৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

DOE aims to get maximum information with minimum expenditure and time, while reducing experimental error.

The experimental unit is the object to which treatments are applied and responses measured.

Suppose a researcher wants to study the effect of different fertilizers on the growth of tomato plants. The researcher has 30 tomato plants and wants to apply three types of fertilizer (A, B, and C), with 10 plants receiving each fertilizer type.

The experimental unit in this case is each individual tomato plant because the treatments (fertilizers) are applied at the plant level.

Each plant is the basic entity on which the experiment is performed and the response (growth) is measured.

Randomization minimizes systematic error and supports valid statistical inference

Replication ensures that the observed effects are reliable and reduces error variance.

Blocking controls known sources of variability by grouping similar units.

Factorial designs study all factors and their interactions in a single experiment.

It ensures that treatments are assigned without bias to allow valid comparison.

Identifying which factors and responses to measure is the first critical step.

Fisher’s three principles are:

Randomization – avoid bias.
Replication – reduce error, estimate variability.
Local control (blocking) – reduce variation from nuisance factors.

Normality and equal variances are assumptions, not principles.

Randomization ensures each experimental unit has an equal chance of receiving any treatment.

This prevents hidden biases (e.g., assigning better soil plots to a certain fertilizer).

CRD works best when experimental units are homogeneous (like identical pots).

RCBD controls one nuisance (soil blocks).

LSD controls two nuisances (rows & columns).

RCBD groups similar experimental units into blocks (e.g., soil type, gender).

Within each block, treatments are randomized.

Advantage: removes variability due to the block factor.

LSD arranges treatments so that each treatment appears once per row and once per column.

This simultaneously controls two nuisance factors (e.g., teachers = rows, classrooms = columns).

Replication means repeating each treatment multiple times.

This reduces the influence of random error and allows variance estimation.

LSD requires a square arrangement.

For t treatments, there are t rows and t columns.

So for 4 treatments → 4×4 square.

Fixed effects: specific levels chosen (e.g., 3 fertilizer types). Inference limited to those levels.

Random effects: levels sampled randomly (e.g., machines chosen from a factory). Inference applies to the population.

If treatments were not randomized, block characteristics (e.g., soil fertility) could bias treatment results.

Randomization inside each block prevents systematic bias.

CRD works only when units are very similar.

If there is natural variation among units (e.g., different soil fertility), CRD fails to control it.

That’s why RCBD or LSD is often preferred in real-life field experiments.

Source: Design and Analysis of Experiment Montogomery.

k = 4 (fertilizers), r = 5 → total N = 20.

df_total = N – 1 = 20 – 1 = 19.

df_treatments = k – 1 = 4 – 1 = 3.

df_error = df_total – df_treatments = 19 – 3 = 16.

a = 3 (treatments), b = 4 (blocks).

Total observations N = ab = 12.

df_total = N – 1 = 12 – 1 = 11.

df_treatments = a – 1 = 3 – 1 = 2.

df_blocks = b – 1 = 4 – 1 = 3.

df_error = (a–1)(b–1) = 2 × 3 = 6.

a = 4 → df_treat = 3.

b = 3 → df_error = (a–1)(b–1) = 3×2 = 6.

F = MSA / MSE = 25/5 = 5.0.

Since 5.0 > 4.76, reject H₀ → treatments significantly differ.

a = 4, b = 3, n = 2. 

Total = abn = 4×3×2 = 24.

a = 5, b = 4 → N = 20.

df_error = (a–1)(b–1) = 4 × 3 = 12.

MSE = SSE / df_error = 100 / 12 ≈ 8.33

a = 3, n = 6 → N = 18.

df_treat = a – 1 = 2.

df_error = N – a = 18 – 3 = 15.

MSA = SSA/df_treat = 90/2 = 45.

MSE = SSE/df_error = 60/15 = 4.

F = MSA / MSE = 45/4 = 11.25

Random vs fixed: If factor levels are intentionally chosen and inference is limited to those specific levels, use fixed effects.

If levels are a random sample from a larger population and you want to generalize (i.e., estimate population variance component), use a random-effects model.

Here, the 6 machines are randomly selected from 100 and the goal is to estimate machine-to-machine variability for the whole factory → random effects is appropriate.

The model will estimate a variance component for machines (σ²_machine) and the residual variance; these variance components are used for broader inference and prediction.

Why A is wrong: Fixed-effects would treat only these six levels as of interest, not generalize.

Why C & D are wrong: Two-way fixed with interaction is not the natural choice here unless there is a second factor and replication per cell design; ANCOVA is for covariate adjustment, not for modeling random sampling of factor levels.

Relative efficiency (RE)
= (MSE in CRD / MSE in RCBD) × 100.
= (12 / 9) × 100 = 133%.

So, RCBD is 33% more efficient than CRD.

Key idea: Blocking reduces error → smaller MSE → higher efficiency.

a = 6 (treatments), b = 5 (blocks).

Total observations N = ab = 30.

df_total = N – 1 = 29.

df_treat = a – 1 = 5.

df_block = b – 1 = 4.

df_error = (a–1)(b–1) = 5×4 = 20.

For LSD with t treatments:

Total observations = t².
df_total = t²–1.
df_treatments = t–1.
df_rows = t–1.
df_columns =t–1.
df_error = (t²–1)–3(t–1).

Here t = 5:
N = 25,
df_total = 24.
df_treat = 4,
df_rows = 4,
df_cols = 4.
df_error = 24 – (4+4+4) = 12.

For LSD with t treatments → need t × t units.

