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A block design assigns treatments to experimental units partitioned into blocks. Its incidence counts $n_{ij}$ record how often treatment $i$ occurs in block $j$. Balance and connectedness determine which treatment contrasts are estimable and how much adjustment for blocks costs.

Design of experiments chooses experimental units, treatments, replication, blocks in experimental design and randomization so that scientifically useful treatment contrasts can be estimated with meaningful standard errors. Allocation determines which measurements supply independent treatment information.

At equally replicated quantitative treatment levels, evaluations of orthogonal polynomials generate pairwise orthogonal treatment contrasts, using the replication-weighted inner product. Five equally spaced levels give four contrast directions representing degrees one through four, besides the constant direction. This decomposes treatment variation without assuming the response has low polynomial degree.

In an R formula, I(...) protects ordinary arithmetic from the formula operators: the predictor is the numerical variable $z=x-165$, rather than an instruction to remove a formula term. Predictor centering leaves fitted values and the height slope unchanged but places the intercept at a meaningful height. It often reduces the correlation between intercept and slope estimates.

With $M_i=1_{\{i=2\}}$, treatment contrasts use female as the reference level in a regression factor. The fitted normal linear model isThe estimates and their standard errors areThe intercept estimates female mean weight at height 165. At that same height the male fitted mean is $58.4711+4.3294=62.8005$. Holding height fixed, the fitted male mean exceeds the female mean by 4.3294 weight units. Within either sex, one additional height unit increases fitted mean weight by 0.6211 weight units. If height is in centimetres and weight in kilograms, these interpretations use centimetres and kilograms accordingly. A standard error for the male intercept would require the covariance between ${\hat{\beta }}_0$ and ${\hat{\beta }}_M$, which is not printed.

The prediction equation is $\hat{Y}=58.4711+4.3294M+0.6211(x-165)$. Predictions for individual students also have residual uncertainty, estimated by $s=5.710$, beyond uncertainty in the fitted mean.

Among patients still alive at the same time after surgery, chemotherapy has an estimated lung-cancer death rate 3.71 times that of radiotherapy, assuming proportional cause-specific hazards. The estimated 95% confidence interval for this hazard ratio is $(1.583,8.697)$. Thus the fit suggests a higher instantaneous lung-cancer death rate on chemotherapy, with appreciable uncertainty in its size.

The null hypothesis of no treatment effect on that hazard is $H_0:{\gamma }_1=0$ against ${\gamma }_1\ne 0$. Its Wald test uses $Z=1.3113/0.4346\simeq 3.018$, approximately standard normal under the null hypothesis, with two-sided $p=0.00255$. This is evidence against equal lung-cancer death hazards in the fitted model. It is not a statement that five-year lung-cancer death probability is multiplied by 3.71 or that survival duration is divided by 3.71. The cumulative incidence function depends on both cause-specific hazards, and the model needs proportional-hazards and independent observation-censoring assumptions. In a randomised comparison the assigned treatment contrast is protected against baseline confounding, but the instantaneous comparison concerns the groups who have survived to the time in question; it does not identify a treatment effect within a common latent survivor subgroup.

An orthogonal block design has treatment contrasts orthogonal to block contrasts after removing the grand mean. This is a statement about vectors on the experimental units, with the usual inner product; the two full spaces are not orthogonal because both contain the constant vector.

For a counting criterion, let $n_{ij}$ count experimental units receiving treatment $i$ in block $j$, let $r_i={\sum }_j{n}_{ij}$, let $k_j={\sum }_i{n}_{ij}$, and let $n={\sum }_i{r}_i$. The inner product of the centered indicators for treatment $i$ and block $j$ is $n_{ij}-r_i{k}_j/n$. ConsequentlyIn particular, equal-sized blocks must contain each treatment in the same proportion. Under an additive block-and-treatment model, adjustment for blocks then does not change the fitted treatment contrasts.

If the biological cells are initially interchangeable, randomly allocate them to five groups of ten for the days. Within each day, independently choose a uniform distribution on a finite set over assignments having two occurrences of each dose level to the ten experimental units. One implementation is a random permutation of a list containing two labels of each level; the duplicate labels give every admissible assignment the same number of underlying permutations.

