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In a two-factor design with four corner observations and $n_0$ independent center replicates, the difference between the corner mean and center mean has variance ${\sigma }^2(1/4+1/n_0)$. Under a full quadratic regression surface it estimates ${\beta }_{11}+{\beta }_{22}$; the linear and interaction terms average to zero at the corners. With $n_0\ge 2$, the center pure error estimate makes its squared standardized contrast an $F_{1,n_0-1}$ test under a first-order mean model. Zero contrast does not exclude curvature, since the two squared-term coefficients may cancel.

A central composite design augments a factorial portion with axial points $\pm \alpha e_j$ and replicated centre points. It supports a full quadratic response model: intercept, linear terms, cross-products and squared coordinates. Centre-point replication estimates pure error. Choosing the axial distance appropriately gives a rotatable design.

Lack of fit is a mismatch between a regression model's mean function and the true mean relationship. Replicated predictor configurations can separate its sum of squares from pure error. A residual-versus-fitted plot can suggest curvature or changing variance even without a formal replication-based test.

For $n$ independent normal responses of common variance at $g$ distinct predictor settings, a rank-$p$ normal linear model has residual degrees $n-p$. The saturated setting-mean model has $g$ parameters and pure error degrees $n-g$. Splitting orthogonal residual spaces gives lack of fit degrees $g-p$. Under the fitted mean model, the two squared lengths divided by the common variance are independent chi-squared variables; their ratio of mean squares has the F-distribution shown. This test is unavailable without repeated settings or positive lack-of-fit degrees.

For this fraction, $SCW=-1$ and $SCMT=1$, so their product is $WMT=-1$. The signed defining relation isThe complete factorial contrast alias classes areThe five main effects and $ST$ occupy six distinct alias classes, and none aliases with the intercept. Every undesired two-factor term in these classes is assumed negligible, and all terms of order at least three are also assumed negligible. Thus design (b) is suitable under the stated interaction assumptions. Its intercept and six requested columns are mutually orthogonal, with $X^{T}X=8I_7$, so each coded regression coefficient is ${\hat{\beta }}_A=8^{-1}{\sum }_{i=1}^8{\chi }_A(x_i)Y_i$. The corresponding high-minus-low main effect is $2{\hat{\beta }}_A$.

There is one residual degree of freedom, but no replicated setting and therefore no separate pure error estimate. Suitability here means estimability under the stated model; it does not protect estimates from active interactions that the assumptions exclude.

For the general normal linear model, minimizing $\Vert Y-X\beta {\Vert }^2$ gives the least-squares normal equations $X^{T}X\hat{\beta }=X^{T}Y$. Full column rank makes $X^{T}X$ invertible, soThe mean and covariance follow by substituting $Y=X\beta +ϵ$ and using $\mathbb{E}ϵ=0$, $\mathrm{Cov}(ϵ)={\sigma }^{2}I$. Normal errors also give the exact multivariate normal distribution of this estimator.

Use coded experimental variablesTheir origin is the middle of the allowed rectangle, and the stated endpoints become $\pm 1$. The four corners and $n_0$ center runs give mutually orthogonal columns $1,x_1,x_2$, withWriting ${\bar{Y}}_0=n_0^{-1}{\sum }_{i=5}^{4+n_0}{Y}_i$ when $n_0>0$, the ordinary least squares estimates areTheir variances are ${\sigma }^2/(4+n_0)$, ${\sigma }^2/4$, ${\sigma }^2/4$, with zero covariances.

The repeated center runs estimate pure error without requiring the fitted mean surface to be correct:They also permit a center-point curvature contrast. Let ${\bar{Y}}_F=(Y_1+Y_2+Y_3+Y_4)/4$ and $\hat{\gamma }={\bar{Y}}_F-{\bar{Y}}_0$. Under the first-order model, $\mathbb{E}\hat{\gamma }=0$ and $\mathrm{Var}(\hat{\gamma })={\sigma }^2(1/4+1/n_0)$. Under a quadratic surface, its expectation is ${\beta }_{11}+{\beta }_{22}$, since the linear and $x_1{x}_2$ terms average to zero. Thusunder the first-order normal model. With only one center run there is no pure-error degree of freedom. A zero center contrast does not exclude all curvature, because the two quadratic coefficients can cancel. The corners also estimate an interaction contrast; the full Lack-of-fit F-test compares the first-order fit with the five setting means, with two lack-of-fit and $n_0-1$ pure-error degrees of freedom.