Here t = 4 → 4×4 = 16 units (observations).

df_total = 15, partitioned into: treatment = 3, row = 3, column = 3, error = 6.

In a factorial design, total treatment combinations = (number of levels of A) × (number of levels of B).

Here: A = 3 levels, B = 4 levels → 3×4 = 12.

CRD works when experimental units are homogeneous.

Lab conditions (identical rats, pots, etc.) → differences are mainly due to treatments.

In heterogeneous units (fields, schools, factories), CRD is not ideal.

RCBD is chosen when one nuisance source exists.

Blocks group similar units → treatment differences are then tested within blocks.

Example: Fertilizers tested across different soil fertility blocks.

LSD controls two sources of variation simultaneously (rows & columns).

Example: Testing teaching methods across teachers (rows) and classrooms (columns) while still comparing treatments.

ANCOVA = ANOVA + regression.

Used when there is a quantitative covariate (e.g., baseline measurement, age, GPA) that may affect the outcome.

Adjusts treatment means by removing variation due to covariates → increases precision.

Factorial design is efficient: studies multiple factors in one experiment.

It also detects interaction effects, which cannot be seen in one-factor-at-a-time studies.

A significant F-test in ANOVA only tells us that at least one group mean is different from the others.

It does not pinpoint where those differences lie.

To identify which specific pairs of means are different, post-hoc tests (like Tukey's HSD) are required.

Options a, c, and d are incorrect:
a) is false (it doesn't prove all are different),
c) describes an assumption, not a direct limitation of the result's interpretation, and
d) is false (ANOVA can handle unequal sample sizes, though homogeneity of variance becomes more critical).

The core utility of the F-test is its ability to compare the "signal" (variance between group means) to the "noise" (variance within the groups).

If the between-group variance is significantly larger than the within-group variance, we conclude the group means are not all equal.

Option a is false (the F-test is not robust to all violations),
c is false (the F-statistic itself is not an effect size; eta-squared or omega-squared are), and
d is false (it tests for differences, not directionality like a correlation coefficient).

The key weakness of a CRD is that it relies entirely on randomization to control for extraneous variation.

If there is a known source of heterogeneity (e.g., a fertility gradient in a field, or skill level differences in subjects), the CRD does not actively control for it.

This variation gets lumped into the experimental error, potentially masking true treatment effects.

Option a is true but not the major limitation;
c is false (other designs can be more powerful);
d is false.

An interaction effect indicates that the effect of one factor is not consistent across all levels of another factor.

For example, high fertilizer might increase yield only when irrigation is applied, but have no effect when it is not.

This is the most important concept in factorial experiments.

Option a describes main effects, not the interaction.

The Latin Square Design controls for two sources of nuisance variation (rows and columns).

However, a critical assumption is that these sources of variation do not interact with each other or with the treatments.

If such interactions exist, the design and its analysis can be invalidated.

Option a is false (it controls for two);
c is true but not the key statistical limitation;
d is false (it is typically more efficient).

By removing the variability due to blocks from the experimental error, the RCBD aims to produce a smaller, more precise estimate of the random error (MSE).

A smaller MSE leads to a larger F-statistic for testing treatments, making the test more sensitive (powerful).

The notation indicates the number of factors and their levels: 2 x 3 x 2 means three factors.

The first factor has 2 levels, the second has 3 levels, and the third has 2 levels.

The total number of treatment combinations is the product: 2 * 3 * 2 = 12.

The known source of variability is the driver.

To control for this, we can use a block design where each block consists of a single driver using all three additives (in a random order).

This removes driver-to-driver variability from the experimental error.

Blocking on the car (b) is incorrect because the cars are identical; the nuisance factor is the driver.

While factorial designs are highly efficient for understanding interactions, a major practical limitation is "combinatorial explosion."

As the number of factors and/or their levels increases, the total number of runs (treatment combinations) increases multiplicatively, which can be costly and time-consuming.

Option c is false (factorial designs are more efficient);
d is false (they are specifically used to estimate interactions).

Both options b and c are valid strategies.

A data transformation (like log, square root) can often stabilize variances across groups.

Alternatively, one can abandon the ANOVA framework altogether and use a non-parametric method that does not rely on the assumption of equal variances.

Option a might help but does not directly fix the violation.

While often stated as "normality of the data," the key assumption for the validity of the F-test is that the residuals (the differences between the observed values and their group means) are normally distributed.

The F-test is reasonably robust to mild violations of this assumption, especially with large sample sizes.

A non-significant interaction means that the effect of one factor is roughly the same at every level of the other factor(s).

In other words, the combined effect of the factors is simply the sum of their individual effects; they act independently.

This allows for the straightforward interpretation of the main effects.

An RCBD controls for one source of nuisance variation (e.g., a field gradient in one direction).

A Latin Square Design is used when there are two such sources of variation that can be arranged in rows and columns (e.g., a field gradient in two perpendicular directions, or effects of rows and columns in a factory).

The F-statistic is calculated as MS~between~ / MS~within~. MS~within~ is the denominator.

A large MS~within~ means there is a lot of variability within the treatment groups (high "noise"), which will lead to a smaller F-value, making it more difficult to detect a significant difference between the group means (the "signal").

The primary advantage of a CRD is its simplicity.

It is straightforward to randomize treatments to units and the subsequent statistical analysis (one-way ANOVA) is simple.

This makes it a good choice when experimental units are homogenous and no major known sources of variation are present.

Option a is false (blocking designs provide higher precision);
c is false;
d is not necessarily true.