# lm

##### Fitting Linear Models

`lm`

is used to fit linear models.
It can be used to carry out regression,
single stratum analysis of variance and
analysis of covariance (although `aov`

may provide a more
convenient interface for these).

- Keywords
- regression

##### Usage

`lm(formula, data, subset, weights, na.action, method = "qr", model = TRUE, x = FALSE, y = FALSE, qr = TRUE, singular.ok = TRUE, contrasts = NULL, offset, ...)`

##### Arguments

- formula
- an object of class
`"formula"`

(or one that can be coerced to that class): a symbolic description of the model to be fitted. The details of model specification are given under ‘Details’. - data
- an optional data frame, list or environment (or object
coercible by
`as.data.frame`

to a data frame) containing the variables in the model. If not found in`data`

, the variables are taken from`environment(formula)`

, typically the environment from which`lm`

is called. - subset
- an optional vector specifying a subset of observations to be used in the fitting process.
- weights
- an optional vector of weights to be used in the fitting
process. Should be
`NULL`

or a numeric vector. If non-NULL, weighted least squares is used with weights`weights`

(that is, minimizing`sum(w*e^2)`

); otherwise ordinary least squares is used. See also ‘Details’, - na.action
- a function which indicates what should happen
when the data contain
`NA`

s. The default is set by the`na.action`

setting of`options`

, and is`na.fail`

if that is unset. The ‘factory-fresh’ default is`na.omit`

. Another possible value is`NULL`

, no action. Value`na.exclude`

can be useful. - method
- the method to be used; for fitting, currently only
`method = "qr"`

is supported;`method = "model.frame"`

returns the model frame (the same as with`model = TRUE`

, see below). - model, x, y, qr
- logicals. If
`TRUE`

the corresponding components of the fit (the model frame, the model matrix, the response, the QR decomposition) are returned. - singular.ok
- logical. If
`FALSE`

(the default in S but not in R) a singular fit is an error. - contrasts
- an optional list. See the
`contrasts.arg`

of`model.matrix.default`

. - offset
- this can be used to specify an
*a priori*known component to be included in the linear predictor during fitting. This should be`NULL`

or a numeric vector of length equal to the number of cases. One or more`offset`

terms can be included in the formula instead or as well, and if more than one are specified their sum is used. See`model.offset`

. - ...
- additional arguments to be passed to the low level regression fitting functions (see below).

##### Details

Models for `lm`

are specified symbolically. A typical model has
the form `response ~ terms`

where `response`

is the (numeric)
response vector and `terms`

is a series of terms which specifies a
linear predictor for `response`

. A terms specification of the form
`first + second`

indicates all the terms in `first`

together
with all the terms in `second`

with duplicates removed. A
specification of the form `first:second`

indicates the set of
terms obtained by taking the interactions of all terms in `first`

with all terms in `second`

. The specification `first*second`

indicates the *cross* of `first`

and `second`

. This is
the same as `first + second + first:second`

.

If the formula includes an `offset`

, this is evaluated and
subtracted from the response.

If `response`

is a matrix a linear model is fitted separately by
least-squares to each column of the matrix.

See `model.matrix`

for some further details. The terms in
the formula will be re-ordered so that main effects come first,
followed by the interactions, all second-order, all third-order and so
on: to avoid this pass a `terms`

object as the formula (see
`aov`

and `demo(glm.vr)`

for an example).

A formula has an implied intercept term. To remove this use either
`y ~ x - 1`

or `y ~ 0 + x`

. See `formula`

for
more details of allowed formulae.

Non-`NULL`

`weights`

can be used to indicate that different
observations have different variances (with the values in
`weights`

being inversely proportional to the variances); or
equivalently, when the elements of `weights`

are positive
integers $w_i$, that each response $y_i$ is the mean of
$w_i$ unit-weight observations (including the case that there are
$w_i$ observations equal to $y_i$ and the data have been
summarized).

`lm`

calls the lower level functions `lm.fit`

, etc,
see below, for the actual numerical computations. For programming
only, you may consider doing likewise.

All of `weights`

, `subset`

and `offset`

are evaluated
in the same way as variables in `formula`

, that is first in
`data`

and then in the environment of `formula`

.

##### Value

- coefficients
- a named vector of coefficients
- residuals
- the residuals, that is response minus fitted values.
- fitted.values
- the fitted mean values.
- rank
- the numeric rank of the fitted linear model.
- weights
- (only for weighted fits) the specified weights.
- df.residual
- the residual degrees of freedom.
- call
- the matched call.
- terms
- the
`terms`

object used. - contrasts
- (only where relevant) the contrasts used.
- xlevels
- (only where relevant) a record of the levels of the factors used in fitting.
- offset
- the offset used (missing if none were used).
- y
- if requested, the response used.
- x
- if requested, the model matrix used.
- model
- if requested (the default), the model frame used.
- na.action
- (where relevant) information returned by
`model.frame`

on the special handling of`NA`

s. In addition, non-null fits will have components

`lm`

returns an object of `class`

`"lm"`

or for
multiple responses of class `c("mlm", "lm")`

.The functions `summary`

and `anova`

are used to
obtain and print a summary and analysis of variance table of the
results. The generic accessor functions `coefficients`

,
`effects`

, `fitted.values`

and `residuals`

extract
various useful features of the value returned by `lm`

.An object of class `"lm"`

is a list containing at least the
following components:`assign`

,
`effects`

and (unless not requested) `qr`

relating to the linear
fit, for use by extractor functions such as `summary`

and
`effects`

.
##### Note

Offsets specified by `offset`

will not be included in predictions
by `predict.lm`

, whereas those specified by an offset term
in the formula will be.

