Error z value Pr(>|z|)    
#> trtK   0.1393     0.4275   0.326    0.745    
#> trtM   1.7814     0.2226   8.002 1.22e-15 ***
#> trtN   1.9028     0.2142   8.881  < 2e-16 ***
#> trtO   1.7147     0.2275   7.537 4.80e-14 ***
#> trtP   1.4829     0.2463   6.020 1.74e-09 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Correlation of Fixed Effects:
#>      trtK  trtM  trtN  trtO 
#> trtM 0.185                  
#> trtN 0.192 0.369            
#> trtO 0.181 0.347 0.361      
#> trtP 0.167 0.321 0.333 0.314
## Fixed effects:
##      Estimate Std. Error z value Pr(>|z|)    
## trtK   0.1393     0.4275   0.326    0.745    
## trtM   1.7814     0.2226   8.002 1.22e-15 ***
## trtN   1.9028     0.2142   8.881  < 2e-16 ***
## trtO   1.7147     0.2275   7.537 4.80e-14 ***

# }