Why Generate Tables From Model Objects?
A regression table is a compact report of selected results from one or more fitted models. It may display coefficient estimates and their standard errors, the number of observations used, and a small set of fit summaries. The central practice in this tutorial is to build that table from the fitted model object, rather than typing values into a table by hand.
A fitted model object is the software result created when a specified model is estimated from data. Depending on the software and model type, it can contain coefficient estimates, an estimated variance-covariance matrix, residual information, the model formula, and details about the observations used. Table-generating functions extract selected information from that object and format it consistently. This reduces transcription errors, but it does not decide which results are appropriate to report or establish that the model is suitable.
What Goes Into a Regression Table?
A useful starting point is to distinguish coefficient-level quantities from model-level quantities. A coefficient row usually contains an estimate and an uncertainty measure, commonly a standard error. A model-level section may include the sample size and selected fit summaries. Keep the table focused: a value belongs in it because it helps readers understand the fitted model, not merely because the software returned it.
| Table entry | Level | What it summarizes |
|---|---|---|
| Coefficient estimate | Coefficient | The estimated value of one model coefficient |
| Standard error | Coefficient | Estimated sampling uncertainty for that coefficient, under the model and standard-error method used |
| Sample size, \(n\) | Model | The number of observations used to fit that model |
| \(R^2\) | Model | For ordinary least squares with an intercept, the proportion of observed outcome variation accounted for by the fitted regression in the estimation data |
| Residual standard error | Model | An estimate of the typical residual scale, in the outcome's units |
The same label can refer to different quantities in different software or model families, so check the model class and the extraction method. For example, the ordinary least-squares definition of \(R^2\) should not be silently treated as a universal fit measure for every regression model. Report only summaries whose meaning is clear for the model being shown.
Worked Example: Building a Table for One Linear Model
Worked Example: Building a Table for One Linear Model
Suppose a hypothetical data set contains five observations on an outcome and one quantitative predictor. An ordinary least-squares model with an intercept has been fitted. The model object reports an intercept estimate of 4.20 with standard error 0.735, a predictor estimate of 0.500 with standard error 0.596, a regression sum of squares of 1.90, a residual sum of squares of 8.10, and \(n=5\). The predictor has a centered sum of squares of 7.60.
First, verify the model-level summaries rather than copying them uncritically. The total sum of squares is the regression sum of squares plus the residual sum of squares:
For this ordinary least-squares model with an intercept, \(R^2=SSR/SST\). Therefore,
There is one predictor, so \(k=1\). The residual degrees of freedom are \(n-k-1=5-1-1=3\). The residual standard error is the square root of the residual mean square:
The reported coefficient standard errors are also consistent with these summaries. The mean squared error is \(2.70\). With centered predictor values whose sum of squares is \(7.60\), the slope standard error is \(\sqrt{2.70/7.60}=0.5960\). With the predictor centered, the intercept standard error is \(\sqrt{2.70/5}=0.7348\). Rounding to three significant figures gives the reported values.
A compact table generated from the model object could therefore show:
| Term | Estimate | Standard error |
|---|---|---|
| Intercept | 4.20 | 0.735 |
| Predictor | 0.500 | 0.596 |
| Model summary | Value | |
| Observations | 5 | |
| \(R^2\) | 0.190 | |
| Residual standard error | 1.64 |
The ratio \(1.90/10.00\) is the reported \(R^2\), not its complement. The residual proportion is \(8.10/10.00=0.810\), and \(1-R^2=0.810\). Keeping regression and residual sums of squares distinct prevents a common reversal.
How the Extraction Works
Many statistical tools provide separate functions for coefficient-level and model-level results. In R, for example, a common workflow uses broom::tidy(fit) to extract coefficient rows and broom::glance(fit) to extract model-level summaries. A table-making function can then combine those results and format the chosen columns. Other software and packages use different function names, but the principle is the same: use the fitted object as the source, and make the extraction and formatting steps part of the analysis script.
A coefficient extractor may return columns such as the term name, estimate, standard error, test statistic, and p-value. A model-summary extractor may return \(n\), \(R^2\), or other fit measures. These returned fields are not a requirement to display every field. Select the columns in the script, label them clearly, and keep the model object and table-making code together so the table can be regenerated after a rerun.
Save the fitted object under a clear name after applying the documented analysis sample and model specification.
Use the software's model-aware extraction method for coefficient results and model-level summaries.
Choose the terms and summaries needed for the report, then apply consistent rounding and labels.
Compare its entries with the fitted object and confirm that the sample size and fit statistics match the intended model.
