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A dependent variable has its value specified by a function of one or more independent variables. In $y=x^2$, choosing the function input $x$ determines the function output $y$; equal values of $y$ can nevertheless arise from different inputs.

In an ordinary differential equation such as $dy/dx=2x$, $x$ is the independent variable and $y$ is the dependent variable. The unknown object being sought is a function $y(x)$, rather than one numerical value of $y$. Its derivative describes how the dependent variable changes with the independent variable.

An Errors-in-variables model distinguishes a latent true predictor from its noisy measurement and models both the measurement and the response. For example, $W_i\mid U_i\sim N(U_i,s^2)$ and $Y_i\mid U_i\sim N(a+b{U}_i,v^2)$, with a population distribution for $U_i$, propagate predictor uncertainty into inference on $b$. Treating $W_i$ as exact can create attenuation bias from classical measurement error. Identifiability requires appropriate information about the measurement process or latent predictor distribution.

An Euler-Cauchy equation has powers of the independent variable matching the derivative orders, for example $a{x}^2{y}^{\prime \prime }+bx{y}^{\prime }+cy=0$. On $x>0$, trying $y=x^r$ gives the algebraic equation $ar(r-1)+br+c=0$. Distinct roots give two power solutions; a repeated root $r$ gives $x^r$ and $x^r\log x$.

The substitution $t=\log x$ explains the recipe: $x{y}^{\prime }=Y_t$ and $x^2{y}^{\prime \prime }=Y_{tt}-Y_t$, so the equation becomes a constant-coefficient differential equation. For $x^2{y}^{\prime \prime }-x{y}^{\prime }+y=0$, the repeated root is one and the solution is $y=x(A+B\log x)$. Intervals crossing zero require separate attention because zero is a singular point of a differential equation.

A variable whose levels define a comparison or predictor of interest. In an experiment it can be assigned or manipulated; calling an observed predictor an independent variable does not establish a causal effect. The measured outcome is the response variable.

A linear least-squares projection is the affine predictor that minimizes mean squared error. For real random variables $X,Y$ with finite second moments and $\mathrm{Var}(X)>0$, minimizing $\mathbb{E}(Y-a-bX{)}^2$ givesThe residual $R=Y-a-bX$ satisfies $\mathbb{E}R=0$ and $\mathbb{E}(RX)=0$. Consequently it is orthogonal, in the inner product $⟨U,V⟩=\mathbb{E}(UV)$, to the span of the constant variable and $X$. For any other affine predictor $a^{\prime }+b^{\prime }X$, expanding the square yieldsbecause the cross term vanishes. This proves optimality and uniqueness of the predicted random variable. The residual variance is the Schur complement covariance. No normal distribution is needed. For several predictors, the same orthogonality argument gives the normal equations for Linear least squares; a singular predictor covariance can make coefficients nonunique while leaving the projected prediction unique.

A measurement error is the difference between a recorded measurement $W$ and the quantity $U$ intended to be measured. The additive model $W=U+e$ separates that error $e$ from genuine variation in $U$. A shared offset is a systematic error; fluctuations between repeated measurements are random measurement error. Repeating a measurement can reduce independent Random measurement error, but does not remove a shared systematic error.

measurement error in a predictor can bias a regression slope even when it has mean zero. Suppose $Y=a+bU+\epsilon$, where $e$ is uncorrelated with $U$ and $\epsilon$, and $\epsilon$ is uncorrelated with $U$. A regression of $Y$ on the recorded $W$ has population slopeThus equal predictor and error variances halve the slope: this is attenuation bias from classical measurement error. In contrast, independent mean-zero error added only to $Y$ increases residual variance without changing that population slope. An errors-in-variables model describes the measurement mechanism explicitly rather than treating noisy predictors as exact.

In an additive mean model, the partial residual for component $j$ subtracts every fitted term except that component. It equals the full regression residual plus the fitted component. Plotting it against that component's predictor shows the conditional pattern after accounting for other fitted effects, and supplies the response for a backfitting update.

The two independent variables are social-cue congruence, with congruent and incongruent levels, and cue-to-response delay, with short ($250$ ms) and long ($750$ ms) levels. The response variable is reproduction reaction time, measured in seconds. Thus the known treatment structure is a $2\times 2$ factorial design, potentially estimating two main effects and their interaction.

TODO: identify the allocation from the missing supplementary sheet or protocol. The eighty recruited participants do not establish whether either factor is within-subjects or between-subjects, whether all observations were retained, or how trials were summarized. The design cannot responsibly be called fully repeated-measures, fully independent-groups or mixed without that information.

For finite second moments and $\mathrm{Var}(X)>0$, the residual variance after affine projection onto one predictor $X$ is $\mathrm{Var}(Y)-\mathrm{Cov}(X,Y{)}^2/\mathrm{Var}(X)$, a covariance Schur complement.