##### Using time series

Considerable care is needed when using `lm`

with time series. Unless `na.action = NULL`

, the time series attributes are
stripped from the variables before the regression is done. (This is
necessary as omitting `NA`

s would invalidate the time series
attributes, and if `NA`

s are omitted in the middle of the series
the result would no longer be a regular time series.) Even if the time series attributes are retained, they are not used to
line up series, so that the time shift of a lagged or differenced
regressor would be ignored. It is good practice to prepare a
`data`

argument by `ts.intersect(..., dframe = TRUE)`

,
then apply a suitable `na.action`

to that data frame and call
`lm`

with `na.action = NULL`

so that residuals and fitted
values are time series.

##### References

Chambers, J. M. (1992)
*Linear models.*
Chapter 4 of *Statistical Models in S*
eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

Wilkinson, G. N. and Rogers, C. E. (1973)
Symbolic descriptions of factorial models for analysis of variance.
*Applied Statistics*, **22**, 392--9.

##### See Also

`summary.lm`

for summaries and `anova.lm`

for
the ANOVA table; `aov`

for a different interface.

The generic functions `coef`

, `effects`

,
`residuals`

, `fitted`

, `vcov`

.

`predict.lm`

(via `predict`

) for prediction,
including confidence and prediction intervals;
`confint`

for confidence intervals of *parameters*.

`lm.influence`

for regression diagnostics, and
`glm`

for **generalized** linear models.

The underlying low level functions,
`lm.fit`

for plain, and `lm.wfit`

for weighted
regression fitting.

More `lm()`

examples are available e.g., in
`anscombe`

, `attitude`

, `freeny`

,
`LifeCycleSavings`

, `longley`

,
`stackloss`

, `swiss`

.

`biglm`

in package \href{https://CRAN.R-project.org/package=#1}{\pkg{#1}}biglmbiglm for an alternative
way to fit linear models to large datasets (especially those with many
cases).

##### Examples

`library(stats)`

```
require(graphics)
## Annette Dobson (1990) "An Introduction to Generalized Linear Models".
## Page 9: Plant Weight Data.
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
group <- gl(2, 10, 20, labels = c("Ctl","Trt"))
weight <- c(ctl, trt)
lm.D9 <- lm(weight ~ group)
lm.D90 <- lm(weight ~ group - 1) # omitting intercept
anova(lm.D9)
summary(lm.D90)
opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0))
plot(lm.D9, las = 1) # Residuals, Fitted, ...
par(opar)
### less simple examples in "See Also" above
```

*Documentation reproduced from package stats, version 3.2.1, License: Part of R 3.2.1*

### Community examples

**richie@datacamp.com**at Jan 17, 2017 stats v3.3.1

`lm()` takes a formula and a data frame. See [`formula()`](https://www.rdocumentation.org/packages/stats/topics/formula) for how to contruct the first argument. ```{r} (model_with_intercept <- lm(weight ~ group, PlantGrowth)) (model_without_intercept <- lm(weight ~ group - 1, PlantGrowth)) ``` You get more information about the model using [`summary()`](https://www.rdocumentation.org/packages/stats/topics/summary.lm) ```{r} (model_without_intercept <- lm(weight ~ group - 1, PlantGrowth)) summary(model_without_intercept) ``` Diagnostic plots are available; see [`plot.lm()`](https://www.rdocumentation.org/packages/stats/topics/plot.lm) for more examples. ```{r} (model_without_intercept <- lm(weight ~ group - 1, PlantGrowth)) layout(matrix(1:6, nrow = 2)) plot(model_without_intercept, which = 1:6) ``` You can predict new values; see [`predict()`](https://www.rdocumentation.org/packages/stats/topics/predict) and [`predict.lm()`](https://www.rdocumentation.org/packages/stats/topics/predict.lm) . ```{r} (model_without_intercept <- lm(weight ~ group - 1, PlantGrowth)) predictions <- data.frame(group = levels(PlantGrowth$group)) predictions$weight <- predict(model_without_intercept, predictions) predictions # Plot predictions against the data boxplot(weight ~ group, PlantGrowth, ylab = "weight") points(weight ~ group, predictions, col = "red") ``` There are many methods available for inspecting `lm` objects. ```{r} (model_without_intercept <- lm(weight ~ group - 1, PlantGrowth)) confint(model_without_intercept) anova(model_without_intercept) residuals(model_without_intercept) fitted(model_without_intercept) influence(model_without_intercept) methods(class = "lm") ```