Linear Regression

The Linear Regression tab performs ordinary least squares (OLS) linear regression analysis. OLS finds the coefficient vector $\hat\beta = (X'X)^{-1}X'Y$ that minimizes the residual sum of squares. See Regression Analysis Fundamentals for the mathematical background.

For count data, binary outcomes, or other non-normal response variables, use Generalized Linear Model (GLM) instead.

Basic Usage

Opening Linear Regression

Select Analysis > Linear Regression (OLS)... from the menu bar to open a new Linear Regression tab.

Setting Up Variables

Variable setup

Dataset selects the dataset to analyze.

Response Variable (Y) selects the response variable. Numeric (int64, float64) and boolean columns are available. Boolean values are treated as 0/1. Date and datetime columns, and columns whose scale is set to nominal or ordinal, cannot be selected.

Predictor Variables (X) selects predictor variables using checkboxes. The same columns as for the response variable are selectable; columns with nominal or ordinal scale and date/datetime columns are grayed out. To use string or other categorical variables, convert them to numeric dummy variables using the Dummy Coding tab first (see Notes).

Include intercept toggles the intercept term. Enabled by default.

Confidence Level sets the confidence level used for the interval estimates in the coefficients table and the Prediction & Confidence Intervals table. The default is 95; values from 50 to 99.99 are accepted.

Click the Run Analysis button to run the analysis.

Understanding Results

Model Summary

Analysis results

Displays overall model fit statistics.

Metric	Description
R-squared	Proportion of variance explained by the model (0 to 1)
Adj. R-squared	R-squared adjusted for the number of predictors
RMSE	Standard deviation of residuals (measure of prediction error)
N (observations)	Number of observations used in the analysis

If rows containing missing or invalid values were excluded, the number of excluded rows is displayed.

Coefficients

Displays regression coefficients for each variable.

Column	Description
Variable	Variable name (intercept shown as "(Intercept)")
Estimate	Estimated coefficient $\hat\beta$
Std. Error	Standard error $\sqrt{\operatorname{diag}\bigl(\hat\sigma^2 (X'X)^{-1}\bigr)}$
Lower N% / Upper N%	Confidence interval $\hat\beta \pm t_{\alpha/2,\, n-p} \times \operatorname{SE}(\hat\beta)$ , where N is the selected confidence level (any value from 50 to 99.99)
Std. Coef.	Standardized coefficient $\hat\beta_j \times s_{X_j} / s_Y$ for comparing effect sizes across variables (shown as "-" for the intercept)
VIF	Variance Inflation Factor (see Multicollinearity)

If the errors are normally distributed, the $t$ -distribution-based confidence intervals have exact coverage regardless of sample size.

The coefficients table can be saved as a dataset using the Save as Dataset button for export to CSV. You must save the model first (using the Save Model button). Linking the coefficient dataset to a specific model means that deleting the model also deletes the derived coefficient dataset and any report element that references it, and refitting the model updates the dataset contents to reflect the new fit.

The saved dataset contains Variable, Estimate, Std. Error, Lower N%, Upper N%, Std. Coef., and VIF.

Interpreting Coefficients

Coefficients are directly interpretable on the response scale.

Continuous predictor: Holding other variables constant, a one-unit increase in $X_j$ changes $E[Y]$ by $\hat\beta_j$
Dummy variable: Represents the difference in $E[Y]$ relative to the reference category
Intercept: $E[Y]$ when all predictors are zero
Standardized coefficient (Std. Coef.): Expressed in standard deviation units, enabling direct comparison of effect sizes across variables with different scales

Model Fit

Metric	Description
Residual Deviance	Residual sum of squares $\text{RSS} = \sum(y_i - \hat y_i)^2$
Null Deviance	Total sum of squares $\text{TSS} = \sum(y_i - \bar y)^2$
AIC	Akaike Information Criterion $\text{AIC} = -2\ell + 2k$ , where $k$ is the total number of estimated parameters ( $p$ regression coefficients plus $\sigma^2$ , i.e. $p + 1$ ). Lower is better
BIC	Bayesian Information Criterion $\text{BIC} = -2\ell + k \ln n$ . Penalizes complexity more strongly than AIC. BIC is consistent for model selection: when the true model is among the candidates, the probability of selecting it converges to 1 as the sample size increases

ANOVA Table

ANOVA table

Evaluates each predictor's contribution through analysis of variance. Switch between Type I and Type III using radio buttons.

Type I (Sequential): Computes sum of squares in the order variables were entered. Results depend on variable entry order.
Type III (Partial): Computes sum of squares as if each variable were entered last. Results are independent of variable entry order.

Column	Description
Source	Source of variation
Sum Sq	Sum of squares
DF	Degrees of freedom
Mean Sq	Mean square (Sum Sq / DF)
partial η²	Partial eta-squared (SS_effect / (SS_effect + SS_residual)). Shown for Type III only
partial ω²	Partial omega-squared, a bias-adjusted effect size estimator. Displayed as 0 when the estimate is negative. Shown for Type III only

Prediction & Confidence Intervals

Displays predicted values and interval estimates for each observation.

Column	Description
Fitted	Predicted value
CI Lower N% / CI Upper N%	Confidence interval for the mean response
PI Lower N% / PI Upper N%	Prediction interval for individual observations

The confidence interval (CI) represents the precision of the estimated mean, while the prediction interval (PI) represents the range for a new individual observation. PI is always wider than CI. N is the selected confidence level (any value from 50 to 99.99).

When exceeding 100 rows, only the first 50 rows are displayed. Click Show all N rows to display all rows.

The prediction intervals table can also be saved as a dataset using the Save as Dataset button. As with the coefficients table, you must save the model first (using the Save Model button).

