When the block size is less than the number of treatments to be tested is known as an incomplete block design (IBD). Yates introduced incomplete block designs to eliminate the heterogeneity when the number of treatments becomes very large.

It is known that the precision of the estimate of a treatment effect depends on the number of replications of the treatment, that is, the larger the number of replications, the more the precision. A similar criterion holds for the precision of estimating the difference between two treatment effects. If two treatments occur together in a block, then we say that these are replicated once in that block.

Different patterns of values of the numbers of replications or different pairs of treatments in a design have given rise to different types of incomplete block designs.

The randomized block designs in which every treatment is not present in every block then these designs are known as randomized incomplete block designs. The choice of incomplete block designs depends on factors such as the number of treatments.

### Examples of Incomplete Block Designs

**Example 1:** When the set of treatments is larger than the block size, we use incomplete block designs. Suppose, we want to test the quality of six tires on a given car only 4 tires can be tested, such a block would be incomplete, as it is not possible to test all 6 tires on a given car at once.

**Example 2:** Consider a study comparing the effectiveness of three fertilizers ($A$, $B$, and $C$) on crop yield. If there are 12 experimental plots, a BIBD with 4 blocks of 3 plots each could be used. Each fertilizer would appear in 4 blocks, and each pair of fertilizers would appear together in 2 blocks.

**Example 3:** A pharmaceutical company wants to compare the effectiveness of four new drugs for treating a disease. Due to ethical considerations, patients cannot receive all four drugs. An IBD can be used to assign the drugs to different groups of patients, ensuring that each drug is tested against a variety of patient characteristics.

Using an IBD, the experimenter can control for variability between plots while still comparing the effects of the fertilizers.

### Types of Incomplete Block Designs

The following are types of incomplete block design:

**Balanced Incomplete Block Designs (BIBDs):**
- Each treatment appears in an equal number of blocks.
- Each block contains an equal number of experimental units.
- Every pair of treatments appears together in an equal number of blocks.

**Partially Balanced Incomplete Block Designs (PBIBDs):**
- Similar to BIBDs but with a more relaxed constraint on the number of times pairs of treatments appear together.

**Cyclic Designs:**
- A special type of BIBD where the treatments are arranged in a cyclic order within each block.

### Advantages and Disadvantages of IBDs

**Advantages**:

**Reduced Experiment Size:** IBD can require fewer experimental units compared to complete block designs.
**Feasibility:** IBD can be more practical when it is difficult or impossible to apply all treatments to every experimental unit.

**Disadvantages**:
**Increased Complexity:** Analysis can be more complex compared to complete block designs.
**Reduced Efficiency:** May not be as efficient as complete block designs in terms of precision.

### Applications of IBD

**Agricultural Experiments:** Testing different crop varieties or fertilizer treatments.
**Industrial Experiments:** Evaluating different manufacturing processes or materials.
**Medical Research:** Comparing the effectiveness of different treatments for a disease.

### Analysis of IBD

**Hypothesis Testing:** Testing hypotheses about the effects of treatments.
**Estimation of Treatment Effects:** Estimating the differences between treatment effects.
**Analysis of Variance (ANOVA):** IBD Can be used to assess the effects of treatments and blocks.
**Least Squares Estimation:** IBD is used to estimate treatment effects and block effects.
**Tukey’s HSD:** IBD can be used for multiple comparisons to identify significant differences between treatments.

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