Blocking in Experiments

Additional Topic
Math 215

Yurk

Experiments with two factors

A factor is a categorical variable that we think may explain some of the variability in the response variable
A treatment factor is a factor that we want to evaluate to see if there is an effect
A blocking factor is a factor that we believe will have an effect
The treatment factor is the factor of interest
The blocking factor is used to reduce the unexplained variability in the response

An experiment with two factors may have two treatment factors (more on this later)
Or it may have a treatment factor and a blocking factor

Experimental Designs

Randomize complete block design (RCBD)

1 treatment factor with \(t\) levels, 1 blocking factor with \(b\) levels
Each block has exactly \(t\) experimental units (one for each treatment level)
Experimental units in same block expected to respond similarly if treated similarly
Matched pair design: \(t=2\)
Repeated measures design: Each block is a single individual, subject to all \(t\) treatments

Factorial design (more on this later)

More general than RCBD
Two or more factors
One or more observations per cell (replication)

Strawberries

Does changing the air in which strawberries are stored affect the firmness of the berries?
Data from Smith and Skog (1992) ¹
Inspired by an example from Tintle et al. (2020)

Response: Firmness (force in N to pierce berry)
Treatment factor: Storage with 3 levels (\(t=3\)):
- “Control” (not stored)
- “Air” (21% O₂, 0.04% CO₂)
- “ModifiedAir” (15% CO₂, 18% O₂)
Blocking factor: Variety with 5 levels (\(b=5\)): “Allstar”, “Bounty”, “Kent”, “Selva”, “Vesper”
Stored berries were stored at \(0.5^{\circ}\)C for two days
Three clamshells of each variety randomly assigned to treatments

Statistical Model

\[y_{ij}=\mu+\tau_i+\rho_j+\varepsilon_{ij}\]

\(\mu\) is the overall mean
\(\tau_i\) is the differential effect of treatment \(i\)
\(\rho_j\) is the differential effect of block \(j\)
\(\varepsilon_{ij}\sim N(0,\sigma^2)\) represents the error
This is an additive model: the treatment is assumed to have the same effect in each block

Hypothesis Test

We expect firmness to vary according to variety
However, we are interested in the effect of the treatment (air type)
We conduct a hypothesis test for the treatment variable:
- \(H_0: \tau_1 = \tau_2 = \tau_3 = 0\)
- \(H_A:\) At least one \(\tau_i\) is different

EDA

Storage
Variety
Both

Strawberry firmness for different storage conditions.

Firmness for different strawberry varieities.

Firmness for different strawberry varieities with different storage conditions.

ANOVA Table (Not acconting for Variety)

First we consider ANOVA without accounting for Variety
We are unable to detect a treatment effect
Much of the variability in firmness is due to differences between varieties and is left unexplained

lm(Firmness ~ Storage, data = strawberries) |>
  anova() |>
  tidy()

# A tibble: 2 × 6
  term         df sumsq meansq statistic p.value
  <chr>     <int> <dbl>  <dbl>     <dbl>   <dbl>
1 Storage       2  11.2   5.59     0.995   0.398
2 Residuals    12  67.4   5.62    NA      NA

ANOVA Table (Accounting for variety)

This time we account for Variety
Now the treatment effect is apparent
We reject \(H_0\) and conclude that there is convincing evidence of a treatment effect.

lm(Firmness ~ Variety + Storage, data = strawberries) |>
  anova() |>
  tidy()

# A tibble: 3 × 6
  term         df sumsq meansq statistic    p.value
  <chr>     <int> <dbl>  <dbl>     <dbl>      <dbl>
1 Variety       4 63.5  15.9        32.4  0.0000547
2 Storage       2 11.2   5.59       11.4  0.00455  
3 Residuals     8  3.93  0.491      NA   NA

Pairwise Comparisons

We can follow up with pairwise comparisons between different treatment levels
Adjust for multiple comparisons

aov(Firmness ~ Variety + Storage, data = strawberries) |>
  TukeyHSD("Storage")

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = Firmness ~ Variety + Storage, data = strawberries)

$Storage
                    diff        lwr      upr     p adj
Control-Air         0.10 -1.1659048 1.365905 0.9723992
ModifiedAir-Air     1.88  0.6140952 3.145905 0.0070490
ModifiedAir-Control 1.78  0.5140952 3.045905 0.0095586

There is a significant difference between the mean firmness for “Modified Air” and “Air” and between “Modified Air” and “Control”

Scope of Inference

Because this was an experiment, we can conclude that the difference in storage method caused the difference in firmness
Storing strawberries in air enriched in CO₂ increases firmness compared to not storing the berries at all or storing them in normal air

Choosing varieties with large variability in firmness (large variation between blocks) broadens the scope of inference
For example, we could have controlled unexplained variability by focusing on a single variety
Then our conclusions would be limited to that single variety