If the first-order model is adequate and $\hat{b}=({\hat{\beta }}_1,{\hat{\beta }}_2{)}^T\ne 0$, steepest ascent in response surface methodology uses the unit coded direction $d=\hat{b}/\Vert \hat{b}\Vert$. The Cauchy-Schwarz inequality proves that it maximizes the predicted directional derivative. Run sequential settings $x=hd$ for increasing feasible $h$, corresponding to physical increments $(5h{d}_1,2.5h{d}_2)$ from the center. Stop or reduce the steps when measured yield ceases to improve, and refit locally near promising settings. Respect the allowed rectangle; if a boundary is reached, investigate feasible boundary directions rather than extrapolating outside it. A near-zero gradient offers no first-order ascent direction and calls for further modeling.

If curvature is indicated, add the four axial settings $(1,0),(-1,0),(0,1),(0,-1)$, together with further center replication. This face-centered central composite design stays inside the allowed physical ranges and separates the two squared coordinates, which the corners alone cannot do. Fit the full quadratic regression surfaceWrite its fitted quadratic part as $x^{T}Bx$, where $B_{11}={\hat{\beta }}_{11}$, $B_{22}={\hat{\beta }}_{22}$ and $B_{12}=B_{21}={\hat{\beta }}_{12}/2$. Its gradient and Hessian are $\hat{b}+2Bx$ and $2B$. If $B$ is nonsingular, the stationary setting isIf $B$ is negative definite and $x_{\ast }$ is feasible, it is the fitted maximum. Otherwise compare feasible stationary settings, optima on each edge obtained from the restricted one-variable quadratic, and the corners. A singular or poorly determined $B$ calls for attention to ridge directions and more runs. Convert any chosen coded setting back using ${\xi }_1=95+5x_1$, ${\xi }_2=37.5+2.5x_2$, and confirm its yield experimentally. The fitted local surface guides the search; it does not establish an untested physical optimum.

Code each two-level factor as a sign $x_j\in \{-1,1\}$. A factorial contrast indexed by a factor subset $A$ has sign column ${\chi }_A(x)={\prod }_{j\in A}{x}_j$, with the empty subset giving the intercept. Construct a regular fractional factorial design by choosing $k$ independent words $G_1,\dots ,G_k$ and retaining the settings satisfying ${\chi }_{G_i}(x)=s_i$, where each $s_i\in \{-1,1\}$. In binary exponent notation these are $k$ independent linear equations over ${\mathbb{F}}_2$, leaving $m-k$ freely chosen coordinates and exactly $2^{m-k}$ settings. Dependence of the chosen words would fail to give the stated fraction size.

The words generated by multiplying the $G_i$ form the defining contrast subgroup $\mathcal{H}$, of order $2^k$. Multiplication cancels a repeated factor, so word multiplication is symmetric difference of subsets. On the fraction, ${\chi }_H=s_H$ is constant for every $H\in \mathcal{H}$, and henceTo see that these are all the aliases, average ${\chi }_A{\chi }_B={\chi }_{AB}$ over the fraction. If $AB\notin \mathcal{H}$, this is a nonconstant character of its freely varying binary coordinates; pairing settings that reverse that character makes the average zero. If $AB\in \mathcal{H}$, the product is constant. Thus the aliasing in a fractional factorial design classes are exactly the cosets of $\mathcal{H}$. For a half fraction, $\mathcal{H}=\{I,G\}$ with $G\ne I$, and every contrast column is aliased with exactly the other word $AG$, with a plus or minus sign according to the chosen fraction. This includes the defining word paired with the intercept.

For the six-factor experiment, take the 32-run half fractionIts resolution of a fractional factorial design is VI. Every main effect is aliased with a five-factor interaction, and every two-factor interaction with a four-factor interaction. In particular, no two required low-order effects are aliased with one another. Under the stated negligible-higher-interaction assumption, all six main effects and the fourteen required two-factor interactions are estimable before introducing blocks.

To form four equal days, classify runs by the two signsTheir product is $x_B{x}_C{x}_F$. The three nonconstant block contrasts therefore coincide with the treatment-column alias classesOnly $AF$ among the main effects and two-factor interactions is lost to block confounding in a factorial design; the other two block directions involve only negligible three-factor interactions.

