--- title: "Models" --- # From Designs to Models This semester we started with mechanisms we could name. - A random sample told us how observations entered a study. - A randomized experiment told us how treatment was assigned. - Conditioning let us separate two random stages and put them back together. Those mechanisms supplied probability distributions for our estimators. They told us what would change if we repeated the study and what would remain fixed. In many applications nobody literally rolls a die to sample a person, and nobody randomizes the variables we want to compare. Statistical analysis still proceeds by describing the data **as if** a repeatable random process had generated them. That description is a model. A model is not a claim that every detail of the world is random. It is a claim that a particular imagined repetition preserves the features that matter for the question and reproduces the estimator's uncertainty well enough to calibrate it. # Two Jobs a Model Can Do Consider $$ Y_i=m(X_i)+\epsilon_i, \qquad \E[\epsilon_i\mid X_i]=0. $$ The same notation can be used for two different jobs. 1. **Description.** The function $m(x)$ is the actual conditional mean $\E[Y_i\mid X_i=x]$. We use it to describe or predict the population. 2. **Approximation for inference.** A simpler function $m$ stands in for a more complicated relationship so we can calculate an estimate and its uncertainty. Those jobs should not be confused. A line can be a poor literal description of a curve and still provide a useful summary. But an interval around the line's slope is calibrated for that summary only if the imagined random process gets the relevant variation right. # A Deliberately Wrong Line The black curve in the next figures is $\sin(w)$. The red line is a linear fit. The purple and green ribbons show two different covariate distributions. The dashed red line is fitted from the displayed sample; the solid red line is the large-sample best linear approximation. :::: {.columns} ::: {.column} ![A linear fit with 100 observations.](../figs/least-squares-linear-sin-n100.pdf){#fig-model-n100} ::: ::: {.column} ![A linear fit with 10,000 observations.](../figs/least-squares-linear-sin-n10000.pdf){#fig-model-n10000} ::: :::: With 100 observations, the fitted line moves noticeably from sample to sample. With 10,000 observations, the dashed fitted line is close to the solid best line. More data removed **fitting error**: uncertainty about which line is best. More data did not make a line equal a sine curve. The gap between the best line and the curve is **modeling error**. It remains even when the sample is enormous. Write the total error schematically as $$ \widehat\theta-\theta =\underbrace{(\widehat\theta-\theta_{\mathcal M})}_{\text{fitting error}} +\underbrace{(\theta_{\mathcal M}-\theta)}_{\text{modeling error}}, $$ where $\theta_{\mathcal M}$ is the answer supplied by the best approximation in the chosen model $\mathcal M$. Fitting error usually shrinks with sample size. Modeling error need not. Consequently an interval can become very narrow around the wrong answer. ::: {.callout-note} ### Figure Provenance The two panels reproduce the plotting path in the older Lab 10 source. The source did not set a seed; this reconstruction uses seed 285. The $n=10{,}000$ run continues the random-number stream after the $n=100$ run. No plotted layer was added, removed, or restyled. ::: # Bias Does Not Shrink Like Standard Error ![A reconstructed comparison of fixed misspecification bias and shrinking standard errors.](../figs/error-vs-n.pdf){#fig-model-error-vs-n} In @fig-model-error-vs-n, the blue and green series are standard errors. They fall as the sample grows. The red series is the absolute bias of an unweighted fit for a target population whose covariate distribution differs from the sampling population. It does not fall toward zero. This candidate reconstructs a missing older figure from the preserved simulation ingredients. The original generator is absent. The eight sample sizes, 500 replications, logarithmic axes, and styling are therefore disclosed reconstruction choices rather than recovered course-source facts. The qualitative comparison is the source-backed point: more data can make random error small while systematic error stays put. ::: {.callout-exercise} ### More Certain About What? Suppose an estimator has bias $0.10$ for every sample size, while its standard error falls from $0.08$ to $0.01$. What improves with more data, and what does not? ::: ::: {.callout-solution} The estimator becomes more repeatable: its sampling distribution gets narrower. Its center remains $0.10$ away from the target. More data increases precision but does not repair the modeling error. ::: # Variance Models and Calibration Even when the mean description is useful, the spread can change with the covariates: $$ Y_i=m(X_i)+\sigma(X_i)\epsilon_i. $$ A homoskedastic calculation replaces $\sigma(X_i)$ by one constant $\sigma$. If the actual spread differs greatly across $X$, that shortcut can give the wrong standard error even when the fitted mean is the summary we want. This is why point estimation and calibration are separate obligations. - A point estimator asks which population summary the fitted object approaches. - A standard-error estimator asks whether the reported spread matches repeated estimates under the relevant random process. Robust standard errors and design-based variance formulas try to preserve the second obligation without claiming constant variance. # The Bootstrap Is Also a Model The bootstrap treats the empirical distribution---the sample with probability $1/n$ on each observed row---as a stand-in for the population. A bootstrap sample is a random sample from that empirical population. This does not make the bootstrap assumption-free. It assumes that resampling rows recreates the variation relevant to the estimator. - For independent sampling units, resampling units can reproduce sampling variability. - If treatment was randomized while the sample was fixed, rerandomizing treatment represents a different mechanism from resampling rows. - If observations arrive in households, classrooms, or time series, resampling individual rows can destroy dependence that the real repetition would preserve. The question is never merely “did we bootstrap?” It is “what population and what repetition did this bootstrap imagine?” # When the Model Breaks A useful model deliberately ignores detail. It fails when an ignored detail controls the answer or its uncertainty. - **Selection:** the observed units do not represent the target population. - **Confounding:** treatment groups differ in their untreated potential outcomes. - **Dependence:** units move together, so row-by-row variation is too small. - **Heavy tails or rare events:** a modest sample does not reveal the events that dominate the estimator. - **Misspecified structure:** a line, constant effect, or common variance erases a feature the target depends on. Checking calibration by simulation is useful only when the simulation includes the failure one is worried about. Simulating from the fitted model and recovering the fitted model checks the calculation under its own assumptions; it does not test those assumptions against the world. # The Pragmatic View A model is a bet that certain details do not matter for a stated purpose. To evaluate that bet, ask four questions: 1. What is fixed and what is random in the imagined repetition? 2. Which population quantity does the estimator approach? 3. Which features determine its standard error? 4. What real feature, if omitted, would change the target or the calibration? The first semester built answers from explicit sampling and randomization designs. The next semester will work with regression models directly. The designs remain the reference point: a model earns an inferential interpretation by reproducing the part of a credible design that the question needs, not by being literally true in every detail.