Randomize the processing order within each day as well, while pairing each preparation with its immediate measurement. Randomize dose within day, not merely the names of the days. This restricted randomization preserves the planned balance while protecting treatment contrasts against systematic order effects. Conceal the dose labels from the assessor where practical.

A completely randomized design uses $2m$ eligible animals, randomly selecting $m$ to receive $P$ and assigning the remainder to $Q$. Use comparable follow-up and assess the same response variable in both groups. Its advantage is simplicity and freedom from previous-treatment carryover effects; its disadvantage is that between-animal variation enters the residual and can make a treatment contrast imprecise.

A randomized complete block design with matched pairs first forms $m$ pairs using pre-treatment characteristics such as initial disease severity, age or breed. Independently choose which animal in each pair receives $P$, with its partner receiving $Q$. The experimental units are animals; pairs are blocks. The average within-pair difference estimates the treatment contrast, and positive within-pair similarity can reduce its variance. Its advantage is control of known heterogeneity; its disadvantage is the need for useful matching, with fewer residual statistical degrees of freedom and little gain if the matching variables are uninformative. Do not construct pairs using outcomes observed after assignment.

A two-period crossover design randomly assigns half the animals to sequence $PQ$ and half to $QP$. Each animal receives both treatments in separate periods, with a scientifically justified interval between them and the same outcome assessment after each period. Animal blocks remove persistent between-animal differences, while the two sequences balance treatment against period. Its advantage is potentially high precision from within-animal comparisons; its disadvantage is vulnerability to carryover effects, changing disease state and irreversible effects. It is suitable only when comparing the treatments in both periods remains meaningful and residual effects of the first treatment are adequately controlled. A two-period crossover design does not by itself disentangle arbitrary treatment-specific carryover effects.

The orchard residual mean square in ANOVA is on the original tree-response scale: it equals 30 times the corresponding residual mean square in ANOVA for orchard means. Thus the estimated variance of one orchard's sample mean is $240/30=8$.

Each pruning marginal sample mean averages four independent orchard means, giving variance $8/4=2$. Two different pruning marginals use disjoint orchards, so the variance of their estimated treatment contrast isEach spray marginal sample mean uses six orchards. Similarly,The corresponding standard errors are $2$ and $\sqrt{8/3}$ in the units of weight per tree. These compare per-tree marginal sample means, averaging equally over the other factor, even when an interaction term is fitted. Comparing orchard totals instead would multiply these variances by $30^2$.

Report the six treatment-combination means and their uncertainty, together with useful marginal treatment contrasts, standard errors and confidence intervals. Means are needed to interpret direction and practical importance, while the fitted interaction term determines whether one factor's effects should be reported separately at levels of the other. For the pairwise marginal comparisons in part (c), ordinary model-based 95% Student t confidence intervals use $t_{6,0.975}$ times the corresponding standard error; a simultaneous set of comparisons needs an appropriate adjustment.

Also describe orchard selection, actual randomization, treatment delivery, numbers of trees observed, missing data and response units. Check orchard-level residuals for unequal variance, outliers or systematic patterns using a residual-versus-fitted plot and an appropriate distributional diagnostic. Explain that twelve orchards provide independent treatment replication; 360 tree measurements provide subsampling precision. No numerical treatment means or numerical confidence intervals can be recovered from the supplied sums of squares in ANOVA alone.

A regression factor is a categorical variable encoded by indicator columns or treatment contrasts in a statistical model. A factor with $k$ observed levels normally contributes $k-1$ statistical degrees of freedom when an intercept is present. The reference level in a regression factor has coefficient zero under treatment contrasts. Coding a year as a factor permits arbitrary year effects rather than imposing a linear trend.

A treatment contrast is a linear contrast of cell means comparing treatment responses, with coefficients adding to zero. Pairwise differences and marginal factorial differences are examples. Its variance depends on the true allocation units and their covariance matrix.