Sampling and Design of Experiments

1. SRS, stratified and systematic sampling

Exam focus: Know when SRSWOR beats SRSWR (the finite population correction), and that proportional allocation beats SRS while Neyman allocation beats proportional.

Lecture 53: Selection of Sample — Simple Random Sampling

IIT Kanpur — NPTEL

SRS with and without replacement, and why the finite population correction appears.

Lecture 22: Sampling Techniques-I

NPTEL — IIT Roorkee

Stratified and systematic sampling alongside SRS: when stratification actually reduces variance.

Lec 9: Sampling Strategies

NPTEL — IIT Guwahati

A compact comparison of the standard strategies — good revision before the exam.

2. CRD, RBD, LSD essentials

Exam focus: Degrees of freedom are the most-asked item: CRD N − k, RBD (k−1)(r−1), LSD (p−1)(p−2). Blocking removes a nuisance source and costs error degrees of freedom.

Lecture 38: Introduction to Design of Experiment

NPTEL — IIT Kharagpur

Why designs exist: randomisation, replication, local control — the vocabulary the questions assume.

Lecture 39: Randomized Block Design

NPTEL — IIT Kharagpur

RBD and its ANOVA table; note where the (t−1)(b−1) error degrees of freedom come from.

Lec 26: Randomised Block Design (RBD)

NPTEL — IIT Roorkee

A second pass on RBD with worked computation — useful if the first treatment felt fast.

Design of experiments essentials free

The three basic designs

Design	Model	Error d.f.	Controls
CRD	$y_{ij} = μ + τ_{i} + ε_{ij}$	N − k	nothing
RBD	$y_{ij} = μ + τ_{i} + β_{j} + ε_{ij}$	(k−1)(r−1)	one nuisance factor
LSD (p×p)	$y_{ij k} = μ + τ_{i} + ρ_{j} + γ_{k} + ε$	(p−1)(p−2)	two nuisance factors

In an LSD the number of treatments equals the number of rows and columns (p), and $p^{2}$ units are used. If the error d.f. is 42, then (p−1)(p−2) = 42 gives p = 8, so treatments have 7 d.f.

ANOVA table shape

Total d.f. = N − 1, split into treatment, block(s) and error. The F statistic is MS_treatment/MS_error with (k−1, error) degrees of freedom.

Principles

Randomisation removes bias; replication gives an error estimate; local control (blocking) removes a known source of variability.
Blocking increases precision only if the blocks really differ — otherwise you have merely lost error degrees of freedom.
Relative efficiency of RBD to CRD = (MSE_CRD/MSE_RBD), adjusted for d.f.

Factorial experiments

$A 2^{n}$ factorial estimates main effects and interactions; confounding sacrifices a high-order interaction to reduce block size. Effects are estimated by contrasts of the treatment totals.

Missing plot

For one missing value in an RBD: ŷ = (rB + kT − G)/((r−1)(k−1)), and the error d.f. drops by one.

Key takeaways

Memorise the three error d.f. formulas — they are asked directly.
LSD: p treatments, $p^{2}$ , two nuisance directions.
Blocking trades error d.f. for reduced error variance; it only pays if blocks differ.