# Multiple linear regression calculator
Paste a numeric data matrix and fit an OLS multiple regression: coefficients, p-values, confidence intervals, R², adjusted R² and F-test.
## Run it
- **CLI:** `gizza tool multiple-regression "1,6
2,8
3,11
4,14
5,18
6,20"`
- **Web:** https://gizza.ai/tools/multiple-regression/
- **Agents:** machine-readable descriptor (parameters JSON Schema) at https://gizza.ai/tools/multiple-regression/tool.json
## Inputs
- `data` — Data matrix (one observation per line; columns split on comma, tab, semicolon or space) _(field)_
- `response` — Response column (Y): last, first, or a 1-based number _(field)_
- `labels` — Column names (optional, comma-separated, in data order) _(field)_
- `intercept` — Fit an intercept (uncheck to force a zero Y-intercept) _(field)_
- `conf_level` — Confidence level for the intervals _(field)_
- `format` — Output _(field)_
## Output
- Regression summary (text)
## Query parameters
Open the tool pre-filled and auto-run via URL:
- `data` — Data matrix (one observation per line; columns split on comma, tab, semicolon or space)
- `response` — Response column (Y): last, first, or a 1-based number
- `labels` — Column names (optional, comma-separated, in data order)
- `intercept` — Fit an intercept (uncheck to force a zero Y-intercept)
- `conf_level` — Confidence level for the intervals
- `format` — Output
Example: `https://gizza.ai/tools/multiple-regression/?data=1%2C6%0A2%2C8%0A3%2C11%0A4%2C14%0A5%2C18%0A6%2C20&response=last&labels=sqft%2Crooms%2Cprice&intercept=true&conf_level=0.95&format=text`
---
## About this tool
This multiple linear regression calculator fits an **ordinary least-squares (OLS)** model
`y = b₀ + b₁·x₁ + … + bₖ·xₖ` to a data matrix you paste in — one observation per line, columns
separated by commas, tabs, semicolons or spaces. By default the **last column** is the response
(dependent variable) *y* and every other column is a predictor; point `response` at `first` or a
1-based column number to pick a different Y. Give the columns real names with **labels** so the
coefficient table and equation read in your terms (`sqft`, `rooms`, `price`) instead of `v1`, `v2`, `v3`.
For each coefficient — including the intercept — you get the **estimate**, its **standard error**,
the **t-statistic**, the two-tailed **p-value** (H₀: the coefficient is zero) and a **confidence
interval** at your chosen level. At the model level it reports **R²** and **adjusted R²** (which
penalises extra predictors), the **residual standard error** with its degrees of freedom, and the
overall **F-test** with its p-value — the joint test that at least one predictor matters. Switch the
output to **JSON** to additionally get the per-observation **fitted values** and **residuals** for
diagnostics. Everything runs in your browser; your data never leaves the page.
### Worked example
Input (`data`, with `labels = sqft,rooms,price`, so `price` is the response):
```
1,1,6
2,1,8
3,2,11
4,2,14
5,3,18
6,3,20
```
Output:
```
price = 2 + 2.333333·sqft + 1.333333·rooms
Coefficients:
term estimate std error t p 95% CI
(Intercept) 2 0.559384 3.575424 0.037461 [0.219859, 3.780141]
sqft 2.333333 0.3849 6.062178 0.009007 [1.108218, 3.558449]
rooms 1.333333 0.930949 1.432222 0.247544 [-1.62904, 4.295707]
```
*(exact numbers are shown in full on the page)* — with **R² = 0.995638**, **adjusted R² = 0.99273**
and an **F-statistic of 342.375 on 2 and 3 DF (p = 0.000288)**, both predictors are strongly
significant and the model explains almost all of the variation in `price`.
### Controls
- **Response column** — `last` (default, rightmost), `first`, or a 1-based column index; every other
column becomes a predictor.
- **Column names** — optional comma-separated labels in data order; default `v1, v2, …`.
- **Fit an intercept** — on by default. Uncheck to force a zero Y-intercept (regression through the
origin); this also switches R² to the uncentered definition, as statistical software does.
- **Confidence level** — for the coefficient intervals (default `0.95`); the two-tailed significance
level is α = 1 − level.
- **Output** — a formatted summary, or JSON that adds the fitted values and residuals.
### Limits and edge cases
- You need **more observations than parameters** (rows > predictors + intercept), otherwise the
residual degrees of freedom would be zero or negative and the fit is rejected.
- Predictors must not be **perfectly collinear** (one an exact linear combination of others) — the
normal-equations matrix is then singular and the model is not identifiable, so the tool errors
instead of returning unstable numbers.
- Every value must be a **finite number** and every row must have the **same number of columns**;
blank lines are ignored.
- All predictors are treated as **numeric** — encode categorical variables as dummy (0/1) columns
and apply any log/sqrt transforms yourself before pasting.
- p-values and confidence intervals assume the usual OLS conditions (independent, homoscedastic,
roughly normal residuals); check the residuals (JSON output) before trusting them.
## FAQ
How do I choose which column is the dependent variable (Y)?
By default the **last (rightmost) column** is the response and everything to its left is a
predictor — the convention most calculators use. Set the response field to `first` for the first
column, or to a **1-based column number** (`2` = the second column) to use any other. Whichever
column you pick becomes *y*; all remaining columns are entered as predictors together.
What is the difference between R² and adjusted R²?
R² is the share of the variance in *y* explained by the model, from 0 to 1. It can only go **up**
when you add predictors, even useless ones, so it rewards a bigger model unfairly. Adjusted R²
corrects for that by penalising each extra predictor relative to the sample size, so it can fall
when a predictor adds nothing. Use adjusted R² when comparing models with different numbers of
predictors.
What does the p-value on each coefficient mean?
It is the two-tailed probability, under the null hypothesis that the true coefficient is **zero**,
of seeing a t-statistic at least as extreme as the one observed. A small p-value (say below your α
= 1 − confidence level) is evidence the predictor has a non-zero effect *holding the other
predictors fixed*. The overall **F-test** is the joint version: it asks whether the predictors
together explain more than chance would.
Why does forcing a zero intercept change my R²?
With an intercept, R² is computed against the mean of *y* (centered total sum of squares). When you
force the line through the origin there is no intercept to absorb the mean, so software — this tool
included — switches to the **uncentered** total sum of squares. The two R² values are not
comparable, and a no-intercept R² is often much higher for reasons that have nothing to do with fit
quality, so interpret it with care.
Why did I get a "perfectly collinear" or "not enough observations" error?
OLS needs the predictor matrix to be full rank. **Perfect collinearity** — one predictor being an
exact linear combination of the others (e.g. a column that is another times a constant, or dummy
columns that sum to 1) — makes the system unsolvable, so the tool stops instead of printing
meaningless coefficients. You also need **more rows than parameters**; with as many parameters as
observations the model would fit perfectly with zero residual degrees of freedom and no way to
estimate uncertainty. Add rows or drop a redundant predictor.
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