Worked Example: Comparing Two Fitted Models
Worked Example: Comparing Two Fitted Models
Consider two hypothetical ordinary least-squares models for the same outcome. Model A uses one predictor and Model B uses that predictor plus a second predictor. Both were fitted to the same 120 observations. Their retained model objects report the following values: Model A has \(R^2=0.28\) and residual standard error 5.40; Model B has \(R^2=0.34\) and residual standard error 5.20. The coefficient extractors return the estimates and standard errors shown below.
| Term | Model A estimate | Model A SE | Model B estimate | Model B SE |
|---|---|---|---|---|
| Intercept | 18.2 | 2.10 | 17.6 | 2.20 |
| Predictor 1 | 1.35 | 0.310 | 1.10 | 0.340 |
| Predictor 2 | — | — | 0.620 | 0.280 |
A model-summary section can show 120 observations for each model, \(R^2\) values of 0.28 and 0.34, and residual standard errors of 5.40 and 5.20, respectively. For an ordinary least-squares model with an intercept, the residual standard error uses \(\sqrt{SSE/(n-k-1)}\). Model A has residual degrees of freedom \(120-1-1=118\), while Model B has \(120-2-1=117\). Thus the two residual standard errors need not be computed with the same denominator, even though the models use the same number of observations.
The table-maker should extract each model's estimates and summaries from its own object. It should not assume that the row order is identical in both coefficient outputs: terms should be aligned by their names, not simply placed side by side by position. The dash indicates that Predictor 2 is not a term in Model A; it does not represent an estimated zero coefficient.
Model Assumptions and the Meaning of Uncertainty
Generating a standard error from a model object does not make that standard error automatically appropriate. For ordinary least squares, a familiar model is \(Y_i=\beta_0+\beta_1x_{i1}+\cdots+\beta_kx_{ik}+\epsilon_i\), with errors having mean zero, constant variance, and independence across observations. Normal errors support exact small-sample \(t\)-based inference; other standard-error methods may rely on different conditions. The model form and the chosen uncertainty method should be appropriate to the data and analysis.
A standard error is an estimated measure of the sampling variability of an estimator under the specified model and method. It is not the spread of individual outcomes and is not a guarantee that assumptions hold. If a table uses robust or otherwise adjusted standard errors, identify that method rather than allowing readers to assume the default. This tutorial concerns constructing the table; it does not replace checking the model or justifying the inferential method.
Worked Example: Preventing a Sample-Size Mismatch
Worked Example: Preventing a Sample-Size Mismatch
Suppose two hypothetical models have been fitted to data with missing values. Model A uses 96 observations, while Model B uses 91 because its additional predictor is missing for five records. The model objects report those sample sizes, alongside their own coefficient estimates, standard errors, and fit summaries.
The reporting script should extract \(n\) separately from each object. A suitable model-summary row is therefore “Observations” with 96 under Model A and 91 under Model B. Copying 96 into both columns because it was the original data-set size would be incorrect: it would describe records available before model fitting, not the observations used by each fitted model.
Before publishing, compare the table entries with the fitted objects. If the table says Model B used 96 observations but the object reports 91, investigate the discrepancy and regenerate the table. Do not edit the table cell to force agreement with a preferred number. The object is the source for what the model actually used; the analysis record should explain why the samples differ.
Common Mistakes and Quality Checks
Programmatic extraction improves reproducibility only when the extraction is correctly connected to the intended objects. It can faithfully generate a wrong or misleading table if the wrong model was passed to the function, the wrong columns were selected, or labels were attached to the wrong columns. A short validation step is therefore part of table generation, not an optional cosmetic review.
- Typing results by hand. The wrong reasoning is that a small table is easy to copy accurately. The correction is to extract values from the model object and rerun the table code whenever the model changes.
- Confusing \(SSR/SST\) with \(SSE/SST\). For ordinary least squares with an intercept, \(R^2=SSR/SST=1-SSE/SST\). The residual fraction \(SSE/SST\) is not \(R^2\).
- Using one sample size for every model. Different models can use different analysis samples. Extract \(n\) separately and show the model-specific values.
- Assuming returned output is automatically report-ready. Software may return statistics that are unnecessary, ambiguously named, or not comparable across models. Select and label only quantities whose meaning is suitable for the report.
- Ignoring rounding and alignment. Apply a consistent rounding rule and align coefficient rows by term names. Rounding should change displayed precision, not the underlying calculations.
A final check is to rerun the script from the saved analysis objects and inspect the rendered table. Confirm that the model labels, term rows, standard errors, sample sizes, and selected summaries correspond to the intended fits. The table should not depend on edits made only in a word processor or spreadsheet.
Check Your Understanding
Answer these questions using the definitions and checks in this tutorial.
- What is the difference between a coefficient-level table entry and a model-level entry? Give one example of each.
- An ordinary least-squares model with an intercept has \(SSR=24\) and \(SST=80\). Calculate \(R^2\), and state the corresponding residual fraction \(SSE/SST\).
- Two fitted models report sample sizes of 74 and 69. What sample sizes should appear in a model-comparison table, and why?
- Why should coefficient rows from two model objects be aligned by term names rather than by their positions in extracted output?
- A table's standard errors were extracted correctly, but the model's assumptions and chosen standard-error method have not been checked. What does successful extraction establish, and what does it not establish?