Saving and Diagnostics

Model saving

Save analysis results to the project and view diagnostic plots.

Saving the Model

Enter a model name in the Model Name field and click Save Model. The model name defaults to the format "Linear Regression: Y ~ X1 + X2".

If an existing model with the same configuration (dataset, response variable, predictor variables, and intercept setting) exists, a confirmation dialog for overwriting is displayed.

Data Generated for Diagnostics

After saving the model, opening the Residual Diagnostics tab for the first time via View Diagnostics creates a derived dataset that adds diagnostic columns to the original data.

Column	Symbol	Description
`fitted_values`	$\hat y_i$	Predicted values
`deviance_residuals`	$e_i = y_i - \hat y_i$	Residuals
`standardized_residuals`	$r_i^* = e_i / (\hat\sigma\sqrt{1 - h_i})$	Standardized residuals
`sqrt_abs_std_residuals`	$\sqrt{\lvert r_i^* \rvert}$	Square root of the absolute standardized residuals (used in the Scale-Location plot)
`leverage`	$h_i$	Leverage (diagonal of the hat matrix)
`cooks_distance`	$D_i$	Cook's Distance

Diagnostics and Details

After saving the model, two buttons appear:

View Model Details - Opens the Model Detail tab showing detailed model information. Changing the Confidence Level input recomputes the Wald confidence intervals and column headers in place from the saved coefficients and standard errors (the saved value is not modified). Use the Add to Report button to add the coefficients table to a report.
View Diagnostics - Opens the Residual Diagnostics tab showing residual diagnostic plots

Residual Diagnostics

View Diagnostics opens four diagnostic plots for assessing the OLS assumptions:

OLS assumptions:

Linearity - A linear relationship exists between the response and predictor variables
Normality - Residuals follow a normal distribution
Homoscedasticity - The variance of residuals is constant and does not depend on fitted values
Independence - Residuals are independent of each other (not directly verifiable through diagnostic plots). For hierarchical or clustered data, consider using GLMM with random effects. For time series data with autocorrelation, consider using ARIMA (available in the Analysis menu)

Residual diagnostics

The diagnostic plots use standardized residuals (internally studentized residuals) $r_i^*$ . See Regression Analysis Fundamentals for the formulas.

Residuals vs Fitted

Plots residuals $e_i$ against fitted values $\hat y$ . With a well-specified model, residuals scatter randomly around zero.

Curved pattern: Nonlinear effects of predictors may be missing
Funnel-shaped pattern: Possible heteroscedasticity (check Scale-Location plot for details)

Normal Q-Q Plot

Plots standardized residual $r_i^*$ quantiles against theoretical normal quantiles. Points fall along the diagonal when residuals are normally distributed. Deviation at both ends indicates heavy tails; an S-shaped departure indicates skewness.

Scale-Location

Plots $\sqrt{|r_i^*|}$ against fitted values. Constant variance produces an even horizontal spread. A funnel-shaped or upward-trending pattern indicates heteroscedasticity. When heteroscedasticity is present, coefficient estimates remain unbiased but standard errors become inaccurate, so the actual coverage of confidence intervals can deviate from the nominal level.

Residuals vs Leverage

Plots standardized residuals $r_i^*$ against leverage $h_i$ . Cook's (1977) distance contours ( $D = 0.5$ : orange dashed, $D = 1.0$ : red dashed) are overlaid. Cook suggested comparing $D_i$ to the 50th percentile of $F_{p,\, n-p}$ , which is typically near 1. This is not a formal rejection threshold but a guide for comparing relative influence among observations.

Leverage: Measures how far an observation's predictor values are from others. $2p/n$ is the conventional threshold for high leverage
Cook's Distance: Combines leverage and residual size into a single influence measure

Observations outside the contour lines may substantially change the model estimates if removed.

For observations with leverage extremely close to 1 ( $h_i \ge 0.999$ ), the denominators of the standardized residual and Cook's Distance approach zero and the computation becomes numerically unstable, so MIDAS displays both as 0. This does not mean the observation has little influence; it indicates that the leverage itself is extreme. Check the horizontal axis of the plot.

Point Selection

Click or rectangle-select data points on any plot to display details (fitted values, residuals, leverage, Cook's Distance, etc.) in a table below the plots. Selection state is synchronized across all four plots.

Notes

Using Categorical Variables

To use string or other categorical variables as predictors, convert them to numeric dummy variables using the Dummy Coding tab before running the analysis. Boolean columns need no conversion and are treated as 0/1.

Automatic Exclusion of Missing and Invalid Values

Rows containing missing values (null), non-numeric values, or infinity are automatically excluded from the analysis. The number of excluded rows is displayed in the Model Summary. This is listwise deletion. See Missing Data Mechanisms for conditions under which it yields valid estimates.

Multicollinearity

When predictors are highly correlated, coefficient estimates become unstable. If VIF (Variance Inflation Factor) is large for any variable in the coefficients table, consider removing redundant variables or combining correlated predictors. See Regression Analysis Fundamentals for details on VIF.

Sample Size and Normality

Whether the coverage probability of the confidence intervals matches the nominal level in finite samples depends on the normality of errors. With large samples, the central limit theorem keeps the coverage approximately correct, but for small samples, verify residual normality using the Q-Q plot. The required sample size depends on the true error distribution, so no universal threshold applies.

References

Cook, R. D. (1977). Detection of influential observation in linear regression. Technometrics, 19(1), 15-18. https://www.jstor.org/stable/1268249

Next steps

Residual Diagnostics - Verify model assumptions
Generalized Linear Model (GLM) - Modeling count data and binary outcomes