Here is an explicit construction of every run without an ambiguous generator convention. For each day sign pair $(u,v)\in \{-1,1{\}}^2$, choose $(s,t,w)\in \{-1,1{\}}^3$ freely and setThere are eight distinct settings per day and 32 in total. Direct multiplication gives $ABCDEF=1$, $AF=u$ and $ABC=v$. Conversely those three equations solve $C,E,F$ uniquely from $A,B,D$, so there are no missing or repeated settings. Assign the four sign pairs randomly to the four days, and randomize the eight runs within each day.

Distinct nonaliased factorial contrasts are orthogonal over the fraction. None of the 20 required low-order contrast columns belongs to a block class, so all are also orthogonal to the three block-effect columns. Their joint design matrix, including an intercept, therefore has rank $1+3+6+14=24$. The analysis of variance partition isThe eight residual directions are the remaining unconfounded three-factor alias classes. They estimate error only under the negligible-higher-interaction assumption; they are not a separate replicated pure error stratum. This provides the requested comparisons within exactly four days of eight runs.

For the initial normal linear model, the design matrix has rows $(1,x_1,x_2,x_3)$. The eight factorial corners have balanced signs and orthogonal coordinate columns; the six centre points contribute only to the intercept. ConsequentlyUsing the least-squares normal equations gives the least-squares estimatorIt is a linear transformation of independent normal responses, so its sampling distribution isThe four estimated coefficients are independent, since they are jointly normal with diagonal covariance matrix.

Adding the axial settings produces a central composite design: eight factorial corners, six axial points and six replicated centre points. The axial distance $1.682$ is the rounded value of $8^{1/4}=1.6817928\dots$. With that exact distance it is a rotatable design for a full quadratic model. Indeed, sign symmetry annihilates odd moments, all coordinates have the same second moment, and the fourth moments satisfyRotational symmetry requires the first to be three times the second, giving ${\alpha }^4=8$. Using the printed rounded distance makes rotatability approximate, rather than mathematically exact.

The full quadratic regression mean has ten coefficients:The expanded design matrix has rank ten. To verify this rather than just count parameters, suppose such a quadratic vanishes at every setting. Its centre value sets its constant term to zero; subtracting the values at opposite axial points sets every linear coefficient to zero, and adding them then sets every squared-coordinate coefficient to zero. On the factorial corners, orthogonality of the three pair-product columns forces their coefficients to zero as well. Thus all ten columns are independent, leaving $20-10=10$ residual statistical degrees of freedom.

There are $15$ distinct settings: eight corners, six axial points and the centre. The six centre replicates give the only pure error, with $6-1=5$ degrees and sum of squares $859.33$. The saturated setting-mean model has $15$ parameters, so the lack of fit beyond the quadratic model has $15-10=5$ degrees. For any setting $r$ with $n_r$ replicates, the identitysplits the residual sum of squares into pure error and lack of fit; the cross term vanishes because the deviations from the setting mean sum to zero. Summing givesUnder a correct quadratic mean with independent errors of constant normal variance, the pure-error and lack-of-fit projections are orthogonal, hence their sums of squares are independent with respective distributions ${\sigma }^2{\chi }_5^2$ and ${\sigma }^2{\chi }_5^2$. The Lack-of-fit F-test therefore usesThis is below the given $10\%$ and $5\%$ critical values $3.45$ and $5.05$. There is no significant lack of fit at either level. That is failure to reject this quadratic mean, not proof that it is exact or that a maximum has already been located. The pure-error denominator also assumes the error variance at the centre represents the common variance throughout the experiment.

The data do not cross every age with every gender: only age 21 has both genders, while age 30 has only women and age 40 only men. Thus a gender effect separate from age can be estimated under an additive model, but age-by-gender interactions cannot all be estimated or checked. There is only one observation in each recorded combination of age, gender, policy, and points, so there is no within-cell replication for separating pure error from lack of fit. No sampling method or additional risk characteristics are supplied, limiting comparisons between policyholder categories.

There is also a source discrepancy: the printed table gives $268$ for one comprehensive premium, whereas the R data vector gives $368$ at the corresponding position. The supplied regression output is treated as referring to the R data. This should be resolved against the original data before substantive use.